Abstract
We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let \(G\) be a reductive group over a number field \(F\) which admits discrete series representations at infinity. Let \(^{L}G=\widehat{G} \rtimes \mathrm{Gal}(\bar{F}/F)\) be the associated \(L\)group and \(r:{}^L G\rightarrow \mathrm{GL}(d,\mathbb {C})\) a continuous homomorphism which is irreducible and does not factor through \(\mathrm{Gal}(\bar{F}/F)\). The families under consideration consist of discrete automorphic representations of \(G(\mathbb {A}_F)\) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato–Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321–331, 1987) and Serre (J Am Math Soc 10(1):75–102, 1997). As an application we study the distribution of the lowlying zeros of the associated family of \(L\)functions \(L(s,\pi ,r)\), assuming from the principle of functoriality that these \(L\)functions are automorphic. We find that the distribution of the \(1\)level densities coincides with the distribution of the \(1\)level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz–Sarnak heuristics. We provide a criterion based on the Frobenius–Schur indicator to determine this symmetry type. If \(r\) is not isomorphic to its dual \(r^\vee \) then the symmetry type is unitary. Otherwise there is a bilinear form on \(\mathbb {C}^d\) which realizes the isomorphism between \(r\) and \(r^\vee \). If the bilinear form is symmetric (resp. alternating) then \(r\) is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).
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1 Introduction
The nontrivial zeros of automorphic \(L\)functions are of central significance in modern number theory. Problems on individual zeros, such as the Riemann hypothesis (GRH), are elusive. There is however a theory of the statistical distribution of zeros in families. The subject has a long and rich history. A unifying modern viewpoint is that of a comparison with a suitably chosen model of random matrices: the Katz–Sarnak heuristics. There are both theoretical and numerical evidences for this comparison. Comprehensive results in the function field case [59] have suggested an analogous picture in the number field case as explained in [60]. In a large number of cases, and with high accuracy, the distribution of zeros of automorphic \(L\)functions coincide with the distribution of eigenvalues of random matrices. See [37, 85] for numerical investigations and conjectures and see [40, 49, 50, 53, 68, 82, 84] and the references therein for theoretical results. «Schémas en groupes»
The concept of families is central to modern investigations in number theory. We want to study in the present paper certain families of automorphic representations over number fields in a very general context. The families under consideration are obtained from the discrete spectrum by imposing constraints on the local components at archimedean and nonarchimedean places and by applying Langlands global functoriality principle.
Our main result is a Sato–Tate equidistribution theorem for these families (Theorem 1.3). As an application of this main result we can give some evidence towards the Katz–Sarnak heuristics [60] in general and establish a criterion for the random matrix model attached to families, i.e. for the symmetry type.
1.1 Sato–Tate theorem for families
The original Sato–Tate conjecture is about an elliptic curve \(E\), assumed to be defined over \(\mathbb {Q}\) for simplicity. The number of points in \(E(\mathbb {F}_p)\) for almost all primes \(p\) (with good reduction) gives rise to an angle \(\theta _p\) between \(\pi \) and \(\pi \). The conjecture, proved in [7], asserts that if \(E\) does not admit complex multiplication then \(\{\theta _p\}\) are equidistributed according to the measure \(\frac{2}{\pi } \sin ^2\theta d\theta \). In the context of motives a generalization of the Sato–Tate conjecture was formulated by Serre [96].
To speak of the automorphic version of the Sato–Tate conjecture, let \(G\) be a connected split reductive group over \(\mathbb {Q}\) with trivial center and \(\pi \) an automorphic representation of \(G(\mathbb {A})\). Here \(G\) is assumed to be split for simplicity (however we stress that our results are valid without even assuming that \(G\) is quasisplit; see Sect. 5 below for details). The triviality of center is not serious as it essentially amounts to fixing central character. Let \(T\) be a maximal split torus of \(G\). Denote by \(\widehat{T}\) its dual torus and \(\Omega \) the Weyl group. As \(\pi =\otimes '_v \pi _v\) is unramified at almost all places \(p\), the Satake isomorphism identifies \(\pi _p\) with a point on \(\widehat{T}/\Omega \). The automorphic Sato–Tate conjecture should be a prediction about the equidistribution of \(\pi _p\) on \(\widehat{T}/\Omega \) with respect to a natural measure (supported on a compact subset of \(\widehat{T}/\Omega \)). It seems nontrivial to specify this measure in general. The authors do not know how to do it without invoking the (conjectural) global \(L\)parameter for \(\pi \). The automorphic Sato–Tate conjecture is known in the limited cases of (the restriction of scalars of) \(\mathrm{GL}_1\) and \(\mathrm{GL}_2\) [6, 7]. In an ideal world the conjecture should be closely related to Langlands functoriality.
In this paper we consider the Sato–Tate conjecture for a family of automorphic representations, which is easier to state and prove but still very illuminating. Our working definition of a family \(\{\mathcal {F}_k\}_{k\geqslant 1}\) is that each \(\mathcal {F}_k\) consists of all automorphic representations \(\pi \) of \(G(\mathbb {A})\) of level \(N_k\) with \(\pi _\infty \) cohomological of weight \(\xi _k\), where \(N_k\in \mathbb {Z}_{\geqslant 1}\) and \(\xi _k\) is an irreducible algebraic representation of \(G\), such that either

(1)
(level aspect) \(\xi _k\) is fixed, and \(N_k\rightarrow \infty \) as \(k\rightarrow \infty \) or

(2)
(weight aspect) \(N_k\) is fixed, and \(m(\xi _k)\rightarrow \infty \) as \(k\rightarrow \infty \),
where \(m(\xi _k)\in \mathbb {R}_{\geqslant 0}\) should be thought of as the minimal distance of the highest weight of \(\xi _k\) to root hyperplanes. (See Sect. 6.4 below for the precise definition.) Note that each \(\mathcal {F}_k\) has finite cardinality and \(\mathcal {F}_k\rightarrow \infty \) as \(k\rightarrow \infty \). (For a technical reason \(\mathcal {F}_k\) is actually allowed to be a multiset. Namely the same representation can appear multiple times, for instance more than its automorphic multiplicity.) In principle we could let \(\xi _k\) and \(N_k\) vary simultaneously but decided not to do so in the current paper in favor of transparency of arguments. For instance families of type (i) and (ii) require somewhat different ingredients of proof in establishing the Sato–Tate theorem for families, and the argument would be easier to understand if we separate them. It should be possible to treat the mixed case (where both \(N_k\) and \(\xi _k\) vary) by combining techniques in the two cases (i) and (ii).
Let \(\widehat{T}_c\) be the maximal compact subtorus of the complex torus \(\widehat{T}\). The quotient \(\widehat{T}_c/\Omega \) is equipped with a measure \(\widehat{\mu }^{\mathrm {ST}}\), to be called the Sato–Tate measure, coming from the Haar measure on a maximal compact subgroup of \(\widehat{G}\) (of which \(\widehat{T}_c\) is a maximal torus). The following is a rough version of our result on the Sato–Tate conjecture for a family.
Theorem 1.1
Suppose that \(G(\mathbb {R})\) has discrete series representations. Let \(\{\mathcal {F}_k\}_{k\geqslant 1}\) be a family in the level aspect (resp. weight aspect) as above. Let \(\{p_k\}\) be a strictly increasing sequence of primes such that \(N_k\) (resp. \(\xi _k\)) grows faster than any polynomial in \(p_k\) in the sense that \(\dfrac{\log p_k}{\log N_k} \rightarrow 0\) (resp. \(\dfrac{\log p_k}{\log m(\xi _k)} \rightarrow 0\)) as \(k\rightarrow \infty \). Assume that the members of \(\mathcal {F}_k\) are unramified at \(p_k\) for every \(k\). Then the Satake parameters \(\{\pi _{p_k}:\pi \in \mathcal {F}_k\}_{k\geqslant 1}\) are equidistributed with respect to \(\widehat{\mu }^{\mathrm {ST}}\).
To put things in perspective, we observe that there are three kinds of statistics about the Satake parameters of \(\{\pi _{p_k}:\pi \in \mathcal {F}_k\}_{k\geqslant 1}\) depending on how the arguments vary.

(i)
Sato–Tate: \(\mathcal {F}_k\) is fixed (and a singleton) and \(p_k\rightarrow \infty \).

(ii)
Sato–Tate for a family: \(\mathcal {F}_k\rightarrow \infty \) and \(p_k\rightarrow \infty \).

(iii)
Plancherel: \(\mathcal {F}_k\rightarrow \infty \) and \(p_k\) is a fixed prime.
The Sato–Tate conjecture in its original form is about equidistribution in case (i) whereas our Theorem 1.1 is concerned with case (ii). The last item is marked as Plancherel since the Satake parameters are expected to be equidistributed with respect to the Plancherel measure (again supported on \(\widehat{T}_c/\Omega \)) in case (iii). This has been shown to be true under the assumption that \(G(\mathbb {R})\) admits discrete series in [99]. We derive Theorem 1.1 from an error estimate (depending on \(k\)) on the difference between the Plancherel distribution at \(p\) and the actual distribution of the Satake parameters at \(p_k\) in \(\mathcal {F}_k\). This estimate (see Theorem 1.3 below) refines the main result of [99] and is far more difficult to prove in that several nontrivial bounds in harmonic analysis on reductive groups need to be justified.
1.2 Families of \(L\)functions
An application of Theorem 1.1 is to families of \(L\)functions. We are able to verify to some extent the heuristics of Katz and Sarnak [60] and determine the symmetry type, see Sect. 1.3 below. In this subsection we define the relevant families of \(L\)functions and record some of their properties.
Let \(r:{}^{L} G \rightarrow \mathrm{GL}(d,\mathbb {C})\) be a continuous \(L\)homomorphism. We assume the Langlands functoriality principle: for all \(\pi \in \mathcal {F}_k\) there exists an isobaric automorphic representation \(\Pi =r_* \pi \) of \(\mathrm{GL}(d,\mathbb {A})\) which is the functorial lift of the automorphic representation \(\pi \) of \(G(\mathbb {A})\), see Sect. 4.3 for a review of the concept of isobaric representations and Sect. 10 for the precise statement of the hypothesis. This hypothesis is only used in Theorem 1.5, Sects. 11 and 12. By the strong multiplicity one theorem \(\Pi \) is uniquely determined by all but finitely many of its local factors \(\Pi _v=r_* \pi _v\).
To an automorphic representation \(\Pi \) on \(\mathrm{GL}(d,\mathbb {A})\) we associate its principal \(L\)function \(L(s,\Pi )\). By definition \(L(s,\pi ,r)=L(s,\Pi )\). By the theory of Rankin–Selberg integrals or by the integral representations of Godement–Jacquet, \(L(s,\Pi )\) has good analytic properties: analytic continuation, functional equation, growth in vertical strips. In particular we know the existence and some properties of its nontrivial zeros, such as the Weyl’s law (Sect. 4.4).
We denote by \(\mathfrak {F}_k=r_*\mathcal {F}_k\) the set of all such \(\Pi =r_* \pi \) for \(\pi \in \mathcal {F}_k\). Since the strong multiplicity one theorem implies that \(\Pi \) is uniquely determined by its \(L\)function \(L(s,\Pi )\). We simply refer to \(\mathfrak {F}=r_*\mathcal {F}\) as a family of \(L\) functions.
In general there are many ways to construct interesting families of \(L\)functions. In a recent manuscript [87], Sarnak attempts to sort out these constructions into a comprehensive framework and proposes^{Footnote 1} a working definition (see also [67]). The families of \(L\)functions under consideration in the present paper fit well into that framework. Indeed they are harmonic families in the sense that their construction involves inputs from local and global harmonic analysis. Other types of families include geometric families constructed as Hasse–Weil \(L\)functions of arithmetic varieties and Galois families associated to families of Galois representations.
1.3 Criterion for the symmetry type
Katz and Sarnak [60] predict that one can associate a symmetry type to a family of \(L\)functions. By definition the symmetry type is the random matrix model which is conjectured to govern the distribution of the zeros. There is a long and rich history for the introduction of this concept.
Hilbert and Pólya suggested that there might be a spectral interpretation of the zeros of the Riemann zeta function. Nowadays strong evidence for the spectral nature of the zeros of \(L\)functions comes from the function field case: zeros are eigenvalues of the Frobenius acting on cohomology. This is exemplified by the equidistribution theorem of Deligne and the results of Katz and Sarnak [59] on the distribution of the lowlying eigenvalues in geometric families.
In the number field case the first major result towards a spectral interpretation is the pair correlation of high zeros of the Riemann zeta function by Montgomery. Developments then include Odlyzko’s extensive numerical study and the determination of the \(n\)level correlation by Hejhal and Rudnick and Sarnak [86]. The number field analogue of the Frobenius eigenvalue statistics of [59] concerns the statistics of lowlying zeros.
More precisely [60] predicts that the lowlying zeros of families of \(L\)functions are distributed according to a determinantal point process associated to a random matrix ensemble. This will be explained in more details in Sects. 1.5 and 1.6 below. We shall distinguish between the three determinantal point processes associated to the unitary, symplectic and orthogonal ensembles.^{Footnote 2} Accordingly the symmetry type associated to a family \(\mathfrak {F}\) is defined to be unitary, symplectic or orthogonal (see Sect. 1.6 for typical results).
Before entering into the details of this theory in Sect. 1.5 below, we state here our criterion for the symmetry type of the harmonic families \(r_*\mathcal {F}\) defined above. We recall in Sect. 6.8 the definition of the Frobenius–Schur indicator \(s(r)\in \left\{ 1,0,1\right\} \) associated to an irreducible representation. We shall prove that the symmetry type is determined by \(s(r)\). This is summarized in the following which may be viewed as a refinement of the Katz–Sarnak heuristics.
Criterion 1.2
Let \(r:{}^LG\rightarrow \mathrm{GL}(d,\mathbb {C})\) be a continuous \(L\)homomorphism which is irreducible and nontrivial when restricted to \(\widehat{G}\). Consider the family \(r_*\mathcal {F}\) of automorphic \(L\)functions of degree \(d\) as above.
 (i):

If \(r\) is not isomorphic to its dual \(r^\vee \) then \(s(r)=0\) and the symmetry type is unitary.
 (ii):

Otherwise there is a bilinear form on \(\mathbb {C}^d\) which realizes the isomorphism between \(r\) and \(r^\vee \). By Schur lemma it is unique up to scalar and is either symmetric or alternating. If it is symmetric then \(r\) is real, \(s(r)=1\) and the symmetry type is symplectic. If it is alternating then \(r\) is quaternionic, \(s(r)=1\) and the symmetry type is orthogonal.
We note that the conditions that \(r\) be irreducible and nontrivial when restricted to \(\widehat{G}\) are optimal. If \(r\) were trivial when restricted to \(\widehat{G}\) then \(L(s,\pi ,r)\) would be constant and equal to a single Artin \(L\)function and the lowlying zeros would correspond to the eventual vanishing of this Artin \(L\)function at the central point (which is a different problem). Also the universality exhibited in our criterion may be compared with the GUE universality of the high zeros of [86].
If \(r\) were reducible then the \(L\)functions would factor as a product \(L(s,\pi ,r_1)L(s,\pi ,r_2)\). Suppose that both \(r_1\) and \(r_2\) are irreducible and nontrivial when restricted to \(\widehat{G}\). If \(r_1=r_2\) then clearly the distribution of zeros will be as before but with multiplicity two. If \(r_1\not \simeq r_2\) then we expect that the zeros will follow the distribution of the independent superposition of the two random matrix ensembles attached to \(r_1\) and \(r_2\). In other words the zeros of \(L(s,\pi ,r_1)\) are uncorrelated to the zeros of \(L(s,\pi ,r_2)\), and one could verify this using the methods of this paper to some extent. In particular we expect no repulsion between the respective sequences of zeros.
It would be interesting to study families of automorphic representations over a function field \(k=\mathbb {F}_q(X)\) of a curve \(X\). To our knowledge the Katz–Sarnak heuristics for such families are not treated in the literature, except in the case of \(G=GL(1)\) where harmonic families coincide with the geometric families treated by Katz–Sarnak (e.g. Dirichlet \(L\)series with quadratic character are the geometric families of hyperelliptic curves in [59, §10]). Over function fields our criterion has the following interpretation. We consider families of automorphic representations \(\pi \) of \(G(\mathbb {A}_k)\); for simplicity we suppose that each automorphic representations \(\pi \) of \(G(\mathbb {A}_k)\) in the family \(\mathcal {F}\) is attached to an irreducible \(\ell \)adic representation \(\rho :\mathrm{Gal}(k^{sep}/k) \rightarrow {}^LG\). Then \(r_*\pi \) is attached to the Galois representation \(r\circ \rho \), and corresponds to a constructible \(\ell \)adic sheaf \(F\) of dimension \(d\) on the curve \(X\). The zeros of the \(L\)function \(L(s,\pi ,r)\) are the eigenvalues of Frobenius on the first cohomology, more precisely the numerator of the \(L\)function \(L(s,\pi ,r)\) is
If \(s(r)=1\) [resp. \(s(r)=1\)] then there is an alternating (resp. symmetric) pairing on the sheaf \(F\). The natural pairing on \(H^1(X,F)\) induced by the cup product is symmetric (resp. alternating) and invariant by the action of Frobenius. Thus the zeros of \(L(s,\pi ,r)\) are the eigenvalues of an orthogonal (resp. symplectic) matrix. This is in agreement with the assertion (ii) of our Criterion 1.2. We also note the related situation [58].
Known analogies between \(L\)functions and their symmetries over number fields and function fields are discussed in [60, §4]. Overall we would like propose Criterion 1.2 and its analogue for geometric families as an answer to the question mark in the entry 6A of Table 2 in [60].
1.4 Automorphic Plancherel density theorem with error bounds
We explain a more precise version of the theorem and method of proof for the Sato–Tate theorem for families (Sect. 1.1). The key is to bound the error terms when we approximate the distribution of local components of automorphic representations in a family with the Plancherel measure.
For simplicity of exposition let us assume that \(G\) is a split reductive group over \(\mathbb {Q}\) with trivial center as in Sect. 1.1. A crucial hypothesis is that \(G(\mathbb {R})\) admits an \(\mathbb {R}\)anisotropic maximal torus [in which case \(G(\mathbb {R})\) admits discrete series representations]. Let \(\mathcal {A}_{\mathrm {disc}}(G)\) denote the set of isomorphism classes of discrete automorphic representations of \(G(\mathbb {A})\). We say that \(\pi \in \mathcal {A}_{\mathrm {disc}}(G)\) has level \(N\) and weight \(\xi \) if \(\pi \) has a nonzero fixed vector under the adelic version of the full level \(N\) congruence subgroup \(K(N)\subset G(\mathbb {A}^\infty )\) and if \(\pi _\infty \otimes \xi \) has nonzero Lie algebra cohomology. In this subsection we make a further simplifying hypothesis that \(\xi \) has regular highest weight, in which case \(\pi _\infty \) as above must be a discrete series representation. (In the main body of this paper, the latter assumption on \(\xi \) is necessary only for the results in Sects. 9.6–9.8, where more general test functions are considered)
Define \(\mathcal {F}=\mathcal {F}(N,\xi )\) to be the finite multiset consisting of \(\pi \in \mathcal {A}_{\mathrm {disc}}(G)\) of level \(N\) and weight \(\xi \), where each such \(\pi \) appears in \(\mathcal {F}\) with multiplicity
This quantity naturally occurs as the dimension of the \(\pi \)isotypical subspace in the cohomology of the locally symmetric space for \(G\) of level \(N\) with coefficient defined by \(\xi \). The main motivation for allowing \(\pi \) to appear \(a_{\mathcal {F}}(\pi )\) times is to enable us to compute the counting measure below with the trace formula.
Let \(p\) be a prime number. Write \(G(\mathbb {Q}_p)^{\wedge }\) for the unitary dual of irreducible smooth representations of \(G(\mathbb {Q}_p)\). The unramified (resp. unramified and tempered) part of \(G(\mathbb {Q}_p)^{\wedge }\) is denoted \(G(\mathbb {Q}_p)^{\wedge ,\mathrm {ur}}\) [resp. \(G(\mathbb {Q}_p)^{\wedge ,\mathrm {ur},\mathrm {temp}}\)]. There is a canonical isomorphism
The unramified Hecke algebra of \(G(\mathbb {Q}_p)\) will be denoted \(\mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))\). There is a map from \(\mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))\) to the space of continuous functions on \(\widehat{T}_c/\Omega \):
There are two natural measures supported on \( \widehat{T}_c/\Omega \). The Plancherel measure \(\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p\), dependent on \(p\), is defined on \(G(\mathbb {Q}_p)^{\wedge ,\mathrm {ur}}\) and naturally arises in local harmonic analysis. The Sato–Tate measure \(\widehat{\mu }^{\mathrm {ST}}\) on \( \widehat{T}_c/\Omega \) is independent of \(p\) and may be extended to \(G(\mathbb {Q}_p)^{\wedge ,\mathrm {ur}} \) by zero. Both \(\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p\) and \(\widehat{\mu }^{\mathrm {ST}}\) assign volume 1 to \( \widehat{T}_c/\Omega \). There is yet another measure \(\widehat{\mu }^{\mathrm {count}}_{\mathcal {F},p}\) on \(G(\mathbb {Q}_p)^{\wedge ,\mathrm {ur}}\), which is the averaged counting measure for the \(p\)components of members of \(\mathcal {F}\). Namely
where \(\delta _{\pi _p}\) denotes the Dirac delta measure supported at \(\pi _p\). [Each \(\pi \in \mathcal {A}_{\mathrm {disc}}(G)\) contributes \(a_{\mathcal {F}}(\pi )\) times to the above sum.] Our primary goal is to bound the difference between \(\widehat{\mu }^{\mathrm {pl},\mathrm {ur}}_p\) and \(\widehat{\mu }^{\mathrm {count}}_{\mathcal {F},p}\). [Note that our definition of \(\widehat{\mu }^{\mathrm {count}}_{\mathcal {F},p}\) in the main body will be a little different from (1.2) but asymptotically the same, see Remark 9.9.]
In order to quantify error bounds, we introduce a filtration \(\{\mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))^{\leqslant \kappa }\}_{\kappa \in \mathbb {Z}_{\geqslant 0}}\) on \(\mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))\) as a complex vector space. The filtration is increasing, exhaustive and depends on a noncanonical choice. Roughly speaking, \(\mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))^{\leqslant \kappa }\) is like the span of all monomials of degree \(\leqslant \kappa \) when \(\mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))\) is identified with (a subalgebra of) a polynomial algebra. For each \(\xi \), it is possible to assign a positive integer \(m(\xi )\) in terms of the highest weight of \(\xi \). When we say that weight is going to infinity, it means that \(m(\xi )\) grows to \(\infty \) in the usual sense.
The main result on error bounds alluded to above is the following. (See Theorems 9.16 and 9.19 for the precise statements and Remarks 9.18 and 9.21 for an explicit choice of constants.) A uniform bound on orbital integrals, cf. (1.9) below, enters the proof of (ii) [but not (i)].
Theorem 1.3
Let \(\mathcal {F}=\mathcal {F}(N,\xi )\) be as above. Consider a prime \(p\), an integer \(\kappa \geqslant 1\), and a function \(\phi _p\in \mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))^{\leqslant \kappa }\) such that \(\phi _p\leqslant 1\) on \(G(\mathbb {Q}_p)\).
 (i):

(level aspect) Suppose that \(\xi \) remains fixed. There exist constants \(A_{\mathrm {lv}},B_{\mathrm {lv}},C_{\mathrm {lv}}>0\) depending only on \(G\) such that for any \(p\), \(\kappa \), \(\phi _p\) as above and for any \(N\) coprime to \(p\),
$$\begin{aligned} \widehat{\mu }^{\mathrm {count}}_{\mathcal {F},p}(\widehat{\phi }_p)\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p(\widehat{\phi }_p) = O(p^{A_{\mathrm {lv}}+B_{\mathrm {lv}}\kappa }N^{C_{\mathrm {lv}}}). \end{aligned}$$  (ii):

(weight aspect) Fix a level \(N\). There exist constants \(A_{\mathrm {wt}},B_{\mathrm {wt}},C_{\mathrm {wt}}>0\) and a lower bound \(c>0\) depending only on \(G\) such that for any \(p\geqslant c\), \(\kappa \), \(\phi _p\) as above with \((p,N)=1\) and for any \(\xi \),
$$\begin{aligned} \widehat{\mu }^{\mathrm {count}}_{\mathcal {F},p}(\widehat{\phi }_p)\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p(\widehat{\phi }_p) = O(p^{A_{\mathrm {wt}}+B_{\mathrm {wt}}\kappa }m(\xi )^{C_{\mathrm {wt}}}). \end{aligned}$$
Let \(\{ \mathcal {F}_k=\mathcal {F}(N_k,\xi _k)\}_{k\geqslant 1}\) be either kind of family in Sect. 1.1, namely either \(N_k\rightarrow \infty \) and \(\xi _k\) is fixed or \(N_k\) is fixed and \(\xi _k\rightarrow \infty \). When applied to \(\{ \mathcal {F}_k\}_{k\geqslant 1}\), Theorem 1.3 leads to the equidistribution results in the following corollary [cf. cases (ii) and (iii) in the paragraph below Theorem 1.1]. Indeed, (i) of the corollary is immediate. Part (ii) is easily derived from the fact that \(\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p\) weakly converges to \(\widehat{\mu }^{\mathrm {ST}}\) as \(p\rightarrow \infty \). Although the unramified Hecke algebra at \(p\) gives rise to only regular functions on the complex variety \(\widehat{T}_c/\Omega \), it is not difficult to extend the results to continuous functions on \(\widehat{T}_c/\Omega \). (See Sects. 9.6–9.8 for details.)
Corollary 1.4
Keep the notation of Theorem 1.3. Let \(\widehat{\phi }\) be a continuous function on \(\widehat{T}_c/\Omega \). In view of (1.1) \(\widehat{\phi }\) can be extended by zero to a function \(\widehat{\phi }_p\) on \(G(\mathbb {Q}_p)^{\wedge ,\mathrm {ur}}\) for each prime \(p\).
 (i):

(Automorphic Plancherel density theorem [99])
$$\begin{aligned} \lim _{k\rightarrow \infty } \widehat{\mu }^{\mathrm {count}}_{\mathcal {F}_k,p}(\widehat{\phi }_p) = \widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p(\widehat{\phi }_p). \end{aligned}$$  (ii):

(Sato–Tate theorem for families) Let \(\{p_k\}_{k\geqslant 1}\) be a sequence of primes tending to \(\infty \). Suppose that \(\dfrac{\log p_k}{\log N_k} \rightarrow 0\) (resp. \(\dfrac{\log p_k}{\log m(\xi _k)} \rightarrow 0\)) as \(k\rightarrow \infty \) if \(\xi _k\) (resp. \(N_k\)) remains fixed as \(k\) varies. Then
$$\begin{aligned} \lim _{k\rightarrow \infty } \widehat{\mu }^{\mathrm {count}}_{\mathcal {F}_k,p_k}(\widehat{\phi }_{p_k}) = \widehat{\mu }^{\mathrm {ST}}(\widehat{\phi }). \end{aligned}$$
Theorem 1.3 and Corollary 1.4 remain valid if any finite number of primes are simultaneously considered in place of \(p\) or \(p_k\). Moreover (i) of the corollary holds true for more general (and possibly ramified) test functions \(\widehat{\phi }_p\) on \(G(\mathbb {Q}_p)^{\wedge }\) thanks to Sauvageot’s density theorem. It would be interesting to quantify the error bounds in this generality. Finally the above results should be compared with the proposition 4 in [97] and the theorem 1 in [78] for modular forms on \(\mathrm{GL}(2)\). We also note [90] for Maass forms (which are not considered in the the present paper).
1.5 Random matrices
We provide a brief account of the theory of random matrices. The reader will find more details in Sect. 11.1 and extensive treatments in [59, 74].
The Gaussian unitary ensemble and Gaussian orthogonal ensemble were introduced by Wigner in the study of resonances of heavy nucleus. The Gaussian symplectic ensemble was introduced later by Dyson together with his circular ensembles. In this paper we are concerned with the ensembles attached to compact Lie groups which are introduced by Katz–Sarnak and occur in the statistics of \(L\)functions. (See [39] for the precise classification of these ensembles attached to different Riemannian symmetric spaces.)
One considers eigenvalues of matrices in compact groups \(\mathcal {G}(N)\) of large dimension endowed with the Haar probability measure. We have three symmetry types \(\mathcal {G}={{\mathrm{SO}}}(even)\) (resp. \(\mathcal {G}={{\mathrm{U}}}\), \(\mathcal {G}={{\mathrm{USp}}}\)); the notation says that for all \(N\geqslant 1\), the groups are \(\mathcal {G}(N)={{\mathrm{SO}}}(2N)\) [resp. \(\mathcal {G}(N)={{\mathrm{U}}}(N)\) and \(\mathcal {G}(N)={{\mathrm{USp}}}(2N)\)].
For all matrices \(A \in \mathcal {G}(N)\) we have an associated sequence of normalized angles
For example in the case \(\mathcal {G}={{\mathrm{U}}}\), the eigenvalues of \(A\in {{\mathrm{U}}}(N)\) are given by \(e(\tfrac{\vartheta _j}{N})=e^{2i\pi \vartheta _j/N}\) for \(1\leqslant j\leqslant N\). The normalization is such that the mean spacing of the \((\vartheta _i)\) in (1.3) is about one.
For each \(N\geqslant 1\) these angles \((\vartheta _i)_{1\leqslant i\leqslant N}\) are correlated random variables (a point process). By the Weyl integration formula their joint density is proportional to
The parameter \(\beta \) is a measure of the repulsion between nearby eigenvalues. We have that \(\beta =1\) (resp. \(\beta =2\), \(\beta =4\)) for \(\mathcal {G}={{\mathrm{SO}}}(even)\) (resp. \(\mathcal {G}={{\mathrm{U}}}\), \(\mathcal {G}={{\mathrm{USp}}}\)).
A fundamental result of Gaudin–Mehta and Dyson, which has been extended to the above ensembles by Katz–Sarnak, is that when \(N\rightarrow \infty \) the distribution of the angles \((\vartheta _i)_{1\leqslant i\leqslant N}\) converges to a determinantal point process.^{Footnote 3} The kernel of the limiting point process when \(\mathcal {G}={{\mathrm{U}}}\) is given by the Dyson sine kernel
The kernel for \(\mathcal {G}={{\mathrm{SO}}}(even)\) is \(K_+(x,y)=K(x,y)+K(x,y)\) and the kernel for \(\mathcal {G}={{\mathrm{USp}}}\) is \(K_(x,y)=K(x,y)K(x,y)\).
In particular this means that there is a limiting \(1\) level density \(W(\mathcal {G})\) for the angles \((\vartheta _i)_{1\leqslant i \leqslant N}\) as \(N\rightarrow \infty \) (see also Proposition 11.1). It is given by the following formulas:
1.6 Lowlying zeros
We can now state more precisely our results on families of \(L\)functions. Let \(\mathfrak {F}=r_*\mathcal {F}\) be a family of \(L\)functions as defined above in Sects. 1.1–1.2.
For all \(\Pi \in \mathfrak {F}_k\) we denote by \(\rho _j(\Pi )\), the zeros of the completed \(L\)function \(\Lambda (s,\Pi )\), where \(j\in \mathbb {Z}\). We write \(\rho _j(\Pi )=\frac{1}{2}+i\gamma _j(\Pi )\) and therefore \(\frac{1}{2} < \mathrm {Re} \gamma _j(\Pi ) < \frac{1}{2}\) for all \(j\). By the functional equation \(\Lambda (\frac{1}{2} + i\gamma ,\Pi )=0\) if and only if \(\Lambda (\frac{1}{2} + i\overline{\gamma },\Pi )=0\). We do not assume the GRH that would further imply \(\gamma _j(Pi)\in \mathbb {R}\) for all \(j\).
In the case that \(\Pi \) is selfdual the zeros occur in complex pairs, namely \(L(\frac{1}{2}+i\gamma ,\Pi )=0\) if and only if \(\Lambda (\frac{1}{2} i\gamma ,\Pi )=0\).
Following Iwaniec–Sarnak we associate an analytic conductor \(C(\mathfrak {F}_k)\geqslant 1\) to the family, see Sects. 4.2 and 11.5. We assume from now that the family is in the weight aspect, so that for each \(k\geqslant 1\), all of \(\Pi \in \mathfrak {F}_k\) share the same archimedean factor \(\Pi _\infty \) and we can set \(C(\mathfrak {F}_k):=C(\Pi _\infty )\). (For families in the level aspect we obtain similar results, see Sect. 11). Note that \(C(\mathfrak {F}_k)\rightarrow \infty \) and furthermore we shall make the assumption that \(\log C(\mathfrak {F}_k) \asymp \log m(\xi _k)\) as \(k\rightarrow \infty \).
For a given \(\Pi \in \mathfrak {F}_k\) the number of zeros \(\gamma _j(\Pi )\) of bounded height is \(\asymp \) \(\log C(\mathfrak {F}_k)\). The lowlying zeros of \(\Lambda (s,\Pi )\) are those within distance \(O(1/\log (C(\mathfrak {F}_k))\) to the central point; heuristically there are a bounded number of lowlying zeros for a given \(\Pi \in \mathfrak {F}_k\), although this can only be proved on average over the family. For a technical reason related to the fact that the explicit formula counts both the zeros and poles of \(\Lambda (s,\Pi )\) (Sect. 4.4), we make an hypothesis on the occurrence of poles of \(\Lambda (s,\Pi )\) for \(\Pi \in \mathfrak {F}_k\), see Hypothesis 11.2.
The statistics of lowlying zeros of the family are studied via the functional
where \(\Phi \) is a Paley–Wiener function. This is the \(1\)level density for the family \(\mathfrak {F}_k\). Choosing \(\Phi \) as a smooth approximation of the characteristic function of an interval \([a,b]\), the sum (1.6) should be thought as a weighted count of all the zeros of the family lying in \([a,b]\):
We want to compare the asymptotic as \(k\rightarrow \infty \) with the limiting \(1\)level density of normalized angles (1.3) of the random matrix ensembles described in Sect. 1.5 above.
Theorem 1.5
Let \(r:{}^LG:\rightarrow \mathrm{GL}(d,\mathbb {C})\) be a continuous \(L\)homomorphism which is irreducible and nontrivial when restricted to \(\widehat{G}\). There exists \(\delta >0\) depending on \(\mathfrak {F}\) such that the following holds. Let \(\mathfrak {F}=r_*\mathcal {F}\) be a family of \(L\)functions in the weight aspect as in Sect. 1.2, assuming the functoriality conjecture as in Hypothesis 10.1. Assume Hypothesis 11.2 concerning the poles of \(\Lambda (s,\Pi )\) for \(\Pi \in \mathfrak {F}_k\). Then for all PaleyWiener functions \(\Phi \) whose Fourier transform \(\widehat{\Phi }\) has support in \((\delta ,\delta )\):
 (i):

there is a limiting \(1\)level density for the lowlying zeros, namely there is a density \(W(x)\) such that
$$\begin{aligned} \lim _{k\rightarrow \infty } D(\mathfrak {F}_k;\Phi ) = \int _{\infty }^{\infty } \Phi (x)W(x)dx; \end{aligned}$$  (ii):

the density \(W(x)\) is determined by the Frobenius–Schur indicator of the irreducible representation \(r\). Precisely,
$$\begin{aligned} W={\left\{ \begin{array}{ll} W({{\mathrm{SO}}}(even)),&{} \text {if }s(r)=1,\\ W({{\mathrm{U}}}),&{} \text {if }s(r)=0,\\ W({{\mathrm{USp}}}),&{} \text {if }s(r)=1. \end{array}\right. } \end{aligned}$$(1.7)
The constant \(\delta >0\) depends on the family \(\mathfrak {F}\), in other words it depends on the group \(G\), the \(L\)morphism \(r:{}^LG \rightarrow \mathrm{GL}(d,\mathbb {C})\) and the limit of the ratio \(\dfrac{\log C(\mathfrak {F}_k)}{\log m(\xi _k)}\). Its numerical value is directly related to the numerical values of the exponents in the error term occurring in Theorem 1.3. Although we do not attempt to do so in the present paper, it is interesting to produce a value of \(\delta \) that is as large as possible, see [53] for the case of \(\mathrm{GL}(2)\). This would require sharp bounds for orbital integrals as can be seen from the outline below. A specific problem would be to optimize the exponents \(a,b,e\) in (1.9). (In fact we can achieve \(e=1\), see Sect. 1.7 below.)
Our proofs of Theorems 1.3 and 1.5 are effective in the sense that each constant and each exponent in the statements of the estimates could, in principle, be made explicit. Finally we note that, refining the work of E. Royer, Cogdell and Michel [31] have studied the question of distribution of \(L\)values at the edge in the case of symmetric powers of \(\mathrm{GL}(2)\) and noted in that context the relevance of the indicator \(s(r)\).
1.7 Outline of proofs
A wide range of methods are used in the proof. Among them are the ArthurSelberg trace formula, the analytic theory of \(L\)functions, representation theory and harmonic analysis on \(p\)adic and real groups, and random matrix theory.
The first main result of our paper is Theorem 1.3, proved in Sect. 9. We already pointed out after stating the theorem that the Sato–Tate equidistribution for families (Corollary 1.4) is derived from Theorem 1.3 and the fact that the Plancherel measure tends to the Sato–Tate measure as the residue characteristic is pushed to \(\infty \).
Let us outline the proof of the theorem. In fact we restrict our attention to part (ii), as (i) is handled by a similar method and only simpler to deal with. Thus we consider \({\mathcal {F}}\) with fixed level and weight \(\xi \), where \(\xi \) is regarded as a variable. Our starting point is to realize that for \(\phi _p\in C^\infty _c(G(\mathbb {Q}_p))\), we may interpret \(\widehat{\mu }^{\mathrm {count}}_{{\mathcal {F}},p}(\widehat{\phi }_p)\) in terms of the spectral side of the trace formula for \(G\) evaluated against the function \(\phi _p\phi ^{\infty ,p}\phi _\infty \in C^\infty _c(G(\mathbb {A}))\) for a suitable \(\phi ^{\infty ,p}\) (depending on \({\mathcal {F}}\) and \(p\); note that \(p\) is allowed to vary) and an Euler–Poincaré function \(\phi _\infty \) at \(\infty \) (depending on \(\xi \)). Applying the trace formula, which has a simple form thanks to \(\phi _\infty \), we get a geometric expansion for \(\widehat{\mu }^{\mathrm {count}}_{{\mathcal {F}},p}(\widehat{\phi }_p)\):
where \(a'_{M,\gamma }\in \mathbb {C}\) is a coefficient encoding a certain volume associated with the connected centralizer of \(\gamma \) in \(M\) and \(\phi ^\infty _{M}\) is the constant term of \(\phi ^\infty \) along (a parabolic subgroup associated with) \(M\). The Plancherel formula identifies the term for \(M=G\) and \(\gamma =1\) with \(\widehat{\mu }^{\mathrm {pl}}_p(\widehat{\phi }_p)\), which basically dominates the right hand side.
The proof of Theorem 1.3 (ii) boils down to bounding the other terms on the right hand side of (1.8). Here is a rough explanation of how to analyze each component there. The first summation is finite and controlled by \(G\), so we may as well look at the formula for each \(M\). There are finitely many conjugacy classes in the second summation for which the summand is nonzero. The number of such conjugacy classes may be bounded by a power of \(p\) where the exponent of \(p\) depends only on \(\kappa \) (measuring the “complexity” of \(\phi _p\)). The term \(a'_{M,\gamma }\), when unraveled, involves a special value of some Artin \(L\)function. We establish a bound on the special value which suffices to deal with \(a'_{M,\gamma }\). The last term \(\frac{\Phi ^G_M(\gamma ,\xi )}{\dim \xi }\) can be estimated by using a character formula for the stable discrete series character \(\Phi ^G_M(\gamma ,\xi )\) as well as the dimension formula for \(\xi \). It remains to take care of \(O^{M(\mathbb {A}^\infty )}_\gamma (\phi ^\infty _{M})\). This turns out to be the most difficult task since Theorem 1.3 asks for a bound that is uniform as the residue characteristic varies.
We are led to prove that there exist \(a,b,e> 0\), depending only on \(G\), such that for almost all \(q\),
for all semisimple \(\gamma \) and all \(\phi _q\) with \(\phi _q\in {\mathcal {H}}^{\mathrm {ur}}(M(\mathbb {Q}_q))^{\leqslant \kappa }\) and \(\phi _q\leqslant 1\), where \(D^M(\cdot )\) denotes the Weyl discriminant. The justification of (1.9) takes up the whole of Sect. 7. The problem already appears to be deep for the unit elements of unramified Hecke algebras in which case one can take \(\kappa =0\). (By a different argument based on arithmetic motivic integration, Cluckers, Gordon, and Halupczok establish a stronger uniform bound with \(e=1\). This work is presented in Appendix B.) At the (fixed) finite set of primes where wild ramification occurs, the problem comes down to bounding the orbital integral \(O^{M(\mathbb {Q}_q)}_\gamma (\phi _q)\) for fixed \(q\) and \(\phi _q\). It is deduced from the Shalika germ theory that the orbital integral is bounded by a constant, if normalized by the Weyl discriminant \(D^M(\gamma )^{1/2}\), as \(\gamma \) runs over the set of semisimple elements. See Appendix A by Kottwitz for details.
We continue with Theorem 1.5. The proof relies heavily on Theorem 1.3. The connection between the two statements might not be immediately apparent.
A standard procedure based on the explicit formula (see Sect. 4) expresses the sum (1.6) over zeros of \(L\)function as a sum over prime numbers of Satake parameters. The details are to be found in Sect. 12, and the result is that \(D(\mathfrak {F}_k,\Phi )\) can be approximated by
Here \(\phi _p\in \mathcal {H}^{\mathrm {ur}}(G(\mathbb {Q}_p))^{\leqslant \kappa }\) is suitably chosen such that \(\widehat{\phi }_p(\pi _p)\) is a sum of powers of the Satake parameters of \(r_*\pi \) (see Sects. 2 and 3). The integer \(\kappa \) may be large but it depends only on \(r\) so should be considered as fixed. Also the sum is over unramified primes. We have \(\log C(\mathcal {F}_k) \asymp \log m(\xi _k)\) (see Sects. 10 and 11). We deduce that the sum is supported on those primes \(p\leqslant m(\xi _k)^{A\delta }\) where \(A\) is a suitable constant and \(\delta \) is as in Theorem 1.5.
We apply Theorem 1.3 which has two components: the main term and the error term. We begin with the main term which amounts to substituting \(\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p(\widehat{\phi }_p)\) for \(\widehat{\mu }^{\mathrm {count}}_{\mathcal {F}_k,p}(\widehat{\phi }_p)\) in (1.10). Unlike \(\widehat{\mu }^{\mathrm {count}}_{\mathcal {F}_k,p}\), this term is purely local, thus simpler. Indeed \(\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p(\widehat{\phi }_p)\) can be computed explicitly for low rank groups, see e.g. [48] for all the relevant properties of the Plancherel measure. However we want to establish Theorem 1.5 in general so we proceed differently.
Using certain uniform estimates by Kato [57], we can approximate \(\widehat{\mu }^{\mathrm {pl,\mathrm {ur}}}_p(\widehat{\phi }_p)\) by a much simpler expression that depends directly on the restriction of \(r\) to \(\widehat{G} \rtimes W_{\mathbb {Q}_p}\). Then a pleasant computation using the Cebotarev equidistribution theorem, Weyl’s unitary trick and the properties of the Frobenius–Schur indicator shows that the sum over primes of this main term contribute \(\frac{s(r)}{2}\Phi (0)\) to (1.10). This exactly reflects the identities (1.7) in the statement (ii) of Theorem 1.5.
We continue with the error term \(O(p^{A_\mathrm {wt}+B_\mathrm {wt}\kappa }m(\xi _k)^{C_\mathrm {wt}})\) which we need to insert in (1.10). We can see the reasons why the proof of Theorem 1.5 requires the full force of Theorem 1.3 and its error term: the polynomial control by \(p^{A_\mathrm {wt}+B_\mathrm {wt}\kappa }\) implies that the sum over primes is at most \(m(\xi _k)^{D\delta }\) for some \(D>0\); the power saving \(m(\xi _k)^{C_\mathrm {wt}}\) is exactly what is needed to beat \(m(\xi _k)^{D\delta }\) when \(\delta \) is chosen small enough.
1.8 Notation
We distinguish the letter \(\mathcal {F}\) for families of automorphic representations on general reductive groups and \(\mathfrak {F}=r_*\mathcal {F}\) for the families of automorphic representations on \(\mathrm{GL}(d)\).
Let us describe in words the significance of various constants occurring in the main statements. We often use the convention to write multiplicative constants in lowercase letters and constants in the exponents in uppercase or greek letters.

The exponent \(\beta \) from Lemma 2.6 is such that for all \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d)\) of degree at most \(\kappa \), the pullback \(r^*\phi \) is of degree at most \(\leqslant \beta \kappa \).

The exponent \(b_G\) from Lemma 2.14 controls a bound for the constant term \(\left \phi _M(1)\right \) for all Levi subgroups \(M\) and \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(G)\) of degree at most \(\kappa \).

The exponent \(0<\theta <\frac{1}{2}\) is a nontrivial bound towards RamanujanPetersson for \(\mathrm{GL}(d,\mathbb {A})\).

The integer \(i\geqslant 1\) in Corollary 6.9 is an upperbound for the ramification of the Galois group \(\mathrm{Gal}(E/F)\).

The constants \(B_\Xi \) and \(c_\Xi \) in Lemma 8.4 and \(A_3,B_3\) in Proposition 8.7 control the number of rational conjugacy classes intersecting a small open compact subgroup.

The integer \(u_G\geqslant 1\) in Lemma 8.11 is a uniform upper bound for the number of \(G(F_v)\)conjugacy classes in a stable conjugacy class.

The integer \(n_G\geqslant 0\) is the minimum value for the dimension of the unipotent radical of a proper parabolic subgroup of \(G\) over \(\overline{F}\).

The constant \(c>0\) is a bound for the number of connected components \(\pi _0(Z(\widehat{I}_\gamma )^{\Gamma })\) in Corollary 8.12.

The exponents \(A_\mathrm {lv},B_\mathrm {lv},C_\mathrm {lv}>0\) in Theorem 9.16 (see also Theorem 1.3) and \(A_\mathrm {wt},B_\mathrm {wt},C_\mathrm {wt}>0\) in Theorem 9.19.

For families in the weight aspect, the constant \(\eta >0\) which may be chosen arbitrary small enters in the condition (11.5) that the dominant weights attached to \(\xi _k\) stay away from the walls.

The exponent \(C_{\text {pole}}>0\) in the Hypothesis 11.2 concerning the density of poles of \(L\)functions.

The exponents \(0<C_1 < C_2\) control the analytic conductor \(C(\mathfrak {F}_k)\) of the families in the weight aspect [Inequality (11.7)] and \(0<C_3 < C_4\) in the level aspect (Hypothesis 11.4).

The constant \(\delta >0\) in Theorem 11.5 controls the support of the Fourier transform \(\widehat{\Phi }\) of the test function \(\Phi \).

The constant \(c(f)>0\) depending on the test function \(f\) is a uniform upper bound for normalized orbital integrals \(D^G(\gamma )^{\frac{1}{2}} O_\gamma (f)\) (Appendix A).
Several constants are attached directly to the group \(G\) such as the dimension \(d_G=\dim G\), the rank \(r_G=\mathrm{rk}G\), the order of the Weyl group \(w_G=\left \Omega \right \), the degree \(s_G\) of the smallest extension of \(F\) over which \(G\) becomes split. Also in Lemma 2.14 the constant \(b_G\) gives a bound for the constant terms along Levi subgroups. The constants \(a_G,b_G, e_G\) in Theorem 7.3 gives a uniform bound for certain orbital integrals. In general we have made effort to keep light and consistent notation throughout the text.
In Sect. 6 we will choose a finite extension \(E/F\) which splits maximal tori of subgroups of \(G\). The degree \(s_G^{\mathrm{spl}}=[E:F]\) will be controlled by \(s_G^{\mathrm{spl}}\leqslant s_Gw_G\) (see Lemma 6.5), while the ramification of \(E/F\) will vary. In Sect. 5 we consider the finite extension \(F_1/F\) such that \(\mathrm{Gal}(\overline{F}/F)\) acts on \(\widehat{G}\) through the faithful action of \(\mathrm{Gal}(F_1/F)\). For example if \(G\) is a nonsplit inner form of a split group then \(F_1=F\). In Sect. 12 we consider a finite extension \(F_2/F_1\) such that the representation \(r\) factors through \(\widehat{G}\rtimes \mathrm{Gal}(F_2/F)\). For a general \(G\), there might not be any direct relationship between the extensions \(E/F\) and \(F_2/F_1/F\).
1.9 Structure of the paper
For a quick tour of our main results and the structure of our arguments, one could start reading from Sect. 9 after familiarizing oneself with basic notation, referring to earlier sections for further notation and basic facts as needed.
The first Sects. 2 and 3 are concerned with harmonic analysis on reductive groups over local fields, notably the Satake transform, \(L\)groups and \(L\)morphisms, the properties of the Plancherel measure and the Macdonald formula for the unramified spectrum. We establish bounds for truncated Hecke algebras and for character traces that will play a role in subsequent chapters. In Sect. 4 we recall various analytic properties of automorphic \(L\)functions on \(\mathrm{GL}(d)\) and notably isobaric sums, bounds towards Ramanujan–Petersson and the socalled explicit formula for the sum of the zeros. Section 5 introduces the Sato–Tate measure for general groups and Sato–Tate equidistribution for Satake parameters and for families. The next Sect. 6 gathers various background materials on orbital integral, the Gross motive and Tamagawa measure, discrete series characters and Euler–Poincaré functions, and Frobenius–Schur indicator. We establish bounds for special values of the Gross motive which will enter in the geometric side of the trace formula.
In Sect. 7 we establish a uniform bound for orbital integrals of the type (1.9). In Sect. 8 we establish various bounds on conjugacy classes and level subgroups. How these estimates enter in the trace formula has been detailed in the outline above.
Then we are ready in Sect. 9 to establish our main result, an automorphic Plancherel theorem for families with error terms and its application to the Sato–Tate theorem for families. The theorem is first proved for test functions on the unitary dual coming from Hecke algebras by orchestrating all the previous results in the trace formula. Then our result is improved to allow more general test functions, either in the input to the Sato–Tate theorem or in the prescribed local condition for the family, by means of Sauvageot’s density theorem.
The last three Sects. 10, 11 and 12 concern the application to lowlying zeros. In complete generality we need to rely on Langlands global functoriality and other hypothesis that we state precisely. These unproven assumptions are within reach in the context of endoscopic transfer and we will return to it in subsequent works.
Appendix A by Kottwitz establishes the boundedness of normalized orbital integrals from the theory of Shalika germs. Appendix B by Cluckers–Gordon–Halupczok establishes a strong form of (1.9) with \(e=1\) by using recent results in arithmetic motivic integration.
2 Satake transforms
2.1 \(L\)groups and \(L\)morphisms
We are going to recall some definitions and facts from [9, §1, §2] and [62, §1]. Let \(F\) be a local or global field of characteristic 0 with an algebraic closure \(\overline{F}\), which we fix. Let \(W_F\) denote the Weil group of \(F\) and set \(\Gamma :=\mathrm{Gal}(\overline{F}/F)\). Let \(H\) and \(G\) be connected reductive groups over \(F\). Let \((\widehat{B},\widehat{T},\{X_{\alpha }\}_{\alpha \in \Delta ^{\vee }})\) be a splitting datum fixed by \(\Gamma \), from which the \(L\)group
is constructed. An \(L\)morphism \(\eta :{}^L H\rightarrow {}^L G\) is a continuous map commuting with the canonical surjections \({}^L H\rightarrow W_F\) and \({}^L G\rightarrow W_F\) such that \(\eta _{\widehat{H}}\) is a morphism of complex Lie groups. A representation of \({}^L G\) is by definition a continuous homomorphism \({}^L G\rightarrow \mathrm{GL}(V)\) for some \(\mathbb {C}\)vector space \(V\) with \(\dim V<\infty \) such that \(r_{\widehat{G}}\) is a morphism of complex Lie groups. Clearly giving a representation \({}^L G\rightarrow \mathrm{GL}(V)\) is equivalent to giving an \(L\)morphism \({}^L G\rightarrow {}^L \mathrm{GL}(V)\).
Let \(f:H\rightarrow G\) be a normal morphism, which means that \(f(H)\) is a normal subgroup of \(G\). Then it gives rise to an \(L\)morphism \({}^L G \rightarrow {}^L H\) as explained in [9, 2.5]. In particular, there is a \(\Gamma \)equivariant map \(Z(\widehat{G})\rightarrow Z(\widehat{H})\), which is canonical (independent of the choice of splittings). Thus an exact sequence of connected reductive groups over \(F\)
gives rise to a \(\Gamma \)equivariant exact sequence of \(\mathbb {C}\)diagonalizable groups
2.2 Satake transform
From here throughout this section, let \(F\) be a finite extension of \(\mathbb {Q}_p\) with integer ring \(\mathcal {O}\) and a uniformizer \(\varpi \). Set \(q:=\mathcal {O}/\varpi \mathcal {O}\). Let \(G\) be an unramified group over \(F\) and \(B=TU\) be a Borel subgroup decomposed into the maximal torus and the unipotent radical in \(B\). Let \(A\) denote the maximal \(F\)split torus in \(T\). Write \(\Phi _F\) (resp. \(\Phi \)) for the set of all \(F\)rational roots (resp. all roots over \(\overline{F}\)) and \(\Phi _F^+\) (resp. \(\Phi ^+\)) for the subset of positive roots. Choose a smooth reductive model of \(G\) over \(\mathcal {O}\) corresponding to a hyperspecial point on the apartment for \(A\). Set \(K:=G(\mathcal {O})\). Denote by \(X_*(A)^+\) the subset of \(X_*(A)\) meeting the closed Weyl chamber determined by \(B\), namely \(\lambda \in X_*(A)^+\) if \(\alpha (\lambda )\geqslant 0\) for all \(\alpha \in \Phi _F^+\). Denote by \(\Omega _F\) (resp. \(\Omega \)) the \(F\)rational Weyl group for \((G,A)\) (resp. the absolute Weyl group for \((G,T\))), and \(\rho _F\) (resp. \(\rho \)) the half sum of all positive roots in \(\Phi ^+_F\) (resp. \(\Phi ^+\)). A partial order \(\leqslant \) is defined on \(X_*(A)\) (resp. \(X_*(T)\)) such that \(\mu \leqslant \lambda \) if \(\lambda \mu \) is a linear combination of \(F\)rational positive coroots (resp. positive coroots) with nonnegative coefficients. The same order extends to a partial order \(\leqslant _{\mathbb {R}}\) on \(X_*(A)\otimes _\mathbb {Z}\mathbb {R}\) and \(X_*(T)\otimes _\mathbb {Z}\mathbb {R}\) defined analogously.
Let \(F^{\mathrm {ur}}\) denote the maximal unramified extension of \(F\). Let \(\mathrm{Fr}\) denote the geometric Frobenius element of \(\mathrm{Gal}(F^{\mathrm {ur}}/F)\). Define \(W_F^{\mathrm {ur}}\) to be the unramified Weil group, namely the subgroup \(\mathrm{Fr}^{\mathbb {Z}}\) of \(\mathrm{Gal}(F^{\mathrm {ur}}/F)\). Since \(\mathrm{Gal}(\overline{F}/F)\) acts on \(\widehat{G}\) through a finite quotient of \(\mathrm{Gal}(F^{\mathrm {ur}}/F)\), one can make sense of \({}^L G^{\mathrm {ur}}:=\widehat{G}\rtimes W_F^{\mathrm {ur}}\).
Throughout this section we write \(G\), \(T\), \(A\) for \(G(F)\), \(T(F)\), \(A(F)\) if there is no confusion. Define \({\mathcal {H}}^{\mathrm {ur}}(G):=C^\infty _c(K\backslash G / K)\) and \({\mathcal {H}}^{\mathrm {ur}}(T):=C^\infty _c(T(F)/T(F)\cap K)\). The latter is canonically isomorphic to \({\mathcal {H}}^{\mathrm {ur}}(A):=C^\infty _c(A(F)/A(\mathcal {O}))\) via the inclusion \(A\hookrightarrow T\). We can further identify
where the last \(\mathbb {C}\)algebra isomorphism matches \(\lambda \in X_*(A)\) with \({\mathbf {1}}_{\lambda (\varpi )(A\cap K)}\in {\mathcal {H}}^{\mathrm {ur}}(A)\). Let \(\lambda \in X_*(A)\). Write
The sets \(\{\tau ^G_\lambda \}_{\lambda \in X_*(A)^+}\) and \(\{\tau ^A_\lambda \}_{\lambda \in X_*(A)^+}\) are bases for \({\mathcal {H}}^{\mathrm {ur}}(G)\) and \({\mathcal {H}}^{\mathrm {ur}}(A)^{\Omega _F}\) as \(\mathbb {C}\)vector spaces, respectively. Consider the map
composed with \({\mathcal {H}}^{\mathrm {ur}}(T)\simeq {\mathcal {H}}^{\mathrm {ur}}(A)\) above. The composite map induces a \(\mathbb {C}\)algebra isomorphism
called the Satake isomorphism. We often write just \({\mathcal {S}}\) for \({\mathcal {S}}^G\). We note that in general \({\mathcal {S}}\) does not map \(\tau ^G_\lambda \) to \(\tau ^A_{\lambda }\).
Another useful description of \({\mathcal {H}}^{\mathrm {ur}}(G)\) is through representations of \({}^L G^{\mathrm {ur}}\). (The latter notion is defined as in Sect. 2.1). Write \((\widehat{G}\rtimes \mathrm{Fr})_{\mathrm{ssconj}}\) for the set of \(\widehat{G}\)conjugacy classes of semisimple elements in \(\widehat{G}\rtimes \mathrm{Fr}\). Consider the set
Define \(\mathbb {C}[\mathrm{ch}({}^L G^{\mathrm {ur}})]\) to be the \(\mathbb {C}\)algebra generated by \(\mathrm{ch}({}^L G^{\mathrm {ur}})\) in the space of functions on \((\widehat{G}\rtimes \mathrm{Fr})_{\mathrm{ssconj}}\). For each \(\lambda \in X_*(A)^+\) define the quotient
which exists as an element of \(\mathbb {C}[X_*(A)]^{\Omega _F}\) and is unique. (One may view \(\chi _\lambda \) as the analogue in the disconnected case of the irreducible character of highest weight \(\lambda \), cf. proof of Lemma 2.1 below.) Then \(\{\chi _\lambda \}_{\lambda \in X_*(A)^+}\) is a basis for \(\mathbb {C}[X_*(A)]^{\Omega _F}\) as a \(\mathbb {C}\)vector space, cf. [57, p. 465]. (Another basis was given by \(\tau ^A_\lambda \)’s above.) There is a canonical \(\mathbb {C}\)algebra isomorphism
determined as follows (see [9, Prop 6.7] for detail): for each irreducible \(r\), \(\mathrm{tr}\,r_{\widehat{T}}\) is shown to factor through \(\widehat{T}\rightarrow \widehat{A}\) (induced by \(A\subset T\)). Hence \(\mathrm{tr}\,r_{\widehat{T}}\) can be viewed as an element of \(\mathbb {C}[X^*(\widehat{A})]=\mathbb {C}[X_*(A)]\), which can be seen to be invariant under \(\Omega _F\). Define \({\mathcal {T}}(\mathrm{tr}\,r)\) to be the latter element.
Let \(r_0\) be an irreducible representation of \(\widehat{G}\) of highest weight \(\lambda _0\in X^*(\widehat{T})^+=X_*(T)^+\). The group \(W_F^{\mathrm {ur}}\) acts on \(X^*(\widehat{T})^+\). Write \(\mathrm {Stab}(\lambda _0)\subset W^{\mathrm {ur}}_F\) for the stabilizer subgroup for \(\lambda _0\), which has finite index (since a finite power of \(\mathrm{Fr}\) acts trivially on \(\widehat{G}\) and thus also on \(\widehat{T}\)). Put \(r:=\mathrm{Ind}_{\widehat{G}\rtimes \mathrm {Stab}(\lambda _0)}^{^L G^{\mathrm {ur}}} r_0\) and \(\lambda :=\sum _{\sigma \in W^{\mathrm {ur}}_F/\mathrm {Stab}(\lambda _0)} \sigma \lambda _0 \in X_*(A)^+\). Clearly \(r\) and \(\lambda \) depend only on the \(W_F^{\mathrm {ur}}\)orbit of \(\lambda _0\). Put \(i(\lambda _0):=[ W^{\mathrm {ur}}_F:\mathrm {Stab}(\lambda _0)]\).
Lemma 2.1
 (i):

Suppose that \(r\) and \(\lambda \) are obtained from \(r_0\) and \(\lambda _0\) as above. Then
$$\begin{aligned} {\mathcal {T}}(\mathrm{tr}\,r)=\chi _\lambda . \end{aligned}$$(2.5)  (ii):

In general for any irreducible representation \(r':{}^L G^{\mathrm {ur}}\rightarrow \mathrm{GL}_d(\mathbb {C})\) such that \(r'(W^{\mathrm {ur}}_F)\) has relatively compact image, let \(r_0\) be any irreducible subrepresentation of \(r'_{\widehat{G}}\). Let \(r\) be obtained from \(r_0\) as above. Then for some \(\zeta \in \mathbb {C}^\times \) with \(\zeta =1\),
$$\begin{aligned} \mathrm{tr}\,r'=\zeta \cdot \mathrm{tr}\,r. \end{aligned}$$
Proof
Let us prove (i). For any \(i\geqslant 1\), let \({}^L G_{i}\) denote the finite \(L\)group \(\widehat{G}\rtimes \mathrm{Gal}(F_{i}/F)\) where \(F_{i}\) is the degree \(i\) unramified extension of \(F\) in \(\overline{F}\). It is easy to see that \(r(\mathrm{Fr}^{i(\lambda _0)})\) is trivial and that \(r=\mathrm{Ind}^{{}^L G_{i(\lambda _0)}}_{\widehat{G}} r_0\). Then (2.5) amounts to Kostant’s character formula for a disconnected group [61, Thm 7.5] applied to \({}^L G_{i(\lambda _0)}\). As for (ii), let \(\lambda _0\) and \(\lambda \) be as in the paragraph preceding the lemma. Let \(j\geqslant 1\) be such that \(G\) becomes split over a degree \(j\) unramified extension of \(F\). (Recall that \(G\) is assumed to be unramified.) By twisting \(r'\) by a unitary character of \(W^{\mathrm {ur}}_F\) one may assume that \(r'\) factors through \({}^L G_j\). Then both \(r\) and \(r'\) factor through \({}^L G_j\) and are irreducible constituents of \(\mathrm{Ind}^{{}^L G_{j}}_{\widehat{G}} r_0\). From this it is easy to deduce that \(r'\) is a twist of \(r\) by a finite character of \(W^{\mathrm {ur}}_F\) of order dividing \(j\). Assertion (ii) follows. \(\square \)
Each \(\lambda \in X_*(A)^+\) determines \(s_{\lambda ,\mu }\in \mathbb {C}\) such that
where only finitely many \( s_{\lambda ,\mu }\) are nonzero. In fact Theorem 1.3 of [57] identifies \(s_{\lambda ,\mu }\) with \(K_{\lambda ,\mu }(q^{1})\) defined in (1.2) of that paper, cf. §4 of [48]. In particular \(s_{\lambda ,\lambda }\ne 0\) and \(s_{\lambda ,\mu }\ne 0\) unless \(\mu \leqslant \lambda \). The following information will be useful in Sect. 2.7.
Lemma 2.2
Let \(\lambda ,\mu \in X_*(A)^+\). Suppose that \(\lambda \star _w \mu :=w(\lambda +\rho _F) (\mu +\rho _F)\) is nontrivial for all \(w\in \Omega _F\). For \(\kappa \in X_*(A)\) let \(p(\kappa )\in \mathbb {Z}_{\geqslant 0}\) be the number of tuples \((c_{\alpha ^\vee })_{\alpha ^\vee \in (\Phi ^{\vee }_F)^+}\) with \(c_{\alpha ^\vee }\in \mathbb {Z}_{\geqslant 0}\) such that \(\sum _{\alpha ^\vee } c_{\alpha ^\vee }\cdot \alpha ^\vee = \kappa \). Then
Proof
It is easy to see from the description of \(K_{\lambda ,\mu }(q^{1})\) in [57, (1.2)] that
The definition of \(\widehat{{\fancyscript{P}}}\) in [57, (1.1)] shows that \(0\leqslant \widehat{{\fancyscript{P}}}(\kappa ;q^{1})\leqslant p(\kappa ) q^{1}\) if \(\kappa \ne 0\). \(\square \)
2.3 Truncated unramified Hecke algebras
Set \(n:=\dim T\) and \(X_*(T)_{\mathbb {R}}:=X_*(T)\otimes _\mathbb {Z}\mathbb {R}\). Choose an \(\mathbb {R}\)basis \({\mathcal {B}}=\{e_1,\ldots ,e_{n}\}\) of \(X_*(T)_\mathbb {R}\). For each \(\lambda \in X_*(T)_\mathbb {R}\), written as \(\lambda =\sum _{i=1}^{n} a_i(\lambda ) e_i\) for unique \(a_i(\lambda )\in \mathbb {R}\), define
When there is no danger of confusion, we will simply write \(\cdot _{{\mathcal {B}}}\) or even \(\cdot \) instead of \(\cdot _{{\mathcal {B}}}\), and similarly for \(\Vert \cdot \Vert _{{\mathcal {B}}}\). It is clear that \(\Vert \cdot \Vert _{{\mathcal {B}}}\) is \(\Omega \)invariant and that \(\lambda _1+\lambda _2_{{\mathcal {B}}}\leqslant \lambda _1_{{\mathcal {B}}}+\lambda _2_{{\mathcal {B}}}\) for all \(\lambda _1,\lambda _2\in X_*(T)\). When \(\kappa \in \mathbb {Z}_{\geqslant 0}\), define
It is simply written as \({\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant \kappa }\) when the choice of \({\mathcal {B}}\) is clear.
Lemma 2.3
Let \({\mathcal {B}}\) and \({\mathcal {B}}'\) be two \(\mathbb {R}\)bases of \(X_*(T)_\mathbb {R}\). Then there exist constants \(c_1,c_2,B_1,B_2,B_3,B_4>0\) such that for all \(\lambda \in X_*(T)_\mathbb {R}\),
 (i):

\(c_1 \lambda _{{\mathcal {B}}'} \leqslant \lambda _{{\mathcal {B}}} ~\,\leqslant c_2 \lambda _{{\mathcal {B}}'}\),
 (ii):

\(B_1 \lambda _{{\mathcal {B}}} \leqslant \Vert \lambda \Vert _{{\mathcal {B}}} ~\,\leqslant B_2 \lambda _{{\mathcal {B}}}\) for all \( \lambda \in X_*(T)_\mathbb {R}\),
 (iii):

\(B_3 \Vert \lambda \Vert _{{\mathcal {B}}'} \leqslant \Vert \lambda \Vert _{{\mathcal {B}}} ~\,\leqslant B_4 \Vert \lambda \Vert _{{\mathcal {B}}'}\) for all \( \lambda \in X_*(T)_\mathbb {R}\) and
 (iv):

\({\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant B_4^{1}\kappa ,{\mathcal {B}}'}\subset {\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant \kappa ,{\mathcal {B}}}\subset {\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant B_3^{1}\kappa ,{\mathcal {B}}'}\).
Proof
Let us verify (i). As the roles of \({\mathcal {B}}\) and \({\mathcal {B}}'\) can be changed, it suffices to prove the existence of \(c_2\). For this, it suffices to take \(c_2=\sup _{\lambda _{{\mathcal {B}}}\leqslant 1} \lambda _{{\mathcal {B}}'}\). The latter is finite since \(\cdot _{{\mathcal {B}}'}\) is a continuous function on the set of \(\lambda \) such that \(\lambda _{{\mathcal {B}}}\leqslant 1\), which is compact. Part (ii) is obtained by applying the lemma to the bases \({\mathcal {B}}'=\omega {\mathcal {B}}\) for all \(\omega \in \Omega \). Let us check (iii). Let \(B_1,B_2>0\) (resp. \(B'_1,B'_2>0\)) be the constants of (ii) for the basis \({\mathcal {B}}\) (resp. \({\mathcal {B}}'\)). Then
and similarly \(\Vert \lambda \Vert _{{\mathcal {B}}}\leqslant c_2B_2(B'_1)^{1}\Vert \lambda \Vert _{{\mathcal {B}}'}\). Finally (iv) immediately follows from (iii). \(\square \)
It is natural to wonder whether the definition of truncation in (2.7) changes if a different basis \(\{\tau ^G_\lambda \}\) or \(\{\chi _\lambda \}\) is used. We assert that it changes very little in a way that the effect on \(\kappa \) is bounded by a \(\kappa \)independent constant. To ease the statement define \({\mathcal {H}}_i^{\mathrm {ur}}(G)^{\leqslant \kappa ,{\mathcal {B}}}\) for \(i=1\) (resp. \(i=2\)) to be the \(\mathbb {C}\)subspace of \({\mathcal {H}}^{\mathrm {ur}}(G)\) generated by \({\mathcal {S}}^{1}(\tau ^A_\lambda )\) (resp. \({\mathcal {S}}^{1}(\chi _\lambda )\)) for \(\lambda \in X_*(A)^+\) with \(\Vert \lambda \Vert _{{\mathcal {B}}}\leqslant \kappa \).
Lemma 2.4
There exists a constant \(C\geqslant 1\) such that for every \(\kappa \in \mathbb {Z}_{\geqslant 0}\) and for any \(i,j\in \{\emptyset , 1,2\}\),
Proof
It is enough to prove the lemma for a particular choice of \({\mathcal {B}}\) by Lemma 2.3. So we may assume that \({\mathcal {B}}\) extends the set of simple coroots in \(\Phi ^\vee \) by an arbitrary basis of \(X_*(Z(G))_{\mathbb {R}}\). Again by Lemma 2.3 the proof will be done if we show that each of the following generates the same \(\mathbb {C}\)subspace:

(i)
the set of \(\tau ^G_{\lambda }\) for \(\lambda \in X_*(A)^+\) with \(\lambda _{{\mathcal {B}}}\leqslant \kappa \),

(ii)
the set of \({\mathcal {S}}^{1}(\tau ^A_\lambda )\) for \(\lambda \in X_*(A)^+\) with \(\lambda _{{\mathcal {B}}}\leqslant \kappa \),

(iii)
the set of \({\mathcal {S}}^{1}(\chi _\lambda )\) for \(\lambda \in X_*(A)^+\) with \(\lambda _{{\mathcal {B}}}\leqslant \kappa \).
It suffices to show that the matrices representing the change of bases are “upper triangular” in the sense that the \((\lambda ,\lambda )\) entries are nonzero and \((\lambda ,\mu )\) entries are zero unless \(\lambda \geqslant \mu \). (Note that \(\lambda \geqslant \mu \) implies \(\lambda _{{\mathcal {B}}}\geqslant \mu _{{\mathcal {B}}}\) by the choice of \({\mathcal {B}}\).) We have remarked below (2.3) that \(s_{\lambda ,\mu }\)’s have this property, accounting for (i)\(\leftrightarrow \)(iii). For (ii)\(\leftrightarrow \)(iii) the desired property can be seen directly from (2.3) by writing \(\chi _\lambda \) in terms of \(\tau ^A_\mu \)’s. \(\square \)
2.4 The case of \(\mathrm{GL}_d\)
The case \(G=\mathrm{GL}_d\) is considered in this subsection. Let \(A=T\) be the diagonal maximal torus and \(B\) the group of upper triangular matrices. For \(1\leqslant i\leqslant d\), take \(Y_i\in X_*(A)\) to be \(y\mapsto \mathrm{diag}(1,\ldots ,1,y,1,\ldots ,1)\) with \(y\) in the \(i\)th place. One can naturally identify \(X_*(A)\simeq \mathbb {Z}^d\) such that the images of \(Y_i\) form the standard basis of \(\mathbb {Z}^d\). Then \(\Omega _F\) is isomorphic to \({\fancyscript{S}}_d\), the symmetric group in \(d\) variables acting on \(\{Y_1,\ldots ,Y_d\}\) via permutation of indices. We have the Satake isomorphism
For an alternative description let us introduce standard symmetric polynomials \(X_1,\ldots ,X_d\) by the equation in a formal \(Z\)variable \((ZY_1)\ldots (ZY_d)=Z^dX_1Z^{d1}+\cdots + (1)^{d} X_d\). Then
Let \(\kappa \in \mathbb {Z}_{\geqslant 0}\). Define \({\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d)^{\leqslant \kappa }\), or simply \({\mathcal {H}}^{\leqslant \kappa }_d\), to be the preimage under \({\mathcal {S}}\) of the \(\mathbb {C}\)vector space generated by
The following is standard (cf. [48]).
Lemma 2.5
Let \(r\in \mathbb {Z}_{\geqslant 1}\). Let \(\lambda _r:=(r,0,0,\ldots ,0)\in X_*(A)^+\). Then
for \(c_{\lambda _r,\mu }\in \mathbb {C}\) with \(c_{\lambda _r,\lambda _r}=q^{r(1d)/2}\), where the sum runs over the set of \(\mu \in X_*(T)^+\) such that \(\mu \leqslant \lambda _r\). In particular,
2.5 \(L\)morphisms and unramified Hecke algebras
Assume that \(H\) and \(G\) are unramified groups over \(F\). Let \(\eta :{}^L H\rightarrow {}^L G\) be an unramified \(L\)morphism, which means that it is inflated from some \(L\)morphism \({}^L H^{\mathrm {ur}}\rightarrow {}^L G^{\mathrm {ur}}\) (the notion of \(L\)morphism for the latter is defined as in Sect. 2.1). There is a canonically induced map \(\mathrm{ch}({}^L G^{\mathrm {ur}})\rightarrow \mathrm{ch}({}^L H^{\mathrm {ur}})\). Via (2.2) and (2.4), the latter map gives rise to a \(\mathbb {C}\)algebra map \(\eta ^*:{\mathcal {H}}^{\mathrm {ur}}(G)\rightarrow {\mathcal {H}}^{\mathrm {ur}}(H)\).
We apply the above discussion to an unramified representation
Viewing \(r\) as an \(L\)morphism \({}^L G\rightarrow {}^L \mathrm{GL}_d\), we obtain
Lemma 2.6
Let \({\mathcal {B}}\) be an \(\mathbb {R}\)basis of \(X_*(T)_\mathbb {R}\). There exists a constant \(\beta >0\) (depending on \({\mathcal {B}}\), \(d\) and \(r\)) such that for all \(\kappa \in \mathbb {Z}_{\geqslant 0}\), \(r^*({\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d)^{\leqslant \kappa })\subset {\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant \beta \kappa ,{\mathcal {B}}}\) .
Proof
Thanks to Lemma 2.3, it is enough to deal with a particular choice of \({\mathcal {B}}\). Choose \({\mathcal {B}}\) by extending the set \(\Delta ^{\vee }\) of simple coroots, and write \({\mathcal {B}}=\Delta ^{\vee }\coprod {\mathcal {B}}_0\). We begin by proving the following claim: let \(\lambda _1,\lambda _2\in X_*(A)^+\) and expand the convolution product
where only \(\mu \in X_*(A)^+\) such that \(\mu \leqslant _\mathbb {R}\lambda _1+\lambda _2\) contribute (cf. [18, p. 148]). Only finitely many terms are nonzero. Then the claim is that
To check the claim, consider \(\mu =\sum _{e\in {\mathcal {B}}} a_e(\mu ) \cdot e \) and \(\lambda _1+\lambda _2 = \sum _{e\in {\mathcal {B}}} a_e(\lambda _1+\lambda _2) \cdot e\), where the coefficients are in \(\mathbb {R}\). The conditions \(\mu \leqslant _{\mathbb {R}} \lambda _1+\lambda _2\) and \(\mu \in X_*(T)_{\mathbb {R},+}\) imply that \(a_e(\mu )=a_e(\lambda _1+\lambda _2)\) if \(e\in {\mathcal {B}}_0\) and \(0\leqslant a_e(\mu )\leqslant a_e(\lambda _1+\lambda _2)\) if \(e\in \Delta ^{\vee }\). Hence \(\mu _{{\mathcal {B}}}\leqslant \lambda _1+\lambda _2_{{\mathcal {B}}}\).
We are ready to prove the lemma. It is explained in Lemma 2.4 and the remark below it that there exists a constant \(\beta _1>0\) which is independent of \(\kappa \) such that every \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d)^{\leqslant \kappa }\) can be written as a \(\mathbb {C}\)linear combination of
Each element above can be rewritten in terms of the symmetric polynomials \(X_i\)’s of Sect. 2.4: first, \(X_d^{\beta _1\kappa }\) times \(\sum _{\sigma \in {\fancyscript{S}}_d} Y_{\sigma (1)}^{a_1}Y_{\sigma (2)}^{a_2}\ldots Y_{\sigma (d)}^{a_d}\) is a symmetric polynomial of degree \(\leqslant 2\beta _1\kappa \), which in turn is a polynomial in \(X_1,\ldots ,X_d\) of degree \(\leqslant 2\beta _1\kappa \). We conclude that every \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d)^{\leqslant \kappa }\) is in the span of monomials
For each \(1\leqslant i\leqslant d\), write \(r^*(X_i)\) [resp. \(r^*(X_i^{1})\)] as a linear combination of \(\tau ^G_{\lambda _{i,j}}\) (resp. \(\tau ^G_{\lambda ^_{i,j}}\)) with nonzero coefficients. Define \(\beta _0\) to be the maximum among all possible \(\lambda _{i,j}\) and \(\lambda ^_{i,j}\). The above claim \(r^*(X_1^{b_1}X_1^{b_2}\ldots X_d^{b_d})\) as in (2.8) is in the \(\mathbb {C}\)span of \(\tau ^G_\mu \) satisfying
So the above span contains \(r^*(\phi )\) for \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d)^{\leqslant \kappa }\). By Lemma 2.3 there exists a constant \(B_2>0\) such that \(\Vert \mu \Vert _{{\mathcal {B}}}\leqslant B_2\mu _{{\mathcal {B}}}\) for every \(\mu \in X_*(T)\). Hence the lemma holds true with \(\beta :=2B_2d\beta _0\beta _1\). \(\square \)
The map \(r\) also induces a functorial transfer for unramified representations
uniquely characterized by \(\mathrm{tr}\,r_*(\pi )(\phi )=\mathrm{tr}\,\pi (r^*\phi )\) for all \(\pi \in \mathrm{Irr}^{\mathrm {ur}}(G(F))\) and \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(\mathrm{GL}_d(F))\).
2.6 Partial Satake transform
Keep the assumption that \(G\) is unramified over \(F\). Let \(P\) be an \(F\)rational parabolic subgroup of \(G\) with Levi \(M\) and unipotent radical \(N\) such that \(B=TU\) is contained in \(P\). Let \(\Omega _M\) (resp. \(\Omega _{M,F}\)) denote the absolute (resp. \(F\)rational) Weyl group for \((M,T)\). A partial Satake transform is defined as [cf. (2.1)]
It is well known that \({\mathcal {S}}^G={\mathcal {S}}^M\circ {\mathcal {S}}^G_M\). More concretely, \({\mathcal {S}}^G_M\) is the canonical inclusion \(\mathbb {C}[X_*(A)]^{\Omega _{M,F}}\hookrightarrow \mathbb {C}[X_*(A)]^{\Omega _{F}}\) if \({\mathcal {H}}^{\mathrm {ur}}(M)\) and \({\mathcal {H}}^{\mathrm {ur}}(G)\) are identified with the source and the target via \({\mathcal {S}}^G\) and \({\mathcal {S}}^M\), respectively. Since \(T\) is a common maximal torus of \(M\) and \(G\), an \(\mathbb {R}\)basis \({\mathcal {B}}\) of \(X_*(T)_\mathbb {R}\) determines truncations on \({\mathcal {H}}^{\mathrm {ur}}(M)\) and \({\mathcal {H}}^{\mathrm {ur}}(G)\).
Lemma 2.7
For any \(\kappa \in \mathbb {Z}_{\geqslant 0}\), \({\mathcal {S}}^G_M({\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant \kappa ,{\mathcal {B}}})\subset {\mathcal {H}}^{\mathrm {ur}}(M)^{\leqslant \kappa ,{\mathcal {B}}}\).
Proof
It is enough to note that \(\Vert \lambda \Vert _{{\mathcal {B}},M}\leqslant \Vert \lambda \Vert _{{\mathcal {B}},G}\) for all \(\lambda \in X_*(A)\), which holds since the \(\Omega _{M}\)orbit of \(\lambda \) is contained in the \(\Omega \)orbit of \(\lambda \).\(\square \)
Remark 2.8
Let \(\eta :{}^L M\rightarrow {}^L G\) be the embedding of [9, §3], well defined up to \(\widehat{G}\)conjugacy. Then \({\mathcal {S}}^G_M\) coincides with \(\eta ^*:{\mathcal {H}}^{\mathrm {ur}}(G)\rightarrow {\mathcal {H}}^{\mathrm {ur}}(M)\) of Sect. 2.5
2.7 Some explicit test functions
Assume that \(r:{}^L G=\widehat{G}\rtimes W_F\rightarrow \mathrm{GL}_d(\mathbb {C})\) is an irreducible representation arising from an unramified \(L\)morphism \({}^L G^{\mathrm {ur}} \rightarrow {}^L \mathrm{GL}_d^{\mathrm {ur}}\) such that \(r(W_F)\) is relatively compact. For later applications it is useful to study the particular element \(r^*(Y_1+\cdots +Y_d)\) in \({\mathcal {H}}^{\mathrm {ur}}(G)\).
Lemma 2.9
Let \(\phi =r^*(Y_1+\cdots +Y_d)\). Then
 (i):

Suppose that \(r:{}^L G^{\mathrm {ur}}\rightarrow \mathrm{GL}_d(\mathbb {C})\) does not factor through \(W^{\mathrm {ur}}_F\) (or equivalently that \(r_{\widehat{G}}\) is not the trivial representation). Then
$$\begin{aligned} \phi (1)\leqslant \Omega _F \max _{w\in \Omega _F} p(\lambda \star _w 0)\cdot q^{1}. \end{aligned}$$  (ii):

Suppose that \(r_{\widehat{G}}\) is trivial. Then \(\phi (1)=r(\mathrm{Fr})\).
Proof
Let us do some preparation. By twisting \(r\) by an unramified unitary character of \(W_F\) (viewed as a character of \(^L G\)) we may assume that \(r=\mathrm{Ind}^{^L G_j}_{\widehat{G}} r_0\) for some irreducible representation \(r_0\) of \(\widehat{G}\), cf. the proof of Lemma 2.1 (ii). Let \(\lambda _0\) be the highest weight of \(r_0\) and define \(\lambda \in X_*(A)^+\) as in the paragraph preceding Lemma 2.1. The lemma tells us that \({\mathcal {S}}(\phi )=\zeta \chi _\lambda \in \mathbb {C}[X_*(A)]^{\Omega _F}\) with \(\zeta =1\).
In the case of (ii), \(r\) is just an unramified unitary character of \(W_F\) (with \(d=1\)), and it is easily seen that \(\chi _\lambda =\tau ^A_0\), \(\zeta =r(\mathrm{Fr})\), and so \(\phi (1)=r(\mathrm{Fr})\). Let us put ourselves in the case (i) so that \(\lambda \ne 0\). Note that \(\phi (1)\) is just the coefficient of \(\tau ^G_0\) when \(\phi =\zeta {\mathcal {S}}^{1}(\chi _\lambda )\) is written with respect to the basis \(\{\tau ^G_\mu \}\). Such a coefficient equals \(\zeta \cdot s_{\lambda ,0}\) according to (2.6), so \(\phi (1)=  s_{\lambda ,0}\). Now Lemma 2.2 concludes the proof. [Observe that \(\lambda \star _w 0\ne 0\) whenever \(0\ne \lambda \in X_*(A)^+\).] \(\square \)
2.8 Examples in the split case
When \(G\) is split, it is easy to see that \(\mathbb {C}[\mathrm{ch}({}^L G^{\mathrm {ur}})]\) is canonically identified with \(\mathbb {C}[\mathrm{ch}(\widehat{G})]\) which is generated by finite dimensional characters in the space of functions on \(\widehat{G}\). So we may use \(\mathbb {C}[\mathrm{ch}(\widehat{G})]\) in place of \(\mathbb {C}[\mathrm{ch}({}^L G^{\mathrm {ur}})]\).
Example 2.10
(When \(G={{\mathrm{Sp}}}_{2n}\), \(n\geqslant 1\))
Take \(r:\widehat{G}={{\mathrm{SO}}}_{2n+1}(\mathbb {C})\hookrightarrow \mathrm{GL}_{2n+1}(\mathbb {C})\) to be the standard representation. Then
is mapped to \(\mathrm{tr}\,(r)\in \mathbb {C}[\mathrm{ch}({{\mathrm{SO}}}_{2n+1})]\) and
is mapped to \(\mathrm{tr}\,(r)\in \mathbb {C}[\mathrm{ch}({{\mathrm{SO}}}_{2n+1})]\). Then \(\mathrm{Sym}^2(V)\) breaks into \(\mathbb {C}\) and an irreducible representation of \(\widehat{G}\) of highest weight \((2,0,\ldots ,0)\) in the standard parametrization. When \(n>1\), \(\wedge ^2(V)\) is irreducible of highest weight \((1,1,0,\ldots ,0)\). When \(n=1\), \(\wedge ^2(V)\simeq V^\vee \), i.e. isomorphic to \((\mathrm{Std})^\vee \). (See [41, §19.5].) Let us systematically write \(\Lambda _\lambda \) for the irreducible representation of \({{\mathrm{SO}}}_{2n+1}\) with highest weight \(\lambda \). Then
if \(n\geqslant 2\). If \(n=1\), the same is true if \(\Lambda _{(1,1,0,\ldots ,0)}\) is replaced with \(\Lambda _{(1)}\). For \(i=1,2\), define
Then one computes
where \(\mu _{\lambda }\) is the cocharacter of a maximal torus given by \(\lambda \) in the standard parametrization. In particular, \(\phi ^{(1)}(1)=0\) and \(\phi ^{(2)}(1)=1\).
Example 2.11
(When \(G={{\mathrm{SO}}}_{2n}\), \(n\geqslant 2\))
Take \(r:\widehat{G}={{\mathrm{SO}}}_{2n}(\mathbb {C})\hookrightarrow \mathrm{GL}_{2n}(\mathbb {C})\) to be the standard representation. Similarly as before, \(\mathrm{Sym}^2(V)\) breaks into \(\mathbb {C}\) and an irreducible representation of \(\widehat{G}\) of highest weight \((2,0,\ldots ,0)\). When \(n>1\), \(\wedge ^2(V)\) is irreducible of highest weight \((1,1,0,\ldots ,0)\). When \(n=1\), \(\wedge ^2(V)\simeq \mathbb {C}\). (See [41, §19.5].) The same formulas as (2.10) hold in this case. Defining
we can compute \(\phi ^{(1)}\), \(\phi ^{(2)}\) and see that \(\phi ^{(1)}(1)=0\) and \(\phi ^{(2)}(1)=1\).
Example 2.12
(When \(G={{\mathrm{SO}}}_{2n+1}\))
Take \(r:\widehat{G}={{\mathrm{Sp}}}_{2n}(\mathbb {C})\hookrightarrow \mathrm{GL}_{2n}(\mathbb {C})\) to be the standard representation. Then
is mapped to \(\mathrm{tr}\,(r\circ \mathrm{Std})\in \mathbb {C}[\mathrm{ch}(Sp_{2n})]\) and Then
is mapped to \(\mathrm{tr}\,(r\circ \mathrm{Std})\in \mathbb {C}[\mathrm{ch}(Sp_{2n})]\). If \(n\geqslant 2\) then \(\wedge ^2(V)\) breaks into \(\mathbb {C}\) and an irreducible representation of \(\widehat{G}\) of highest weight \((1,1,0,\ldots ,0)\). (See [41, §17.3].) We have
As in Example 2.10, \(\Lambda \) designates a highest weight representation (now of \({{\mathrm{Sp}}}_{2n}\)). Define \(\phi ^{(i)}\) as in (2.11). By a similar computation as above, \(\phi ^{(1)}(1)=0\), \(\phi ^{(2)}(1)=1\).
2.9 Bounds for truncated unramified Hecke algebras
Let \(F\), \(G\), \(A\), \(T\) and \(K\) be as in Sect. 2.2. Throughout this subsection, an \(\mathbb {R}\)basis \({\mathcal {B}}\) of \(X_*(T)_{\mathbb {R}}\) will be fixed once and for all. Denote by \(\rho \in X^*(T)\otimes _\mathbb {Z}\frac{1}{2}\mathbb {Z}\) half the sum of all \(\alpha \in \Phi ^+\).
Lemma 2.13
For any \(\mu \in X_*(A)\), \([K\mu (\varpi )K:K]\leqslant q^{d_G+r_G+\langle \rho ,\mu \rangle }\).
Proof
Let \({{\mathrm{vol}}}\) denote the volume for the Haar measure on \(G(F)\) such that \({{\mathrm{vol}}}(K)=1\). Let \(I\subset K\) be an Iwahori subgroup of \(G(F)\). Then \(I=(I\cap U)(I\cap T)(I\cap \overline{U})\). We follow the argument of [106, pp. 241–242], freely using Waldspurger’s notation. Our \(I\), \(U\), \(\overline{U}\), and \(T\) will play the roles of his \(H\), \(U_0\), \(\overline{U}_0\) and \(M_0\), respectively. For all \(m\in \overline{M}_0^+\) (in his notation), it is not hard to verify that \(c'_{U_0}(m)=c_{\overline{U}_0}(m)=c_{M_0}(m)=1\). Then Waldspurger’s argument shows
Finally observe that \([K:I]\leqslant G(\mathbb {F}_q)\leqslant q^{d_G}(1+\frac{1}{q})^{r_G}\leqslant q^{d_G+r_G}\). (The middle inequality is easily derived from Steinberg’s formula. cf. [47, (3.1)].) \(\square \)
The following lemma will play a role in studying the level aspect in Sect. 9.
Lemma 2.14
Let \(M\) be an \(F\)rational Levi subgroup of \(G\). There exists a constant \(b_G> 0\) (depending only on \(G\)) such that for all \(\kappa \in \mathbb {Z}_{>0}\) and all \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(G)^{\leqslant \kappa ,{\mathcal {B}}}\) such that \(\phi \leqslant 1\), we have \(\phi _{M}(1)= O(q^{d_G+r_G+b_G\kappa })\) (the implicit constant being independent of \(\kappa \) and \(\phi \)), where we put \(\phi _M:={\mathcal {S}}^G_M(\phi )\).
Proof
When \(M=G\), the lemma is obvious (with \(b_G=0\)). Henceforth we assume that \(M\subsetneq G\). In view of Lemma 2.3, it suffices to treat one \(\mathbb {R}\)basis \({\mathcal {B}}\). Fix a \(\mathbb {Z}\)basis \(\{e_1,\ldots ,e_{\dim A}\}\) of \(X_*(A)\), and choose any \({\mathcal {B}}\) which extends that \(\mathbb {Z}\)basis. It is possible to write
for \(a_\mu \leqslant 1\). Thus
For each \(\mu \), \(K\mu (\varpi )K\) is partitioned into left \(K\)cosets. On each coset \(\gamma K\),
Hence, together with Lemma 2.13,
Write \(b_{0}\) for the maximum of \(\langle \rho ,e_i\rangle \) for \(i=1,\ldots ,\dim A\). Take \(b_G:=b_{0}\dim A+2 \dim A\). If \(\Vert \mu \Vert \leqslant \kappa \) then \(\mu =\sum _{i=1}^{\dim A} a_ie_i\) for \(a_i\in \mathbb {Z}\) with \(\kappa \leqslant a_i\leqslant \kappa \). Hence the right hand side is bounded by \((2\kappa +1)^{\dim A} q^{d_G+r_G+b_{0}\kappa \dim A}\leqslant q^{d_G+r_G+b_G\kappa }\) since \(2\kappa +1\leqslant 2^{2\kappa }\leqslant q^{2\kappa }\). \(\square \)
An elementary matrix computation shows the bound below, which will be used several times.
Lemma 2.15
Let \(s=\mathrm{diag}(s_1,\ldots ,s_m)\in \mathrm{GL}_m(\overline{F}_v)\) and \(u=(u_{ij})_{i,j=1}^{m}\in \mathrm{GL}_m(\overline{F}_v)\). Define \(v_{\min }(u):=\min _{i,j} v(u_{ij})\) and similarly \(v_{\min }(u^{1})\). Then for any eigenvalue \(\lambda \) of \(su\in \mathrm{GL}_m(F_v)\),
Remark 2.16
The lemma will be typically applied when \(u\in \mathrm{GL}_m(\overline{\mathcal {O}}_v)\) where \(\overline{\mathcal {O}}_v\) is the integer ring of \(\overline{F}_v\). In this case \(v_{\min }(u)=v_{\min }(u^{1})=0\).
Proof
Let \(V\) be the underlying \(\overline{F}_v\)vector space with standard basis \(\{e_1,\ldots ,e_m\}\). Let \({\mathcal {B}}_j=\{\mathbf {i}=(i_1,\ldots ,i_j)1\leqslant i_1<\cdots <i_j\leqslant m\}\). Then \(\wedge ^jV\) has a basis \(\{e_{i_1}\wedge \cdots \wedge e_{i_j}\}_{\mathbf {i}\in {\mathcal {B}}_j}\). We claim that
Let us verify this. Let \((u_{\mathbf {i},\mathbf {i}'})_{\mathbf {i},\mathbf {i}'\in {\mathcal {B}}_j}\) denote the matrix entries for the \(u\)action on \(\wedge ^j V\) with respect to the above basis. Observe that \(v(u_{\mathbf {i},\mathbf {i}'}) \geqslant j\cdot v_{\min }(u)\) for all \(\mathbf {i},\mathbf {i}'\in {\mathcal {B}}_j\). Then
The coefficients of the characteristic polynomial for \(su\in \mathrm{GL}_m(F_v)\) are given by \(\mathrm{tr}\,(su\wedge ^j V)\) up to sign. The above claim and an elementary argument with the Newton polygon show that any root \(\lambda \) satisfies \(v(\lambda )\geqslant v_{\min }(u)+ \min _{i} v(s_i)\). Finally, applying the argument so far to \(s^{1}\) and \(u^{1}\), we obtain the upper bound for \(v(\lambda )\). \(\square \)
As before, the smooth reductive model for \(G\) over \(\mathcal {O}\) such that \(G(\mathcal {O})=K\) will still be denoted \(G\).
Lemma 2.17
Let \(\Xi :G\hookrightarrow \mathrm{GL}_m\) be an embedding of algebraic groups over \(\mathcal {O}\). Then there exists a \(GL_m(\mathcal {O})\)conjugate of \(\Xi \) which maps \(A\) (a fixed maximal split torus of \(G\)) into the diagonal maximal torus of \(\mathrm{GL}_m\).
Proof
Note that the maximal \(F\)split torus \(A\) naturally extend to \(A\subset G\) over \(\mathcal {O}\), cf. [103, §3.5]. The representation of \(A\) on a free \(\mathcal {O}\)module of rank \(m\) via \(\Xi \) defines a weight decomposition of \(\mathcal {O}^m\) into free \(\mathcal {O}\)modules. Choose any refinement of the decomposition to write \(\mathcal {O}^m=L_1\oplus \cdots \oplus L_m\), as the direct sum of rank 1 free \(\mathcal {O}\)submodules. Let \(v_i\) be an \(\mathcal {O}\)generator of \(L_i\) for \(1\leqslant i\leqslant m\). Conjugating \(\Xi \) by the matrix representing the change of basis from \(\{v_1,\ldots ,v_m\}\) to the standard basis for \(\mathcal {O}^m\), one can achieve that \(\Xi (A)\) lies in the diagonal maximal torus. \(\square \)
Let \(\gamma \in G(F)\) be a semisimple element and choose a maximal torus \(T_\gamma \) of \(G\) defined over \(F\) such that \(\gamma \in T_\gamma (F)\). Denote by \(\Phi (G,T_\gamma )\) the set of roots for \(T_\gamma \) in \(G\).
Lemma 2.18
Suppose that there exists an embedding of algebraic groups \(\Xi :G\hookrightarrow \mathrm{GL}_m\) over \(\mathcal {O}\). There exists a constant \(B_5>0\) such that for every \(\kappa \in \mathbb {Z}_{\geqslant 0}\), every \(\mu \in X_*(A)\) satisfying \(\Vert \mu \Vert \leqslant \kappa \), every semisimple \(\gamma \in K\mu (\varpi )K\) and every \(\alpha \in \Phi _\gamma \) (for any choice of \(T_\gamma \) as above), we have \(B_5\kappa \leqslant v(\alpha (\gamma ))\leqslant B_5\kappa \). In particular, \(1\alpha (\gamma )\leqslant q^{B_5 \kappa }\).
Remark 2.19
Later \(\Xi \) will be provided by Proposition 8.1.
Proof
We may assume that \(\Xi (A)\) is contained in the diagonal torus of \(\mathrm{GL}_m\), denoted by \(\mathbb {T}\), thanks to Lemma 2.17. Write \(T\) for the maximal torus of \(G\) which is the centralizer of \(A\) so that \(\Xi (T)\subset \mathbb {T}\). We have a surjection \(X^*(\mathbb {T})\twoheadrightarrow X^*(T)\) induced by \(\Xi \). For each \(\alpha \) in the set of roots \(\Phi (G,T)\), we fix a lift \(\widetilde{\alpha }\in X^*(\mathbb {T})\) once and for all. Set \(c_1:=\max _{\alpha \in \Phi (G,T)} \Vert \widetilde{\alpha }\Vert _{\mathrm{GL}_m}\).
Let \(c_2:=\max _{\Vert \mu \Vert \leqslant 1} \Vert \Xi \circ \mu \Vert _{\mathrm{GL}_m}\) where \(\mu \in X_*(A)_{\mathbb {R}}\) runs over elements with \(\Vert \mu \Vert \leqslant 1\). Then for any \(\kappa \in \mathbb {Z}_{\geqslant 0}\), \(\Vert \mu \Vert \leqslant \kappa \) implies \(\Vert \Xi \circ \mu \Vert _{\mathrm{GL}_m}\leqslant c_2\kappa \). Hence \(\Xi (\mu (\varpi ))\) is a diagonal matrix in which each entry \(x\) satisfies \(c_2\kappa \leqslant v(x)\leqslant c_2\kappa \).
We can write \(\gamma =k_1 \mu (\varpi )k_2\) for some \(k_1,k_2\in G(\mathcal {O})\). Then \(\Xi (\gamma )=k'_1 \Xi (\mu (\varpi )) k'_2\) for \(k'_1,k'_2\in \mathrm{GL}_m(\mathcal {O})\), and \(\Xi (\gamma )\) is conjugate to \(\Xi (\mu (\varpi )) k'_2(k'_1)^{1}\). It follows from Lemma 2.15 that for every eigenvalue \(\lambda \) of \(\Xi (\gamma )\), we have \(c_2\kappa \leqslant v(\lambda )\leqslant c_2\kappa \).
Choose any \(T_\gamma \) as above. There exists an isomorphism \(T\simeq T_\gamma \) over \(\overline{F}\) induced by a conjugation action \(t\mapsto g t g^{1}\) given by some \(g\in G(\overline{F})\). The isomorphism is well defined only up to the Weyl group action but induces a bijection from \(\Phi (G,T)\) onto \(\Phi (G,T_\gamma )\). Put \(\mathbb {T}_\gamma :=\Xi (g)\mathbb {T}\Xi (g)^{1}\). The conjugation by \(\Xi (g)\) induces an isomorphism \(\mathbb {T}\simeq \mathbb {T}_\gamma \) over \(\overline{F}\) and a bijection from \(\Phi (\mathrm{GL}_m,\mathbb {T})\) onto \(\Phi (\mathrm{GL}_m,\mathbb {T}_\gamma )\). Let \(\alpha _\gamma \in \Phi (G,T_\gamma )\) (resp. \(\widetilde{\alpha }_\gamma \in \Phi (\mathrm{GL}_m,\mathbb {T}_\gamma )\)) denote the image of \(\alpha \) (resp. \(\widetilde{\alpha }\)) under the bijections. By construction, the composition \(T_\gamma \simeq T\mathop {\rightarrow }\limits ^{\Xi } \mathbb {T}\simeq \mathbb {T}_\gamma \) coincides with the restriction of \(\Xi \) to \(T_\gamma \). Hence the induced map \(X^*(\mathbb {T}_\gamma )\rightarrow X^*(T_\gamma )\) maps \(\widetilde{\alpha }_\gamma \) to \(\alpha _\gamma \).
Using the isomorphisms \(\mathbb {T}_\gamma (\overline{F})\simeq \mathbb {T}(\overline{F})\simeq (\overline{F}^\times )^m\), let \((\lambda _1,\ldots ,\lambda _m)\in (\overline{F}^\times )^m\) be the image of \(\Xi (\gamma )\) under the composition isomorphism. We may write \(\widetilde{\alpha }_\gamma \) as a character \((\overline{F}^\times )^m\rightarrow \overline{F}^\times \) given by \((t_1,\ldots ,t_m)\mapsto t_1^{a_1}\ldots t_m^{a_m}\) with \(a_1,\ldots ,a_m\in \mathbb {Z}\) such that \(c_1\leqslant a_i\leqslant c_1\) for every \(1\leqslant i\leqslant m\). We have
so \(v(\alpha _\gamma (\gamma ))=\sum _{i=1}^m a_i v(\lambda _i)\). Hence \(m c_1c_2\kappa \leqslant v(\alpha _\gamma (\gamma ))\leqslant m c_1c_2\kappa \), proving the first assertion of the lemma. From this the last assertion is obvious. \(\square \)
Remark 2.20
Suppose that \(F\) runs over the completions of a number field \(\mathbf {F}\) at nonarchimedean places \(v\), that \(G\) over \(F\) comes from a fixed reductive group \(\mathbf {G}\) over \(\mathbf {F}\), and that \(\Xi \) comes from an embedding \(\mathbf {G}\hookrightarrow GL_m\) over the integer ring of \(\mathbf {F}\) (at least for every \(v\) where \(\mathbf {G}\) is unramified). Then \(B_5\) of the lemma can be chosen to be independent of \(v\) (and dependent only on the data over \(\mathbf {F}\)). This is easy to see from the proof.
3 Plancherel measure on the unramified spectrum
3.1 Basic setup and notation
Let \(F\) be a finite extension of \(\mathbb {Q}_p\). Suppose that \(G\) is unramified over \(F\). Fix a hyperspecial subgroup \(K\) of \(G\). Recall the notation from the start of Sect. 2.2. In particular \(\Omega \) (resp. \(\Omega _F\)) denotes the Weyl group for \((G_{\overline{F}},T_{\overline{F}})\) [resp. \((G,A)\)]. There is a natural \(\mathrm{Gal}(\overline{F}/F)\)action on \(\Omega \), under which \(\Omega ^{\mathrm{Gal}(\overline{F}/F)}=\Omega _F\). (See [9, §6.1].) Since \(G\) is unramified, \(\mathrm{Gal}(\overline{F}/F)\) factors through a finite unramified Galois group. Thus there is a welldefined action of \(\mathrm{Fr}\) on \(\Omega \), and \(\Omega ^{\mathrm{Fr}}=\Omega _F\).
The unitary dual \(G(F)^{\wedge }\) of \(G(F)\), or simply \(G^\wedge \) if there is no danger of ambiguity, is equipped with Fell topology. (This notation should not be confused with the dual group \(\widehat{G}\)). Let \(G^{\wedge ,\mathrm {ur}}\) denote the unramified spectrum in \(G^{\wedge }\), and \(G^{\wedge ,\mathrm {ur},\text {temp}}\) its tempered subspectrum. The Plancherel measure \(\widehat{\mu }^{\mathrm {pl}}\) on \(G^{\wedge }\) is supported on the tempered spectrum \(G^{\wedge ,\text {temp}}\). The restriction of \(\widehat{\mu }^{\mathrm {pl}}\) to \(G^{\wedge ,\mathrm {ur}}\) will be written as \(\widehat{\mu }^{\mathrm {pl,ur}}\). The latter is supported on \(G^{\wedge ,\mathrm {ur},\text {temp}}\). HarishChandra’s Plancherel formula (cf. [106]) tells us that \(\widehat{\mu }^{\mathrm {pl}}(\widehat{\phi })=\phi (1)\) for all \(\phi \in {\mathcal {H}}(G(F))\). In particular, \(\widehat{\mu }^{\mathrm {pl,ur}}(\widehat{\phi })=\phi (1)\) for all \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(G(F))\).
3.2 The unramified tempered spectrum
An unramified \(L\)parameter \(W^{\mathrm {ur}}_F\rightarrow {}^L G^{\mathrm {ur}}\) is defined to be an \(L\)morphism \({}^L H^{\mathrm {ur}}\rightarrow {}^L G^{\mathrm {ur}}\) (Sect. 2.5) with \(H=\{1\}\). Two such parameters \(\varphi _1\) and \(\varphi _2\) are considered equivalent if \(\varphi _1=g\varphi _2 g^{1}\) for some \(g\in \widehat{G}\). Consider the following sets:

(i)
Irreducible unramified representations \(\pi \) of \(G(F)\) up to isomorphism.

(ii)
Group homomorphisms \(\chi :T(F)/T(F)\cap K \rightarrow \mathbb {C}^\times \) up to \(\Omega _{F}\)action.

(iii)
Unramified \(L\)parameters \(\varphi :W^{\mathrm {ur}}_F\rightarrow {}^L G^{\mathrm {ur}}\) up to equivalence.

(iv)
Elements of \((\widehat{G}\rtimes \mathrm{Fr})_{\mathrm{ssconj}}\); this set was defined in Sect. 2.2.

(v)
\(\Omega ^{\mathrm{Fr}}\)orbits in \(\widehat{T}/(\mathrm{Fr}\mathrm{id})\widehat{T}\).

(vi)
\(\Omega _F\)orbits in \(\widehat{A}\).

(viii)
\(\mathbb {C}\)algebra morphisms \(\theta :{\mathcal {H}}^{\mathrm {ur}}(G)\rightarrow \mathbb {C}\).
Let us describe canonical maps among them in some directions.

(i) \(\rightarrow \) (vii) Choose any \(0\ne v\in \pi ^K\). Define \(\theta (\phi )\) by \(\theta (\phi )v=\int _{G(F)} \phi (g)\pi (g)vdg\).

(ii) \(\rightarrow \) (i) \(\pi \) is the unique unramified subquotient of \(\mathrm{n{\text {}}ind}^{G(F)}_{B(F)} \chi \).

(ii) \(\leftrightarrow \) (vi) Induced by \(\mathrm{Hom}(T(F)/T(F)\cap K , \mathbb {C}^\times )\simeq \mathrm{Hom}(A(F)/A(F)\cap K , \mathbb {C}^\times )\)
$$\begin{aligned} \simeq \mathrm{Hom}(X_*(A),\mathbb {C}^\times )\!\simeq \! \mathrm{Hom}(X^*(\widehat{A}),\mathbb {C}^\times )\!\simeq \! X_*(\widehat{A})\otimes _\mathbb {Z}\mathbb {C}^\times \!\simeq \!\widehat{A}\qquad \end{aligned}$$(3.1)where the second isomorphism is induced by \(X_*(A)\rightarrow A(F)\) sending \(\mu \) to \(\mu (\varpi )\).

(iii) \(\rightarrow \) (iv) Take \(\varphi (\mathrm{Fr})\).

(v) \(\rightarrow \) (iv) Induced by the inclusion \(t\mapsto t\rtimes \mathrm{Fr}\) from \(\widehat{T}\) to \( \widehat{G}\rtimes \mathrm{Fr}\).

(v) \(\rightarrow \) (vi) Induced by the surjection \(\widehat{T}\twoheadrightarrow \widehat{A}\), which is the dual of \(A\hookrightarrow T\). (Recall \(\Omega ^{\mathrm{Fr}}=\Omega _F\).)

(vii) \(\rightarrow \) (vi) Via \({\mathcal {S}}:{\mathcal {H}}^{\mathrm {ur}}(G)\simeq \mathbb {C}[X^*(\widehat{A})]^{\Omega _F}\), \(\theta \) determines an element of [cf. (3.1)]
$$\begin{aligned} \Omega _F\backslash \mathrm{Hom}(X^*(\widehat{A}),\mathbb {C}^\times )\simeq \Omega _F\backslash \widehat{A}. \end{aligned}$$
Lemma 3.1
Under the above maps, the sets corresponding to (i)–(vii) are in bijection with each other.
Proof
See §6, §7 and §10.4 of [9]. \(\square \)
Let \(F'\) be the finite unramified extension of \(F\) such that \(\mathrm{Gal}(\overline{F}/F)\) acts on \(\widehat{G}\) through the faithful action of \(\mathrm{Gal}(F'/F)\). Write \({}^L G_{F'/F}:=\widehat{G}\rtimes \mathrm{Gal}(F'/F)\). Let \(\widehat{K}\) be a maximal compact subgroup of \(\widehat{G}\) which is \(\mathrm{Fr}\)invariant. Denote by \(\widehat{T}_c\) (resp. \(\widehat{A}_c\)) the maximal compact subtorus of \(\widehat{T}\) (resp. \(\widehat{A}\)).
Lemma 3.2
The above bijections restrict to the bijections among the sets consisting of the following objects.
 \(\mathrm{(i)}_t\) :

irreducible unramified tempered representations \(\pi \) of \(G(F)\) up to isomorphism.
 \(\mathrm{(ii)}_t\) :

unitary group homomorphisms \(\chi :T(F)/T(F)\cap K \rightarrow U(1)\) up to \(\Omega _{F}\)action.
 \(\mathrm{(iii)}_t\) :

unramified \(L\)parameters \(\varphi :W^{\mathrm {ur}}_F\rightarrow {}^L G^{\mathrm {ur}}\) with bounded image up to equivalence.
 \(\mathrm{(iv)}_t\) :

\(\widehat{G}\)conjugacy classes in \(\widehat{K}\rtimes \mathrm{Fr}\) (viewed in \({}^L G_{F'/F}\)).
 \(\mathrm{(iv)}^{'}_{t}\) :

\(\widehat{K}\)conjugacy classes in \(\widehat{K}\rtimes \mathrm{Fr}\) (viewed in \(\widehat{K}\rtimes \mathrm{Gal}(F'/F)\)).
 \(\mathrm{(v)}_t\) :

\(\Omega ^{\mathrm{Fr}}\)orbits in \(\widehat{T}_c/(\mathrm{Fr}\mathrm{id})\widehat{T}_c\).
 \(\mathrm{(vi)}_t\) :

\(\Omega _F\)orbits in \(\widehat{A}_c\).
[The boundedness in (iii)\(_t\) means that the projection of \(\mathrm{Im\,}\varphi \) into \({}^L G_{F'/F}\) is contained in a maximal compact subgroup of \({}^L G_{F'/F}\).]
Proof
(i)\(_t\) \(\leftrightarrow \) (ii)\(_t\) is standard and (iii)\(_t\) \(\leftrightarrow \) (iv)\(_t\) is obvious. Also straightforward is (ii)\(_t\) \(\leftrightarrow \)(vi)\(_t\) in view of (3.1).
Let us show that (v)\(_t\) \(\leftrightarrow \) (vi)\(_t\). Choose a topological isomorphism of complex tori \(\widehat{T}\simeq (\mathbb {C}^\times )^{d}\) with \(d=\dim T\). Using \(\mathbb {C}^\times \simeq U(1)\times \mathbb {R}^\times _{>0}\), we can decompose \(\widehat{T}=\widehat{T}_c\times \widehat{T}_{nc}\) such that \(\widehat{T}_{nc}\) is carried over to \((\mathbb {R}^\times _{>0})^d\) under the isomorphism. The decomposition of \(\widehat{T}\) is canonical in that it is preserved under any automorphism of \(\widehat{T}\). By the same reasoning, there is a canonical decomposition \(\widehat{A}=\widehat{A}_c\times \widehat{A}_{nc}\) with \(\widehat{A}_{nc}\simeq (\mathbb {R}^\times _{>0})^{\dim A}\). The canonical surjection \(\widehat{T}\rightarrow \widehat{A}\) carries \(\widehat{T}_c\) onto \(\widehat{A}_c\) and \(\widehat{T}_{nc}\) onto \(\widehat{A}_{nc}\). [This reduces to the assertion in the case of \(\mathbb {C}^\times \), namely that any maps \(U(1)\rightarrow \mathbb {R}^\times _{>0}\) and \(\mathbb {R}^\times _{>0}\rightarrow U(1)\) induced by an algebraic map \(\mathbb {C}^\times \rightarrow \mathbb {C}^\times \) of \(\mathbb {C}\)tori are trivial. This is easy to check.] Therefore the isomorphism \(\widehat{T}/(\mathrm{Fr}\mathrm{id})\widehat{T}\rightarrow \widehat{A}\) of Lemma 3.2 induces an isomorphism \(\widehat{T}_c/(\mathrm{Fr}\mathrm{id})\widehat{T}_c\rightarrow \widehat{A}_c\) (as well as \(\widehat{T}_{nc}/(\mathrm{Fr}\mathrm{id})\widehat{T}_{nc}\rightarrow \widehat{A}_{nc}\)).
Next we show that (iv)\(_t\) \(\leftrightarrow \) (v)\(_t\). It is clear that \(t\mapsto t\rtimes \mathrm{Fr}\) maps (v)\(_t\) into (iv)\(_t\). Since (v)\(_t\) and (iv)\(_t\) are the subsets of (v) and (iv), which are in bijective correspondence, we deduce that (v)\(_t\) \(\rightarrow \) (iv)\(_t\) is injective. To show surjectivity, pick any \(k\in \widehat{K}\). There exists \(t\in \widehat{T}\) such that the image of \(t\) in (iv) corresponds under (iv) \(\leftrightarrow \) (v) to the \(\widehat{G}\)conjugacy class of \(\widehat{k}\rtimes \mathrm{Fr}\). It is enough to show that we can choose \(t\in \widehat{T}_c\). Consider the subgroup \(\widehat{T}_c(t)\) of
generated by \(\widehat{T}_c/(\mathrm{Fr}\mathrm{id})\widehat{T}_c\) and the image of \(t\). The isomorphism (iv)\(\leftrightarrow \)(v) maps \(\widehat{T}_c(t)\) into (v)\(_t\) by the assumption on \(t\). Since (v)\(_t\) form a compact set, the group \(\widehat{T}_c(t)\) must be contained in a compact subset of \(\widehat{T}/(\mathrm{Fr}\mathrm{id})\widehat{T}\). This forces the image of \(t\) in \(\widehat{T}_{nc}/(\mathrm{Fr}\mathrm{id})\widehat{T}_{nc}\) to be trivial. (Indeed, the latter quotient is isomorphic as a topological group to a quotient of \(\mathbb {R}^{\dim T}\) modulo an \(\mathbb {R}\)subspace via the exponential map. So any subgroup generated by a nontrivial element is not contained in a compact set.) Therefore \(t\) can be chosen in \(\widehat{T}_c\).
It remains to verify that (iv)\(_t\), (iv)\('_t\) and (v)\(_t\) are in bijection. Clearly (iv)\('_t\) \(\rightarrow \) (iv)\(_t\) is onto. As we have just seen that (iv)\(_t\) \(\leftrightarrow \) (v)\(_t\), it suffices to observe that (v)\(_t\) \(\rightarrow \) (iv)\('_t\) is onto, which is a standard fact [for instance in the context of the (twisted) Weyl integration formula for \(\widehat{K}\rtimes \mathrm{Fr}\)]. \(\square \)
3.3 Plancherel measure on the unramified spectrum
Lemma 3.2 provides a bijection \(G^{\wedge ,\mathrm {ur},\text {temp}}\simeq \Omega _F\backslash \widehat{A}_c\), which is in fact a topological isomorphism. The Plancherel measure \(\widehat{\mu }^{\mathrm {pl,ur}}\) on \(G^{\wedge ,\mathrm {ur}}\) is supported on \(G^{\wedge ,\mathrm {ur},\text {temp}}\). We would like to describe its pullback measure on \(\widehat{A}_c\), to be denoted \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{0}\). Note that \(\widehat{A}_c\) is topologically isomorphic to \(\widehat{T}_c/(\mathrm{Fr}\mathrm{id})\widehat{T}_c\). (This is induced by the natural surjection \(\widehat{T}_c \twoheadrightarrow \widehat{A}_c\).) Fix a measure \(d\overline{t}\) on the latter which is a push forward from a Haar measure on \(\widehat{T}_c\).
Proposition 3.3
The measure \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{0}\) pulled back to \(\widehat{T}_c/(\mathrm{Fr}\mathrm{id})\widehat{T}_c\) is
for some constant \(C\in \mathbb {C}^\times \), depending on the normalization of Haar measures. Here \(t\in \widehat{T}_c\) is any lift of \(\overline{t}\). (The right hand side is independent of the choice of \(t\).)
Proof
The formula is due to Macdonald [72]. For our purpose, it is more convenient to follow the formulation as in the conjecture of [98, p. 281] (which also discusses the general conjectural formula of the Plancherel measure due to Langlands). By that conjecture (known in the unramified case),
where \(C'\in \mathbb {C}^\times \) is a constant, \(\sigma (\overline{t}):T(F)\rightarrow \mathbb {C}^\times \) is the character corresponding to \(\overline{t}\) [via (ii) \(\leftrightarrow \) (v) of Lemma 3.1], and \(r:{}^L T \rightarrow \mathrm{GL}(\mathrm{Lie}\,({}^L U))\) is the adjoint representation. Here \({}^L U\) is the \(L\)group of \(U\) (viewed in \({}^L B\)). By unraveling the local \(L\)factors, obtain
Finally, observe that \(\det (1q^{s}\mathrm{ad}(t\rtimes \mathrm{Fr})\mathrm{Lie}\,(\widehat{T})/\mathrm{Lie}\,(\widehat{T}^{\mathrm{Fr}}))\) is independent of \(\overline{t}\) (and \(t\)). Therefore the right hand sides are the same up to constant in (3.2) and the proposition. \(\square \)
Remark 3.4
Note that the choice of a Haar measure on \(G(F)\) determines the measure \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{0}\). For example if the Haar measure on \(G(F)\) assigns volume 1 to \(K\) then \(G^{\wedge ,\mathrm {ur},\text {temp}}\) has total volume 1 with respect to \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{0}(\overline{t})\) as implied by the Plancherel formula for \({\mathbf {1}}_K\). Hence the product \(C\cdot d\overline{t}\).
4 Automorphic Lfunctions
According to Langlands conjectures, the most general \(L\)functions should be expressible as products of the principal \(L\)functions \(L(s,\Pi )\) associated to cuspidal automorphic representations \(\Pi \) of \(\mathrm{GL}(d)\) over number fields (for varying \(d\)). The analytic properties and functional equation of such \(L\)functions were first established by Godement–Jacquet for general \(d\geqslant 1\). This involves the Godement–Jacquet integral representation. The other known methods are the Rankin–Selberg integrals, the doubling method and the Langlands–Shahidi method. The purpose of this section is to recall these analytic properties and to setup notation. More detailed discussions may be found in [32, 55, 75], [86, §2] and [52, §5].
In this section and some of the later sections we use the following notation.

\(F\) is a number field, i.e. a finite extension of \(\mathbb {Q}\).

\(G\) is a connected reductive group over \(F\) (not assumed to be quasisplit).

\(Z=Z(G)\) is the center of \(G\).

\(\mathcal {V}_F\) (resp. \(\mathcal {V}_F^\infty \)) is the set of all (resp. all finite) places of \(F\).

\(S_\infty :=\mathcal {V}_F\backslash \mathcal {V}_F^\infty \).

\(A_{G}\) is the maximal \(F\)split subtorus in the center of \({\mathrm {Res}}_{F/\mathbb {Q}} G\), and \(A_{G,\infty }:=A_G(\mathbb {R})^0\).
4.1 Automorphic forms
Let \(\chi :A_{G,\infty }\rightarrow \mathbb {C}^\times \) be a continuous homomorphism. Denote by \(L^2_\chi (G(F)\backslash G(\mathbb {A}_F))\) the space of all functions \(f\) on \(G(\mathbb {A}_F)\) which are squareintegrable modulo \(A_{G,\infty }\) and satisfy \(f(g\gamma z)=\chi (z)f(\gamma )\) for all \(g\in G(F)\), \(\gamma \in G(\mathbb {A}_F)\) and \(z\in A_{G,\infty }\). There is a spectral decomposition into discrete and continuous parts
where the last sum is a Hilbert direct sum running over the set of all irreducible representations of \(G(\mathbb {A}_F)\) up to isomorphism. Write \({\mathcal {AR}}_{\mathrm{disc},\chi }(G)\) for the set of isomorphism classes of all irreducible representations \(\pi \) of \(G(\mathbb {A}_F)\) such that \(m_{\mathrm{disc},\chi }(\pi )>0\). Any \(\pi \in {\mathcal {AR}}_{\mathrm{disc},\chi }(G)\) is said to be a discrete automorphic representation of \(G(\mathbb {A}_F)\). If \(\chi \) is trivial (in particular if \(A_{G,\infty }=\{1\}\)) then we write \(m_{\mathrm{disc}}\) for \(m_{\mathrm{disc},\chi }\).
The above definitions allow a modest generalization. Let \(\mathfrak {X}_G\) be a closed subgroup of \(Z(\mathbb {A}_F)\) containing \(A_{G,\infty }\) and \(\omega :Z(\mathbb {A}_F)\cap \mathfrak {X}_G\backslash \mathfrak {X}_G\rightarrow \mathbb {C}^\times \) be a continuous (quasi)character. Then \(L^2_{\omega }\), \(L^2_{\mathrm{disc},\omega }\), \(m_{\mathrm{disc},\omega }\) etc can be defined analogously. In fact the ArthurSelberg trace formula applies to this setting. (See [4, Ch 3.1].)
For the rest of Sect. 4 we are concerned with \(G=\mathrm{GL}(d)\). Take \(\mathfrak {X}_G=Z(\mathbb {A}_F)\) so that \(\omega \) is a quasicharacter of \(Z(F)\backslash Z(\mathbb {A}_F)\). Note that \(A_{G,\infty }=Z(F_\infty )^\circ \) in this case. We denote by \(\mathcal {A}_\omega (\mathrm{GL}(d,F))\) the space consisting of automorphic functions on \(\mathrm{GL}(d,F)\backslash \mathrm{GL}(d,\mathbb {A}_F)\) which satisfy \(f(zg)=\omega (z)f(g)\) for all \(z\in Z(\mathbb {A}_F)\) and \(g\in \mathrm{GL}(d,\mathbb {A}_F)\) (see Borel and Jacquet [10] for the exact definition and the growth condition). We denote by \(\mathcal {A}_{{\mathrm {cusp}},\omega }(\mathrm{GL}(d,F))\) the subspace of cuspidal functions (i.e. the functions with vanishing period against all nontrivial unipotent subgroups).
An automorphic representation \(\Pi \) of \(\mathrm{GL}(d,\mathbb {A}_F)\) is by definition an irreducible admissible representation of \(\mathrm{GL}(d,\mathbb {A}_F)\) which is a constituent of the regular representation on \(\mathcal {A}_\omega (\mathrm{GL}(d,F))\). Then \(\omega \) is the central character of \(\Pi \). The subspace \(\mathcal {A}_{{\mathrm {cusp}},\omega }(\mathrm{GL}(d,F))\) decomposes discretely and an irreducible component is a cuspidal automorphic representation. The notion of cuspidal automorphic representations is the same if the space of cuspidal functions in \(L^2_{\omega }(GL(d,F)\backslash GL(d,\mathbb {A}_F))\) is used in the definition in place of \(\mathcal {A}_{{\mathrm {cusp}},\omega }(\mathrm{GL}(d,F))\), cf. [10, §4.6].
When \(\omega \) is unitary we can work with the completed space \(L_\omega ^2(\mathrm{GL}(d,F)\backslash \mathrm{GL}(d,\mathbb {A}_F))\) of squareintegrable functions modulo \(Z(\mathbb {A}_F)\) and with unitary automorphic representations. Note that a cuspidal automorphic representation is unitary if and only if its central character is unitary. We recall the Langlands decomposition of \(L_\omega ^2(\mathrm{GL}(d,F)\backslash \mathrm{GL}(d,\mathbb {A}_F))\) into the cuspidal, residual and continuous spectra. What will be important in the sequel is the notion of isobaric representations which we review in Sect. 4.3.
In the context of \(L\)functions, the functional equation involves the contragredient representation \(\widetilde{\Pi }\). An important fact is that the contragredient of a unitary automorphic representation of \(\mathrm{GL}(d,\mathbb {A}_F)\) is isomorphic to its complex conjugate.
4.2 Principal \(L\)functions
Let \(\Pi =\otimes _v \Pi _v\) be a cuspidal automorphic representation of \(\mathrm{GL}(d,\mathbb {A}_F)\) with unitary central character. The principal \(L\)function associated to \(\Pi \) is denoted
The Euler product is absolutely convergent when \({{\mathrm{\mathfrak {R}e}}}s>1\). The completed \(L\)function is denoted \(\Lambda (s,\Pi )\), the product now running over all places \(v\in \mathcal {V}_F\). For each finite place \(v\in \mathcal {V}_F^\infty \), the inverse of the local \(L\)function \(L(s,\Pi _v)\) is a Dirichlet polynomial in \(q_v^{s}\) of degree \(\leqslant d\). Write
Note that when \(\Pi _v\) is unramified, \(\alpha _i(\Pi _v)\) is nonzero for all \(i\) and corresponds to the eigenvalues of a semisimple conjugacy class in \(\mathrm{GL}_d(\mathbb {C})\) associated to \(\Pi _v\), but when \(\Pi _v\) is ramified the Langlands parameters are more sophisticated and we allow some (or even all of) of the \(\alpha _i(\Pi _v)\) to be equal to zero. In this way we have a convenient notation for all local \(L\)factors.
For each archimedean \(v\), the local \(L\)function \(L(s,\Pi _v)\) is a product of \(d\) Gamma factors
where \(\Gamma _\mathbb {R}(s):=\pi ^{s/2}\Gamma (s/2)\) and \(\Gamma _\mathbb {C}(s):=2(2\pi )^{s}\Gamma (s)\). Note that \(\Gamma _{\mathbb {C}}(s)=\Gamma _\mathbb {R}(s)\Gamma _\mathbb {R}(s+1)\) by the doubling formula, so when \(v\) is complex, \(L(s,\Pi _v)\) may as well be expressed as a product of \(2d\) \(\Gamma _\mathbb {R}\) factors.
The completed \(L\)function \(\Lambda (s,\Pi ):=L(s,\Pi )\prod _{v\infty }L(s,\Pi _v)\) has the following analytic properties. It has a meromorphic continuation to the complex plane. It is entire except when \(d=1\) and \(\Pi =\left .\right ^{it}\) for some \(t\in \mathbb {R}\), in which case \(L(s,\Pi )=\zeta _F(s+it)\) is (a shift of) the Dedekind zeta function of the ground field \(F\) with simple poles at \(s=it\) and \(s=1it\). It is bounded in vertical strips and satisfies the functional equation
where \(\epsilon (s,\Pi )\) is the epsilon factor and \(\widetilde{\Pi }\) is the contragredient automorphic representation. The epsilon factor has the form
for some positive integer \(q(\Pi )\in \mathbb {Z}_{\geqslant 1}\) and root number \(\epsilon (\Pi )\) of modulus one.
Note that \(q(\Pi )=q(\widetilde{\Pi })\), \(\epsilon (\widetilde{\Pi })=\overline{\epsilon (\Pi )}\) and for all \(v\in \mathcal {V}_F\), \(L(s,\widetilde{\Pi }_v)=\overline{L(\overline{s},\Pi _v)}\). For instance this follows from the fact [42] that \(\widetilde{\Pi }\) is isomorphic to the complex conjugate \(\overline{\Pi }\) (obtained by taking the complex conjugate of all forms in the vector space associated to the representation \(\Pi \)).
The conductor \(q(\Pi )\) is the product over all finite places \(v\in \mathcal {V}^\infty _F\) of the conductor \(q(\Pi _v)\) of \(\Pi _v\). Recall that \(q(\Pi _v)\) equals one whenever \(\Pi _v\) is unramified. It is convenient to introduce as well the conductor of admissible representations at archimedean places. When \(v\) is real we let \(C(\Pi _v)=\prod \nolimits ^d_{i=1} (2+\left \mu _i(\Pi _v)\right )\) and when \(v\) is complex we let \(C(\Pi _v)=\prod \nolimits ^d_{i=1} (2+\left \mu _i(\Pi _v)\right ^2)\). Then we let \(C(\Pi )\) be the analytic conductor which is the product of all the local conductors
Note that \(C(\Pi )\geqslant 2\) always.
There is \(0\leqslant \theta <\frac{1}{2}\) such that
for all archimedean \(v\) (resp. finite \(v\)) and \(1\leqslant i\leqslant d\). When \(\Pi _v\) is unramified we ask for
The value \(\theta =\frac{1}{2} \frac{1}{d^2+1}\) is admissible by an argument of Serre and Luo–Rudnick–Sarnak based on the analytic properties of the Rankin–Selberg convolution \(L(s,\Pi \times \widetilde{\Pi })\). Note that for all \(v\), the local \(L\)functions \(L(s,\Pi _v)\) are entire on \({{\mathrm{\mathfrak {R}e}}}s>\theta \) and this contains the central line \({{\mathrm{\mathfrak {R}e}}}s=\frac{1}{2}\).
The generalized Ramanujan conjecture asserts that all \(\Pi _v\) are tempered (see [88] and the references herein). This is equivalent to having \(\theta =0\) in the inequalities (4.4) and (4.5). In particular we expect that when \(\Pi _v\) is unramified, \(\left \alpha _i(\Pi _v)\right =1\).
4.3 Isobaric sums
We need to consider slightly more general \(L\)functions associated to noncuspidal automorphic representations on \(\mathrm{GL}(d,\mathbb {A}_F)\). These \(L\)functions are products of the \(L\)functions associated to cuspidal representations and studied in the previous Sect. 4.2. Closely related to this construction it is useful to introduce, following Langlands [70], the notion of isobaric sums of automorphic representations. The concept of isobaric representations is natural in the context of \(L\)functions and the Langlands functoriality conjectures.
Let \(\Pi \) be an irreducible automorphic representation of \(\mathrm{GL}(d,\mathbb {A}_F)\). Then a theorem of Langlands [10] states that there are integers \(r\geqslant 1\) and \(d_1,\ldots ,d_r\geqslant 1\) with \(d=d_1+\cdots +d_r\) and cuspidal automorphic representations \(\Pi _1,\ldots ,\Pi _r\) of \(\mathrm{GL}(d_1,\mathbb {A}_F),\cdots ,\mathrm{GL}(d_r,\mathbb {A}_F)\) such that \(\Pi \) is a constituent of the induced representation of \(\Pi _1\otimes \cdots \otimes \Pi _r\) (from the Levi subgroup \(\mathrm{GL}(d_1)\times \cdots \times \mathrm{GL}(d_r)\) of \(\mathrm{GL}(d)\)). A cuspidal representation is unitary when its central character is unitary. When all of \(\Pi _j\) are unitary then \(\Pi \) is unitary. But the converse is not true: note that even if \(\Pi \) is unitary, the representation \(\Pi _j\) need not be unitary in general.
We recall the generalized strong multiplicity one theorem of Jacquet and Shalika [54]. Suppose \(\Pi \) and \(\Pi '\) are irreducible automorphic representations of \(\mathrm{GL}(d,\mathbb {A}_F)\) such that \(\Pi _v\) is isomorphic to \(\Pi '_v\) for almost all \(v\in \mathcal {V}_F\) (we say that \(\Pi \) and \(\Pi '\) are weakly equivalent) and suppose that \(\Pi \) (resp. \(\Pi '\)) is a constituent of the induced representation of \(\Pi _1\otimes \cdots \otimes \Pi _r\) (resp. \(\Pi '_1\otimes \cdots \otimes \Pi '_{r'}\)). Then \(r=r'\) and up to permutation the sets of cuspidal representations \(\left\{ \Pi _j\right\} \) and \(\left\{ \Pi '_j\right\} \) coincide. Note that this generalizes the strong multiplicity one theorem of PiatetskiShapiro which corresponds to the case where \(\Pi \) and \(\Pi '\) are cuspidal.
Conversely suppose \(\Pi _1,\ldots , \Pi _r\) are cuspidal representations of \(\mathrm{GL}(d_1,\mathbb {A}_F), \ldots , \mathrm{GL}(d_r,\mathbb {A}_F)\). Then from the theory of Eisenstein series there is a unique constituent of the induced representation of \(\Pi _1\otimes \cdots \otimes \Pi _r\) whose local components coincide at each place \(v\in \mathcal {V}_F\) with the Langlands quotient of the local induced representation [70, §2]. It is denoted \(\Pi _1\boxplus \cdots \boxplus \Pi _r\) and called an isobaric representation (automorphic representations which are not isobaric are called anomalous). The above results of Langlands and Jacquet–Shalika may now be summarized by saying that an irreducible automorphic representation of \(\mathrm{GL}(d,\mathbb {A}_F)\) is weakly equivalent to a unique isobaric representation.
We now turn to \(L\)functions. The completed \(L\)function associated to an isobaric representation \(\Pi =\Pi _1\boxplus \cdots \boxplus \Pi _r\) is by definition
All notation from the previous subsection will carry over to \(\Lambda (s,\Pi )\). Namely we have the local \(L\)factors \(L(s,\Pi _v)\), the local Satake parameters \(\alpha _i(\Pi _v)\) and \(\mu _i(\Pi _v)\), the epsilon factor \(\epsilon (s,\Pi )\), the root number \(\epsilon (\Pi )\), the local conductors \(q(\Pi _v)\), \(C(\Pi _v)\) and the analytic conductor \(C(\Pi )\). The Euler product converges absolutely for \({{\mathrm{\mathfrak {R}e}}}s\) large enough.
One important difference concerns the bounds for local Satake parameters. Even if we assume that \(\Pi \) has unitary central character the inequalities (4.4) may not hold. We shall therefore require a stronger condition on \(\Pi \).
Proposition 4.1
Let \(\Pi \) be an isobaric representation of \(\mathrm{GL}(d,\mathbb {A}_F)\). Assume that the archimedean component \(\Pi _\infty \) is tempered. Then the bounds towards Ramanujan are satisfied. Namely there is a positive constant \(\theta <\frac{1}{2}\) such that for all \(1\leqslant i\leqslant d\) and all archimedean (resp. nonarchimedean) places \(v\),
Proof
Let \(\Pi =\Pi _1\boxplus \cdots \boxplus \Pi _r\) be the isobaric decomposition with \(\Pi _j\) cuspidal. Then we will show that all \(\Pi _j\) have unitary central character, which implies Proposition 4.1.
By definition we have that \(\Pi _\infty \) is a Langlands quotient of the induced representation of \(\Pi _{1\infty } \otimes \cdots \otimes \Pi _{r\infty }\). Since \(\Pi _\infty \) is tempered, this implies that all \(\Pi _{j\infty }\) are tempered, and in particular have unitary central character. Then the (global) central character of \(\Pi _j\) is unitary as well. \(\square \)
Remark 4.2
In analogy with the local case, an isobaric representation \(\Pi _1 \boxplus \cdots \boxplus \Pi _r\) where all cuspidal representations \(\Pi _j\) have unitary central character is called “tempered” in [70]. This terminology is fully justified only under the generalized Ramanujan conjecture for \(\mathrm{GL}(d,\mathbb {A}_F)\). To avoid confusion we use the adjective “tempered” for \(\Pi =\otimes _v \Pi _v\) only in the strong sense that the local representations \(\Pi _v\) are tempered for all \(v\in \mathcal {V}_F\).
Remark 4.3
In the proof of Proposition 4.1 we see the importance of the notion of isobaric representations and Langlands quotients. For instance a discrete series representation of \(\mathrm{GL}(2,\mathbb {R})\) is a constituent (but not a Langlands quotient) of an induced representation of a nontempered character of \(\mathrm{GL}(1,\mathbb {R})\times \mathrm{GL}(1,\mathbb {R})\).
4.4 An explicit formula
Let \(\Pi \) be a unitary cuspidal representation of \(\mathrm{GL}(d,\mathbb {A}_F)\). Let \(\rho _{j}(\Pi )\) denote the zeros of \(\Lambda (s,\Pi )\) counted with multiplicities. These are also the nontrivial zeros of \(L(s,\Pi )\). The method of Hadamard and de la Vallée Poussin generalizes from the Riemann zeta function to automorphic \(L\)functions, and implies that \(0<{{\mathrm{\mathfrak {R}e}}}\rho _j(\Pi )<1\) for all \(j\). There is a polynomial \(p(s)\) such that \(p(s)\Lambda (s,\Pi )\) is entire and of order \(1\) (\(p(s)=1\) except when \(d=1\) and \(\Pi =\left .\right ^{it}\), in which case we choose \(p(s)=(sit)(1its)\)).
The Hadamard factorization shows that there are \(a=a(\Pi )\) and \(b=b(\Pi )\) such that
The product is absolutely convergent in compact subsets away from the zeros \(\rho _j(\Pi )\). The functional equation implies that
The number of zeros of bounded imaginary part is bounded above uniformly:
Changing \(\Pi \) into \(\Pi \otimes \left .\right ^{it}\) we have an analogous uniform estimate for the number of zeros with \(\left {{\mathrm{\mathfrak {I}m}}}\rho _j(\Pi )T\right \leqslant 1\) (in particular this is \(\ll _\Pi \log T\)).
Let \(N(T,\Pi )\) be the number of zeros with \(\left {{\mathrm{\mathfrak {I}m}}}\rho _j(\Pi )\right \leqslant T\). Then the following estimate holds uniformly in \(T\geqslant 1\) (Weyl’s law):
The error term could be made uniform in \(\Pi \), see [52, §5.3] for more details.^{Footnote 4} The main term can be interpreted as the variation of the argument of \(C(\Pi )^{s/2}L(s,\Pi _\infty )\) along certain vertical segments.
We are going to discuss an explicit formula [see (4.8) below] expressing a weighted sum over the zeros of \(\Lambda (s,\Pi )\) as a contour integral. It is a direct consequence of the functional Eq. (4.2) and Cauchy formula. The explicit formula is traditionally stated using the Dirichlet coefficients of the \(L\)function \(L(s,\Pi )\). For our purpose it is more convenient to maintain the Euler product factorization.
Definte \(\gamma _j(\Pi )\) by \(\rho _j(\Pi )=\frac{1}{2}+i\gamma _j(\Pi )\). We know that \(\left {{\mathrm{\mathfrak {I}m}}}\gamma _j(\Pi )\right <\frac{1}{2}\) and under the GRH, \(\gamma _j(\Pi )\in \mathbb {R}\) for all \(j\).
It is convenient to denote by \(\frac{1}{2}+ ir_j(\Pi )\) the (eventual) poles of \(\Lambda (s,\Pi )\) counted with multiplicity. We have seen that poles only occur when \(\Pi =\left .\right ^{it}\) in which case the poles are simple and \(\left\{ r_j(\Pi )\right\} =\left\{ t+\frac{i}{2},t\frac{i}{2}\right\} \).
The above discussion applies with little change to isobaric representations. If we also assume that \(\Pi _\infty \) is tempered then we have seen in the proof of Proposition 4.1 that \(\Pi =\Pi _1 \boxplus \cdots \boxplus \Pi _r\) with \(\Pi _i\) unitary cuspidal representations of \(\mathrm{GL}(d_i,\mathbb {A})\) for all \(1\leqslant i\leqslant r\). In particular the bounds towards Ramanujan apply and \(\left {{\mathrm{\mathfrak {I}m}}}\gamma _j(\Pi )\right <\frac{1}{2}\) for all \(j\).
Let \(\Phi \) be a Paley–Wiener function whose Fourier transform
has compact support. Note that \(\Phi \) may be extended to an entire function on \(\mathbb {C}\).
Proposition 4.4
Let \(\Pi \) be an isobaric representation of \(\mathrm{GL}(d,\mathbb {A})\) satisfying the bounds towards Ramanujan (4.4). With notation as above and for \(\sigma >\frac{1}{2}\), the following identity holds
There is an important remark about the explicit formula that we will use frequently. Therefore we insert it here before going into the proof. The line of integration in (4.8) is away from the zeros and poles because \(\sigma >1/2\). In particular the line of integration cannot be moved to \(\sigma =0\) directly. But we can do the following which is a natural way to produce the sum over primes. First we replace \(\Lambda (s,\Pi )\) by its Euler product which is absolutely convergent in the given region (\({{\mathrm{\mathfrak {R}e}}}s>1\)). Then for each of the term we may move the line of integration to \(\sigma =0\) because we have seen that \(\frac{L'}{L}(s,\Pi _v)\) has no pole for \({{\mathrm{\mathfrak {R}e}}}s > \theta \). Thus we have
The latter expression is convenient to use in practice. The integral in the righthand side of (4.9) is absolutely convergent because \(\Phi \) is rapidly decreasing and the sum over \(v\in \mathcal {V}_F\) is actually finite since the support of \(\widehat{\Phi }\) is compact. ^{Footnote 5}
Proof
The first step is to work with the Mellin transform rather than the Fourier transform. Namely we set
Note that \(H\) is an entire function which is rapidly decreasing on vertical strips. This justifies all shifting of contours below.
We form the integral
We shift the contour to \({{\mathrm{\mathfrak {R}e}}}s=1\) crossing zeros and eventual poles of \(\dfrac{\Lambda '}{\Lambda }\) inside the critical strip. The sum over the zeros reads
and the sum over the poles reads
Note that since \(\epsilon (s,\Pi )=\epsilon (\Pi ) q(\Pi )^{\frac{1}{2}s}\) we have
We obtain as consequence of the functional Eq. (4.2) that
Now we observe that
and also
and
Since \(\Lambda (s,\widetilde{\Pi })=\overline{\Lambda (\overline{s},\Pi )}\) this concludes the proof of the proposition by collecting all the terms above. Precisely this yields the formula when \(\sigma =3/2\), and then we can make \(\sigma >1/2\) arbitrary by shifting the line of integration. \(\square \)
We conclude this section with a couple of remarks on symmetries. The first observation is that the functional equation implies that if \(\rho \) is a zero (resp. pole) of \(\Lambda (s,\Pi )\) then so is \(1\bar{\rho }\) (reflexion across the central line). Thus the set \(\left\{ \gamma _j(\Pi )\right\} \) (resp. \(\left\{ r_j(\Pi )\right\} \)) is invariant by the reflexion across the real axis (namely \(\gamma \) goes into \(\overline{\gamma }\)). Note that this is compatible with the GRH which predicts that \({{\mathrm{\mathfrak {R}e}}}\rho _j(\Pi )=\frac{1}{2}\) and \(\gamma _j(\Pi )\in \mathbb {R}\).
Assuming \(\Phi \) is realvalued the explicit formula is an identity between real numbers. Indeed the Schwartz reflection principle gives \(\Phi (s)=\overline{\Phi (\overline{s})}\) for all \(s\in \mathbb {C}\). Because of the above remark the sum over the zeros (resp. poles) in (4.8) is a real; the integrand is realvalued as well for all \(r\in (\infty ,\infty )\).
The situation when \(\Pi \) is selfdual occurs often in practice. The zeros \(\gamma _j(\Pi )\) satisfy another symmetry which is the reflexion across the origin. Assuming \(\Pi \) is cuspidal and nontrivial there is no pole. The explicit formula (4.8) simplifies and may be written
5 Sato–Tate equidistribution
Let \(G\) be a connected reductive group over a number field \(F\) as in the previous section. The choice of a \(\mathrm{Gal}(\overline{F}/F)\)invariant splitting datum \((\widehat{B},\widehat{T},\{X_{\alpha }\}_{\alpha \in \Delta ^{\vee }})\) as in Sect. 2.1 induces a composite map \(\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{Out}(\widehat{G})\hookrightarrow \mathrm{Aut}(\widehat{G})\) with open kernel. Let \(F_1\) be the unique finite extension of \(F\) in \(\overline{F}\) such that
5.1 Definition of the Sato–Tate measure
Set \(\Gamma _1:=\mathrm{Gal}(F_1/F)\). Let \(\widehat{K}\) be a maximal compact subgroup of \(\widehat{G}\) which is \(\Gamma _1\)invariant. (It is not hard to see that such a \(\widehat{K}\) exists, cf. [2].) Set \(\widehat{T}_{c}:=\widehat{T}\cap \widehat{K}\). (The subscript \(c\) stands for “compact” as it was in Sect. 3.3.) Denote by \(\Omega _c\) the Weyl group for \((\widehat{K},\widehat{T}_c)\).
Let \(\theta \in \Gamma _1\). Define \(\Omega _{c,\theta }\) to be the subset of \(\theta \)invariant elements of \(\Omega _c\). Consider the topological quotient \(\widehat{K}^\natural _\theta \) of \(\widehat{K}\rtimes \theta \) by the \(\widehat{K}\)conjugacy equivalence relation. Set \(\widehat{T}_{c,\theta }:=\widehat{T}_c/(\theta \mathrm{id})\widehat{T}_c\). Note that the action of \(\Omega _{c,\theta }\) on \(\widehat{T}_c\) induces an action on \(\widehat{T}_{c,\theta }\). The inclusion \(\widehat{T}_c\hookrightarrow \widehat{K}\) induces a canonical topological isomorphism (cf. Lemma 3.2)
The Haar measure on \(\widehat{K}\) (resp. on \(\widehat{T}_c\)) with total volume 1 is written as \(\mu _{\widehat{K}}\) (resp. \(\mu _{\widehat{T}_c}\)). Then \(\mu _{\widehat{K}}\) on \(\widehat{K}\rtimes \theta \) induces the quotient measure \(\mu _{\widehat{K}^\natural _\theta }\) (so that for any continuous function \(f^\natural \) on \(\widehat{K}_\theta ^\natural \) and its pullback \(f\) on \(\widehat{K}\), \(\int f^\natural \mu _{\widehat{K}^\natural _\theta }=\int f \mu _{\widehat{K}}\)) thus also a measure \(\mu _{\widehat{T}_c,\theta }\) on \(\widehat{T}_{c,\theta }\).
Definition 5.1
The \(\theta \)Sato–Tate measure \(\widehat{\mu }^{\mathrm {ST}}_\theta \) on \(\widehat{T}_{c,\theta }/\Omega _{c,\theta }\) is the measure transported from \(\mu _{\widehat{K}^\natural _\theta }\) via (5.1).
Lemma 5.2
Let \(\widehat{\mu }^{\mathrm {ST}}_{\theta ,0}\) denote the measure on \(\widehat{T}_{c,\theta }\) pulled back from \(\widehat{\mu }^{\mathrm {ST}}_\theta \) on \(\widehat{T}_{c,\theta }/\Omega _{c,\theta }\) [so that \(\int f \widehat{\mu }^{\mathrm {ST}}_{\theta ,0}= \int \overline{f} \widehat{\mu }^{\mathrm {ST}}_\theta \) for every continuous \(\overline{f}\) on \(\widehat{T}_{c,\theta }/\Omega _{c,\theta }\) and its pullback \(f\)]. Then
where \(D_\theta (t)={\det (1\mathrm{ad}(t\rtimes \theta )\mathrm{Lie}\,(\widehat{K})/\mathrm{Lie}\,(\widehat{T}^\theta _c))}\) and \(t\) signifies a parameter on \(\widehat{T}_{c,\theta }\).
Proof
The twisted Weyl integration formula tells us that for a continuous \(f:\widehat{K}\rightarrow \mathbb {C}\),
Notice that \(\widehat{K}_{t\theta }\) is the twisted centralizer group of \(t\) in \(\widehat{K}\) (or, the centralizer group of \(t\theta \) in \(\widehat{K}\)). On the right hand side, \( \mu _{\widehat{T}_{c,\theta }}\) is used for integration. When \(f\) is a pullback from \(\widehat{K}^\natural _{\theta }\), the formula simplifies as
and the left hand side is equal to \(\int _{\widehat{T}_{c,\theta }} f(t) \widehat{\mu }^{\mathrm {ST}}_{\theta ,0}\) by definition. \(\square \)
5.2 Limit of the Plancherel measure versus the Sato–Tate measure
Let \(\theta ,\tau \in \Gamma _1\). Then clearly \(\Omega _{c,\theta }=\Omega _{c,\tau \theta \tau ^{1}}\), \(\widehat{K}^\natural _\theta \simeq \widehat{K}^\natural _{\tau \theta \tau ^{1}}\) via \(k\mapsto \tau (k)\) and \(\widehat{T}_{c,\theta }\simeq \widehat{T}_{c,\tau \theta \tau ^{1}}\) via \(t\mapsto \tau (t)\). Accordingly \(\widehat{\mu }^{\mathrm {ST}}_\theta \) and \(\widehat{\mu }^{\mathrm {ST}}_{\theta ,0}\) are identified with \(\widehat{\mu }^{\mathrm {ST}}_{\tau \theta \tau ^{1}}\) and \(\widehat{\mu }^{\mathrm {ST}}_{\tau \theta \tau ^{1},0}\), respectively.
Fix once and for all a set of representatives \({\fancyscript{C}}(\Gamma _1)\) for conjugacy classes in \(\Gamma _1\). For \(\theta \in {\fancyscript{C}}(\Gamma _1)\), denote by \([\theta ]\) its conjugacy class. For each finite place \(v\) such that \(G\) is unramified over \(F_v\), the geometric Frobenius \(\mathrm{Fr}_v\in \mathrm{Gal}(F_v^{\mathrm {ur}}/F_v)\) gives a welldefined conjugacy class \([\mathrm{Fr}_v]\) in \(\Gamma _1\). The set of all finite places \(v\) of \(F\) where \(G\) is unramified is partitioned into
such that \(v\in \mathcal {V}_F(\theta )\) if and only if \([\mathrm{Fr}_v]=[\theta ]\).
For each finite place \(v\) of \(F\), the unitary dual of \(G(F_v)\) and its Plancherel measure are written as \(G^{\wedge }_v\) and \(\widehat{\mu }^{\mathrm {pl}}_v\). Similarly adapt the notation of Sect. 3.1 by appending the subscript \(v\). Now fix \(\theta \in {\fancyscript{C}}(\Gamma _1)\) and suppose that \(G\) is unramified at \(v\) and that \(v\in \mathcal {V}_F(\theta )\). We choose \(\overline{F}\hookrightarrow \overline{F}_v\) such that \(\mathrm{Fr}_v\) has image \(\theta \) in \(\Gamma _1\) (rather than some other conjugate). This rigidifies the identification in the second map below. (If \(\mathrm{Fr}_v\) maps to \(\tau \theta \tau ^{1}\) then the second map is twisted by \(\tau \).)
By abuse of notation let \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v}\) [a measure on \(G(F_v)^{\wedge ,\mathrm {ur},\text {temp}}\)] also denote the transported measure on \(\widehat{T}_{c,\theta }/\Omega _{c,\theta }\). Let \(C_v\) denote the constant of Proposition 3.3, which we normalize such that \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,0}\) has total volume 1. Note that \(\widehat{\mu }^{\mathrm {ST}}_{\theta }\) also has total volume 1.
Proposition 5.3
Fix any \(\theta \in {\fancyscript{C}}(\Gamma _1)\). As \(v\rightarrow \infty \) in \(\mathcal {V}_F(\theta )\), we have weak convergence \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v}\rightarrow \widehat{\mu }^{\mathrm {ST}}_{\theta }\) as \(v\rightarrow \infty \).
Proof
It is enough to show that \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,0}\rightarrow \widehat{\mu }^{\mathrm {ST}}_{\theta ,0}\) on \(\widehat{T}_{c,\theta }\) as \(v\) tends to \(\infty \) in \(\mathcal {V}_F(\theta )\). Consider the measure \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,1}:=C_v^{1}\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,0}\). It is clear from the formula of Proposition 3.3 that \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,1}\rightarrow \widehat{\mu }^{\mathrm {ST}}_{\theta }\) as \(v\rightarrow \infty \) in \(\mathcal {V}_F(\theta )\). In particular, the total volume of \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,1}\) tends to 1, hence \(C_v\rightarrow 1\) as \(v\rightarrow \infty \) in \(\mathcal {V}_F(\theta )\). We conclude that \(\widehat{\mu }^{\mathrm {pl,ur,temp}}_{v,0}\rightarrow \widehat{\mu }^{\mathrm {ST}}_{\theta ,0}\) as desired. \(\square \)
Remark 5.4
The above proposition was already noticed by Sarnak for \(G=SL(n)\) in [90, §4].
5.3 The generalized Sato–Tate problem
Let \(\pi \) be a cuspidal^{Footnote 6} tempered automorphic representation of \(G(\mathbb {A}_F)\) satisfying
Hypothesis. The conjectural global \(L\)parameter \(\varphi _\pi \) for \(\pi \) has Zariski dense image in \({}^L G_{F_1/F}\).
Of course this hypothesis is more philosophical than practical. The global Langlands correspondence between (\(L\)packets of) automorphic representations and global \(L\)parameters of \(G(\mathbb {A}_F)\) is far from established. A fundamental problem here is that global \(L\)parameters cannot be defined unless the conjectural global Langlands group is defined. (Some substitutes have been proposed by Arthur in the case of classical groups. The basic idea is that a cuspidal automorphic representation of \(\mathrm{GL}_n\) can be put in place of an irreducible \(n\)dimensional representation of the global Langlands group.) Nevertheless, the above hypothesis can often be replaced with another condition, which should be equivalent but can be stated without reference to conjectural objects. For instance, when \(\pi \) corresponds to a Hilbert modular form of weight \(\geqslant 2\) at all infinite places, one can use the hypothesis that it is not a CM form (i.e. not an automorphic induction from a Hecke character over a CM field).
Let us state a general form of the Sato–Tate conjecture. Let \(q_v\) denote the cardinality of the residue field cardinality at a finite place \(v\) of \(F\). Define \(\mathcal {V}_F(\theta ,\pi )^{\leqslant x}:=\{v\in \mathcal {V}_F(\theta ,\pi ): q_v\leqslant x\}\) for \(x\in \mathbb {R}_{\geqslant 1}\).
Conjecture 5.5
Assume the above hypothesis. For each \(\theta \in {\fancyscript{C}}(\Gamma _1)\), let \(\mathcal {V}_F(\theta ,\pi )\) be the subset of \(v\in \mathcal {V}_F(\theta )\) such that \(\pi _v\) is unramified. Then \(\{\pi _v\}_{v\in \mathcal {V}_F(\theta ,\pi )}\) are equidistributed according to \(\widehat{\mu }^{\mathrm {ST}}_\theta \). More precisely
The above conjecture is deemed plausible in that it is essentially a consequence of the Langlands functoriality conjecture at least when \(G\) is (an inner form of) a split group. Namely if we knew that the \(L\)function \(L(s,\pi ,\rho )\) for any irreducible representation \({}^L G\rightarrow \mathrm{GL}_d\) were a cuspidal automorphic \(L\)function for \(\mathrm{GL}_d\) then the desired equidistribution is implied by Theorem 1 of [92, AppA.2].
Remark 5.6
In general when the above hypothesis is dropped, it is likely that \(\pi \) comes from an automorphic representation on a smaller group than \(G\). [If \(\varphi _\pi \) factors through an injective \(L\)morphism \({}^L H_{F_1/F}\rightarrow {}^L G_{F_1/F}\) then the Langlands functoriality predicts that \(\pi \) arises from an automorphic representation of \(H(\mathbb {A}_F)\).] Suppose that the Zariski closure of \(\mathrm{Im\,}(\varphi _\pi )\) in \({}^L G_{F_1/F}\) is isomorphic to \({}^L H_{F_1/F}\) for some connected reductive group \(H\) over \(F\). (In general the Zariski closure may consist of finitely many copies of \({}^L H_{F_1/F}\).) Then \(\{\pi _v\}_{v\in \mathcal {V}_F(\theta ,\pi )}\) should be equidistributed according to the Sato–Tate measure belonging to \(H\) in order to be consistent with the functoriality conjecture.
One can also formulate a version of the conjecture where \(v\) runs over the set of all finite places where \(\pi _v\) are unramified by considering conjugacy classes in \({}^L G_{F_1/F}\) rather than those in \(\widehat{G}\rtimes \theta \) for a fixed \(\theta \). For this let \(\widehat{K}^\natural \) denote the quotient of \(\widehat{K}\) by the equivalence relation coming from the conjugation by \(\widehat{K}\rtimes \Gamma _1\). Since \(\widehat{K}^\natural \) is isomorphic to a suitable quotient of \(\widehat{T}_c\), the Haar measure on \(\widehat{K}\) gives rise to a measure, to be denoted \(\widehat{\mu }^{\mathrm {ST}}\), on the quotient of \(\widehat{T}_c\). Let \(\mathcal {V}_F(\pi )^{\leqslant x}\) (where \(x\in \mathbb {R}_{\geqslant 1}\)) denote the set of finite places of \(F\) such that \(\pi _v\) are unramified and \(q_v\leqslant x\). By writing \(v\rightarrow \infty \) we mean that \(q_v\) tends to infinity.
Conjecture 5.7
Assume the above hypothesis. Then as \(x\rightarrow \infty \) the set \(\{\pi _v:v\in \mathcal {V}_F(\pi )^{\leqslant x}\}\) is equidistributed according to \(\widehat{\mu }^{\mathrm {ST}}_{\theta }\). Namely
Remark 5.8
Unlike Conjecture 5.5 it is unnecessary to choose embeddings \(\overline{F}\hookrightarrow \overline{F}_v\) to rigidify (5.2) since the ambiguity in the rigidification is absorbed in the conjugacy classes in \({}^L G_{F_1/F}\). The formulation of Conjecture 5.7 might be more suitable than the previous one in the motivic setting where we would not want to fix \(\overline{F}\hookrightarrow \overline{F}_v\). The interested reader may compare Conjecture 5.7 with the motivic Sato–Tate conjecture of [96, 13.5].
The next subsection will discuss the analogue of Conjecture 5.5 for automorphic families. Conjecture 5.7 will not be considered any more in our paper. It is enough to mention that the analogue of the latter conjecture for families of algebraic varieties makes sense and appears to be interesting.
5.4 The Sato–Tate conjecture for families
The Sato–Tate conjecture has been proved for Hilbert modular forms in [6, 7]. Analogous equidistribution theorems in the function field setting are due to Deligne and Katz. (See [59, Thm9.2.6] for instance.) Despite these fantastic developments, we have little unconditional theoretical evidence for the Sato–Tate conjecture for general reductive groups over number fields. On the other hand, it has been noticed that the analogue of the Sato–Tate conjecture for families of automorphic representations is more amenable to attack. Indeed there was some success in the case of holomorphic modular forms and Maass forms [34, 53, 83, 97]. The conjecture has the following coarse form, which should be thought of as a guiding principle rather than a rigorous conjecture. Compare with some precise results in Sect. 9.7.
Heuristic 5.9
Let \(\{{\mathcal {F}}_k\}_{k\geqslant 1}\) be a “general” sequence of finite families of automorphic representations of \(G(\mathbb {A}_F)\) such that \({\mathcal {F}}_k\rightarrow \infty \) as \(k\rightarrow \infty \). Then \(\{\pi _v\in {\mathcal {F}}_k\}\) are equidistributed according to \(\widehat{\mu }^{\mathrm {ST}}_{\theta }\) as \(k\) and \(v\) tend to infinity subject to the conditions that \(v\in \mathcal {V}_F(\theta )\) and that all members of \({\mathcal {F}}_k\) are unramified at \(v\).
We are not going to make precise what “general” means, but merely remark that it should be the analogue of the condition that the hypothesis of Sect. 5.3 holds for the “generic fiber” of the family when the family has a geometric meaning (see also [87]). In practice one would verify the conjecture for many interesting families while simply ignoring the word “general”. Some relation between \(k\) and \(v\) holds when taking limit: \(k\) needs to grow fast enough compared to \(v\) (or more precisely \(\left {\mathcal {F}}_k\right \) needs to grow fast enough compared to \(q_v\)).
It is noteworthy that the unpleasant hypothesis of Sect. 5.3 can be avoided for families. Also note that the temperedness assumption is often unnecessary due to the fact that the Plancherel measure is supported on the tempered spectrum. This is an indication that most representations in a family are globally tempered, which we will return to in a subsequent work.
Later we will verify the conjecture for many families in Sect. 9.7 as a corollary to the automorphic Plancherel theorem proved earlier in Sect. 9. Our families arise as the sets of all automorphic representations with increasing level or weight, possibly with prescribed local conditions at finitely many fixed places.
6 Background materials
This section collects background materials in the local and global contexts. Sections 6.1 and 6.3 are concerned with \(p\)adic groups while Sects. 6.4, 6.5 and 6.8 are with real and complex Lie groups. The rest is about global reductive groups.
6.1 Orbital integrals and constant terms
We introduce some notation in the \(p\)adic context.

\(F\) is a finite extension of \(\mathbb {Q}_p\) with integer ring \(\mathcal {O}\) and multiplicative valuation \(\cdot \).

\(G\) is a connected reductive group over \(F\).

\(A\) is a maximal \(F\)split torus of \(G\), and put \(M_0:=Z_G(A)\).

\(K\) is a maximal compact subgroup of \(G\) corresponding to a special point in the apartment for \(A\).

\(P=MN\) is a parabolic subgroup of \(G\) over \(F\), with \(M\) and \(N\) its Levi subgroup and unipotent radical, such that \(M\supset M_0\).

\(\gamma \in G(F)\) is a semisimple element. (The case of a nonsemisimple element is not needed in this paper.)

\(I_\gamma \) is the neutral component of the centralizer of \(\gamma \) in \(G\). Then \(I_\gamma \) is a connected reductive group over \(F\).

\(\mu _G\) (resp. \(\mu _{I_\gamma }\)) is a Haar measure on \(G(F)\) (resp. \(I_\gamma (F)\)).

\(\frac{\mu _G}{\mu _{I_\gamma }}\) is the quotient measure on \(I_\gamma (F)\backslash G(F)\) induced by \(\mu _G\) and \(\mu _{I_\gamma }\).

\(\phi \in C^\infty _c(G(F))\).

\(D^G(\gamma ):=\prod _{\alpha } 1\alpha (\gamma )\) for a semisimple \(\gamma \in G(F)\), where \(\alpha \) runs over the set of roots of \(G\) (with respect to any maximal torus in the connnected centralizer of \(\gamma \) in \(G\)) such that \(\alpha (\gamma )\ne 1\). Let \(M\) be an \(F\)rational Levi subgroup of \(G\). For a semisimple \(\gamma \in G(F)\), we define \(D^G_M(\gamma )\) similarly by further excluding those \(\alpha \) in the set of roots of \(M\).
Define the orbital integral
When the context is clear, we use \(O_\gamma (\phi )\) as a shorthand notation.
We recall the theory of constant terms (cf. [105, p. 236]). Choose Haar measures \(\mu _K\), \(\mu _M\), \(\mu _N\), on \(K\), \(M(F)\), \(N(F)\), respectively, such that \(\mu _G=\mu _K\mu _M\mu _N\) holds with respect to \(G(F)=KM(F)N(F)\). Define the (normalized) constant term \(\phi _M\in C^\infty _c(M(F))\) by
Although the definition of \(\phi _M\) involves not only \(M\) but \(P\), the following lemma shows that the orbital integrals of \(\phi _M\) depend only on \(M\) by the density of regular semisimple orbital integrals, justifying our notation.
Lemma 6.1
For all \((G,M)\)regular semisimple \(\gamma \in M(F)\),
Proof
[105, Lem 9]. (Although the lemma is stated for regular elements \(\gamma \in G\), it suffices to require \(\gamma \) to be \((G,M)\)regular. See Lemma 8 of loc. cit.) \(\square \)
It is standard that the definition and facts we have recollected above extend to the adelic case. (Use [63, §§7–8], for instance). We will skip rewriting the analogous definition in the adelic setting.
Now we restrict ourselves to the local unramified case. Suppose that \(G\) is unramified over \(F\). Let \(B\subset P \subset G\) be Borel and parabolic subgroups defined over \(F\). Write \(B=TU\) and \(P=MN\) where \(T\) and \(M\) are Levi subgroups such that \(T\subset M\) and \(U\) and \(N\) are unipotent radicals.
Lemma 6.2
Let \(\phi \in {\mathcal {H}}^{\mathrm {ur}}(G)\). Then \({\mathcal {S}}^G_M(\phi )=\phi _M\), in particular \({\mathcal {S}}^G(\phi )= {\mathcal {S}}^M({\mathcal {S}}^G_M \phi )=\phi _T\).
Proof
Straightforward from (2.1) and (6.1). \(\square \)
6.2 Gross’s motives
Now let \(F\) be a finite extension of \(\mathbb {Q}\) (although Gross’s theory applies more generally). Let \(G\) be a connected reductive group over \(F\) and consider its quasisplit inner form \(G^*\). Let \(T^*\) be the centralizer of a maximal \(F\)split torus of \(G^*\). Denote by \(\Omega \) the Weyl group for \((G^*,T^*)\) over \(\overline{F}\). Set \(\Gamma =\mathrm{Gal}(\overline{F}/F)\). Gross [47] attaches to \(G\) an Artin–Tate motive
with coefficients in \(\mathbb {Q}\). Here \((1d)\) denotes the Tate twist. The Artin motive \(\mathrm{Mot}_{G,d}\) (denoted \(V_d\) by Gross) may be thought of as a \(\Gamma \)representation on a \(\mathbb {Q}\)vector space whose dimension is \(\dim \mathrm{Mot}_{G,d}\). Define
to be the Artin \(L\)value of \(L(s,\mathrm{Mot}_G)\) at \(s=0\). We recall some properties of \(\mathrm{Mot}_G\) from Gross’s article.
Proposition 6.3
 (i):

\(\mathrm{Mot}_{G,d}\) is selfdual for each \(d\geqslant 1\).
 (ii):

\(\sum _{d\geqslant 1} \dim \mathrm{Mot}_{G,d} = r_G=\mathrm{rk}G\).
 (iii):

\(\sum _{d\geqslant 1} (2d1)\dim \mathrm{Mot}_{G,d} = \dim G\).
 (iv):

\(\Omega =\prod _{d\geqslant 1} d^{\dim \mathrm{Mot}_{G,d}}\).
 (vi):

If \(T^*\) splits over a finite extension \(E\) of \(F\) then the \(\Gamma \)action on \(\mathrm{Mot}_G\) factors through \(\mathrm{Gal}(E/F)\).
The Artin conductor \({\mathfrak f}(\mathrm{Mot}_{G,d})\) is defined as follows. Let \(F'\) be the fixed field of the kernel of the Artin representation \(\mathrm{Gal}(\overline{F}/F)\rightarrow \mathrm{GL}(V_d)\) associated to \(\mathrm{Mot}_{G,d}\). For each finite place \(v\) of \(F\), let \(w\) be any place of \(F'\) above \(v\). Let \(\Gamma (v)_i:=\mathrm{Gal}(F'_w/F_v)_i\) (\(i\geqslant 0\)) denote the \(i\)th ramification subgroups. Set
which is an integer independent of the choice of \(w\). Write \({\mathfrak p}_v\) for the prime ideal of \(\mathcal {O}_F\) corresponding to \(v\). If \(v\) is unramified in \(E\) then \(f(G_v,d)=0\). Thus the product makes sense in the following definition.
Let \(E\) be the splitting field of \(T^*\) (which is an extension of \(F\)) and set \(s^{\mathrm{spl}}_G:=[E:F]\).
Lemma 6.4
For every finite place \(v\) of \(F\),
Proof
Let \(F'\), \(w\) and \(V_d\) be as in the preceding paragraph. Then \(F\subset F'\subset E\). Set \(s_v:=[F'_w:F_v]\) so that \(s_v\leqslant s^{\mathrm{spl}}_G\). The case \(s_v=1\) is obvious (in which case \(f(G_v,d)=0\)), so we may assume \(s_v\geqslant 2\). From (6.2) and Corollary 6.9 below,
\(\square \)
Recall that \(w_G=\Omega \) is the cardinality of the absolute Weyl group. Let \(s_G\) be the degree of the smallest extension of \(F\) over which \(G\) becomes split. The following useful lemma implies in particular that \(s^{\mathrm{spl}}_G\leqslant w_Gs_G\).
Lemma 6.5
[56, Lem 2.2] For any maximal torus \(T\) of \(G\) defined over \(F\), there exists a finite Galois extension \(E\) of \(F\) such that \([E:F]\leqslant w_Gs_G\) and \(T\) splits over \(E\).
6.3 Lemmas on ramification
This subsection is meant to provide an ingredient of proof (namely Corollary 6.9) for Lemma 6.4.
Fix a prime \(p\). Let \(E\) and \(F\) be finite extensions of \(\mathbb {Q}_p\) with uniformizers \(\varpi _E\) and \(\varpi _F\), respectively. Normalize valuations \(v_E:E^\times \rightarrow \mathbb {Z}\) and \(v_F:F^\times \rightarrow \mathbb {Z}\) such that \(v_E(\varpi _E)=v_F(\varpi _F)=1\). Write \(e_{E/F}\in \mathbb {Z}_{\geqslant 1}\) for the ramification index and \({\mathfrak D}_{E/F}\) for the different. For a nonzero principal ideal \({\mathfrak a}\) of \(\mathcal {O}_E\), we define \(v_E({\mathfrak a})\) to be \(v_E(a)\) for any generator \(a\) of \({\mathfrak a}\). This is well defined.
Lemma 6.6
Let \(E\) be a totally ramified Galois extension of \(F\) with \([E:F]=p^n\) for \(n\geqslant 0\). Then
Remark 6.7
In fact the inequality is sharp. There are totally ramified extensions \(E/F\) for which the above equality holds as shown by Öre. See also [95, §1] for similar results.
Proof
The lemma is trivial when \(n=0\). Next assume \(n=1\) but allow \(E/F\) to be a nonGalois extension. Let \(f(x)=\sum _{i=0}^p a_i x^i\in \mathcal {O}_F[x]\) (with \(a_p=1\) and \(v_F(a_i)\geqslant 1\) for \(i<p\)) be the Eisenstein polynomial having \(\varpi _E\) as a root. By [94, III.6, Cor 2], \(v_E({\mathfrak D}_{E/F})=v_E(f'(\varpi _E))\). The latter equals
This prepares us to tackle the case of arbitrary \(n\). Choose a sequence of subextensions \(E=F_0\supset F_1\supset \cdots \supset F_n=F\) such that \([F_m:F_{m+1}]=p\) (where \(F_m/F_{m+1}\) may not be a Galois extension). By above, \(v_{F_m}({\mathfrak D}_{F_m/F_{m+1}})\leqslant e_{F_m/\mathbb {Q}_p}+p1\) for \(0\leqslant m\leqslant n1\). Hence
\(\square \)
Lemma 6.8
Let \(E\) be a finite Galois extension of \(F\). Then
Proof
Let \(E^t\) (resp. \(E^{\mathrm {ur}}\)) be the maximal tame (resp. unramified) extension of \(F\) in \(E\). Then \(v_{E^t}({\mathfrak D}_{E^t/E^{\mathrm {ur}}})=[E^t:E^{\mathrm {ur}}]1\) by [94, III.6, Prop 13]. Clearly \(v_{E^{ur}}({\mathfrak D}_{E^{\mathrm {ur}}/F})=0\). Together with Lemma 6.6, we obtain
\(\square \)
Corollary 6.9
Let \(E\) be a finite Galois extension of \(F\). Then the \(i\)th ramification group \(\mathrm{Gal}(E/F)_{i}\) is trivial for \(i=[E:F](1+e_{F/\mathbb {Q}_p} \log _p [E:F])1\).
Proof
In the notation of section IV.1 of [94], we have \(\mathrm{Gal}(E/F)_{m}=1\) by definition if \(m=\max _{1\ne s\in \mathrm{Gal}(E/F)} i_G(s)\). But the proposition 4 in that section implies that \(m\leqslant v_E({\mathfrak D}_{E/F})\), so Lemma 6.8 finishes the proof.
6.4 Stable discrete series characters
In Sects. 6.4 and 6.5 we specialize to the situation of real groups.

\(G\) is a connected reductive group over \(\mathbb {R}\).

\(A_{G,\infty }=A_G(\mathbb {R})^0\) where \(A_G\) is the maximal split torus in the center of \(G\).

\(K_\infty \) is a maximal compact subgroup of \(G(\mathbb {R})\) and \(K'_\infty :=K_\infty A_{G,\infty }\).

\(q(G):=\frac{1}{2} \dim _{\mathbb {R}} G(\mathbb {R})/K'_\infty \in \mathbb {Z}_{\geqslant 0}\).

\(T\) is an \(\mathbb {R}\)elliptic maximal torus in \(G\). (Assume that such a \(T\) exists.)

\(B\) is a Borel subgroup of \(G\) over \(\mathbb {C}\) containing \(T\).

\(I_\gamma \) denotes the connected centralizer of \(\gamma \in G(\mathbb {R})\).

\(\Phi ^+\) (resp. \(\Phi \)) is the set of positive (resp. all) roots of \(T\) in \(G\) over \(\mathbb {C}\).

\(\Omega \) is the Weyl group for \((G,T)\) over \(\mathbb {C}\), and \(\Omega _c\) is the compact Weyl group.

\(\rho :=\frac{1}{2}\sum _{\alpha \in \Phi ^+} \alpha \).

\(\xi \) is an irreducible finite dimensional algebraic representation of \(G(\mathbb {R})\).

\(\lambda _\xi \in X^*(T)\) is the \(B\)dominant highest weight for \(\xi \).

\(m(\xi ):=\min _{\alpha \in \Phi ^+} \langle \lambda _\xi +\rho ,\alpha \rangle \). We always have \(m(\xi )>0\).

\(\Pi _\mathrm{disc}(\xi )\) is the set of irreducible discrete series representations of \(G(\mathbb {R})\) with the same infinitesimal character and the same central character as \(\xi \). [This is an \(L\)packet for \(G(\mathbb {R})\).]

\(D_\infty ^G(\gamma ):=\prod _{\alpha } 1\alpha (\gamma )\) for \(\gamma \in T(\mathbb {R})\), where \(\alpha \) runs over elements of \(\Phi \) such that \(\alpha (\gamma )\ne 1\). [If \(\gamma \) is in the center of \(G(\mathbb {R})\), \(D_\infty ^G(\gamma )=1\).]
If \(M\) is a Levi subgroup of \(G\) over \(\mathbb {C}\) containing \(T\), the following are defined in the obvious manner as above: \(\Phi ^+_M\), \(\Phi _M\), \(\Omega _M\), \(\rho _M\), \(D_\infty ^M\). Define \(\Omega ^M:=\{\omega \in \Omega : \omega ^{1}\Phi ^+_M\subset \Phi ^+\}\), which is a set of representatives for \(\Omega /\Omega _M\). For each regular \(\gamma \in T(\mathbb {R})\), let us define (cf. [3, (4.4)])
where \(\Theta _\pi \) is the character function of \(\pi \). It is known that the function \(\Phi ^G_M(\gamma ,\xi )\) continuously extends to an \(\Omega _M\)invariant function on \(T(\mathbb {R})\), thus also to a function on \(M(\mathbb {R})\) which is invariant under \(M(\mathbb {R})\)conjugation and supported on elliptic elements ([3, Lem 4.2], cf. [45, Lem 4.1]). When \(M=G\), simply \(\Phi ^G_M(\gamma ,\xi )= \mathrm{tr}\,\xi (\gamma )\).
We would like to have an upper bound for \(\Phi ^G_M(\gamma ,\xi )\) that we will need in Sect. 9.5. This is a refinement of [99, Lem 4.8].
Lemma 6.10
 (i):

\(\dim \xi =\prod _{\alpha \in \Phi ^+} \frac{\langle \alpha ,\lambda _\xi +\rho \rangle }{\langle \alpha ,\rho \rangle }\).
 (ii):

There exists a constant \(c>0\) independent of \(\xi \) such that for every elliptic \(\gamma \in G(\mathbb {R})\) and \(\xi \),
$$\begin{aligned} \frac{\mathrm{tr}\,\xi (\gamma )}{\dim \xi } \leqslant c \frac{D_\infty ^G(\gamma )^{1/2}}{ m(\xi )^{\Phi ^+\Phi ^+_{I_\gamma }}} . \end{aligned}$$
Proof
Part (i) is the standard Weyl dimension formula. Let us prove (ii). The formula right above the corollary 1.12 in [19] implies that
Note that their \(M\) is our \(I_\gamma \) and that \(\alpha (\gamma )=1\) for all \(\alpha \in \Phi \) and all elliptic \(\gamma \in G(\mathbb {R})\). Hence by (i),
\(\square \)
Lemma 6.11
Assume that \(M\) is a Levi subgroup of \(G\) over \(\mathbb {R}\) containing an elliptic maximal torus. There exists a constant \(c_0>0\) independent of \(\xi \) such that for every elliptic \(\gamma \in M(\mathbb {R})\),
Proof
As the case \(M=G\) is already proved by Lemma 6.10 (ii), we assume that \(M\subsetneq G\). Fix an elliptic maximal torus \(T\subset M\). Since every elliptic element has a conjugate in \(T(\mathbb {R})\) and both sides of the inequality are conjugateinvariant, it is enough to verify the lemma for \(\gamma \in T(\mathbb {R})\). In this proof we borrow some notation and facts from [45, pp. 494–498] as well as [3, pp. 272–274]. For the purpose of proving Lemma 6.11, we may restrict to \(\gamma \in \Gamma ^+\), corresponding to a closed chamber for the root system of \(T(\mathbb {R})\) in \(G(\mathbb {R})\). (See page 497 of [45] for the precise definition.) The proof of [45, Lem 4.1] shows that
where \(\xi ^M_\omega \) is the irreducible representation of \(M(\mathbb {R})\) of highest weight \(\omega (\xi +\rho )\rho _M\). We claim that there is a constant \(c_1>0\) independent of \(\xi \) such that
for all \(\omega \) and \(\xi \). The coefficients \( c(\omega ,\xi )\) can be computed by rewriting the right hand side of [3, (4.8)] as a linear combination of \(\mathrm{tr}\,\xi ^M_\omega (\gamma )\) using the Weyl character formula. In order to verify the claim, it suffices to point out that \(\overline{c}(Q^+_{ys\lambda },R^+_H)\) in Arthur’s (4.8) takes values in a finite set which is independent of \(\xi \) (or \(\tau \) in Arthur’s notation). This is obvious: as \(Q^+_{ys\lambda }\subset \Phi ^\vee \) and \(R^+_H\subset \Phi \), there are finitely many possibilities for \(Q^+_{ys\lambda }\) and \(R^+_H\).
Now by Lemma 6.10 (i),
with \(c_2=(\prod _{\alpha \in \Phi ^+} \langle \alpha ,\rho \rangle ) (\prod _{\alpha \in \Phi _M^+} \langle \alpha ,\rho _M \rangle )^{1}>0\). According to Lemma 6.10 (ii), there exists a constant \(c_3>0\) such that
To conclude the proof, multiply the last two formulas. \(\square \)
6.5 Euler–Poincaré functions
We continue to use the notation of Sect. 6.4. Let \(\overline{\mu }^{\mathrm{EP}}_\infty \) denote the Euler–Poincaré measure on \(G(\mathbb {R})/A_{G,\infty }\) (so that its induced measure on the compact inner form has volume 1). There exists a unique Haar measure \(\mu ^{\mathrm{EP}}_\infty \) on \(G(\mathbb {R})\) which is compatible with \(\overline{\mu }^{\mathrm{EP}}_\infty \) and the standard Haar measure on \(A_{G,\infty }\). Write \(\omega _{\xi }\) for the central character of \(\xi \) on \(A_{G,\infty }\). Let \(\Pi (\omega _{\xi }^{1})\) denote the set of irreducible admissible representations of \(G(\mathbb {R})\) whose central characters on \(A_{G,\infty }\) are \(\omega _{\xi }^{1}\). For \(\pi \in \Pi (\omega _{\xi }^{1})\), define
Clozel and Delorme [21] constructed a bi\(K_\infty \)finite function \(\phi _\xi \in C^\infty (G(\mathbb {R}))\) which transforms under \(A_{G,\infty }\) by \(\omega _{\xi }\) and is compactly supported modulo \(A_{G,\infty }\), such that
The following are wellknown:

\(\chi _{\mathrm{EP}}(\pi \otimes \xi )=0\) unless \(\pi \in \Pi (\omega _{\xi }^{1})\) has the same infinitesimal character as \(\xi ^\vee \).

If the highest weight of \(\xi \) is regular then \(\chi _{\mathrm{EP}}(\pi \otimes \xi )\ne 0\) if and only if \(\pi \in \Pi _{\mathrm{disc}}(\xi ^\vee )\).

If \(\pi \in \Pi (\omega _{\xi }^{1})\) is a discrete series and \(\chi _{\mathrm{EP}}(\pi \otimes \xi )\ne 0\) then \(\pi \in \Pi _{\mathrm{disc}}(\xi ^\vee )\) and \(\chi _{\mathrm{EP}}(\pi \otimes \xi )=(1)^{q(G)}\). More precisely, \(\dim H^i(\mathrm{Lie}\,G(\mathbb {R}),K'_\infty , \pi \otimes \xi )\) equals 1 if \(i=q(G)\) and 0 if not.
6.6 Canonical measures and Tamagawa measures
We return to the global setting so that \(F\) and \(G\) are as in Sect. 6.2. Let \(G_\infty :=({\mathrm {Res}}_{F/\mathbb {Q}}G)\times _\mathbb {Q}\mathbb {R}\), to which the contents of Sects. 6.4 and 6.5 apply. In particular we have a measure \(\mu ^{\mathrm{EP}}_\infty \) on \(G_\infty (\mathbb {R})\). For each finite place \(v\) of \(F\), define \(\mu ^{\mathrm{can}}_v:=\Lambda (\mathrm{Mot}_{G_{v}}^\vee (1))\cdot \omega _{G_v}\) in the notation of [47] where \(\omega _{G_v}\) is the “canonical” Haar measure on \(G(F_v)\) as in §11 of that article. When \(G\) is unramified over \(F_v\), the measure \(\mu ^{\mathrm{can}}_v\) assigns volume 1 to a hyperspecial subgroup of \(G(F_v)\). In particular,
is a welldefined measure on \(G(\mathbb {A}_F)\).
Let \(\overline{\mu }^{\mathrm{Tama}}\) denote the Tamagawa measure on \(G(F)\backslash G(\mathbb {A}_F)/A_{G,\infty }\), so that its volume is the Tamagawa number (cf. [64, p. 629])
The Tamagawa measure \(\mu ^{\mathrm{Tama}}\) on \(G(\mathbb {A}_F)\) of [47] is compatible with \(\overline{\mu }^{\mathrm{Tama}}\) if \(G(F)\) and \(A_{G,\infty }\) are equipped with the pointcounting measure and the Lebesgue measure, respectively. The ratio of two Haar measures on \(G(\mathbb {A}_F)\) is computed as:
Proposition 6.12
[47, 10.5]
The following notion will be useful in that the Levi subgroups contributing to the trace formula in Sect. 9 turn out to be the cuspidal ones.
Definition 6.13
We say that \(G\) is cuspidal if \(G_0:={\mathrm {Res}}_{F/\mathbb {Q}}G\) satisfies the condition that \(A_{G_0}\times _\mathbb {Q}\mathbb {R}\) is the maximal split torus in the center of \(G_0\times _\mathbb {Q}\mathbb {R}\).
Assume that \(G\) is cuspidal, so that \(G(\mathbb {R})/A_{G,\infty }\) contains a maximal \(\mathbb {R}\)torus which is anisotropic.
Corollary 6.14
Proof
It suffices to remark that the Euler–Poincaré measure on a compact Lie group has total volume 1, hence \(\overline{\mu }^{\mathrm{EP}}_\infty (\overline{G}(F_\infty )/A_{G,\infty })=1\).
6.7 Bounds for Artin \(L\)functions
For later use we estimate the \(L\)value \(L(\mathrm{Mot}_G)\) in Corollary 6.14.
Proposition 6.15
Let \(s \geqslant 1\) and \(E\) be a Galois extension of \(F\) of degree \([E:F]\leqslant s\).
 (i):

For all \(\epsilon >0\) there exists a constant \(c=c(\epsilon ,s,F)>0\) which depends only on \(\epsilon \), \(s\) and \(F\) such that the following holds: For all nontrivial irreducible representations \(\rho \) of \(\mathrm{Gal}(E/F)\),
$$\begin{aligned} cd_E^{\epsilon }\leqslant L(1,\rho ) \leqslant c d_E^\epsilon . \end{aligned}$$  (ii):

The same inequalities hold for the residue \(\mathrm {Res}_{s=1} \zeta _E(s)\) of the Dedekind zeta function of \(E\).
 (iii):

There is a constant \(A_1=A_1(s,F)>0\) which depends only on \(s\) and \(F\) such that for all faithful irreducible representation \(\rho \) of \(\mathrm{Gal}(E/F)\),
$$\begin{aligned} d_{E/F}^{A_1} \leqslant \mathbb {N}_{F/\mathbb {Q}}(\mathfrak {f}_\rho ) \leqslant d_{E/F}^{1/\dim (\rho )}, \end{aligned}$$where \(d_{E/F}=\mathbb {N}_{F/\mathbb {Q}}({\mathfrak D}_{E{/}F})\) is the relative discriminant of \(E{/}F\); recall that \(d_E= d_F^{[E:F]} d_{E/F}\).
Proof
The assertion (ii) is Brauer–Siegel theorem [14, Theorem 2]. We also note the implication (i) \(\Rightarrow \) (ii) which follows from the formula
where \(\rho \) ranges over all irreducible representations of \(\mathrm{Gal}(E/F)\).
The proof of assertion (i) is reduced to the \(1\)dimensional case by Brauer induction as in [14]. In this reduction one uses the fact that if \(E'/F'\) is a subextension of \(E/F\) then the absolute discriminant \(d_{E'}\) of \(E'\) divides the absolute discriminant \(d_{E}\) of \(E\). Also we may assume that \(E'/F'\) is cyclic. For a character \(\chi \) of \(\mathrm{Gal}(E'/F')\) we have the convexity bound \(L(1,\chi )\leqslant c d^\epsilon _{E'}\) (Landau). The lower bound for \(L(1,\chi )\) follows from (ii) and the product formula (6.4).
In the assertion (iii) the right inequality follows from the discriminantconductor formula which implies that \(\mathfrak {f}^{\dim (\rho )}_\rho {\mid } {\mathfrak D}_{E/F}\). The left inequality follows from local considerations. Let \(v\) be a finite place of \(F\) dividing \({\mathfrak D}_{E/F}\); since \(\rho \) is faithful, its restriction to the inertia group above \(v\) is nontrivial and therefore \(v\) divides \(\mathfrak {f}_\rho \). Since \(v({\mathfrak D}_{E/F})\) is bounded above by a constant \(A_1(s,F)\) depending only on \([E:F]\leqslant s\) and \(F\) by Lemma 6.8, we have \( v({\mathfrak D}_{E/F}) \leqslant A_1 v(\mathfrak {f}_\rho )\) which concludes the proof. \(\square \)
Corollary 6.16
For all integers \(R, D,s\in \mathbb {Z}_{\geqslant 1}\), and \(\epsilon >0\) there is a constant \(c_1=c_1(\epsilon ,R,D,s,F)>0\) (depending on \(R\), \(D\), \(s\), \(F\) and \(\epsilon \)) with the following property
 (i):

For any \(G\) such that \(r_G\leqslant R\), \(\dim G\leqslant D\), \(Z(G)\) is \(F\)anisotropic, and \(G\) splits over a Galois extension of \(F\) of degree \(\leqslant s\),
$$\begin{aligned} L(\mathrm{Mot}_G)\leqslant c_1 \prod _{d=1 }^{\lfloor \frac{d_G+1}{2}\rfloor } \mathbb {N}_{F/\mathbb {Q}}({\mathfrak f}(\mathrm{Mot}_{G,d}))^{d\frac{1}{2}+\epsilon }. \end{aligned}$$  (ii):

There is a constant \(A_{20}=A_{20}(R,D,s,F)\) such that for any \(G\) as in (i),
$$\begin{aligned} L(\mathrm{Mot}_G)\leqslant c_1 \prod _{v\in \mathrm{Ram}(G)} q_v^{A_{20}}. \end{aligned}$$
The choice \(A_{20}= \frac{(D+1)Rs}{2} \max \limits _{\text {prime }p} (1+e_{F_v/\mathbb {Q}_p}\log s)\) is admissible.
Proof
The functional equation for \(\mathrm{Mot}_G\) reads
where \( \epsilon (\mathrm{Mot}_G)=\Delta _F^{d_G/2}\prod _{d\geqslant 1} \mathbb {N}_{F/\mathbb {Q}}({\mathfrak f}(\mathrm{Mot}_{G,d}))^{d\frac{1}{2}}\).
The (possibly reducible) Artin representation for \(\mathrm{Mot}_{G,d}\) factors through \(\mathrm{Gal}(E/F)\) with \([E:F]\leqslant s\) by the assumption. Let \(A_1=A_1(s,F)\) be as in (iii) of Proposition 6.15. For all \(\epsilon >0\), (i) of Proposition 6.15 implies that there is a constant \(c=c(\epsilon ,s,F)>1\) depending only on \(s\) and \(F\) such that
Formula (7.7) of [47], the first equality below, leads to the following bound since only \(1\leqslant d \leqslant \lfloor \frac{d_G+1}{2}\rfloor \) can contribute in view of Proposition 6.3 (iii).
Set \(c_1(R,D,s,F,\epsilon ):= \Delta _F^{D/2} 2^{[F:\mathbb {Q}]R} \left( \lfloor \frac{D1}{2}\rfloor !\right) ^{R}\). Then we see that
This concludes the proof of (i).
According to Lemma 6.4, the exponent in the right hand side is bounded by
(we have chosen \(\epsilon =\frac{1}{2}\)). The proof of (ii) is concluded by the fact that
see Proposition 6.3 (ii).
Corollary 6.17
Let \(G\) be a connected cuspidal reductive group over \(F\) with anisotropic center. Then there exist constants \(c_2=c_2(G,F)>0\) and \(A_2(G,F)>0\) depending only on \(G\) and \(F\) such that: for any cuspidal \(F\)Levi subgroup \(M\) of \(G\) and any semisimple \(\gamma \in M(F)\) which is elliptic in \(M(\mathbb {R})\),
where \(I^M_\gamma \) denote the connected centralizer of \(\gamma \) in \(M\). The following choice is admissible:
Proof
According to Lemma 6.5, \(s^{\mathrm{spl}}_{ I^M_\gamma }\leqslant w_Gs_G\). Apply Corollary 6.16 for each \(I^M_\gamma \) with \(R=r_G\), \(D=d_G\) and \(s=w_Gs_G\) to deduce the first assertion, which obviously implies the last assertion. Note that \(\mathrm{rk}I^M_\gamma \leqslant r_G\) and that \(\dim I^M_\gamma \leqslant d_G\).
Instead of using the Brauer–Siegel theorem which is ineffective, we could use the estimates by Zimmert [109] for the size of the regulator of number fields. This yields an effective estimate for the constants \(c_2\) and \(c_3\) above, at the cost of enlarging the value of the exponents \(A_1\) and \(A_2\).
6.8 Frobenius–Schur indicator
The Frobenius–Schur indicator is an invariant associated to an irreducible representation. It may take the three values \(1,0,1\). This subsection gathers several wellknown facts and recalls some familiar constructions.
The Frobenius–Schur indicator can be constructed in greater generality but the following setting will suffice for our purpose. We will only consider finite dimensional representations on vector spaces over \(\mathbb {C}\) or \(\mathbb {R}\). The representations are continuous (and unitary) from compact Lie groups or algebraic from linear algebraic groups (these are in fact closely related by the classical “unitary trick” of Hurwitz and Weyl).
Let \(G\) be a compact Lie group and denote by \(\mu \) the Haar probability measure on \(G\). Let \((V,r)\) be a continuous irreducible representation of \(G\). Denote by \(\chi (g)={{\mathrm{Tr}}}(r(g))\) its character.
Definition 6.18
The Frobenius–Schur indicator of an irreducible representation \((V,r)\) of \(G\) is defined by
We have that \(s(r)\in \left\{ 1,0,1\right\} \) always.
Remark 6.19
More generally if \(G\) is an arbitrary group but \(V\) is still finite dimensional, then \(s(r)\) is defined as the multiplicity of the trivial representation in the virtual representation on \({{\mathrm{Sym}}}^2 V  \wedge ^2 V\). This is consistent with the above definition.
Remark 6.20

(i)
Let \((V^\vee ,r^\vee )\) be the dual representation of \(G\) in the dual \(V^\vee \). It is easily seen that \(s(r)=s(r^\vee )\).

(ii)
If \(G=G_1\times G_2\) and \(r\) is the irreducible representation of \(G\) on \(V=V_1\otimes V_2\) where \((V_1,r_1)\) and \((V_2,r_2)\) are irreducible representations of \(G_1\) (resp. \(G_2\)), then \(s(r)=s(r_1)s(r_2)\).
The classical theorem of Frobenius and Schur says that \(r\) is a real, complex or quaternionic representation if and only if \(s(r)=1,0\) or \(1\) respectively. We elaborate on that dichotomy in the following three lemma.
Lemma 6.21
(Real representation) Let \((V,r)\) be an irreducible representation of \(G\). The following assertions are equivalent:
 (i):

\(s(r)=1\);
 (ii):

\(r\) is selfdual and defined over \(\mathbb {R}\) in the sense that \(V\simeq V_0\otimes _\mathbb {R}\mathbb {C}\) for some irreducible representation on a real vector space \(V_0\). (Such an \(r\) is said to be a real representation;)
 (iii):

\(r\) has an invariant real structure. Namely there is a \(G\)invariant antilinear map \(j:V\rightarrow V\) which satisfies \(j^2=1\).
 (vi):

\(r\) is selfdual and any bilinear form on \(V\) that realizes the isomorphism \(r\simeq r^\vee \) is symmetric;
 (v):

\({{\mathrm{Sym}}}^2 V\) contains the trivial representation (then the multiplicity is exactly one).
We don’t repeat the proof here (see e.g. [93]) and only recall some of the familiar constructions. We have a direct sum decomposition
The character of the representation \(V\otimes V\) is \(g\mapsto \chi (g)^2\). By Schur lemma the trivial representation occurs in \(V\otimes V\) with multiplicity at most one. In other words the subspace of invariant vectors of \(V^\vee \otimes V^\vee \) is at most one. Note that this subspace is identified with \({{\mathrm{Hom}}}_G(V,V^\vee )\) which is also the subspace of invariant bilinear forms on \(V\).
The character of the representation \({{\mathrm{Sym}}}^2 V\) (resp. \(\wedge ^2 V\)) is
From that the equivalence of (i) with (v) follows because the multiplicity of the trivial representation in \({{\mathrm{Sym}}}^2 V\) (resp. \(\wedge ^2 V\)) is the mean of its character. The equivalence of (iv) and (v) is clear because a bilinear form on \(V\) is an element of \(V^\vee \otimes V^\vee \) and it is symmetric if and only if it belongs to \({{\mathrm{Sym}}}^2 V^\vee \).
The equivalence of (ii) and (iii) follows from the fact that \(j\) is induced by complex conjugation on \(V_0\otimes _\mathbb {R}\mathbb {C}\) and conversely \(V_0\) is the subspace of fixed points by \(j\). Note that a real representation is isomorphic to its complex conjugate representation because \(j\) may be viewed equivalently as a \(G\)isomorphism \(V\rightarrow \overline{V}\). Since \(V\) is unitary the complex conjugate representation \(\overline{r}\) is isomorphic to the dual representation \(r^\vee \). In assertion (ii) one may note that the endomorphism ring of \(V_0\) is isomorphic to \(\mathbb {R}\).
Lemma 6.22
(Complex representation) Let \((V,r)\) be an irreducible representation of \(G\). The following assertions are equivalent:
 (i):

\(s(r)=0\);
 (ii):

\(r\) is not selfdual;
 (iii):

\(r\) is not isomorphic to \(\overline{r}\); (such an \(r\) is called a complex representation;)
 (iv):

\(V\otimes V\) does not contain the trivial representation.
We note that for a complex representation, the restriction \({\mathrm {Res}}_{\mathbb {C}/\mathbb {R}} V\) (obtained by viewing \(V\) as a real vector space) is an irreducible real representation of twice the dimension of \(V\). Its endomorphism ring is isomorphic to \(\mathbb {C}\).
Lemma 6.23
(Quaternionic/symplectic representation) Let \((V,r)\) be an irreducible representation of \(G\). The following assertions are equivalent:
 (i):

\(s(r)=1\);
 (ii):

\(r\) is selfdual and cannot be defined over \(\mathbb {R}\).
 (iii):

\(r\) has an invariant quaternionic structure. Namely there is a \(G\)invariant antilinear map \(j:V\rightarrow V\) which satisfies \(j^2=1\). (Such an \(r\) is called a quaternionic representation.)
 (iv):

\(r\) is selfdual and the bilinear form on \(V\) that realizes the isomorphism \(r\simeq r^\vee \) is antisymmetric. (Such an \(r\) is said to be a symplectic representation;)
 (v):

\(\bigwedge ^2 V\) contains the trivial representation (the multiplicity is exactly one).
The equivalence of (iii) and (iv) again comes from the fact that \(V\) is unitarizable (because \(G\) is a compact group). In that context the notion of symplectic representation is identical to the notion of quaternionic representation. Note that for a quaternionic representation, the restriction \({\mathrm {Res}}_{\mathbb {C}/\mathbb {R}} V\) is an irreducible real representation of twice the dimension of \(V\). Furthermore its ring of endomorphisms is isomorphic to the quaternion algebra \(\mathbb {H}\). Indeed the endomorphism ring contains the (linear) action by \(i\) because \(V\) is a representation over the complex numbers and together with \(j\) and \(k=ij\) this is the standard presentation of \(\mathbb {H}\).
From the above discussions we see that the Frobenius–Schur indicator can be used to classify irreducible representations over the reals. The endomorphism ring of an irreducible real representation is isomorphic to either \(\mathbb {R},\mathbb {C}\) or \(\mathbb {H}\) and we have described a correspondence with associated complex representations.
7 A uniform bound on orbital integrals
This section is devoted to showing an apparently new result on the uniform bound on orbital integrals evaluated at semisimple conjugacy classes and basis elements of unramified Hecke algebras. Our bound is uniform in the finite place \(v\) of a number field (over which the group is defined), the “size” of (the support of) the basis element for the unramified Hecke algebra at \(v\) as well as the conjugacy class at \(v\).
The main result is Theorem 7.3, which is invoked in Sect. 9.5. The main local input for Theorem 7.3 is Proposition 7.1. The technical heart in the proof of the proposition is postponed to Sect. 7.3, which the reader may want to skip in the first reading. In Appendix B we discuss an alternative approach to Theorem 7.3 via motivic integration.
7.1 The main local result
We begin with a local assertion with a view toward Theorem 7.3 below. Let \(G\) be a connected reductive group over a finite extension \(F\) of \(\mathbb {Q}_p\) with a maximal \(F\)split torus \(A\). As usual \(\mathcal {O}\), \(\varpi \), \(k_F\) denote the integer ring, a uniformizer and the residue field. Let \(\mathbf {G}\) be the Chevalley group for \(G\times _{F} \overline{F}\), defined over \(\mathbb {Z}\). Let \(\mathbf {B}\) and \(\mathbf {T}\) be a Borel subgroup and a maximal torus of \(\mathbf {G}\) such that \(\mathbf {B}\supset \mathbf {T}\). We assume that

\(G\) is unramified over \(F\),

\(\mathrm{char}\,k_F>w_Gs_G\) and \(\mathrm{char}\,k_F\) does not divide the finitely many constants in the Chevalley commutator relations [namely \(C_{ij}\) of (7.34)].
(We assume \(\mathrm{char}\,k_F>w_Gs_G\) to ensure that any maximal torus of \(G\) splits over a finite tame extension, cf. Sect. 7.3 below. The latter assumption on \(\mathrm{char}\,k_F\) depends only on \(\mathbf {G}\).) Fix a smooth reductive model over \(\mathcal {O}\) so that \(K:=G(\mathcal {O})\) is a hyperspecial subgroup of \(G(F)\). Fix a Borel subgroup \(B\) of \(G\) whose Levi factor is the centralizer of \(A\) in \(G\). Denote by \(v:F^\times \rightarrow \mathbb {Q}\) the discrete valuation normalized by \(v(\varpi )=1\) and by \(D^G\) the Weyl discriminant function, cf. (13.1) below. Set \(q_v:=k_F\).
Suppose that there exists a closed embedding of algebraic groups \(\Xi ^{\mathrm{spl}}:\mathbf {G}\hookrightarrow \mathrm{GL}_m\) defined over \(\mathcal {O}\) such that \(\Xi ^{\mathrm{spl}}(\mathbf {T})\) [resp. \(\Xi ^{\mathrm{spl}}(\mathbf {B})\)] lies in the group of diagonal (resp. upper triangular) matrices. This assumption will be satisfied by Lemma 2.17 and Proposition 8.1, or alternatively as explained at the start of Sect. 7.4. The assumption may not be strictly necessary but is convenient to have for some later arguments. In the setup of Sect. 7.2 such a \(\Xi ^{\mathrm{spl}}\) will be chosen globally over \(\mathbb {Z}[1/Q]\) (i.e. away from a certain finite set of primes), which gives rise to an embedding over \(\mathcal {O}\) if \(v\) does not divide \(Q\).
Proposition 7.1
There exist \(a_{G,v},b_{G,v},e_{G,v}\geqslant 0\) (depending on \(F\), \(G\) and \(\Xi ^{\mathrm{spl}}\)) such that

for every semisimple \(\gamma \in G(F)\),

for every \(\lambda \in X_*(A)\) and \(\kappa \in \mathbb {Z}_{\geqslant 0}\) such that \(\Vert \lambda \Vert \leqslant \kappa \),
Remark 7.2
We chose the notation \(a_{G,v}\) etc rather than \(a_{G,F}\) etc in anticipating the global setup of the next subsection where \(F\) is the completion of a number field at the place \(v\).
Proof
For simplicity we will omit the measures chosen to compute orbital integrals when there is no danger of confusion. Let us argue by induction on the semisimple rank \(r^{\mathrm {ss}}_G\) of \(G\). In the rank zero case, namely when \(G\) is a torus, the proposition is true since \(O_\gamma (\tau ^G_\lambda )\) is equal to 0 or 1. Now assume that \(r^{\mathrm {ss}}_G\geqslant 1\) and that the proposition is known for all groups whose semisimple ranks are less than \(r^{\mathrm {ss}}_G\). In the proof we write \(a_G\), \(b_G\), \(e_G\) instead of \(a_{G,v}\), \(b_{G,v}\), \(e_{G,v}\) for simplicity.
Step 1. Reduce to the case where \(Z(G)\) is anisotropic.
Let \(A_G\) denote the maximal split torus in \(Z(G)\). Set \(\overline{G}:=G/A_G\). The goal of Step 1 is to show that if the proposition for \(\overline{G}\) then it also holds for \(G\). We have an exact sequence of algebraic groups over \(\mathcal {O}\): \(1\rightarrow A_G\rightarrow G\rightarrow \overline{G}\rightarrow 1\). By taking \(F\)points one obtains an exact sequence of groups
where the surjectivity is implied by Hilbert 90 for \(A_G\). [In fact \(G(\mathcal {O})\rightarrow \overline{G}(\mathcal {O})\) is surjective since it is surjective on \(k_F\)points and \(G\rightarrow \overline{G}\) is smooth, cf. [63, p. 386], but we do not need this.] For any semisimple \(\gamma \in G(F)\), denote its image in \(\overline{G}(F)\) by \(\overline{\gamma }\). The connected centralizer of \(\overline{\gamma }\) is denoted \(\overline{I}_{\overline{\gamma }}\). There is an exact sequence
We see that \(G(F) \rightarrow \overline{G}(F)\) induces a bijection \(I_\gamma (F)\backslash G(F) \simeq \overline{I}_{\overline{\gamma }}(F)\backslash \overline{G}(F)\). Let \(A\) be a maximal \(F\)split torus of \(G\), and \(\overline{A}\) be its image in \(\overline{G}\). For any \(\lambda \in X_*(A)\), denote its image in \(X_*(\overline{A})\) by \(\overline{\lambda }\). Then
Indeed, this follows from the fact that \(I_\gamma (F)\backslash G(F) \simeq \overline{I}_{\overline{\gamma }}(F)\backslash \overline{G}(F)\) carries \(\frac{\mu ^{\mathrm{can}}_G}{\mu ^{\mathrm{can}}_{I_\gamma }}\) to \(\frac{\mu ^{\mathrm{can}}_{\overline{G}}}{\mu ^{\mathrm{can}}_{\overline{I}_{\overline{\gamma }}}}\). As the proposition is assumed to hold for \(\overline{G}\), the right hand side is bounded by \(q_v^{ a_{\overline{G}}+b_{\overline{G}}\kappa }\cdot D^{\overline{G}}(\gamma )^{e_{\overline{G}}/2}= q_v^{ a_{\overline{G}}+b_{\overline{G}}\kappa }\cdot D^{G}(\gamma )^{e_{\overline{G}}/2}\). Hence the proposition holds for \(G\) if we set \(a_G=a_{\overline{G}}\), \(b_G=b_{\overline{G}}\) and \(e_G=e_{\overline{G}}\). This finishes Step 1.
Step 2. When \(Z(G)\) is anisotropic.
The problem will be divided into three cases depending on \(\gamma \). In each case we find a sufficient condition on \(a_G\), \(b_G\) and \(e_G\) for (7.1) to be true.
Step 21. When \(\gamma \in Z(G)(F)\).
In this case the proposition holds for any \(a_G,b_G,e_G\geqslant 0\) since \(O_\gamma (\tau ^G_\lambda )=0\) or \(1\) and \(D^G(\gamma )=1\).
Step 22. When \(\gamma \) is noncentral and nonelliptic.
Then there exists a nontrivial split torus \(S\subset Z(I_\gamma )\). Set \(M:=Z_G(S)\), which is an \(F\)rational Levi subgroup of \(G\). Then \(I_\gamma \subset M \subsetneq G\). Note that \(\gamma \) is \((G,M)\)regular. Lemma 6.1 reads
By conjugation we may assume without loss of generality that \(\lambda (\varpi )\in M(F)\). (To justify, find \(x\in G(F)\) such that \(xMx^{1}\) contains \(A\). Then \(\lambda (\varpi )\in xM(F)x^{1}\) and \(O^M_\gamma = O^{xMx^{1}}_{x\gamma x^{1}}\).) Moreover by conjugating \(\lambda \) we may assume that \(\lambda \) is \(B\cap M\)dominant. We can write
The ordering in the sum is relative to \(B\cap M\). For any \(m=\mu (\varpi )\), \(c_{\lambda ,\mu }\) is equal to
Lemma 2.13 and the easy inequality \(\langle \rho _P,\mu \rangle \leqslant \langle \rho ,\lambda \rangle \) allow us to deduce that
The sum in (7.3) runs over the set of
such that \(\mu \in (X^*(T)_\mathbb {R})^+\). Here we need to explain \(\delta _G\): If \(\mu \leqslant _{\mathbb {R}} \lambda \) then \(\lambda \mu \) is a linear combination of positive coroots with nonnegative rational coefficients. The denominators of such coefficients under the constraint \(c_{\lambda ,\mu }\ne 0\) are uniformly bounded, where the bound depends on the coroot datum. We write \(\delta _G\) for this bound.
The above condition on \(\mu \) and \(\Vert \lambda \Vert \leqslant \kappa \) imply that \(a_\alpha \leqslant \kappa \). We get, by using the induction hypothesis for \(O^M_\gamma \),
Set
In view of (7.2) it suffices to find \(a_G,b_G,e_G\geqslant 0\) such that
or equivalently
whenever a conjugate of \(\gamma \) lies in \(K\lambda (\varpi )K\). For each \(\alpha \in \Phi \),
where \(b_\Xi \) is the constant \(B_5\) (depending only on \(G\) and \(\Xi \) and not on \(v\)) of Lemma 2.18. Hence
and likewise \(D^M(\gamma )\leqslant q_v^{\Phi _M b_\Xi \kappa /2}\). (We divide the exponents by 2 because it cannot happen simultaneously that \(v(\alpha (\gamma ))<0\) and \(v(\alpha ^{1}(\gamma ))<0\).) Therefore condition (7.4) on \(a_G,b_G,e_G\) is implied by the two conditions
There are only finitely many Levi subgroups \(M\) (up to conjugation) giving rise to the triples \((a_M,b_M,e_M)\). It is elementary to observe that (7.7) holds as long as \(a_G\) and \(b_G\) are sufficiently large while \(e_G\) has any fixed value such that (7.6) holds. We will impose another condition on \(a_G,b_G,e_G\) in Step 23.
Step 23. When \(\gamma \) is noncentral and elliptic in \(G\).
This case is essentially going to be worked out in Sect. 7.3. Let \(Z_1,Z_2\geqslant 0\) be as in Lemma 7.9 below. By (7.11) and Corollary 7.11 below, (7.1) will hold if
We have \(D^G(\gamma )\leqslant q_v^{\Phi b_\Xi \kappa /2}\) thanks to (7.5) (cf. Step 22). So (7.8) (is not equivalent to but) is implied by the combination of the following two inequalities:
The latter two will hold true, for instance, if \(e_G\) has any fixed value greater than or equal to \(2Z_2\) and if \(a_G\) and \(b_G\) are sufficiently large. (We will see in Sect. 7.3 below that \(Z_1\) and \(Z_2\) are independent of \(\lambda \), \(\gamma \) and \(\kappa \).)
Now that we are done with analyzing three different cases, we finish Step 2. For this we use the induction on semisimple ranks (to ensure the existence of \(a_M\), \(b_M\) and \(e_M\) in Step 22) to find \(a_G,b_G,e_G\geqslant 0\) which satisfy the conditions described at the ends of Step 22 and Step 23. We are done with the proof of Proposition 7.1.
7.2 A global consequence
Here we switch to a global setup. Let \(\mathbf {F}\) be a number field. For a finite place \(v\) of \(\mathbf {F}\), let \(k(v)\) denote the residue field and put \(q_v:=k(v)\).

\(G\) is a connected reductive group over \(\mathbf {F}\).

\(\mathrm{Ram}(G)\) is the set of finite places \(v\) of \(\mathbf {F}\) such that \(G\) is ramified at \(\mathbf {F}_v\).

\(\mathbf {G}\) is the Chevalley group for \(G\times _{\mathbf {F}} \overline{\mathbf {F}}\), and \(\mathbf {B}\), \(\mathbf {T}\) are as in Sect. 7.1.

\(\Xi ^{\mathrm{spl}}:\mathbf {G}\hookrightarrow \mathrm{GL}_m\), fixed once and for all, is a closed embedding defined over \(\mathbb {Z}[1/R]\) for a large enough integer \(R\) such that \(\Xi ^{\mathrm{spl}}(\mathbf {T})\) [resp. \(\Xi ^{\mathrm{spl}}(\mathbf {B})\)] lies in the group of diagonal (resp. upper triangular) matrices of \(\mathrm{GL}_m\). The choice of \(R\) depends only on \(\mathbf {G}\) and \(\Xi ^{\mathrm{spl}}\). (We defer to Sect. 7.4 more details and the explanation that there exists such a \(\Xi ^{\mathrm{spl}}\).)

\(S_{\mathrm{bad}}\) is the set of finite places \(v\) such that either \(v\in \mathrm{Ram}(G)\), \(\mathrm{char}\,k(v)\leqslant w_Gs_G\), \(\mathrm{char}\,k(v)\) divides \(R\), or \(\mathrm{char}\,k(v)\) divides at least one of the constants for the Chevalley commutator relations for \(\mathbf {G}\), cf. (7.34) below.
Examining the dependence of various constants in Proposition 7.1 leads to the following main result of this section. For each finite place \(v\notin S_{\mathrm{bad}}\), denote by \(A_v\) a maximal \(\mathbf {F}_v\)split torus of \(G\times _{\mathbf {F}} \mathbf {F}_v\).
Theorem 7.3
There exist \(a_G,b_G\geqslant 0\) and \(e_G\geqslant 1\) (depending on \(\mathbf {F}\), \(G\) and \(\Xi ^{\mathrm{spl}}\)) such that

for every finite \(v\notin S_{\mathrm{bad}}\),

for every semisimple \(\gamma \in G(\mathbf {F}_v)\),

for every \(\lambda \in X_*(A_v)\) and \(\kappa \in \mathbb {Z}_{\geqslant 0}\) such that \(\Vert \lambda \Vert \leqslant \kappa \),
Remark 7.4
It is worth drawing a comparison between the above theorem and Theorem 13.1 proved by Kottwitz. In the latter the test function (in the full Hecke algebra) and the base \(p\)adic field are fixed whereas the main point of the former is to allow the test function (in the unramified Hecke algebra) and the place \(v\) to vary. The two theorems are complementary to each other and will play a crucial role in the proof of Theorem 9.19.
Remark 7.5
In an informal communication Kottwitz and Ngô pointed out that there might be yet another approach based on a geometric argument involving affine Springer fibers, as in [46, §15], which might lead to a streamlined and conceptual proof, as well as optimized values of the constants \(a_G\) and \(b_G\). Appendix B provides an important step in that direction, see Theorem 14.7 which implies that the constants are transferable from finite characteristic to characteristic zero.
Proof
Since the case of tori is clear, we may assume that \(r^{\mathrm {ss}}_G\geqslant 1\). Let \(\theta \in {\fancyscript{C}}(\Gamma _1)\). (Recall the definition of \(\Gamma _1\) and \({\fancyscript{C}}(\Gamma _1)\) from Sects. 5.1 and 5.2.) Our strategy is to find \(a_{G,\theta },b_{G,\theta },e_{G,\theta }\geqslant 0\) which satisfy the requirements (7.7), (7.9), and (7.10) on \(a_{G,v},b_{G,v},e_{G,v}\) at all \(v\in \mathcal {V}_{\mathbf {F}}(\theta )\backslash S_{\mathrm{bad}}\). As for (7.7), we inductively find \(a_{M,\theta },b_{M,\theta },e_{M,\theta }\geqslant 0\) for all local Levi subgroups \(M\) of \(G\) as will be explained below.
We would like to explain an inductive choice of \(a_{M,\theta },b_{M,\theta },e_{M,\theta }\geqslant 0\) for a fixed \(\theta \). To do so we ought to clarify what Levi subgroups \(M\) of \(G\) we consider. Let \(\Delta \) denote the set of \(\mathbf {B}\)positive simple roots for \((\mathbf {G},\mathbf {T})\). Via an identification \(G\times _{\mathbf {F}} \overline{\mathbf {F}}\simeq \mathbf {G}\times _\mathbb {Z}\overline{\mathbf {F}}\) we may view \(\Delta \) as the set of simple roots for \(G\) equipped with an action by \(\Gamma _1\), cf. [9, §1.3]. Note that \({{\mathrm{Frob}}}_v\) acts as \(\theta \in \Gamma _1\) on \(\Delta \) for all \(v\in \mathcal {V}_{\mathbf {F}}(\theta )\backslash S_{\mathrm{bad}}\). According to [9, §3.2], the \(\theta \)stable subsets of \(\Delta \) are in bijection with \(G(\mathbf {F}_v)\)conjugacy classes of \(\mathbf {F}_v\)parabolic subgroups of \(G\). For each \(v\in \mathcal {V}_{\mathbf {F}}(\theta )\backslash S_{\mathrm{bad}}\), fix a Borel subgroup \(B_v\) of \(G\) over \(\mathbf {F}_v\) containing the centralizer \(T_v\) of \(A_v\) in \(G\) so that the following are in a canonical bijection with one another.

\(\theta \)stable subsets \(\Upsilon \) of \(\Delta \)

parabolic subgroups \(P_v\) of \(G\) containing \(B_v\)
Denote by \(P_{\Upsilon ,v}\) the parabolic subgroup corresponding to \(\Upsilon \) and by \(M_{\Upsilon ,v}\) its Levi subgroup containing \(T_v\). Here is an important observation. The constants \(Z_1\), \(Z_2\) (see Remark 7.10 below) and the inequalities (7.7), (7.9), and (7.10) to be satisfied by \(a_{M_{\Upsilon },v},b_{M_{\Upsilon },v},e_{M_{\Upsilon },v}\) depend only on \(\theta \) and not on \(v\in \mathcal {V}_{\mathbf {F}}(\theta )\backslash S_{\mathrm{bad}}\). (We consider the case where \(G\) and \(M\) of those inequalities are \( M_{\Upsilon }\) and a \(\mathbf {F}_v\)Levi subgroup of \(M_{\Upsilon }\), respectively.) Hence we will write \(a_{M_{\Upsilon },\theta },b_{M_{\Upsilon },\theta },e_{M_{\Upsilon },\theta }\geqslant 0\) for these constants. What we need to do is to define them inductively according to the semisimple rank of \(M\) such that (7.7), (7.9), and (7.10) hold true. In particular the desired \(a_{G,\theta },b_{G,\theta },e_{G,\theta }\) will be obtained and the proof will be finished (by returning to the first paragraph in the current proof).
Now the inductive choice of \(a_{M_{\Upsilon },\theta },b_{M_{\Upsilon },\theta },e_{M_{\Upsilon },\theta }\) is easy to make once the choice of \(a_{M_{\Omega },\theta },b_{M_{\Omega },\theta },e_{M_{\Omega },\theta }\) has been made for all \(\Omega \subsetneq \Upsilon \). Indeed, we may choose \(e_{M_{\Omega },\theta }\in \mathbb {Z}_{\geqslant 1}\) to fulfill (7.9) and then choose \(a_{M_{\Omega },\theta },b_{M_{\Omega },\theta }\) to be large enough to verify (7.7) and (7.10). Notice that \(Z_1,Z_2,Z_3\) of (7.10) (which are constructed in Lemma 7.9 below) depend only on the grouptheoretic information of \(M_{\Upsilon }\) (such as the dimension, rank, affine root data, \(\delta _{M_{\Upsilon }}\) of \(M_{\Upsilon }\) as well as an embedding of the Chevalley form of \(M_{\Upsilon }\) into \(GL_d\) coming from \(\Xi ^{\mathrm{spl}}\)) but not on \(v\), cf. Remark 7.10.
In view of Theorem 13.1 and other observations in harmonic analysis, a natural question is whether it is possible to achieve \(e_{G}=1\). This is a deep and difficult question which is of independent interest. It was a pleasant surprise to the authors that the theory of arithmetic motivic integration provides a solution. A precise theorem due to Cluckers, Gordon, and Halupczok is stated in Theorem 14.1 below. It is worth remarking that their method of proof is significantly different from that of this section and also that they make use of Theorem 13.1, the local boundedness theorem. Finally it would be interesting to ask about the analogue in the case of twisted or weighted orbital integrals. Such a result would be useful in the more general situation than the one considered in this paper.
7.3 The noncentral elliptic case
The objective of this subsection is to establish Corollary 7.11, which was used in Step 23 of the proof of Proposition 7.1 above. Since the proof is quite complicated let us guide the reader. The basic idea, going back to Langlands, is to interpret the orbital integral \(O^{G(F)}_{\gamma }(\tau ^G_{\lambda })\) in question as the number of points in the building fixed “up to \(\lambda \)” under the action of \(\gamma \). The set of such points, denoted \(X_F(\gamma ,\lambda )\) below, is finite since \(\gamma \) is elliptic. Then it is shown that every point of \(X_F(\gamma ,\lambda )\) is within a certain distance from a certain apartment, after enlarging the ground field \(F\) to a finite extension. We exploit this to bound \(X_F(\gamma ,\lambda )\) by a ball of an explicit radius in the building. By counting the number of points in the ball (which is of course much more tractable than counting \(X_F(\gamma ,\lambda )\)) we arrive at the desired bound on the orbital integral. The proof presented here is inspired by the beautiful exposition of [66, §§3–5] but uses brute force and crude bounds at several places. We defer some technical lemmas and their proofs to Sect. 7.4 below and refer to them in this subsection but there is no circular logic since no results of this subsection are used in Sect. 7.4.
Throughout this subsection the notation of Sect. 7.1 is adopted and \(\gamma \) is assumed to be noncentral and elliptic in \(G(F)\). (However \(\gamma \) need not be regular.) We assume \(Z(G)\) to be anisotropic over \(F\) as we did in Step 2 of the proof of Proposition 7.1. Then \(I_\gamma (F)\) is a compact group, on which the EulerPoincare measure \(\mu ^{\mathrm{EP}}_{I_\gamma }\) assigns total volume 1. Our aim is to bound \(O^{G(F)}_{\gamma }({\mathbf {1}}_{K\mu (\varpi )K}, \mu ^{\mathrm{can}}_G,\mu ^{\mathrm{can}}_{I_\gamma })\). It follows from [47, Thm 5.5] (for the equality) and Proposition 6.3 that
Thus we may as well bound \(O^{G(F)}_{\gamma }({\mathbf {1}}_{K\mu (\varpi )K}, \mu ^{\mathrm{can}}_G,\mu ^{\mathrm{EP}}_{I_\gamma })\).
Let \(T_\gamma \) be an elliptic maximal torus of \(I_\gamma \) defined over \(F\) containing \(\gamma \). By Lemma 6.5, there exists a Galois extension \(F'/F\) with
such that \(T_\gamma \) is a split torus over \(F'\). Hence \(I_\gamma \) and \(G\) are split groups over \(F'\). Note that \(F'\) is a tame extension of \(F\) under the assumption that \(\mathrm{char}\,k_F>w_Gs_G\). Let \(A'\) be a split maximal torus of \(G\) over \(F'\) such that \(A\times _F F'\subset A'\). Since \(F'\)split maximal tori are conjugate over \(F'\), we find
and fix such a \(y\). Write \(\mathcal {O}'\), \(\varpi '\) and \(v'\) for the integer ring of \(F'\), a uniformizer and the valuation on \(F'\) such that \(v'(\varpi ')=1\). With respect to the integral model of \(G\) over \(\mathcal {O}\) at the beginning of Sect. 7.1, we put \(K':=G(\mathcal {O}')\). A point of \(G(F)/K\) will be denoted \(\overline{x}\) and any of its lift in \(G(F)\) will be denoted \(x\). Let \(\overline{x}_0\in G(F)/K\) [resp. \(\overline{x}'_0\in G(F')/K'\)] denote the element represented by the trivial coset of \(K\) (resp. \(K'\)). Then \(\overline{x}_0\) (resp. \(\overline{x}'_0\)) may be thought of as a base point of the building \({\mathcal {B}}(G(F),K)\) [resp. \({\mathcal {B}}(G(F'),K')\)] and its stabilizer is identified with \(K\) (resp. \(K'\)). There exists an injection
such that \({\mathcal {B}}(G(F),K)\) is the \(\mathrm{Gal}(F'/F)\)fixed points of \({\mathcal {B}}(G(F'),K')\). (This is the case because \(F'\) is tame over \(F\).) The natural injection \(G(F)/K\hookrightarrow G(F')/K'\) coincides with the injection induced by (7.13) on the set of vertices.
Define \(\lambda '\in X_*(A')\) by \(\lambda ':=e_{F'/F} \lambda \) (where \(e_{F'/F}\) is the ramification index of \(F'\) over \(F\)) so that \(\lambda '(\varpi ')=\lambda (\varpi )\) and
For (the fixed \(\gamma \) and) a semisimple element \(\delta \in G(F')\), set
By abuse of notation we write \(\overline{x}^{1} \gamma \overline{x}\in K\lambda (\varpi ) K\) for the condition that \(x^{1}\gamma x\in K\lambda (\varpi ) K\) for some (thus every) lift \(x\in G(F)\) of \(\overline{x}\) and similarly for the condition on \(\overline{x}'\). It is clear that \(X_F(\gamma ,\lambda )\subset X_{F'}(\gamma ,\lambda ')\cap ( G(F)/K)\). By (3.4.2) of [66],
Our goal of bounding the orbital integrals on the left hand side can be translated into a problem of bounding \( X_F(\gamma ,\lambda )\).
Let \(\mathrm{Apt}(A'(F'))\) denote the apartment for \(A'(F')\). Likewise \(\mathrm{Apt}(T_\gamma (F))\) and \(\mathrm{Apt}(T_\gamma (F'))\) are given the obvious meanings. We have \(\overline{x}'_0\in \mathrm{Apt}(A'(F'))\). The metrics on \({\mathcal {B}}(G(F),K)\) and \({\mathcal {B}}(G(F'),K')\) are chosen such that (7.13) is an isometry. The metric on \({\mathcal {B}}(G(F'),K')\) is determined by its restriction to \(\mathrm{Apt}(A'(F'))\), which is in turn pinned down by a (noncanonical choice of) a Weylgroup invariant scalar product on \(X_*(A')\), cf. [103, §2.3]. Henceforth we fix the scalar product once and for all. Scaling the scalar product does not change our main results of this subsection.
Remark 7.6
For any other tame extension \(F''\) of \(F\) and a split maximal torus \(A''\) of \(G\) over \(F''\), we can find an isomorphism \(X_*(A')\) and \(X_*(A'')\) over the composite field of \(F'\) and \(F''\), well defined up to the Weyl group action. So the scalar product on \(X_*(A'')\) is uniquely determined by that on \(X_*(A')\). So we need not choose a scalar product again when considering a different \(\gamma \in G(F)\).
We define certain length functions. Consider an \(F'\)split maximal torus \(A''\) of \(G\) (for instance \(A''=T_\gamma \) or \(A''=A'\)) and the associated set of roots \(\Phi =\Phi (G,A'')\) and the set of coroots \(\Phi ^\vee =\Phi ^\vee (G,A'')\). Let \(l_{\max }(\Phi )\) denote the largest length of a positive coroot in \(\Phi ^\vee \). Note that these are independent of the choice of \(A''\) and completely determined by the previous choice of a Weyl group invariant scalar product on \(X_*(A')\). It is harmless to assume that we have chosen the scalar product such that the longest positive coroot in each irreducible system of \(X_*(A')\) has length \(l_{\max }(\Phi )\).
Fix a Borel subgroup \(B'\) of \(G\) over \(F'\) containing \(A'\) so that \(y^{1}B'y\) is a Borel subgroup containing \(T_\gamma \). Relative to these Borel subgroups we define the subset of positive roots \(\Phi ^+(G,A')\) and \(\Phi ^+(G,T_\gamma )\). Let \(m_{\Xi ^{\mathrm{spl}}}\) be as in Lemma 7.12 below. In order to bound \(X_F(\gamma ,\lambda )\) in (7.15), we control the larger set \(X_{F'}(\delta ,\lambda ')\) by bounding the distance from its points to the apartment for \(A'\).
Lemma 7.7
Let \(\delta \in A'(F')\) and \(\overline{x}'\in G(F')/K'\). Then there exist constants \(C=C(\mathbf {G},\Xi )>0\), \(c_\mathbf {G}>0\), and \(Y=Y(\mathbf {G})\in \mathbb {Z}_{\geqslant 1}\) such that whenever \((\overline{x}')^{1} \delta \overline{x}'\in K'\lambda '(\varpi ') K'\) [i.e. whenever \(\overline{x}'\in X_{F'}(\delta ,\lambda ')\)],
where the left hand side denotes the shortest distance from \(\overline{x}'\) to \(\mathrm{Apt}(A'(F'))\).
Proof of Lemma 7.7
Write \(\overline{x}'=an\overline{x}_0'\) for some \(a\in A'(F')\) and \(n\in N(F')\) using the Iwahori decomposition. As both sides of the above inequality are invariant under multiplication by \(a\), we may assume that \(a=1\). Let \(\lambda _\delta \in X_*(A')\) be such that \(\delta \in \lambda _\delta (\varpi ')A'(\mathcal {O}')\). For each \(\lambda _0\in X_*(A')^+\) recall the definition of \(n_\mathbf {G}(\lambda _0)\) from (2.6). Let \(c_\mathbf {G}>0\) be a constant depending only on \(\mathbf {G}\) such that every \(\lambda _0\in X_*(A')\) satisfies the inequality \(\langle \alpha ,\lambda _0\rangle \leqslant c_\mathbf {G}\Vert \lambda _0\Vert \) for all \(\alpha \in \Phi ^+(G,A')\).

Step 1.
Show that \(\delta ^{1} n^{1}\delta n\in K'\lambda _0(\varpi ')K'\) for some \(\lambda _0\in X_*(A')^+\) such that \(n_\mathbf {G}(\lambda _0)\leqslant (m_{\Xi ^{\mathrm{spl}}}+c_\mathbf {G})e_{F'/F}\kappa \). By Cartan decomposition there exists a \(B'\)dominant \( \lambda _0\in X_*(A')\) such that \(\delta ^{1} n^{1}\delta n\in K'\lambda _0(\varpi ')K'\). The condition on \(\delta \) in the lemma is unraveled as \((x'_0)^{1} n^{1} \delta n x'_0\in K'\lambda '(\varpi ') K'\). So
$$\begin{aligned} \delta ^{1}n^{1} \delta n\in \delta ^{1} K'\lambda '(\varpi ') K' \subset (K'\lambda _\delta ^{1}(\varpi ') K')(K'\lambda '(\varpi ') K'). \end{aligned}$$Let \(w\) be a Weyl group element for \(A'\) in \(G\) such that \(w\lambda _\delta ^{1}\) is \(B'\)dominant. The fact that \(K'\lambda _0(\varpi ')K'\) intersects \((K'\lambda _\delta ^{1}(\varpi ') K')(K'\lambda '(\varpi ') K')\) implies [16, Prop4.4.4.(iii)] that
$$\begin{aligned} \langle \alpha ,\lambda _0\rangle \leqslant \left\langle \alpha ,w\lambda _\delta ^{1}+\lambda '\right\rangle ,\quad \alpha \in \Phi ^+(G,A'). \end{aligned}$$We have \(\langle \alpha ,\lambda '\rangle \leqslant c_\mathbf {G}\Vert \lambda '\Vert \). Note also that
$$\begin{aligned} v'(\alpha (\delta ))\in [m_{\Xi ^{\mathrm{spl}}} \Vert \lambda '\Vert ,m_{\Xi ^{\mathrm{spl}}} \Vert \lambda '\Vert ] \end{aligned}$$(7.16)by Lemma 7.12 since a conjugate of \(\delta \) belongs to \(K'\lambda '(\varpi ')K'\). This implies that
$$\begin{aligned} \left\langle \alpha ,w\lambda _\delta ^{1}\right\rangle = v'(w\alpha ^{1}(\delta ))\leqslant m_{\Xi ^{\mathrm{spl}}}\Vert \lambda '\Vert . \end{aligned}$$On the other hand \(\Vert \lambda '\Vert \leqslant e_{F'/F}\kappa \) according to (7.14). These inequalities imply the desired bound on \(n_\mathbf {G}(\lambda _0)\), which is the maximum of \(\langle \alpha ,\lambda _0\rangle \) over \(\alpha \in \Phi ^+(G,A')\). Before entering Step 2, we notify the reader that we are going to use the convention and notation for the Chevalley basis as recalled in Sect. 7.4 below. In particular \(n\in N(F')\) can be written as [cf. (7.33)]
$$\begin{aligned} n=x_{\alpha _1}(X_{\alpha _1})\ldots x_{\alpha _{\Phi ^+}}(X_{\alpha _{\Phi ^+}}) \end{aligned}$$(7.17)for unique \(X_{\alpha _1},\ldots ,X_{\alpha _{\Phi ^+}}\in F'\).

Step 2.
Show that there exists a constant \({\mathcal {M}}_{\Phi ^+}\geqslant 0\) [explicitly defined in (7.20) below] such that \(v'(X_{\alpha _i})\geqslant {\mathcal {M}}_{\Phi ^+}\) for all \(1\leqslant i\leqslant \Phi ^+\). In our setting we compute
$$\begin{aligned} \delta ^{1}n^{1}\delta n= & {} \delta ^{1} \left( \prod _{i=\Phi ^+}^{1} x_{\alpha _i}(X_{\alpha _i})\right) \delta \prod _{i=1}^{\Phi ^+} x_{\alpha _i}(X_{\alpha _i})\nonumber \\= & {} \left( \prod _{i=\Phi ^+}^{1} x_{\alpha _i}(\alpha ^{1}_i(\delta ) X_{\alpha _i}) \right) \prod _{i=1}^{\Phi ^+} x_{\alpha _i}(X_{\alpha _i}) \nonumber \\ {}= & {} \prod _{i=1}^{\Phi ^+} x_{\alpha _i}\left( (1\alpha ^{1}_i(\delta ))X_{\alpha _i} + P_{\alpha _i} \right) \end{aligned}$$(7.18)where the last equality follows from the repeated use of (7.34) to rearrange the terms. Here \(P_{\alpha _i}\) is a polynomial (which could be zero) in \(\alpha _{j}^{1}(\delta )\) and \(X_{\alpha _j}\) with integer coefficients for \(j<i\). It is not hard to observe from (7.34) that \(P_{\alpha _i}\) has no constant term. As \(i\) varies in \([1,\Phi ^+]\), let \(Y\) denote the highest degree for the nonzero monomial term appearing in \(P_{\alpha _i}\) viewed as a polynomial in either \(\alpha _i^{1}(\delta )\) or \(X_{\alpha _i}\) (but not both).^{Footnote 7} Set \(Y=1\) if \(P_{\alpha _i}=0\). As mentioned above, the positive roots for a given \((\mathbf {G},\mathbf {B},\mathbf {T})\) are ordered once and for all so that \(Y\) depends only on \(\mathbf {G}\) in the sense that for any \(G\) having \(\mathbf {G}\) as its Chevalley form, \(Y\) is independent of the local field \(F\) over which \(G\) is defined. Applying Corollary 7.14 below, we obtain from (7.18) and the condition \(\delta ^{1}n^{1}\delta n\in K'\lambda _0(\varpi ')K'\) that
$$\begin{aligned} v'\left( (1\alpha ^{1}_i(\delta ))X_{\alpha _i} + P_{\alpha _i}\right) \geqslant m_{\mathbf {G}} n_\mathbf {G}(\lambda _0). \end{aligned}$$(7.19)For \(1\leqslant i\leqslant \Phi ^+\), put
$$\begin{aligned} {\mathcal {M}}_i:= & {} \sum _{j=1}^i \left( Y^{ij} (v'(1\alpha _j^{1}(\delta ))+ m_{\mathbf {G}}n_{\mathbf {G}}(\lambda _0))\right) \nonumber \\&+\sum _{j=1}^{i1}Y^j m_{\Xi ^{\mathrm{spl}}} e_{F'/F}\kappa . \end{aligned}$$(7.20)Obviously \(0\leqslant {\mathcal {M}}_1\leqslant {\mathcal {M}}_2\leqslant \cdots \leqslant {\mathcal {M}}_{\Phi ^+}\). We claim that for every \(i\geqslant 1\),
$$\begin{aligned} v'(X_{\alpha _i}) \geqslant {\mathcal {M}}_i. \end{aligned}$$(7.21)When \(i=1\), this follows from (7.19) as \(P_{\alpha _1}=0\). (Use the fact that \(x_{\alpha _1}(a_1X_{\alpha _1})\) commutes with any other \(x_{\alpha _j}(a_jX_{\alpha _j})\) in view of (7.34) since \(\alpha _1\) is a simple root.) Now by induction, suppose that (7.21) is verified for all \(j<i\). By (7.19),
$$\begin{aligned} v'(X_{\alpha _i})+v'\left( 1\alpha ^{1}_i(\delta )\right) \geqslant \min \left( m_{\mathbf {G}} n_\mathbf {G}(\lambda _0),v'(P_{\alpha _i})\right) . \end{aligned}$$Note that \(P_{\alpha _i}\) is the sum of monomials of the form \(\alpha _j^{1}(\delta )^{k_1}X_{\alpha _j}^{k_2}\) with \(j,k_1,k_2\in \mathbb {Z}\) such that \(1\leqslant j<i\) and \(0\leqslant k_1,k_2\leqslant Y\). Each monomial satisfies
$$\begin{aligned}&v'\left( \alpha _j^{1}(\delta )^{k_1}X_{\alpha _j}^{k_2}\right) =k_1v'\left( \alpha _j^{1}(\delta )\right) +k_2v'(X_{\alpha _j})\nonumber \\&\quad \geqslant Ym_{\Xi ^\mathrm{spl}}e_{F'/F}\kappa Y{\mathcal {M}}_{i1}, \end{aligned}$$where the inequality follows from (7.16), (7.14), the induction hypothesis, and the fact that \(0\leqslant {\mathcal {M}}_{j}\leqslant {\mathcal {M}}_{i1}\). Hence
$$\begin{aligned} v'(P_{\alpha _i})\geqslant Ym_{\Xi ^\mathrm{spl}}e_{F'/F}\kappa Y{\mathcal {M}}_{i1}. \end{aligned}$$Now
$$\begin{aligned} v'(X_{\alpha _i})\geqslant & {} \min \left( m_{\mathbf {G}}n_\mathbf {G}(\lambda _0),v'(P_{\alpha _i})\right) v'\left( 1\alpha ^{1}_i(\delta )\right) \\\geqslant & {} m_{\mathbf {G}}n_\mathbf {G}(\lambda _0) Ym_{\Xi ^\mathrm{spl}}e_{F'/F}\kappa Y{\mathcal {M}}_{i1}\\&\quad \left v'\left( 1\alpha ^{1}_i(\delta )\right) \right ={\mathcal {M}}_i, \end{aligned}$$as desired. Now that the claim is verified, we have a fortiori
$$\begin{aligned} v'(X_{\alpha _i})\geqslant {\mathcal {M}}_{\Phi ^+}, \quad \forall 1\leqslant i\leqslant \Phi ^+ . \end{aligned}$$(7.22)For our purpose it suffices to use the following upper bound, which is simpler than \({\mathcal {M}}_{\Phi ^+}\). Note that we used the upper bound on \(n_\mathbf {G}(\lambda _0)\) from Step 1.
$$\begin{aligned} {\mathcal {M}}_{\Phi ^+}\leqslant & {} Y^{\Phi ^+} \sum _{\alpha \in \Phi ^+} \Big ( v'(1\alpha ^{1}(\delta ) + (m_\mathbf {G}m_{\Xi ^{\mathrm{spl}}}+m_\mathbf {G}c_\mathbf {G}\nonumber \\&\quad + m_{\Xi ^{\mathrm{spl}}})e_{F'/F}\kappa \Big ). \end{aligned}$$(7.23) 
Step 3.
Find \(a\in A'(F')\) such that \(a^{1}na\in K'\). We can choose a sufficiently large \(C=C(\mathbf {G},\Xi )>0\), depending only on the Chevalley group \(\mathbf {G}\) and \(\Xi \), and integers \(a^0_{\alpha }\in [C,0]\) for \(\alpha \in \Delta ^+\) such that
$$\begin{aligned} 1\leqslant \sum _{\alpha \in \Delta ^+} (a^0_\alpha ) \langle \beta ,\alpha ^\vee \rangle \leqslant C,\quad \forall \beta \in \Delta ^+. \end{aligned}$$[This is possible because the matrix \((\langle \beta ,\alpha ^\vee \rangle )_{\beta ,\alpha \in \Delta ^+}\) is nonsingular. For instance one finds \(a^0_{\alpha }\in \mathbb {Q}\) satisfying the above inequalities for \(C=1\) and then eliminate denominators in \(a^0_{\alpha }\) by multiplying a large positive integer.] Now put \(a_{\alpha }:={\mathcal {M}}_{\Phi ^+} a^0_{\alpha }\in [C{\mathcal {M}}_{\Phi ^+},0]\) and \(a:=\sum _{\alpha \in \Delta ^+} a_\alpha \alpha ^\vee (\varpi ')\in A'(F')\) so that
$$\begin{aligned} {\mathcal {M}}_{\Phi ^+}\leqslant v(\beta (a))\leqslant C\cdot {\mathcal {M}}_{\Phi ^+}, \quad \forall \beta \in \Delta ^+. \end{aligned}$$(7.24)In fact (7.24) implies that the left inequality holds for all \(\beta \in \Phi ^+\). Hence
$$\begin{aligned}&a^{1}na=\prod _{i=1}^{\Phi ^+} x_{\alpha _i}\left( \alpha _i(a)^{1} X_{\alpha _{i}}\right) \\&\in \prod _{i=1}^{\Phi ^+} U_{\alpha _i,v(X_{\alpha _i})v(\alpha _i(a))}\subset \prod _{i=1}^{\Phi ^+} U_{\alpha _i,{\mathcal {M}}_{\Phi ^+} +v(X_{\alpha _i})}. \end{aligned}$$Here we have written \(U_{\alpha ,m}\) with \(m\in \mathbb {R}\) for the image under the isomorphism \(x_\alpha :F\simeq U_\alpha (F)\) of the set \(\{a\in F:v(a)\geqslant m\}\). In light of (7.21), \({\mathcal {M}}_{\Phi ^+} +v(X_{\alpha _i})\geqslant 0\). Hence \(a^{1}na\in K'\).

Step 4.
Conclude the proof. Step 3 shows that \(a\overline{x}'_0\in \mathrm{Apt}(A'(F'))\) is invariant under the left multiplication action by \(n\) on \({\mathcal {B}}(G(F'),K')\), which acts as an isometry. Recalling that \(\overline{x}'=n\overline{x}'_0\) we have
$$\begin{aligned}&d(\overline{x}', \mathrm{Apt}(A'(F')))\leqslant d(n\overline{x}'_0, a\overline{x}'_0)= d(n\overline{x}'_0, na\overline{x}'_0)\nonumber \\&\quad = d(\overline{x}'_0, a\overline{x}'_0). \end{aligned}$$(7.25)On the other hand, for any \(\overline{x}'\in \mathrm{Apt}(A'(F'))\) and any positive simple coroot \(\alpha ^{\vee }\), we have
$$\begin{aligned} d(\overline{x}', \alpha ^\vee (\varpi ')^{1} \overline{x}')\leqslant l_{\max }(\Phi ). \end{aligned}$$(7.26)Indeed this holds by the definition of \(l_{\max }(\Phi )\) as the left hand side is the length of \(\alpha ^\vee \). Since \(a=\prod _{\alpha \in \Delta ^+} (\alpha ^\vee (\varpi '))^{a_{\alpha }}\) with \(a_\alpha \in [C{\mathcal {M}}_{\Phi ^+},0]\), a repeated use of (7.26), together with a triangle inequality, shows that
$$\begin{aligned} d(\overline{x}'_0, a\overline{x}'_0)\leqslant l_{\max }(\Phi )\cdot C\cdot {\mathcal {M}}_{\Phi ^+}\cdot \Delta ^+. \end{aligned}$$(7.27)Lemma 7.7 follows from (7.25), (7.27), (7.22), (7.23), and \(e_{F'/F}\leqslant [F':F]\leqslant w_G s_G\) as we saw in (7.12).\(\square \)
Since \(\gamma \) is elliptic and \(G\) is anisotropic over \(F\), \(\mathrm{Apt}(T_\gamma (F))\) is a singleton. Let \(\overline{x}_1\) denote its only point. Then the \(\mathrm{Gal}(F'/F)\)action on \(\mathrm{Apt}(T_\gamma (F'))\) has \(\overline{x}_1\) as the unique fixed point. Motivated by Lemma 7.7 we set \({\mathcal {M}}(\gamma ,\kappa )\) to be
and similarly \( {\mathcal {M}}(\delta ,\kappa )\) using \(\alpha \in \Phi (G,A')\) in the sum instead. Note that we are summing over all roots, not just positive roots as in the lemma. This is okay since it will only improve the inequality of the lemma. We do this such that \({\mathcal {M}}(\gamma ,\kappa )={\mathcal {M}}(\delta ,\kappa )\). Indeed the equality is induced by a bijection \(\Phi (G,T_\gamma )\simeq \Phi (G,A')\) coming from any element \(y'\in G(F')\) such that \(A'=y'T_\gamma (y')^{1}\) (for example one can take \(y'=y\)). Define a closed ball in \(G(F)/K\): for \(\overline{z} \in G(F)/K\) and \(R\geqslant 0\),
Lemma 7.8
\(X_F(\gamma ,\lambda )~\subset ~\mathrm {Ball}(\overline{x}_1,{\mathcal {M}}(\gamma ,\kappa )).\)
Proof
As we noted above, \(X_F(\gamma ,\lambda )\subset X_{F'}(\gamma ,\lambda ')=X_{F'}(y^{1}\delta y,\lambda ')\). Lemma 7.7 tells us that
The last implication uses \(\mathrm{Apt}(A'(F'))=y \mathrm{Apt}(T_\gamma (F'))\) (recall \(A'=yT_\gamma y^{1}\)). We have viewed \(\overline{x}\) as a point of \({\mathcal {B}}(G(F'),K')\) via the isometric embedding \({\mathcal {B}}(G(F),K)\hookrightarrow {\mathcal {B}}(G(F'),K')\). In order to prove the lemma, it is enough to check that \(d(\overline{x},\overline{x}_1)\leqslant d(\overline{x},\overline{x}_2)\) for every \(\overline{x}_2\in \mathrm{Apt}(T_\gamma (F'))\). To this end, we suppose that there exists an \(\overline{x}_2\) with
and will draw a contradiction.
As \(\sigma \in \mathrm{Gal}(F'/F)\) acts on \({\mathcal {B}}(G(F'),K')\) by isometry, \(d(\overline{x},\sigma \overline{x}_2)=d(\overline{x},\overline{x}_2)\). As \(\mathrm{Apt}(T_\gamma (F'))\) is preserved under the Galois action, \(\sigma \overline{x}_2\in \mathrm{Apt}(T_\gamma (F'))\). According to the inequality of [103, 2.3], for any \(x,y,z\in {\mathcal {B}}(G(F'),K')\) and for the unique mid point \(m=m(x,y)\in {\mathcal {B}}(G(F'),K')\) such that \(d(x,m)=d(y,m)=\frac{1}{2}d(x,y)\),
Consider the convex hull \({\fancyscript{C}}\) of \({\fancyscript{C}}_0:=\{\sigma \overline{x}_2\}_{\sigma \in \mathrm{Gal}(F'/F)}\). Since \({\fancyscript{C}}_0\) is contained in \(\mathrm{Apt}(T_\gamma (F'))\), so is \({\fancyscript{C}}\). Moreover \({\fancyscript{C}}_0\) is fixed under \(\mathrm{Gal}(F'/F)\), from which it follows that \({\fancyscript{C}}\) is also preserved under the same action. [One may argue as follows. Inductively define \({\fancyscript{C}}_{i+1}\) to be the set consisting of the mid points \(m(x,y)\) for all \(x,y\in {\fancyscript{C}}_{i}\). Then it is not hard to see that \({\fancyscript{C}}_i\) must be preserved under \(\mathrm{Gal}(F'/F)\) and that \(\cup _{i\geqslant 0} {\fancyscript{C}}_i\) is a dense subset of \({\fancyscript{C}}\).] As \({\fancyscript{C}}\) is a compact set, one may choose \(\overline{x}_3\in {\fancyscript{C}}\) which has the minimal distance to \(\overline{x}\) among the points of \({\fancyscript{C}}\). By construction
Applying (7.29) to \((x,y,z)=(\overline{x}_3,\sigma \overline{x}_3,\overline{x})\), where \(\sigma \in \mathrm{Gal}(F'/F)\),
As \(\overline{x}_3,\sigma \overline{x}_3\in {\fancyscript{C}}\), we also have \(m(\overline{x}_3,\sigma \overline{x}_3)\in {\fancyscript{C}}\) by the convexity of \({\fancyscript{C}}\). The choice of \(\overline{x}_3\) ensures that \(d(\overline{x}_3,\overline{x})\leqslant d(m(\overline{x}_3,\sigma \overline{x}_3),\overline{x})\), therefore \(d(\overline{x}_3,\sigma \overline{x}_3)=0\), i.e. \(\overline{x}_3=\sigma \overline{x}_3\). Hence \(\overline{x}_3\) is a \(\mathrm{Gal}(F'/F)\)fixed point of \(\mathrm{Apt}(T_\gamma (F'))\). This implies that \(\overline{x}_3=\overline{x}_1\), but then (7.30) contradicts (7.28). \(\square \)
Lemma 7.9
There exist constants \(Z_1,Z_2\geqslant 0\), independent of \(\gamma \) and \(\lambda \), such that
Remark 7.10
A scrutiny into the defining formulas for \(Z_1\) and \(Z_2\) (as well as \(Z'_1\) and \(Z'_2\)) at the end of the proof reveals that \(Z_1\) and \(Z_2\) depend only on the affine root data, the grouptheoretic constants for \(G\) (and its Chevalley form), and \(\Xi \). An important point is that, in the situation where local data arise from some global reductive group over a number field by localization, the constants \(Z_1\) and \(Z_2\) do not depend on the residue characteristic \(p\) or the \(p\)adic field \(F\) as long as the affine root data remain unchanged. This observation is used in the proof of Theorem 7.3 to establish a kind of uniformity when traveling between places in \(\mathcal {V}(\theta )\backslash S_{\mathrm{bad}}\) for a fixed \(\theta \in {\fancyscript{C}}(\Gamma _1)\) in the notation there.
Proof
To ease notation we write \({\mathcal {M}}\) for \({\mathcal {M}}(\gamma ,\kappa )\) in the proof. Let us introduce some quantities and objects of geometric nature for the building \({\mathcal {B}}(G(F),K)\). Write \(e_{\max }>0\) for the maximum length of the edges of \({\mathcal {B}}(G(F),K)\). For a subset \(S\) of \({\mathcal {B}}(G(F),K)\), define \(\mathrm {Ch}^+(S)\) to be the set of chambers \({\fancyscript{C}}\) of the building such that \({\fancyscript{C}}\cap S\) contains a vertex. Let \(v\in {\mathcal {B}}(G(F),K)\) be a vertex. (We are most interested in the case \(v=\overline{x}_1\).) We put \({\fancyscript{C}}_1(v)\) to be the union of chambers in \(\mathrm {Ch}^+(\{v\})\) and define \({\fancyscript{C}}_{i+1}(v)\) to be the union of chambers in \(\mathrm {Ch}^+({\fancyscript{C}}_i(v))\) for all \(i\in \mathbb {Z}_{\geqslant 1}\) so as to obtain a strictly increasing chain \(\{v\}\subsetneq {\fancyscript{C}}_1(v)\subsetneq {\fancyscript{C}}_2(v)\subsetneq {\fancyscript{C}}_3(v)\subsetneq \cdots \). Denote by \(\mathrm {V}_i(v)\) (resp. \(\mathrm {Ch}_i(v)\)) the set of vertices (resp. chambers) contained in \({\fancyscript{C}}_i(v)\) for \(i\in \mathbb {Z}_{\geqslant 1}\).
Choose any chamber \({\fancyscript{C}}\) in \({\mathcal {B}}(G(F),K)\). Define \({\fancyscript{C}}^+\) to be the union of all chambers in \(\mathrm {Ch}^+({\fancyscript{C}})\). Clearly \({\fancyscript{C}}^+\) is compact and its interior contains the compact subset \({\fancyscript{C}}\). Hence there exists a maximal \(R_G>0\) such that for every point \(y\in {\fancyscript{C}}\) (which may not be a vertex), the ball centered at \(y\) of radius \(R_G\) is contained in \({\fancyscript{C}}^+\). Since the isometric action of \(G(F)\) is transitive on the set of chambers, \(R_G\) does not depend on the choice of \({\fancyscript{C}}\). Moreover the ratio \(l_{\max }(\Phi )/R_G\) does not depend on the choice of metric on the building.
From the definitions we have \(\mathrm {Ball}(\overline{x}_1,R_G)\subset {\fancyscript{C}}_1(\overline{x}_1)\) and deduce recursively that
Take \({\mathcal {M}}'\) to be the integer such that \(\frac{{\mathcal {M}}}{R_G}\leqslant {\mathcal {M}}' < \frac{{\mathcal {M}}}{R_G}+1\) so that in particular
Let us bound \(\mathrm {Ch}_1(v)\) for every vertex \(v\in {\mathcal {B}}(G(F),K)\). The stabilizer of \(v\), denoted by \(\mathrm {Stab}(v)\), acts transitively on \(\mathrm {Ch}_1(v)\). Let \({\fancyscript{C}}\in \mathrm {Ch}_1(v)\). Then
where \(\mathrm {Iw}\) denotes an Iwahori subgroup of \(G(\mathcal {O})\), which is conjugate to \(\mathrm {Stab}({\fancyscript{C}})\). The group \(\mathrm {Stab}(v)\) may not be hyperspecial, but the first inequality follows from the fact that the hyperspecial has the largest volume among all maximal compact subgroups [103, 3.8.2]. See the proof of Lemma 2.13 for the last inequality.
Each chamber contains \(\dim A+1\) vertices as a \(\dim A\)dimensional simplex. Hence for each \(i\geqslant 1\),
On the other hand,
We see that \(\mathrm {Ch}_{i}(\overline{x}_1)\leqslant q_v^{i(d_G+r_G)} (\dim A+1)^{i1}\) and thus
Note that
which can be rewritten in the form
Since \(v(1\alpha (\gamma ))+v(1\alpha ^{1}(\gamma ))\leqslant v(1\alpha (\gamma ))+v(1\alpha ^{1}(\gamma ))+2b_\Xi \kappa \) in view of (7.5), we have
Returning to (7.31) and (7.32),
The proof of Lemma 7.9 is complete once we set \(Z_1\) and \(Z_2\) as follows, the point being that they

\(Z_1:=(Z'_1+2 b_{\Xi } Z'_2)(d_G+2r_G+1)\),

\(Z_2:=Z'_2(d_G+2r_G+1)\).
Corollary 7.11
\(O^{G(F)}_{\gamma }({\mathbf {1}}_{K\lambda (\varpi )K}, \mu _G,\mu ^{\mathrm{EP}}_{I_\gamma })\!\leqslant \! q_v^{r_G(d_G+1)} q_v^{1+Z_1\kappa } D^G(\gamma )^{Z_2}\).
Proof
Follows from (7.15), Lemmas 7.8 and 7.9.
7.4 Lemmas in the split case
This subsection plays a supporting role for the previous subsections, especially Sect. 7.3. As in Sect. 7.2 let \(\mathbf {G}\) be a Chevalley group with a Borel subgroup \(\mathbf {B}\) containing a split maximal torus \(\mathbf {T}\), all over \(\mathbb {Z}\). Let \(\Xi ^{\mathrm{spl}}_{\mathbb {Q}}:\mathbf {G}\hookrightarrow \mathrm{GL}_m\) be a closed embedding of algebraic groups over \(\mathbb {Q}\). Let \(\mathbb {T}\) denote the diagonal maximal torus of \(\mathrm{GL}_m\), \(\mathbb {B}\) the upper triangular Borel subgroup of \(\mathrm{GL}_m\), and \(\mathbb {N}\) the unipotent radical of \(\mathbb {B}\).
Extend \(\Xi ^{\mathrm{spl}}_{\mathbb {Q}}\) to a closed embedding \(\Xi ^{\mathrm{spl}}:\mathbf {G}\hookrightarrow \mathrm{GL}_m\) defined over \(\mathbb {Z}[1/R]\) for some integer \(R\) such that \(\Xi ^{\mathrm{spl}}(\mathbf {T})\) [resp. \(\Xi ^{\mathrm{spl}}(\mathbf {B})\)] lies in the group of diagonal (resp. upper triangular) matrices of \(\mathrm{GL}_m\). To see that this is possible, find a maximal \(\mathbb {Q}\)split torus \(\mathbb {T}'\) of \(\mathrm{GL}_m\) containing \(\Xi ^{\mathrm{spl}}_{\mathbb {Q}}(\mathbf {T})\). Choose any Borel subgroup \(\mathbb {B}'\) over \(\mathbb {Q}\) containing \(\mathbb {T}\). Then there exists \(g\in \mathrm{GL}_m(\mathbb {Q})\) such that the inner automorphism \(\mathrm {Int}(g):\mathrm{GL}_m\rightarrow \mathrm{GL}_m\) by \(\gamma \mapsto g \gamma g^{1}\) carries \((\mathbb {B}',\mathbb {T}')\) to \((\mathbb {B},\mathbb {T})\). Then \(\Xi ^{\mathrm{spl}}_{\mathbb {Q}}\) and \(\mathrm {Int}(g)\) extend over \(\mathbb {Q}\) to over \(\mathbb {Z}[1/R]\) for some \(R\in \mathbb {Z}\), namely at the expense of inverting finitely many primes [basically those in the denominators of the functions defining \(\Xi ^{\mathrm{spl}}_{\mathbb {Q}}\) and \(\mathrm {Int}(g)\)].
Now suppose that \(p\) is a prime not diving \(R\). Let \(F\) be a finite extension of \(\mathbb {Q}_p\) with integer ring \(\mathcal {O}\) and a uniformizer \(\varpi \). The field \(F\) is equipped with a unique discrete valuation \(v_F\) such that \(v_F(\varpi )=1\). Let \(\lambda \in X_*(\mathbf {T})\). We are interested in assertions which work for \(F\) as the residue characteristic \(p\) varies. Lemma 7.12 (resp. Corollary 7.14) below is used in Step 1 (resp. Step 2) of the proof of Lemma 7.7.
Lemma 7.12
There exists \(m_{\Xi ^{\mathrm{spl}}}\in \mathbb {Z}_{>0}\) such that for every \(p\), \(F\) and \(\lambda \) as above and for every semisimple \(\delta \in \mathbf {G}(\mathcal {O})\lambda (\varpi )\mathbf {G}(\mathcal {O})\) (and for any choice of \(T_\delta \) containing \(\delta \)),
Proof
The argument is the same as in the proof of Lemma 2.18. The constant \(m_{\Xi ^{\mathrm{spl}}}\) corresponds to the constant \(B_5\) in that lemma. To see that it is independent of \(p\), \(F\) and \(\lambda \), it suffices to examine the argument and see that the constant depends only on \(\mathbf {G}\), \(\mathbf {B}\), \(\mathbf {T}\) (and the auxiliary choice of \(\widetilde{\alpha }\)’s as in the proof of Lemma 2.17, which is fixed once and for all).
The unipotent radical of \(\mathbf {B}\) is denoted \(\mathbf {N}\). For \(F\) as above, let \(x_0\) be the hyperspecial vertex on the building of \(\mathbf {G}(F)\) corresponding to \(\mathbf {G}(\mathcal {O})\). As usual put \(\Phi ^+:=\Phi ^+(\mathbf {G},\mathbf {T})\) be the set of positive roots with respect to \((\mathbf {B},\mathbf {T})\).
Let us recall some facts about the Chevalley basis. For each \(\alpha \in \Phi ^+\), let \(U_\alpha \) denote the corresponding unipotent subgroup equipped with \(x_\alpha :\mathbb {G}_a \simeq U_\alpha \). Order the elements of \(\Phi ^+\) as \(\alpha _1,\ldots ,\alpha _{\Phi ^+}\) once and for all such that simple roots appear at the beginning. The multiplication map
is an isomorphism of schemes (but not as group schemes) over \(\mathbb {Z}\). This can be deduced from [5, Exp XXII, 5.5.1], which deals with a Borel subgroup of a Chevalley group. In particular (since the ordering on \(\Phi ^+\) is fixed) any \(n\in \mathbf {N}(F)\) can be uniquely written as
for unique \(Y_{\alpha _i}\in \mathbb {G}_a(F)\simeq F\)’s. The Chevalley commutation relation ([20, §III]) has the following form: for all \(1\leqslant i<j\leqslant \Phi ^+\) and all \(Y_{\alpha _i}\in F\)’s,
where \(C_{ij}\) are certain integers (depending on \(\mathbf {G}\)) which we need not know explicitly. It suffices to know that, in the cases of \(F\) we are interested in, the constants \(C_{ij}\) are units in \(\mathcal {O}\) (cf. the assumption in the paragraph preceding Proposition 7.1).
We thank Kottwitz for explaining the proof of the following lemma.
Lemma 7.13
Suppose that the Chevalley group \(\mathbf {G}\) is semisimple and simply connected. Let \(\Omega \subset X^*(\mathbf {T})\) denote the set of fundamental weights and \(\rho ^\vee \in X_*(\mathbf {T})\) the half sum of all positive coroots. Let \(\lambda \in X^*(\mathbf {T})\) and define \(n_0(\lambda ):=\max _{\omega \in \Omega } \langle \omega ,\lambda \rangle \). For every prime \(p\), every \(p\)adic field \(F\), and every cocharacter \(\lambda \in X_*(\mathbf {T})\) as above, the following is true: in terms of the decomposition (7.33), each \(y\in \mathbf {G}(\mathcal {O})\lambda (\varpi )\mathbf {G}(\mathcal {O})\cap \mathbf {N}(F)\) satisfies the inequality
Proof
It suffices to check that
[Here we write \(\varpi ^{2 n_0(\lambda )\rho ^\vee }\) for \((\rho ^\vee (\varpi ))^{2 n_0(\lambda )}\).] Indeed, this implies the desired inequality in the lemma since the decomposition (7.33) is defined over \(\mathcal {O}\).
Let us introduce some notation. For each \(\omega \in \Omega \) let \(V_\omega \) denote the irreducible representation of \(\mathbf {G}(F)\) of highest weight \(\omega \) on an \(F\)vector space. Write \(V_\omega =\oplus _{\mu \in X^*(\mathbf {T})} V_{\omega ,\mu }\) for the weight decomposition. The geometric construction of \(V_\omega \) and its weight decomposition by using flag varieties gives us a natural \(\mathcal {O}\)integral structures \(V_\omega (\mathcal {O})\) in \(V_\omega \) such that \(V_\omega (\mathcal {O})=\oplus _{\mu \in X^*(\mathbf {T})} V_{\omega ,\mu }(\mathcal {O})\), where \(V_{\omega ,\mu }(\mathcal {O}):=V_\omega (\mathcal {O})\cap V_{\omega ,\mu }\). Note that each \(V_\omega \) receives an action of \(\mathbb {G}_m\) via \(\mathbb {G}_m\mathop {\rightarrow }\limits ^{\rho ^\vee }\mathbf {T}\hookrightarrow \mathbf {G}\). We may consider a coarser decomposition \(V_\omega =\oplus _{i\in \mathbb {Z}} V_{\omega ,i}\), where \(V_{\omega ,i}:=\oplus _{\langle \mu ,2\rho ^\vee \rangle =i} V_{\omega ,\mu }\). For any \(\omega \in \Omega \) and \(V=V_\omega \), set \(V_{\geqslant i}:=\oplus _{j\geqslant i} V_j\), \(V_{\geqslant i}(\mathcal {O}):=V_{\geqslant i}\cap V(\mathcal {O})\), and \(V_i(\mathcal {O}):=V_i\cap V(\mathcal {O})\). Observe that \(\mathbf {B}(F)\) preserves the filtration \(\{V_{\geqslant i}\}_{i\in \mathbb {Z}}\) and that \(\mathbf {N}(F)\) acts trivially on \(V_{\geqslant i}/V_{\geqslant i+1}\).
As a preparation, suppose that \(g\in \mathbf {G}(\mathcal {O})\lambda (\varpi )\mathbf {G}(\mathcal {O})\) and let us prove that \(g V_\omega (\mathcal {O})\subset \varpi ^{n_0(\lambda )}V_\omega (\mathcal {O})\) for all \(\omega \in \Omega \). Since \(\mathbf {G}(\mathcal {O})\) stabilizes \(V_\omega (\mathcal {O})\), the latter condition is true if and only if \(\lambda (\varpi )V_\omega (\mathcal {O})\subset \varpi ^{n_0(\lambda )} V_\omega (\mathcal {O})\), which holds if and only if
for all weights \(\mu \) for \(V_\omega \) by considering the weight decomposition. The above inequality for all weights \(\mu \) is equivalent to that for the lowest weight \(\mu \) for \(V_\omega \). Since \(\mu =w_0\omega _\omega \) for the longest Weyl element \(w_0\), the condition is that \(\langle w_0 \omega ,\lambda \rangle \leqslant n_0(\lambda )\) for all \(\omega \). This is verified by the definition of \(n_0(\lambda )\) since \(w_0\) preserves the set \(\Omega \).
Now consider \(\varpi ^{2 n_0(\lambda )\rho ^\vee } (y1) \varpi ^{2 n_0(\lambda )\rho ^\vee }\), where \(y\) is as in the lemma. Since \(\varpi ^{2\rho ^\vee }\) acts on \(V_j\) as \(\varpi ^j\), we see from this and the last paragraph that for all \(\omega \in \Omega \) and \(i\in \mathbb {Z}\),
It follows that \(\varpi ^{2 n_0(\lambda )\rho ^\vee } y \varpi ^{2 n_0(\lambda )\rho ^\vee }\) also preserves \(V_{\omega ,i}(\mathcal {O})\), hence \(V_\omega (\mathcal {O})\). Therefore the element belongs to \({\mathcal {N}}(\mathcal {O})={\mathcal {N}}(F)\cap \mathbf {G}(\mathcal {O})\), concluding the proof of (7.35). \(\square \)
For an arbitrary Chevalley group \(\mathbf {G}\) and \(\lambda \in X_*(\mathbf {T})^+\), define a nonnegative integer
Corollary 7.14
Let \(\mathbf {G}\) be an arbitrary Chevalley group. For every prime \(p\), every \(p\)adic field \(F\), and every cocharacter \(\lambda \in X_*(\mathbf {T})\), there exists a constant \(m_\mathbf {G}>0\) such that the following is true: each \(y\in \mathbf {G}(\mathcal {O})\lambda (\varpi )\mathbf {G}(\mathcal {O})\cap \mathbf {N}(F)\), uniquely decomposed as in (7.33), satisfies the inequality
Proof
The corollary is immediate from the lemma if \(\mathbf {G}\) is semisimple and simply connected. Indeed, define \(n_1(\lambda )\) to be the maximum of \(\langle \alpha ,\lambda \rangle \) as \(\alpha \) runs over \(\Delta ^+\), the set of simple roots. Observe that both the sets \(\Omega \) and \(\Delta ^+\) are bases for \(X^*(\mathbf {T})_\mathbb {Q}\). By using the change of basis matrix, it is easy to deduce from Lemma 7.13 that for some constant \(c>0\) depending only on \(\mathbf {G}\), we have that
for all \(p\), \(F\), \(\lambda \), and \(i\). A fortiori the same holds with \(n_\mathbf {G}(\lambda )\) in place of \(n_1(\lambda )\). The proof is completed by setting \(m_\mathbf {G}:=c \max _{\alpha \in \Phi ^+}\langle \alpha ,\rho ^\vee \rangle \).
It remains to extend from the simply connected case to the general case. As usual write \(\mathbf {G}_{\mathrm{ad}}\) for the adjoint group of \(\mathbf {G}\) and \(\mathbf {G}_{\text {sc}}\) for the simply connected cover of \(\mathbf {G}_{\mathrm{ad}}\). The pair \((\mathbf {B},\mathbf {T})\) induces the Borel pairs \((\mathbf {B}_{\mathrm{ad}},\mathbf {T}_{\mathrm{ad}})\) for \(\mathbf {G}_{\mathrm{ad}}\) and \((\mathbf {B}_{\text {sc}},\mathbf {T}_{\text {sc}})\) for \(\mathbf {G}_{\text {sc}}\). Write \(\Phi ^+_{\mathrm{ad}}\) and \(\Phi ^+_{\text {sc}}\) for the associated sets of roots. Let \(\mathbf {N}_{\mathrm{ad}}\) and \(\mathbf {N}_{\text {sc}}\) denote the unipotent radicals of \(\mathbf {B}_{\mathrm{ad}}\) and \(\mathbf {B}_{\text {sc}}\), respectively. Then the natural maps \(\mathbf {G}\rightarrow \mathbf {G}_{\mathrm{ad}}\) and \(\mathbf {G}_{\text {sc}}\rightarrow \mathbf {G}_{\mathrm{ad}}\) induce isomorphisms \(\mathbf {N}\simeq \mathbf {N}_{\mathrm{ad}}\) and \(\mathbf {N}_{\text {sc}}\simeq \mathbf {N}_{\mathrm{ad}}\) as well as settheoretic bijections \(\Phi ^+\rightarrow \Phi ^+_{\mathrm{ad}}\) and \(\Phi ^+_{\text {sc}}\rightarrow \Phi ^+_{\mathrm{ad}}\). In particular the ordering on \(\Phi ^+\) induces unique orderings on \(\Phi ^+_{\mathrm{ad}}\) and \(\Phi ^+_{\text {sc}}\). With respect to these orderings, the decomposition (7.33) is compatible with the maps \(\mathbf {G}\rightarrow \mathbf {G}_{\mathrm{ad}}\) and \(\mathbf {G}_{\text {sc}}\rightarrow \mathbf {G}_{\mathrm{ad}}\). From all this it follows that the corollary for \(\mathbf {G}_{\text {sc}}\) implies that for \(\mathbf {G}_{\mathrm{ad}}\), and then for \(\mathbf {G}\). \(\square \)
8 Lemmas on conjugacy classes and level subgroups
This section contains several results which are useful for estimating the geometric side of Arthur’s invariant trace formula in the next section.
8.1 Notation and basic setup
Let us introduce some global notation in addition to that at the start of Sect. 4.

\(M_0\) is a minimal \(F\)rational Levi subgroup of \(G\).

\(A_{M_0}\) is the maximal split \(F\)torus in the center of \(M_0\).

\(\mathrm{Ram}(G):=\{v\in \mathcal {V}_F^\infty :\, G \text{ is } \text{ ramified } \text{ at } v\}\).

\(S\subset \mathcal {V}_F^\infty \) is a finite subset, often with a partition \(S=S_0\coprod S_1\).

\(r:{}^L G \rightarrow \mathrm{GL}_d(\mathbb {C})\) is an irreducible continuous representation such that \(r_{\widehat{G}}\) is algebraic.

\(\Xi :G\rightarrow \mathrm{GL}_m\) is a faithful algebraic representation defined over \(F\) (or over \(\mathcal {O}_F\) as explained below)

For any \(\mathbb {C}\)subspace \({\mathcal {H}}'\subset C^\infty _c(G(F_S))\), define
$$\begin{aligned} \mathrm{supp}\,{\mathcal {H}}'=\cup \, \mathrm{supp}\,\phi _S \end{aligned}$$where the union is taken over \(\phi _S\in {\mathcal {H}}'\).

\(q_{S}:=\prod _{v\in S} q_v\) where \(q_v\) is the cardinality of the residue field at \(v\). (Convention: \(q_S=1\) if \(S=\emptyset \).)
For each finite place \(v\in \mathrm{Ram}(G)\) of \(F\), fix a special point \(x_v\) on the building of \(G\) once and for all, where \(x_v\) is required to belong to an apartment corresponding to a maximal \(F_v\)split torus \(A_v\) containing \(A_{M_0}\). The stabilizer \(K_v\) of \(x_v\) is a good special maximal compact subgroup of \(G(F_v)\) (good in the sense of [16]). Set \(K_{M,v}:=K_v\cap M(F_v)\) for each \(F_v\)rational Levi subgroup \(M\) of \(G\) containing \(A_v\). Then \(K_{M,v}\) is a good special maximal compact subgroup of \(M(F_v)\).
It is worth stressing that this article treats a reductive group \(G\) without any hypothesis on \(G\) being split (or quasisplit). To do so, we would like to carefully choose an integral model of \(G\) over \(\mathcal {O}_F\) for convenience and also for clarifying a notion like “level \({\mathfrak n}\) subgroups”. We thank Brian Conrad for explaining us crucial steps in the proof below (especially how to proceed by using the facts from [12]).
Proposition 8.1
The \(F\)group \(G\) extends to a group scheme \({\mathfrak G}\) over \(\mathcal {O}_F\) (thus equipped with an isomorphism \({\mathfrak G}\times _{\mathcal {O}_F} F\simeq G\)) such that

\({\mathfrak G}\times _{\mathcal {O}_F} \mathcal {O}_F[\frac{1}{\mathrm{Ram}(G)}]\) is a reductive group scheme (cf. [32]),

\({\mathfrak G}(\mathcal {O}_v)=K_v\) for all \(v\in \mathrm{Ram}(G)\) (where \(K_v\) are chosen above),

there exists a faithful embedding of algebraic groups \(\Xi :{\mathfrak G}\hookrightarrow \mathrm{GL}_m\) over \(\mathcal {O}_F\) for some \(m\geqslant 1\).
Remark 8.2
If \(G\) is split then \(\mathrm{Ram}(G)\) is empty and the above proposition is standard in the theory of Chevalley groups.
Proof
For any finite place \(v\) of \(F\), we will write \(\mathcal {O}_{(v)}\) for the localization of \(\mathcal {O}_F\) at \(v\) (to be distinguished from the completion \(\mathcal {O}_v\)). As a first step there exists an injective morphism of group schemes \(\Xi _F:G\hookrightarrow GL_{m}\) defined over \(F\) for some \(m\geqslant 1\) ([33, Prop A.2.3]. The schemetheoretic closure \({\mathfrak G}'\) of \(G\) in \(GL_{m'}\) is a smooth affine scheme over \(\mathrm {Spec}\,\mathcal {O}_F[1/S]\) for a finite set \(S\) of primes of \(\mathcal {O}_F\) by arguing as in the first paragraph of [32, §2]. We may assume that \(S\supset \mathrm{Ram}(G)\). By [32, Prop 3.1.9.(1)], by enlarging \(S\) if necessary, we can arrange that \({\mathfrak G}'\) is reductive. For \(v\in \mathrm{Ram}(G)\) we have fixed special points \(x_v\), which give rise to the BruhatTits group schemes \(\widehat{{\mathfrak G}}(v)\) over \(\mathcal {O}_v\). Similarly for \(v\in S\backslash \mathrm{Ram}(G)\), let us choose hyperspecial points \(x_v\) so that the corresponding group schemes \(\widehat{{\mathfrak G}}(v)\) over \(\mathcal {O}_v\) are reductive.
According to [12, Prop D.4,p. 147] the obvious functor from the category of affine \(\mathcal {O}_{(v)}\)schemes to that of triples \((X,\widehat{\mathfrak {X}}(v),f)\) where \(X\) is an affine \(F\)scheme, \(\widehat{\mathfrak {X}}(v)\) is an affine \(\mathcal {O}_v\)scheme and \(f:X\times _F F_v\simeq \widehat{\mathfrak {X}}(v)\times _{\mathcal {O}_v} F_v\) is an equivalence. (The notion of morphisms is obvious in each category.) Thanks to its functorial nature, the same functor defines an equivalence when restricted to group objects in each category. For \(v\in \mathrm{Ram}(G)\), apply this functor to the BruhatTits group scheme \(\widehat{{\mathfrak G}}(v)\) over \(\mathcal {O}_v\) equipped with \(G\times _F F_v\simeq \widehat{{\mathfrak G}}(v)\times _{\mathcal {O}_v} F_v\) to obtain a group scheme \({\mathfrak G}(v)\) over \(\mathcal {O}_{(v)}\).
An argument analogous to that on page 14 of [12] shows that the obvious functor between the following categories is an equivalence: from the category of finitetype \(\mathcal {O}_F\)schemes to that of triples \((X,\{\mathfrak {X}(v)\}_{v\in S}, \{f_v\}_{v\in S})\) where \(X\) is a finitetype \(\mathcal {O}_F[1/S]\)scheme, \(\mathfrak {X}(v)\) is a finitetype \(\mathcal {O}_{(v)}\)scheme and \(f_v:X\times _{\mathcal {O}_F[1/S]} F\simeq \mathfrak {X}(v)\times _{\mathcal {O}_{(v)}} F\) is an isomorphism. Again this induces an equivalence when restricted to group objects in each category. In particular, there exists a group scheme \({\mathfrak G}\) over \(\mathcal {O}_F\) with isomorphisms \({\mathfrak G}\times _{\mathcal {O}_F} \mathcal {O}_F[1/S]\simeq {\mathfrak G}'\) and \({\mathfrak G}\times _{\mathcal {O}_F} \mathcal {O}_{(v)}\simeq {\mathfrak G}(v)\) for \(v\in S\) which are compatible with the isomorphisms between \({\mathfrak G}'\) and \({\mathfrak G}(v)\) over \(F\). By construction \({\mathfrak G}\) satisfies the first two properties of the proposition.
We will be done if \(\Xi _F:G\hookrightarrow GL_m\) over \(F\) extends to an embedding of group schemes over \(\mathcal {O}_F\). It is evident from the construction of \({\mathfrak G}'\) that \(\Xi _F\) extends to \(\Xi ':G\hookrightarrow GL_m\) over \(\mathcal {O}_F[1/S]\). For each \(v\in S\), \(\Xi _F\) extends to \(\Xi (v):{\mathfrak G}(v)\hookrightarrow GL_m\) over \(\mathcal {O}_{v}\) thanks to [17, Prop 1.7.6], which can be defined over \(\mathcal {O}_{(v)}\) using the first of the above equivalences. Then the second equivalence allows us to glue \(\Xi '\) and \(\{\Xi (v)\}_{v\in S}\) to produce an \(\mathcal {O}_F\)embedding \(\Xi :G\hookrightarrow GL_m\). \(\square \)
For each finite place \(v\notin \mathrm{Ram}(G)\), \({\mathfrak G}\) defines a reductive group scheme over \(\mathcal {O}_v\), so \(K_v:={\mathfrak G}(\mathcal {O}_v)\) is a hyperspecial subgroup of \(G(F_v)\). Fix a maximal \(F_v\)split torus \(A_v\) of \(G\) which contains \(A_{M_0}\) such that the hyperspecial point for \(K_v\) belongs to the apartment of \(A_v\). For each Levi subgroup \(M\) of \(G\) whose center is contained in \(A_v\), define a hyperspecial subgroup \(K_{M,v}:=K_v\cap M(F_v)\) of \(M(F_v)\). At such a \(v\notin \mathrm{Ram}(G)\) define \({\mathcal {H}}^{\mathrm {ur}}(G(F_v))\) (resp. \({\mathcal {H}}^{\mathrm {ur}}(M(F_v))\)). The constant term (Sect. 6.1) of a function in \(C^\infty _c(G(F_v))\) (resp. \(C^\infty _c(M(F_v))\)) will be taken relative to \(K_v\) (resp. \(K_{M,v}\)). When \(P=MN\) is a Levi decomposition, we have Haar measures on \(K_v\), \(M(F_v)\) and \(N(F_v)\) such that the product measure equals \(\mu _v^{\mathrm{can}}\) on \(G(F_v)\) (cf. Sect. 6.1) and the Haar measure on \(M(F_v)\) is the canonical measure of Sect. 6.6. In particular when \(G\) is unramified at \(v\),
with respect to the measure on \(N(F_v)\).
Let \({\mathfrak n}\) be an ideal of \(\mathcal {O}_F\) and \(v\) a finite place of \(F\). Let \(v({\mathfrak n})\in \mathbb {Z}_{\geqslant 0}\) be the integer determined by \({\mathfrak n}\mathcal {O}_v=\varpi _v^{v({\mathfrak n})} \mathcal {O}_v\). Define \(K_v(\varpi _v^s)\) to be the MoyPrasad subgroup \(G(F_v)_{x_v,s}\) of \(G(F_v)\) by using Yu’s minimal congruent filtration as in [108] (which is slightly different from the original definition of Moy and Prasad). Yu has shown that \(G(F_v)_{x_v,s}= \ker ({\mathfrak G}(\mathcal {O}_v)\rightarrow {\mathfrak G}(\mathcal {O}_v/\varpi _v^s))\) in [108, Cor 8.8]. Set
to be considered the level \({\mathfrak n}\)subgroup of \(G(\mathbb {A}^{S,\infty })\).
Fix a maximal torus \(T_0\) of \(G\) over \(\overline{F}\) and an \(\mathbb {R}\)basis \({\mathcal {B}}_0\) of \(X_*(T_0)_\mathbb {R}\), which induces a function \(\Vert \cdot \Vert _{{\mathcal {B}}_0,G}:X_*(T_0)_{\mathbb {R}}\rightarrow \mathbb {R}_{\geqslant 0}\) as in Sect. 2.5. For any other maximal torus \(T\), there is an inner automorphism of \(G\) inducing \(T_0\simeq T\), so \(X_*(T)_\mathbb {R}\) has an \(\mathbb {R}\)basis \({\mathcal {B}}\) induced from \({\mathcal {B}}_0\), well defined up to the action by \(\Omega =\Omega (G,T)\). Therefore \(\Vert \cdot \Vert _{{\mathcal {B}},G}:X_*(T)_\mathbb {R}\rightarrow \mathbb {R}_{\geqslant 0}\) is defined without ambiguity. As it depends only on the initial choice of \({\mathcal {B}}_0\) (and \(T_0\)), let us write \(\Vert \cdot \Vert \) for \(\Vert \cdot \Vert _{{\mathcal {B}},G}\) when there is no danger of confusion.
Let \(v\) be a finite place of \(G\), and \(T_v\) a maximal torus of \(G\times _F \overline{F}_v\) (which may or may not be defined over \(F_v\)). Then \(\Vert \cdot \Vert :X_*(T_v)_\mathbb {R}\rightarrow \mathbb {R}_{\geqslant 0}\) is defined without ambiguity via \(T_v\simeq T_0\times _{\overline{F}} \overline{F}_v\) by a similar consideration as above. Now assume that \(G\) is unramified at \(v\). For any maximal \(F_v\)split torus \(A\subset G\) and a maximal torus \(T\) containing \(A\) over \(F_v\), the function \(\Vert \cdot \Vert _{{\mathcal {B}}_0}\) is well defined on \(X_*(T)_\mathbb {R}\) (resp. \(X_*(A)_\mathbb {R}\)) and invariant under the Weyl group \(\Omega \) (resp. \(\Omega _F\)). Hence for every \(v\) where \(G\) is unramified, the Satake isomorphism allows us to define \({\mathcal {H}}^{\mathrm {ur}}(G(F_v))^{\leqslant \kappa }\) as well as \({\mathcal {H}}^{\mathrm {ur}}(M(F_v))^{\leqslant \kappa }\) for every Levi subgroup \(M\) of \(G\) over \(F_v\). When \(G\) is unramified at \(S\), we put \({\mathcal {H}}^{\mathrm {ur}}(G(F_S))^{\leqslant \kappa }:=\otimes _{v\in S}{\mathcal {H}}^{\mathrm {ur}}(G(F_v))^{\leqslant \kappa } \) and define \({\mathcal {H}}^{\mathrm {ur}}(M(F_S))^{\leqslant \kappa }\) similarly.
For the group \(\mathrm{GL}_m\) with any \(m\geqslant 1\), we use the diagonal torus and the standard basis to define \(\Vert \cdot \Vert _{\mathrm{GL}_m}\) on the cocharacter groups of maximal tori of \(\mathrm{GL}_m\) (cf. Sect. 2.4). For \(\Xi :G\hookrightarrow \mathrm{GL}_m\) introduced above, define
8.2 \(z\)Extensions
A surjective morphism \(\alpha :H\rightarrow G\) of connected reductive groups over \(F\) is said to be a \(z\) extension if the following three conditions are satisfied: \(H^{\mathrm {der}}\) is simply connected, \(\ker \alpha \subset Z(H)\), and \(\ker \alpha \) is isomorphic to a finite product \(\prod {\mathrm {Res}}_{F_i/F}\mathrm{GL}_1\) for finite extensions \(F_i\) of \(F\). Writing \(Z:=\ker \alpha \), we often represent such an extension by an exact sequence of \(F\)groups \(1\rightarrow Z \rightarrow H\rightarrow G\rightarrow 1\). By the third condition and Hilbert 90, \(\alpha :H(F)\rightarrow G(F)\) is surjective.
Lemma 8.3
For any \(G\), a \(z\)extension \(\alpha :H\rightarrow G\) exists. Moreover, if \(G\) is unramified outside a finite set \(S\), where \(S_\infty \subset S\subset \mathcal {V}_F\), then \(H\) can be chosen to be unramified outside \(S\).
Proof
It is shown in [76, Prop 3.1] that a \(z\)extension exists and that if \(G\) splits over a finite Galois extension \(E\) of \(F\) then \(H\) can be chosen to split over \(E\). By the assumption on \(G\), it is possible to find such an \(E\) which is unramified outside \(S\). Since the preimage of a Borel subgroup of \(G\) in \(H\) is a Borel subgroup of \(H\), we see that \(H\) is quasisplit outside \(S\). \(\square \)
8.3 Rational conjugacy classes intersecting a small open compact subgroup
Throughout this subsection \(S=S_0\coprod S_1\) is a finite subset of \(\mathcal {V}^\infty _F\) and it is assumed that \(S_0\supset \mathrm{Ram}(G)\). Fix compact subgroups \(U_{S_0}\) and \(U_{\infty }\) of \(G(F_{S_0})\) and \(G(F\otimes _\mathbb {Q}\mathbb {R})\), respectively. Let \({\mathfrak n}\) be an ideal of \(\mathcal {O}_F\) as before, now assumed to be coprime to \(S\), with absolute norm \(\mathbb {N}({\mathfrak n})\in \mathbb {Z}_{\geqslant 1}\).
Lemma 8.4
Let \(U_{S_1}:=\mathrm{supp}\,{\mathcal {H}}^{\mathrm {ur}}(G(F_{S_1}))^{\leqslant \kappa }\). There exists \(c_\Xi >0\) independent of \(S\), \(\kappa \) and \({\mathfrak n}\) (but depending on \(G\), \(\Xi \), \(U_{S_0}\) and \(U_{\infty }\)) such that for all \({\mathfrak n}\) satisfying
the following holds: if \(\gamma \in G(F)\) and \(x^{1}\gamma x\in K^{S,\infty }({\mathfrak n}) U_{S_0} U_{S_1} U_\infty \) for some \(x\in G(\mathbb {A}_F)\) then \(\gamma \) is unipotent.
Proof
Let \(\gamma '=x^{1}\gamma x\). We keep using the embedding \(\Xi :\mathfrak {G}\hookrightarrow \mathrm{GL}_m\) over \(\mathcal {O}_F\) of Proposition 8.1. (For the lemma, an embedding away from the primes in \(S_0\) or dividing \({\mathfrak n}\) is enough.) At each finite place \(v\notin S_0\) and \(v\not \mid {\mathfrak n}\), Lemma 2.17 allows us to find \(\Xi '_v:\mathfrak {G}\hookrightarrow \mathrm{GL}_m\) over \(\mathcal {O}_v\) which is \(\mathrm{GL}_m(\mathcal {O}_v)\)conjugate to \(\Xi \times _{\mathcal {O}_F} F_v\) such that \(\Xi '_v\) sends \(A_v\) into the diagonal torus of \(\mathrm{GL}_m\).
Write \(\det (\Xi (\gamma )(1X))=X^m+a_{m1}(\gamma )X^{m1}+\cdots + a_0(\gamma )\), where \(a_i(\gamma )\in F\) for \(0\leqslant i\leqslant m1\). Our goal is to show that \(a_i(\gamma )=0\) for all \(i\). To this end, assuming \(a_i(\gamma )\ne 0\) for some fixed \(i\), we will estimate \(a_i(\gamma )_v\) at each place \(v\) and draw a contradiction.
First consider \(v\in S_1\). We claim that
for every \(\gamma \) that is conjugate to an element of \(\mathrm{supp}\,{\mathcal {H}}^{\mathrm {ur}}(G(F_{v}))^{\leqslant \kappa }\). To prove the claim we examine the eigenvalues of \(\Xi '_v(\gamma ')\), which is conjugate to \(\gamma \). We know \(\gamma '\) belongs to \(\mathrm{supp}\,{\mathcal {H}}^{\mathrm {ur}}(G(F_v))^{\leqslant \kappa }\), so \(\Xi '_v(\gamma ')\in \mathrm{GL}_m(\mathcal {O}_v)\Xi '_v(\mu (\varpi _v))\mathrm{GL}_m(\mathcal {O}_v)\) for some \(\mu \in X_*(A_v)\) with \(\Vert \mu \Vert \leqslant \kappa \). Then \(\Vert \Xi '_v(\mu )\Vert _{\mathrm{GL}_m}\leqslant B_{\Xi }\kappa \). [A priori this is true for \(B_{\Xi '_v}\) defined as in (8.2), but \(B_{\Xi '_v}=B_{\Xi }\) as \(\Xi '_v\) and \(\Xi \) are conjugate.] Let \(k_1,k_2\in \mathrm{GL}_m(\mathcal {O}_v)\) be such that \(\Xi '_v(\gamma ')=k_1\Xi '_v(\mu (\varpi _v))k_2\). Lemma 2.15 shows that every eigenvalue \(\lambda \) of \(\Xi '_v(\mu (\varpi _v)) k_2k_1\) [equivalently of \(\Xi '_v(\gamma ')\)] satisfies \(v(\lambda )\geqslant B_{\Xi }\kappa \). If \(\lambda \ne 1\), we must have \(v(1\lambda )\geqslant B_{\Xi }\kappa \). This shows that \(v(a_{i}(\gamma ))\geqslant B_{\Xi }i \kappa \) for any \(i\) such that \(a_i(\gamma )\ne 0\). Hence the claim is true.
At infinity, by the compactness of \(U_\infty \), there exists \(c_\Xi >0\) such that
whenever a conjugate of \(\gamma \in G_\infty \) belongs to \(U_\infty \).
Now suppose that \(v\) is a finite place such that \(v\notin S_1\) and \(v\not \mid {\mathfrak n}\). (This includes \(v\in S_0\).) Then a conjugate of \(\Xi (\gamma )\) lies in an open compact subgroup of \(\mathrm{GL}_m(F_v)\). Therefore the eigenvalues of \(\Xi (\gamma )\) are in \(\mathcal {O}_v\) and
Finally at \(v{\mathfrak n}\), we have \(\Xi (x^{1}\gamma x)1\!\in \! \ker (\mathrm{GL}_m(\mathcal {O}_v)\!\rightarrow \!\mathrm{GL}_m(\mathcal {O}_v/\varpi _v^{v({\mathfrak n})}))\). Therefore
Now assume that \(\mathbb {N}({\mathfrak n})\geqslant c_\Xi q_{S_1}^{B_{\Xi }m\kappa }\). We assert that \(a_i(\gamma )=0\) for all \(i\). Indeed, if \(a_i(\gamma )\ne 0\) for some \(i\) then the above inequalities imply that
which is clearly a contradiction. The proof of lemma is finished. \(\square \)
8.4 Bounding the number of rational conjugacy classes
We begin with a basic lemma, which is a quantitative version of the fact that \(F^r\) is discrete in \( \mathbb {A}_F^r\).
Lemma 8.5
Suppose that \(\{\delta _v \in \mathbb {R}_{>0}\}_{v\in \mathcal {V}_F}\) satisfies the following: \(\delta _v=1\) for all but finitely many \(v\) and \(\prod _v \delta _v<2^{S_\infty }\). Let \(\alpha =(\alpha _1,\ldots ,\alpha _r)\in \mathbb {A}_F^r\). Consider the following compact neighborhood of \(\alpha \)
Then \({\mathcal {B}}(\alpha ,\delta )\cap F^r\) has at most one element.
Proof
Suppose \(\beta =(\beta _i)_{i=1}^r,\gamma =(\gamma _i)_{i=1}^r\in {\mathcal {B}}(\alpha ,\delta )\cap F^r\). By triangular inequalities,
for each \(i\). Hence \(\prod _v \beta _{i,v}\gamma _{i,v}_v<1\). Since \(\beta _{i},\gamma _{i}\in F\), the product formula forces \(\beta _{i}=\gamma _{i}\). Therefore \(\beta =\gamma \). \(\square \)
The next lemma measures the difference between \(G(F)\)conjugacy and \(G(\mathbb {A}_F)\)conjugacy.
Lemma 8.6
Let \(X_G\) (resp. \({\fancyscript{X}}_G\)) be the set of semisimple \(G(F)\)(resp. \(G(\mathbb {A}_F)\))conjugacy classes in \(G(F)\). For any \([\gamma ]\in {\fancyscript{X}}_G\), there exist at most \((w_Gs_G)^{r_G+1}\) elements in \(X_G\) mapping to \([\gamma ]\) under the natural surjection \(X_G\rightarrow {\fancyscript{X}}_G\).
Proof
Let \([\gamma ]\in {\fancyscript{X}}_G\) be an element defined by a semisimple \(\gamma \in G(F)\). Denote by \(X_\gamma \) the preimage of \([\gamma ]\) in \(X_G\). There is a natural bijection
Since \(\ker ^1(F,I_\gamma )=\ker ^1(F,Z(\widehat{I}_\gamma ))\) by [62, §4.2], we have \(X_\gamma \leqslant \ker ^1(F,Z(\widehat{I}_\gamma ))\).
Let \(T\) be a maximal torus in \(I_\gamma \) defined over \(F\). Lemma 6.5 tells us that \(T\) becomes split over a finite extension \(E/F\) such that \([E:F]\leqslant w_Gs_G\). Then \(\mathrm{Gal}(\overline{F}/E)\) acts trivially on \(\widehat{T}\) and \(Z(\widehat{I}_\gamma )\). The group \(\ker ^1(E,Z(\widehat{I}_\gamma ))\) consists of continuous homomorphisms \(\mathrm{Gal}(\overline{F}/E)\rightarrow Z(\widehat{I}_\gamma )\) which are trivial on all local Galois groups. Hence \(\ker ^1(E,Z(\widehat{I}_\gamma ))\) is trivial. This and the inflation–restriction sequence show that \(\ker ^1(F,Z(\widehat{I}_\gamma ))\) is the subset of locally trivial elements in \(H^1(\Gamma _{E/F},Z(\widehat{I}_\gamma ))\), where we have written \(\Gamma _{E/F}\) for \(\mathrm{Gal}(E/F)\). In particular,
Let \(d:=\mathrm{Gal}(E/F)\) and denote by \([d]\) the \(d\)torsion subgroup. The long exact sequence arising from \(0\rightarrow Z(\widehat{I}_\gamma )[d] \rightarrow Z(\widehat{I}_\gamma ) \mathop {\rightarrow }\limits ^{d} d(Z(\widehat{I}_\gamma ))\rightarrow 0\) gives rise to an exact sequence
Let \({\varvec{\mu }}_{d}\) denote the order \(d\) cyclic subgroup of \(\mathbb {C}^\times \). Then \(Z(\widehat{I}_\gamma )[d]\hookrightarrow \widehat{T}[d]\simeq ({\varvec{\mu }}_{d})^{\dim T}\). Hence
\(\square \)
For the proposition below, we fix a finite subset \(S_0\subset \mathcal {V}_F^\infty \) containing \(\mathrm{Ram}(G)\). Also fix compact subsets \(U_{S_0}\subset G(F_{S_0})\) and \(U_\infty \subset G(F_\infty )\). As usual we will write \(S\) for \(S_0\coprod S_1\).
Proposition 8.7
Let \(\kappa \in \mathbb {Z}_{\geqslant 0}\). Let \(S_1\subset \mathcal {V}_F^\infty \backslash S_0\) be a finite subset such that \(G\) is unramified at all \(v\in S_1\). Set \(U_{S_1}:=\mathrm{supp}\,{\mathcal {H}}^{\mathrm {ur}}(G(F_{S_1}))^{\leqslant \kappa }\), \(U^{S,\infty }:=\prod _{v\notin S\cup S_\infty } K_v\) and \(U:=U_{S_0}U_{S_1} U^{S,\infty }U_\infty \). Define \({\fancyscript{Y}}_G\) to be the set of semisimple \(G(\mathbb {A}_F)\)conjugacy classes of \(\gamma \in G(F)\) which meet \(U\). Then there exist constants \(A_3,B_3>0\) such that for all \(S_1\) and \(\kappa \) as above,
[In other words, the implicit constant for \(O(\cdot )\) is independent of \(S_1\) and \(\kappa \).]
Remark 8.8
By combining the proposition with Lemma 8.4 we can deduce the following. Under the same assumption but with \(U:= K^{S,\infty }({\mathfrak n}) U_{S_0} U_{S_1} U_\infty \) we have
for some constants \(A,B,C>0\).
Proof
Our argument will be a quantitative refinement of the proof of [63, Prop 8.2].
Step I. When \(G^{\mathrm {der}}\) is simply connected.
Choose a smooth reductive integral model \(\mathfrak {G}\) over \(\mathcal {O}_F[\frac{1}{S_0}]\) for \(G\) and an embedding of algebraic groups \(\Xi :\mathfrak {G}\rightarrow \mathrm{GL}_{m}\) defined over \(\mathcal {O}_F[\frac{1}{S_0}]\) as in Proposition 8.1. Consider
where the latter map assigns the coefficients of the characteristic polynomial, and call the composite map \(\Xi '\). Set \(U':=\Xi '(U)\). Then \(U'\cap F^m<\infty \) since it is discrete and compact. We would like to estimate the cardinality.
Fix \(\{\delta _v\}\) such that \(\delta _v=1\) for all finite places \(v\) and \(\prod _v \delta _v<2^{S_\infty }\) so that the assumption of Lemma 8.5 is satisfied. We will write \({\mathcal {B}}_v(x,r)\) for the ball with center \(x\) and radius \(r\) in \(F_v\). Since \(\Xi \) is defined over \(\mathcal {O}_F[\frac{1}{S_0}]\), clearly \(\Xi (U^{S,\infty })\subset \mathrm{GL}_m(\widehat{\mathcal {O}}_F^{S,\infty })\). Thus
Set \(J^{S,\infty }:=\{0\}\subset (\mathbb {A}_F^{S,\infty })^m\). Similarly as above, \(\Xi '(U_{S_1})\subset (\mathcal {O}_{F,S_1})^m\). By the compactness of \(U_{S_0}\) and \(U_\infty \), there exist finite subsets \(J_{S_0}\subset F_{S_0}\) and \(J_{\infty }\subset F_\infty \) such that
Now we treat the places contained in \(S_1\). Let \(T\) be a maximal torus of \(G\) over \(\overline{F}\). Since the image of the composite map \(T_{\overline{F}}\hookrightarrow G_{\overline{F}} \mathop {\hookrightarrow }\limits ^{\Xi } (\mathrm{GL}_m)_{\overline{F}}\) is contained in a maximal torus of \(\mathrm{GL}_m\), we can choose \(g=(g_{ij})_{i,j=1}^m\in \mathrm{GL}_m(\overline{F})\) such that \(g\Xi (T_{\overline{F}})g^{1}\) sits in the diagonal maximal torus \(\mathbb {T}\) of \(\mathrm{GL}_d\). Fix the choice of \(T\) and \(g\) once and for all (independently of \(S_1\) and \(\kappa \)) until the end of Step I. Set \(v_{\min }(g):=\min _{i,j} v(g_{ij})\) and \(v_{\max }(g):=\max _{i,j} v(g_{ij})\). There exists \(B_6>0\) such that for any \(\mu \in X_*(T)\) with \(\Vert \mu \Vert \leqslant \kappa \), the element \(g\Xi (\mu )g^{1}\in X_*(\mathbb {T})\) satisfies \(\Vert g\Xi (\mu )g^{1}\Vert \leqslant B_6\kappa \). Let \(\gamma _{S_1}=(\gamma _v)_{v\in S_1}\in U_{S_1}\). Each \(\gamma _v\) has the form \(\gamma _v=k_1\mu (\varpi _{v})k_2\) for some \(\Vert \mu \Vert \leqslant \kappa \) and \(k_1,k_2\in G(\mathcal {O}_v)\). Since \(\Xi (G(\mathcal {O}_v))\subset \mathrm{GL}_m(\mathcal {O}_v)\), we see that \(\Xi (\gamma _v)\) is conjugate to \(\Xi (\mu (\varpi _{v}))k'\) in \(\mathrm{GL}_m(F_v)\) for some \(k'\in \mathrm{GL}_m(\mathcal {O}_v)\). Applying Lemma 2.15 to \((g\Xi (\mu (\varpi _{v}))g^{1})( gk'g^{1})\) with \(u=gk'g^{1}\) and noting that \(v_{\min }(u)\geqslant v_{\min }(g)+v_{\min }(g^{1})\), we conclude that each eigenvalue \(\lambda \) of \(\Xi (\gamma _v)\) satisfies
Therefore the coefficients of its characteristic polynomial lie in \(\varpi _v^{m(B_6\kappa +A_4)} \mathcal {O}_v\), where we have set \(A_4:=(v_{\min }(g)+v_{\min }(g^{1}))\geqslant 0\). To put things together, we see that
[A fortiori \(\Xi '(U_{S_1})\subset \prod _{v\in S_1} \prod _{i=1}^m \varpi _v^{i(B_6\kappa +A_4)}\mathcal {O}_v \) holds as well.] The right hand side is equal to the union of \(\prod _{v\in S_1} {\mathcal {B}}_v(\beta _v,1)\), as \(\{\beta _v\}_{v\in S_1}\) runs over \(J_{S_1}=\prod _{v\in S_1} J_v\), where \(J_v\) is a set of representatives for \((\varpi _v^{m(B_6\kappa +A_4)}\mathcal {O}_v/\mathcal {O}_v)^m\). Notice that \(J_{S_1}=q_{S_1}^{m^2(B_6 \kappa +A_4)}\). Finally, we see that
where \(J=J_{S_0}\times J_{S_1}\times J^{S,\infty }\times J_\infty \). Lemma 8.5 implies that
since \(J_{S_0}\cdot J_\infty \) is a constant independent of \(\kappa \) and \(S_1\).
For each \(\beta \in U'\cap F^m\), we claim that there are at most \(m!\) semisimple \(G(\overline{F})\)conjugacy classes in \(G(\overline{F})\) which map to \(\beta \) via \(G(\overline{F})\rightarrow \mathrm{GL}_m(\overline{F})\rightarrow \overline{F}^m\), the map analogous to (8.3). Let us verify the claim. Let \(T'\) and \(\mathbb {T}'\) be maximal tori in \(G\) and \(\mathrm{GL}_m\) over \(\overline{F}\), respectively, such that \(\Xi (T')\subset \mathbb {T}'\). Then the set of semisimple conjugacy classes in \(G(\overline{F})\) [resp. \(\mathrm{GL}_m(\overline{F})\)] is in a natural bijection with \(T'(\overline{F})/\Omega \) [resp. \(\mathbb {T}'(\overline{F})/\Omega _{\mathrm{GL}_m}\)]. The map \(\Xi _{T'}: T'\rightarrow \mathbb {T}'\) induces a map \(T'(\overline{F})/\Omega \rightarrow \mathbb {T}'(\overline{F})/\Omega _{\mathrm{GL}_m}\). Each fiber of the latter map has cardinality at most \(m!\), hence the claim follows.
Fix \(\beta \in U'\cap F^m\). We also fix \(\gamma \in G(F)\) such that \(\Xi '(\gamma )=\beta \). We assume the existence of such a \(\gamma \); otherwise our final bound will only improve. We would like to bound the number of \(G(\mathbb {A}_F)\)conjugacy classes in \(G(F)\) which meet \(U\) and \(G(\overline{F})\)conjugate to \(\gamma \). Let \(\Phi _\gamma \) denote the set of roots over \(\overline{F}\) for any choice of maximal torus \(T_\gamma \) in \(G\). Define \(V'(\gamma )\) to be the set of places \(v\) of \(F\) such that \(v\notin S\cup S_\infty \) and \(\alpha (\gamma )\ne 1\) and \(1\alpha (\gamma )_v<1\) for at least one \(\alpha \in \Phi _\gamma \). Since \(T_\gamma \) splits over an extension of \(F_v\) of degree at most \(w_G s_G\) (Lemma 6.5), \(1\alpha (\gamma )\) belongs to such an extension. Hence the inequality \(1\alpha (\gamma )_v<1\) implies that
Put \(V(\gamma ):=V'(\gamma )\cup S\cup S_\infty \). Clearly \(V(\gamma )<\infty \). Moreover we claim that \(V(\gamma )=O(1)\) (bounded independently of \(\gamma \)). Set
which is finite since \(U_{S_0} U_\infty \) is compact. Then
Thus \(V'(\gamma )=O(1)\) and also \(V(\gamma )=O(1)\).
We are ready to bound the number of \(G(\mathbb {A}_F)\)conjugacy classes in \( G(F)\) which meet \(U\) and are \(G(\overline{F})\)conjugate to \(\gamma \). For any such conjugacy class of \(\gamma '\in G(F)\), [63, Prop 7.1] shows that \(\gamma '\) is \(G(\mathcal {O}_v)\)conjugate to \(\gamma \) whenever \(v\notin V(\gamma )\). Hence the number of \(G(\mathbb {A}_F)\)conjugacy classes of such \(\gamma '\) is at most \(u_G^{V(\gamma )}\), where \(u_G\) is the constant of Lemma 8.11 below.
Putting all this together, we conclude that \({\fancyscript{Y}}_G=O(q_{S_1}^{m^2(B_7 \kappa +A_5)})\) as \(S_1\) and \(\kappa \) vary. The lemma is proved in this case.
Step II: general case.
Now we drop the assumption that \(G^{\mathrm {der}}\) is simply connected. By Lemma 8.3, choose a \(z\)extension
Our plan is to argue as on page 391 of [63] with a specific choice of \({\fancyscript{C}}_H\) and \({\fancyscript{C}}_Z\) below (denoted \(C_H\) and \(C_Z\) by Kottwitz). In order to explain this choice, we need some preparation. If \(v\notin S\cup S_\infty \), choose \(K_{H,v}\) to be a hyperspecial subgroup of \(H(F_v)\) such that \(\alpha (K_{H,v})=K_v\). (Such a \(K_{H,v}\) exists by the argument of [63, p. 386].) We can find compact sets \(U_{H,S_0}\subset H(F_{S_0})\) and \(U_{H,\infty }\) of \(H(F_\infty )\) such that \(\alpha (U_{H,S_0})=U_{S_0}\) and \(\alpha (U_{H,\infty })=U_\infty \). Moreover, in Lemma 8.9 below we prove the following: \(\square \)
Claim
There exists a constant \(\beta >0\) independent of \(\kappa \) and \(S_1\) with the following property: for any \(\kappa \in \mathbb {Z}_{\geqslant 0}\), we can choose an open compact subset \(U_{H,S_1}\subset \mathrm{supp}\,{\mathcal {H}}^{\mathrm {ur}}(H)^{\leqslant \beta \kappa }\) such that \(\alpha (U_{H,S_1})=U_{S_1}\).
Now choose \(U_{Z,S_1}\) to be the kernel of \(\alpha :U_{H,S_1} \rightarrow U_{S_1}\), which is compact and open in \(Z(F_{S_1})\). Then choose a compact set \(U_{Z}^{S_1}\) such that \(U_{Z,S_1}U_{Z}^{S_1}Z(F)=Z(\mathbb {A})^1\). (This is possible since \(Z(F)\backslash Z(\mathbb {A})^1\) is compact.^{Footnote 8}) Set
and set \(U^1_H:=U_H\cap H(\mathbb {A}_F)^1\), \(U^1_Z:=U_Z\cap Z(\mathbb {A}_F)^1\). Let \({\fancyscript{Y}}_H\) be defined as in the statement of Proposition 8.7 (with \(H\) and \(U^1_H\) replacing \(G\) and \(U\)). Then page 391 of [63] shows that the natural map \({\fancyscript{Y}}_H\rightarrow {\fancyscript{Y}}\) is a surjection, in particular \({\fancyscript{Y}}\leqslant {\fancyscript{Y}}_H\). Since \(H^{\mathrm {der}}\) is simply connected, the earlier proof implies that \({\fancyscript{Y}}_H=O(q_{S_1}^{B_7\beta \kappa +A_5})\) for some \(B_7,A_5>0\). (To be precise, apply the earlier proof after enlarging \(U_{H,S_1}\) to \(\mathrm{supp}\,{\mathcal {H}}^{\mathrm {ur}}(H)^{\leqslant \beta \kappa }\) in the definition of \(U_H\). Such a replacement only increases \({\fancyscript{Y}}_H\), so the bound on \({\fancyscript{Y}}_H\) remains valid.) The proposition follows.
We have postponed the proof of a claim in the proof of Step II above, which we justify now. Simple as the lemma may seem, we apologize for not having found a simple proof.
Lemma 8.9
Claim 8.4 above is true.
Proof
As the claim is concerned with places in \(S_1\), which (may vary but) are contained in the set of places where \(G\) is unramified (thus quasisplit), we may assume that \(H\) and \(G\) are quasisplit over \(F\) by replacing \(H\) and \(G\) with their quasisplit inner forms.
Choose a Borel subgroup \(B_H\) of \(H\), whose image \(B=\alpha (B_H)\) is a Borel subgroup of \(G\). The maximal torus \(T_H\subset B_H\) maps to a maximal torus \(T\subset B\) and there is a short exact sequence
The action of \(\mathrm{Gal}(\overline{F}/F)\) on \(X_*(T_H)\) factors through a finite quotient. Let \(\Sigma \) be the quotient of \(\mathrm{Gal}(\overline{F}/F)\) which acts faithfully on \(X_*(T_H)\). If \(v\notin S_0\) then \(G\) is unramified at \(v\), so the geometric Frobenius at \(v\) defines a welldefined conjugacy class, say \({\fancyscript{C}}_v\), in \(\Sigma \). Let \(A_{H,v}\) (resp. \(A_v\)) be the maximal split torus in \(T_H\) (resp. \(T\)) over \(F_v\). Then \(A_{H,v}\hookrightarrow T_H\) and \(A_v\hookrightarrow T\) induce \(X_*(A_{H,v})\simeq X_*(T_H)^{{\fancyscript{C}}_v}\) and \(X_*(A_{v})\simeq X_*(T)^{{\fancyscript{C}}_v}\). We claim that \(X_*(T_H)\rightarrow X_*(T)\) induces a surjective map \(X_*(A_{H,v})\rightarrow X_*(A_{v})\).
Indeed, we have an isomorphism \( X_*(A_{H,v})\simeq T_H(F_v)/T_H(\mathcal {O}_v)\) via \(\mu \mapsto \mu (\varpi _v)\) and similarly \( X_*(A_{v})\simeq T(F_v)/T(\mathcal {O}_v)\). Further, \(\alpha :T_H(F_v)\rightarrow T(F_v)\) is surjective since \(H^1(\mathrm{Gal}(\overline{F}_v/F_v),Z(\overline{F}_v))\) is trivial (as \(Z\) is an induced torus).
Denote by \([\Sigma ]\) the finite set of all conjugacy classes in \(\Sigma \). For \({\fancyscript{C}}\in [\Sigma ]\), choose \(\mathbb {Z}\)bases \({\mathcal {B}}_{H,{\fancyscript{C}}}\) and \({\mathcal {B}}_{{\fancyscript{C}}}\) for \(X_*(T_H)^{{\fancyscript{C}}}\) and \(X_*(T)^{{\fancyscript{C}}}\) respectively. [Note that the \(\mathbb {Z}\)bases \({\mathcal {B}}_H\) for \(X_*(T)\) and \({\mathcal {B}}\) for \(X_*(T_H)\) are fixed once and for all.] An argument as in the proof of Lemma 2.3 shows that there exist constants \(c({\mathcal {B}}_{{\fancyscript{C}}}),c({\mathcal {B}}_{H,{\fancyscript{C}}})>0\) such that for all \(x\in X_*(T_H)^{{\fancyscript{C}}}_{\mathbb {R}}\) and \(y\in X_*(T)^{{\fancyscript{C}}}_{\mathbb {R}}\),
Set \(m_{\fancyscript{C}}:= \max _y (\min _x x_{{\mathcal {B}}_{H,{\fancyscript{C}}}})\), where \(y\in X_*(T)^{{\fancyscript{C}}}\) varies subject to the condition \(y_{{\mathcal {B}}_{{\fancyscript{C}}}}\leqslant 1\) and \(x\in X_*(T_H)^{{\fancyscript{C}}}\) runs over the preimage of \(y\). (It was shown above that the preimage is nonempty.) Then by construction, for every \(y \in X_*(T)^{{\fancyscript{C}}}\), there exists an \(x\) in the preimage of \(y\) such that \(x _{{\mathcal {B}}_{H,{\fancyscript{C}}}} \leqslant m_{{\fancyscript{C}}} y_{{\mathcal {B}}_{{\fancyscript{C}}}}\).
Recall that \(U_{S_1}=\prod _{v\in S_1} U_v\) where \(U_v=\cup _{\mu } K_v \mu (\varpi _v) K_v\), the union being taken over \(\mu \in X_*(T)^{{\fancyscript{C}}_v}\) such that \(\Vert \mu \Vert _{{\mathcal {B}}}\leqslant \kappa \). We have seen that there exists \(\mu _H\in X_*(T_H)^{{\fancyscript{C}}_v}\) mapping to \(\mu \) and \(\mu _H_{{\mathcal {B}}_{H,{\fancyscript{C}}_v}}\leqslant m_{{\fancyscript{C}}_v} \mu _{{\mathcal {B}}_{{\fancyscript{C}}_v}}\). By (8.4),
Take \(\beta :=\max _{{\fancyscript{C}}\in [\Sigma ]} (m_{{\fancyscript{C}}} c({\mathcal {B}}_{H,{\fancyscript{C}}})^{1}c({\mathcal {B}}_{{\fancyscript{C}}}))\). Clearly \(\beta \) is independent of \(S_1\) and \(\kappa \). Notice that \(\Vert \mu _H\Vert _{{\mathcal {B}}_{H}}\leqslant \beta \Vert \mu \Vert _{{\mathcal {B}}}\leqslant \beta \kappa \).
For each \(\mu \in X_*(T)^{{\fancyscript{C}}_v}\) such that \(\Vert \mu \Vert _{{\mathcal {B}}}\leqslant \kappa \), we can choose a preimage \(\mu _H\) of \(\mu \) such that \(\Vert \mu _H\Vert _{{\mathcal {B}}_{H}}\leqslant \beta \kappa \). Take \(U_{H,v}\) to be the union of \(K_{H,v} \mu _H(\varpi _v) K_{H,v}\) for those \(\mu _H\)’s. By construction \(\alpha (U_{H,v})=U_v\). Hence \(U_{H,S_1}:=\prod _{v\in S_1}U_{H,v}\) is the desired open compact subset in the claim of Lemma 8.9. \(\square \)
Corollary 8.10
In the setting of Proposition 8.7, let \(Y_G\) be the set of all semisimple \(G(F)\)conjugacy (rather than \(G(\mathbb {A}_F)\)conjugacy) classes whose \(G(\mathbb {A}_F)\)conjugacy classes intersect \(U\). Then there exist constants \(A_6,B_8>0\) such that \(Y_G=O(q_{S_1}^{B_8\kappa +A_6})\) as \(S_1\) and \(\kappa \) vary.
Proof
Immediate from Lemma 8.6 and Proposition 8.7.
The following lemma was used in Step I of the proof of Proposition 8.7 and will be applied again to obtain Corollary 8.12 below.
Lemma 8.11
Assume that \(G^{\mathrm {der}}\) is simply connected. For each \(v\in \mathcal {V}_F\) and each semisimple \(\gamma \in G(F)\), let \(n_{v,\gamma }\) be the number of \(G(F_v)\)conjugacy classes in the stable conjugacy class of \(\gamma \) in \(G(F_v)\). Then there exists a constant \(u_G\geqslant 1\) (depending only on \(F\) and \(G\)) such that one has the uniform bound \(n_{v,\gamma }\leqslant u_G\) for all \(v\) and \(\gamma \).
Proof
Put \(\Gamma (v):=\mathrm{Gal}(\overline{F}_v/F_v)\). It is a standard fact that \(n_{v,\gamma }\) is the cardinality of \(\ker (H^1(F_v,I_\gamma )\rightarrow H^1(F_v,G))\). By [63], \(H^1(F_v,I_\gamma )\) is isomorphic to the dual of \(\pi _0(Z(\widehat{I}_\gamma )^{\Gamma (v)})\). Hence \(n_{v,\gamma }\leqslant \pi _0(Z(\widehat{I}_\gamma )^{\Gamma (v)}).\) It suffices to show that a uniform bound for \(\pi _0(Z(\widehat{I}_\gamma )^{\Gamma (v)})\) exists.
By Lemma 6.5, there exists a finite Galois extension \(E/F\) with \([E:F]\leqslant w_Gs_G\) such that \(I_{\gamma }\) splits over \(E\). Then \(\mathrm{Gal}(\overline{F}/F)\) acts on \(Z(\widehat{I}_\gamma )\) through \(\mathrm{Gal}(E/F)\). In particular \(\Gamma (v)\) acts on \(Z(\widehat{I}_\gamma )\) through a group of order \(\leqslant w_Gs_G\). Denote the latter group by \(\Gamma (v)'\).
Note that there is a uniform bound on the number of connected components \([Z(\widehat{I}_\gamma ):Z(\widehat{I}_\gamma )^0]\) as \(v\) and \(\gamma \) vary. Indeed it suffices to observe that there are only finitely many isomorphism classes of root data for \(I_\gamma \) over \(\overline{F}\) (hence also for \(\widehat{I}_\gamma \)). This is easily seen from the fact that the roots of \(I_\gamma \) (for a maximal torus containing \(\gamma \)) are exactly the roots \(\alpha \) of \(G\) such that \(\alpha (\gamma )=1\). Write \(Z(\widehat{I}_\gamma )^{0,\Gamma (v)}\) for the \(\Gamma (v)\)invariants in \(Z(\widehat{I}_\gamma )^0\). Since
it is enough to show that \(\pi _0(Z(\widehat{I}_\gamma )^{0,\Gamma (v)})\) is uniformly bounded.
Now consider the set of pairs
consisting of a \(\mathbb {C}\)torus \(\widehat{T}\) with an action by a finite group \(\Delta \). Two pairs \((\Delta ,\widehat{T})\) and \((\Delta ',\widehat{T}')\) are equivalent if there are isomorphisms \(\Delta \simeq \Delta '\) and \(\widehat{T}\simeq \widehat{T}'\) such that the group actions are compatible. Note that
and that \({\fancyscript{T}}\) depends only on \(G\) and \(F\). Clearly \(\pi _0(\widehat{T}^\Delta )\) depends only on the equivalence class of \((\Delta ,\widehat{T})\in {\fancyscript{T}}\). Hence the proof will be complete if \({\fancyscript{T}}\) consists of finitely many equivalence classes.
Clearly there are finitely many isomorphism classes for \(\Delta \) appearing in \({\fancyscript{T}}\). So we may fix \(\Delta \) and prove the finiteness of isomorphism classes of \(\mathbb {C}\)tori with \(\Delta \)action. By dualizing, it is enough to show that there are finitely many isomorphism classes of \(\mathbb {Z}[\Delta ]\)modules whose underlying \(\mathbb {Z}\)modules are free of rank at most \(r_G\). This is a result of [36, §79]. \(\square \)
Corollary 8.12
There exists a constant \(c>0\) (depending only on \(G\)) such that for every semisimple \(\gamma \in G(F)\), \(\pi _0(Z(\widehat{I}_\gamma )^{\Gamma })<c\). (We do not assume that \(G^{\mathrm {der}}\) is simply connected.)
Proof
Suppose that \(G^{\mathrm {der}}\) is simply connected. The proof of Lemma 8.11 shows that \((\mathrm{Gal}(E/F),Z(\widehat{I}_\gamma ))\in {\fancyscript{T}}\) in the notation there, thus there exists \(c>0\) such that \(\pi _0(Z(\widehat{I}_\gamma )^{\Gamma })<c\) for all semisimple \(\gamma \).
In general, let \(1\rightarrow Z\rightarrow H\rightarrow G\rightarrow 1\) be a \(z\)extension over \(F\) so that \(Z\) is a product of induced tori and \(H^{\mathrm {der}}\) is simply connected. Since \(H(F)\twoheadrightarrow G(F)\), we may choose a semisimple \(\gamma _H\) mapping to \(\gamma \). Let \(I_{\gamma _H}\) denote the centralizer of \(\gamma _H\) in \(H\). (Since \(H^{\mathrm {der}}\) is simply connected, \(I_{\gamma _H}\) is connected.) By the previous argument there exists \(c_H>0\) such that \(\pi _0(Z(\widehat{I}_{\gamma _H})^{\Gamma })<c_H\) for any semisimple \(\gamma _H\). The obvious short exact sequence \(1\rightarrow Z\rightarrow I_{\gamma _H}\rightarrow I_\gamma \rightarrow 1\) over \(F\) gives rise (Sect. 2.1) to a \(\Gamma \)equivariant short exact sequence
hence by [62, Cor 2.3],
On the other hand, the inclusions \(Z\rightarrow I_{\gamma _H}\rightarrow H\) induce \(\Gamma \)equivariant maps \(Z(\widehat{H})\rightarrow Z(\widehat{I}_{\gamma _H})\rightarrow \widehat{Z}\). The map \(Z(\widehat{H})\rightarrow Z(\widehat{I}_{\gamma _H})\) is constructed by [63, 4.2], whereas \(Z(\widehat{I}_{\gamma _H})\rightarrow \widehat{Z}\) and \(Z(\widehat{H})\rightarrow \widehat{Z}\) are given by Sect. 2.1. (The distinction comes from the fact that typically \(I_{\gamma _H}\rightarrow H\) is not normal.) The three maps are compatible in the obvious sense. By the functoriality of \(X_*(\cdot )^\Gamma \), there is a natural surjection
The left hand side is finite because it embeds into the finite group \(\pi _0(Z(\widehat{G})^{\Gamma })\), again by [62, Cor 2.3]. Going back to (8.5), we deduce
The proof is complete as the far right hand side is independent of \(\gamma \). \(\square \)
For a cuspidal group and conjugacy classes which are elliptic at infinity, a more precise bound can be obtained by a simpler argument, which would be worth recording here.
Lemma 8.13
Let \(G\) be a cuspidal \(F\)group. For any \(\gamma \in G(F)\) such that \(\gamma \in G(F_\infty )\) is elliptic,
Proof
Via restriction of scalars, we may assume that \(F=\mathbb {Q}\) without losing generality. Let us prove the lemma when \(A_G\) is trivial. By assumption there exists an \(\mathbb {R}\)anisotropic torus \(T\) in \(G(\mathbb {R})\) containing \(\gamma \). Thus \(T\simeq U(1)^{\mathrm{rk}(G)}\) and \(T\hookrightarrow I_\gamma \) over \(\mathbb {R}\). The former tells us that \(\widehat{T}^{\Gamma (\infty )}\simeq \{\pm 1\}^{\mathrm{rk}(G)}\) and the latter gives rise to \( Z(\widehat{I}_\gamma )^{\Gamma (\infty )}\hookrightarrow \widehat{T}^{\Gamma (\infty )}\) [63, §4]. Hence the assertion follows from
In general when \(A_G\) is not trivial, consider the exact sequence of \(\mathbb {Q}\)groups \(1\rightarrow A_G \rightarrow I_\gamma \rightarrow I_\gamma /A_G\rightarrow 1\), whose dual is the \(\Gamma \)equivariant exact sequence of \(\mathbb {C}\)groups
Thanks to [62, Cor 2.3], we obtain the following exact sequence:
Hence \(\pi _0(Z(\widehat{I}_\gamma )^{\Gamma })\leqslant \pi _0(Z(\widehat{I_\gamma /A_G})^\Gamma )\), and the latter is at most \( 2^{\mathrm{rk}(G/A_G)}\) by the preceding argument. \(\square \)
9 Automorphic Plancherel density theorem with error bounds
The local components of automorphic representations at a fixed finite set of primes tend to be equidistributed according to the Plancherel measure on the unitary dual, namely the error tends to zero in a family of automorphic representations (cf. Corollary 9.22 below). The main result of this section (Theorems 9.16, 9.19) is a bound on this error in terms of the primes in the fixed set as well as the varying parameter (level or weight) in the family. A crucial assumption for us is that the group \(G\) is cuspidal (Definition 9.7), which allows the use of a simpler version of the trace formula. For the proof we interpret the problem as bounding certain expressions on the geometric side of the trace formula and apply various technical results from previous sections. One main application is a proof of the Sato–Tate conjecture for families formulated in Sect. 5.4 under suitable conditions on the parameters involved. In turn the result will be applied to the question on lowlying zeros in later sections.
9.1 Sauvageot’s density theorem on unitary dual
We reproduce a summary of Sauvageot’s result [91] from [99, §2.3] as it can be used to effectively prescribe local conditions in our problem. The reader may refer to either source for more detail.
Let \(G\) be a connected reductive group over a number field \(F\). Use \(v\) to denote a finite place of \(F\). When \(M\) is a Levi subgroup of \(G\) over \(F_v\), write \(\Psi _u(M(F_v))\) (resp. \(\Psi (M(F_v))\)) for the real (resp. complex) torus whose points parametrize unitary (complexvalued) characters of \(M(F_v)\) trivial on any compact subgroup of \(M(F_v)\). The normalized parabolic induction of an admissible representation \(\sigma \) of \(M(F_v)\) is denoted \(\mathrm{n{\text {}}ind}^G_M(\sigma )\).
Denote by \({\fancyscript{B}}_c(G(F_v)^{\wedge })\) the space of bounded \(\widehat{\mu }^{\mathrm {pl}}_v\)measurable functions \(\widehat{f}_v\) on \(G(F_v)^{\wedge }\) whose support has compact image in the Bernstein center, which is the set of \(\mathbb {C}\)points of an (infinite) product of varieties. A measure on \(G(F_v)^{\wedge }\) will be thought of as a linear functional on the space \({\fancyscript{F}}(G(F_v)^{\wedge })\) consisting of \(\widehat{f}_v\in {\fancyscript{B}}_c(G(F_v)^{\wedge })\) such that for every \(F_v\)rational Levi subgroup \(M\) of \(G\) and every discrete series \(\sigma \) of \(M(F_v)\),
is a function whose points of discontinuity are contained in a measure zero set. (Here \(\mathrm{n{\text {}}ind}\) denotes the normalized parabolic induction.) Now for any finite set \(S\) of finite places of \(F\), one can easily extend the above definition to \({\fancyscript{F}}(G(F_S)^{\wedge })\) so that \(\widehat{f}_S(\pi _S)\in \mathbb {C}\) makes sense for \(\widehat{f}_S\in {\fancyscript{F}}(G(F_S)^{\wedge })\) and \(\pi _S\in G(F_S)^{\wedge }\). We have a map
as follows from Proposition 9.6 below. HarishChandra’s Plancherel theorem states that
Our notational convention is that \(\widehat{\phi }_S\) often signifies an element in the image of the above map whereas \(\widehat{f}_S\) stands for a general element of \({\fancyscript{F}}(G(F_S)^{\wedge })\). Sauvageot’s theorem allows us to approximate any \(\widehat{f}_S\in {\fancyscript{F}}(G(F_S)^{\wedge })\) with elements of \(C^\infty _c(G(F_S))\).
Proposition 9.1
[91, Thm 7.3] Let \(\widehat{f}_S\in {\fancyscript{F}}(G(F_S)^{\wedge })\). For any \(\epsilon >0\), there exist \(\phi _S,\psi _S\in C^\infty _c(G(F_S))\) such that
Conversely, any \(\widehat{f}_S\in {\fancyscript{B}}_c(\widehat{G(F_S)})\) with the above property belongs to \({\fancyscript{F}}(G(F_S)^{\wedge })\).
Remark 9.2
It is crucial that \(\widehat{f}_S\in {\fancyscript{F}}(G(F_S)^{\wedge })\) has the set of discontinuity in a measure zero set. Otherwise we could take \(\widehat{f}_S\) to be the characteristic function on the set of points of \(G(F_S)^{\wedge }\) which arise as the \(S\)components of some \(\pi \in {\mathcal {AR}}_{\mathrm{disc},\chi }(G)\) with nonzero Lie algebra cohomology. Note that the latter function typically lies outside \({\fancyscript{F}}(G(F_S)^{\wedge })\). The conclusions of Theorems 9.26, 9.27 and Corollary 9.22 are false in general if such an \(\widehat{f}_S\) is placed at \(S_0\). Namely in that case \(\widehat{\mu }_{{\mathcal {F}}_k,S_1}(\widehat{\phi }_{S_1})\) is often far from zero but \(\widehat{\mu }^{\mathrm {pl}}_S(\widehat{\phi }_S)\) always vanishes.
From here until the end of this subsection let us suppose that \(G\) is unramified at \(S\). It will be convenient to introduce \({\mathcal {F}}(G(F_S)^{\wedge ,\mathrm {ur}})\) and its subspace \({\mathcal {F}}(G(F_S)^{\wedge ,\mathrm {ur},\text {temp}})\) in order to state the Sato–Tate theorem in Sect. 9.7. The former (resp. the latter) consists of \(\widehat{f}_S\in {\mathcal {F}}(G(F_S)^{\wedge })\) such that the support of \(\widehat{f}_S\) is contained in \(G(F_S)^{\wedge ,\mathrm {ur}}\) [resp. \(G(F_S)^{\wedge ,\mathrm {ur},\text {temp}}\)]. Denote by \({\mathcal {F}}(\widehat{T}_{c,\theta }/\Omega _{c,\theta })\) the space of bounded \(\widehat{\mu }^{\mathrm {ST}}_\theta \)measurable functions on \(\widehat{T}_{c,\theta }/\Omega _{c,\theta }\) whose points of discontinuity are contained in a \(\widehat{\mu }^{\mathrm {ST}}_\theta \)measure zero set. Define \({\mathcal {F}}(\prod _{v\in S}\widehat{T}_{c,\theta _v}/\Omega _{c,\theta _v})\) in the obvious analogous way. By using the topological Satake isomorphism for tempered spectrum [cf. (5.2)]
and extending by zero outside the tempered spectrum, one obtains
Although the first two \({\mathcal {F}}(\cdot )\) above are defined with respect to different measures \(\prod _{v\in S}\widehat{\mu }^{\mathrm {ST}}_{\theta _v}\) and \(\widehat{\mu }^{\mathrm {pl}}_S\), the isomorphism is justified by the fact that the ratio of the two measures is uniformly bounded above and below by positive constants (depending on \(q_S\)) in view of Proposition 3.3 and Lemma 5.2. Note that the space of continuous functions on \(\prod _{v\in S}\widehat{T}_{c,\theta _v}/\Omega _{c,\theta _v}\) [resp. on \(G(F_S)^{\wedge ,\mathrm {ur},\text {temp}}\)] is contained in the first (resp. second) term of (9.1), and the two subspaces correspond under the isomorphism.
Corollary 9.3
Let \(\widehat{f}_S\in {\mathcal {F}}(G(F_S)^{\wedge ,\mathrm {ur}})\). For any \(\epsilon >0\), there exist \(\phi _S,\psi _S\in {\mathcal {H}}^{\mathrm {ur}}(G(F_S))\) such that (i) \(\widehat{\mu }^{\mathrm {pl}}_S(\widehat{\psi }_S)\leqslant \epsilon \) and (ii) \(\forall \pi _S\in G(F_S)^{\wedge ,\mathrm {ur}}\), \(\widehat{f}_S(\pi _S)  \widehat{\phi }_S(\pi _S)\leqslant \widehat{\psi }_S(\pi _S)\).
Proof
Let \(\phi _S,\psi _S\in C^\infty _c(G(F_S))\) be the functions associated to \(\widehat{f}_S\) as in Proposition 9.1. Then it is enough to replace \(\phi _S\) and \(\psi _S\) with their convolution products with the characteristic function on \(\prod _{v\in S} K_v\).
The following proposition will be used later in Sect. 9.7. For each \(v\in \mathcal {V}_F(\theta )\), the image of \(\widehat{f}\) in \({\mathcal {F}}(G(F_v)^{\wedge ,\mathrm {ur}})\) via (9.1) will be denoted \(\widehat{f}_v\).
Proposition 9.4
Let \(\widehat{f}\in {\mathcal {F}}(\widehat{T}_{c,\theta }/\Omega _{c,\theta })\) and \(\epsilon >0\). There exists an integer \(\kappa \geqslant 1\) and for all places \(v\in \mathcal {V}_F(\theta )\), there are bounded functions \(\phi _{v},\psi _{v}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{v}))^{\leqslant \kappa }\) such that \(\widehat{\mu }^{\mathrm {pl}}_v(\widehat{\psi }_v)\leqslant \epsilon \) and \(\widehat{f_v}(\pi )  \widehat{\phi _v}(\pi ) \leqslant \widehat{\psi _v}(\pi )\) for all \(\pi \in G(F_v)^{\wedge ,\mathrm {ur}}\).
Proof
This is no more than Corollary 9.3 if we only required \(\phi _{v},\psi _{v}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{v}))\) without the superscript \(\leqslant \kappa \). So we may disregard finitely many \(v\) by considering the subset \(\mathcal {V}_F(\theta )^{\geqslant Q}\) of \(\mathcal {V}_F(\theta )\) consisting of \(v\) such that \(q_v\geqslant Q\) for some \(Q>0\). In view of Proposition 5.3, we may choose \(Q\in \mathbb {Z}_{>0}\) that
Fix any \(w\in \mathcal {V}_F(\theta )^{\geqslant Q}\). Corollary 9.3 allows us to find \(\phi _w,\psi '_w\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{w}))\) such that
Let \(\kappa _0\in \mathbb {Z}_{\geqslant 0}\) be such that \(\phi _w,\psi '_w\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{w}))^{\leqslant \kappa _0}\). Now recall that for every \(v\in \mathcal {V}_F(\theta )\) there is a canonical isomorphism [cf. (2.2), Lemma 3.2] between \({\mathcal {H}}^{\mathrm {ur}}(G(F_v))\) and the space of regular functions in the complex variety \(\widehat{T}_{\theta }/\Omega _{\theta }\). Using the latter as a bridge, we may transport \(\phi _w,\psi '_w\) to \(\phi _v,\psi '_v\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{v}))\) for every \(v\in \mathcal {V}_F(\theta )\). Clearly \(\phi _v,\psi '_v\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{v}))^{\leqslant \kappa _0}\) from the definition of Sect. 2.3. Moreover (9.2) and (9.3) imply that for all \(v\in \mathcal {V}_F(\theta )^{\geqslant Q}\),
[Observe that \(\widehat{\mu }^{\mathrm {pl}}_v(\widehat{\psi }'_v)\leqslant 2 \widehat{\mu }^{\mathrm {ST}}_\theta (\widehat{\psi }'_v)=2\widehat{\mu }^{\mathrm {ST}}_\theta (\widehat{\psi }'_w)\leqslant 4\widehat{\mu }^{\mathrm {pl}}_w(\widehat{\psi }'_w)\leqslant \epsilon /2\) to justify the first inequality.]
To achieve the latter inequality for nontempered \(\pi _v\in G(F_v)^{\wedge ,\mathrm {ur}}\), we would like to perturb \(\psi '_v\) in a way independent of \(v\) while not sacrificing the former inequality. Since \(\widehat{f}_v(\pi _v)=0\) for such \(\pi _v\), what we need to establish is that \(\widehat{\phi }_v(\pi _v)\leqslant \widehat{\psi }_v(\pi _v)\) for all nontempered \(\pi _v\in G(F_v)^{\wedge ,\mathrm {ur}}\). To this end, we use the fact that there is a compact subset \(\mathcal {K}\) of \(\widehat{T}_{\theta }/\Omega _{\theta }\) such that \(G(F_v)^{\wedge ,\mathrm {ur}}\) is contained in \(\mathcal {K}\) for every \(v\in \mathcal {V}_F(\theta )\) (cf. [11, Thm XI.3.3]). By using the Weierstrass approximation theorem, we find \(\psi ''_w\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{w}))\) such that
Choose \(\kappa \geqslant \kappa _0\) such that \(\psi ''_w\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{w}))^{\leqslant \kappa }\) and put \(\psi _w:=\psi '_w+\psi ''_w\) so that \(\widehat{\mu }^{\mathrm {pl}}_w(\widehat{\psi }_w)\leqslant \epsilon /4\) and \(\psi _w\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{w}))^{\leqslant \kappa }\). For each \(v\in \mathcal {V}_F(\theta )^{\geqslant Q}\), let \(\psi _v\) denote the transport of \(\psi _w\) just as \(\psi '_v\) was the transport of \(\psi '_w\) in the preceding paragraph. Then \(\widehat{\mu }^{\mathrm {pl}}_v(\widehat{\psi }_v)\leqslant \epsilon \) and \(\psi _v\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{v}))^{\leqslant \kappa }\) as before. Moreover
and for \(\pi _v\in G(F_v)^{\wedge ,\mathrm {ur}}\backslash G(F_v)^{\wedge ,\mathrm {ur},\text {temp}}\),
the last inequality following from \(\widehat{\psi }_v=\widehat{\psi }'_v+\widehat{\psi }''_v\). \(\square \)
Remark 9.5
A more direct approach to (9.3) that wouldn’t involve Corollary 9.3 would be to use Weierstrass approximation to find polynomials \(\phi \) and \(\psi \) on \(\widehat{T}_{c,\theta }/\Omega _{c,\theta }\) of degree \(\leqslant \kappa \) such that \(\widehat{f}  \widehat{\phi }  \leqslant \widehat{\psi }\) and then the isomorphism (9.1) to transport \(\phi \) and \(\psi \) at the place \(v\).
We note [91, Lemme 3.5] that for any \(\phi _v\in C^\infty _c(G(F_v))\) there exists a \(\phi '_v\in C^\infty _c(G(F_v))\) such that \(\widehat{\phi }_v(\pi _v)\leqslant \widehat{\phi '}_v(\pi _v)\) for all \(\pi _v\in G(F_v)^{\wedge }\). This statement is elementary, e.g. it follows from the Dixmier–Malliavin decomposition theorem. In fact we have the following stronger result due to Bernstein [8].
Proposition 9.6
(Uniform admissibility theorem) For any \(\phi _v \in C^\infty _c(G(F_v))\) there exists \(C>0\) such that \(\mathrm{tr}\,\pi (\phi _v)\leqslant C\) for all \(\pi \in G(F_v)^{\wedge }\).
9.2 Automorphic representations and a counting measure
Now consider a string of complex numbers
such that \(a_{{\mathcal {F}}}(\pi )=0\) for all but finitely many \(\pi \). We think of \({\mathcal {F}}\) as a multiset by viewing \(a_{{\mathcal {F}}}(\pi )\) as multiplicity, or more appropriately as a density function with finite support in \({\mathcal {F}}\) as \(a_{{\mathcal {F}}}(\pi )\) is allowed to be in \(\mathbb {C}\). There are obvious meanings when we write \(\pi \in {\mathcal {F}}\) and \({\mathcal {F}}\) (we could have written \(\pi \in \mathrm {supp}\, {\mathcal {F}}\) for the former):
In order to explain our working hypothesis, we recall a definition.
Definition 9.7
Let \(H\) be a connected reductive group over \(\mathbb {Q}\). The maximal \(\mathbb {Q}\)split torus in \(Z(H)\) is denoted \(A_H\). We say \(H\) is cuspidal if \((H/A_H)\times _\mathbb {Q}\mathbb {R}\) contains a maximal \(\mathbb {R}\)anisotropic torus.
If \(H\) is cuspidal then \(H(\mathbb {R})\) has discrete series representations. (We remind the reader that discrete series always mean “relative discrete series” for us, i.e. those whose matrix coefficients are squareintegrable modulo center.) The converse is true when \(H\) is semisimple but not in general. Throughout this section the following will be in effect:
Hypothesis 9.8
\({\mathrm {Res}}_{F/\mathbb {Q}} G\) is a cuspidal group.
Let \(S=S_0\coprod S_1\subset \mathcal {V}_F^\infty \) be a nonempty finite subset and \(\widehat{f}_{S_0}\in {\fancyscript{F}}(G(F_{S_0})^{\wedge })\). (It is allowed that either \(S_0\) or \(S_1\) is empty.) Let

(level) \(U^{S,\infty }\) be an open compact subset of \(G(\mathbb {A}^{S,\infty })\),

(weight) \(\xi =\otimes _{v\infty }\xi _v\) be an irreducible algebraic representation of
$$\begin{aligned} G_\infty \times _\mathbb {R}\mathbb {C}=({\mathrm {Res}}_{F/\mathbb {Q}} G)\times _\mathbb {Q}\mathbb {C}=\prod _{v\infty } G\times _{F,v} \mathbb {C}. \end{aligned}$$
Denote by \(\chi :A_{G,\infty }\rightarrow \mathbb {C}^\times \) the restriction of the central character for \(\xi ^\vee \). Define
Note that \(\widehat{{\mathbf {1}}}_{K_{S_1}}(\pi _{S_1})\) equals 1 if \(\pi _{S_1}\) is unramified and 0 otherwise, and that \(\chi _{\mathrm{EP}}( \pi _\infty \otimes \xi )=0\) unless \(\pi _\infty \) has the same infinitesimal character as \(\xi ^\vee \). The set of \(\pi \) such that \(a_{{\mathcal {F}}}(\pi )\ne 0\) is finite by HarishChandra’s finiteness theorem. Let us define measures \(\widehat{\mu }_{{\mathcal {F}},S_1}\) and \(\widehat{\mu }^\natural _{{\mathcal {F}},S_1}\) associated with \({\mathcal {F}}\) on the unramified unitary dual \(G(F_{S_1})^{\wedge ,\mathrm {ur}}\), motivated by the trace formula. Put \(\tau '(G):=\overline{\mu }^{\mathrm{can},\mathrm{EP}}(G(F)A_{G,\infty }\backslash G(\mathbb {A}_F))\). For any function \(\widehat{f}_{S_1}\) on \(G(F_{S_1})^{\wedge ,\mathrm {ur}}\) which is continuous outside a measure zero set, define
The sum is finite because \(a_{{\mathcal {F}}}\) is supported on finitely many \(\pi \). Now the key point is that the right hand side can be identified with the spectral side of Arthur’s trace formula with the Euler–Poincaré function at infinity as in Sect. 6.5 when \(\widehat{f}_{S_1}=\widehat{\phi }_{S_1}\) for some \(\phi _{S_1}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{S_1}))\) ([3, pp. 267–268], cf. proof of [99, Prop 4.1]). So to speak, if we write \(\phi ^{\infty }=\phi _{S_0}\phi _{S_1}\phi ^{S,\infty }\),
where \(I_{\mathrm{spec}}\) (resp. \(I_{\mathrm{geom}}\)) denotes the spectral (resp. geometric) side Arthur’s the invariant trace formula with respect to the measure \(\mu ^{\mathrm{can},\mathrm{EP}}\). Finally if \(\widehat{f}_{S_0}\) has the property that \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{f}_{S_0})\ne 0\) then put
Remark 9.9
The measure \( \widehat{\mu }^\natural _{{\mathcal {F}},S_1}\) is asymptotically the same as the counting measure
associated with the \(S_1\)components of \({\mathcal {F}}\) (assuming \({\mathcal {F}}\ne 0\)). More precisely if \(\{{\mathcal {F}}_k\}_{\geqslant 1}\) is a family of Sect. 9.3 below, then \(\widehat{\mu }^{\mathrm {count}}_{{\mathcal {F}}_k,S_1}/ \widehat{\mu }^{\natural }_{{\mathcal {F}}_k,S_1}\) is a constant tending to 1 as \(k\rightarrow \infty \) by Corollary 9.25.
Example 9.10
Let \(\pi \in {\mathcal {AR}}_{\mathrm{disc},\chi }(G)\). Suppose that the highest weight of \(\xi \) is regular and that \(S_0=\emptyset \). Then \(\pi \) belongs to \({\mathcal {F}}\) if and only if the following three conditions hold: \((\pi ^{S,\infty })^{U^{S,\infty }}\ne 0\), \(\pi \) is unramified at \(S\), and \(\pi _\infty \in \Pi _{\mathrm{disc}}(\xi ^\vee )\). When \(\pi _\infty \in \Pi _{\mathrm{disc}}(\xi ^\vee )\), (9.4) simplifies as
Example 9.11
Let \(\widehat{f}_{S_0}\) be a characteristic function on some relatively compact \(\widehat{\mu }^{\mathrm {pl}}_S\)measurable subset \(\widehat{U}_{S_0}\subset G(F_{S_0})^{\wedge }\). Assume that \(S_0\) is large enough such that \(G\) and all members of \({\mathcal {F}}\) are unramified outside \(S_0\). Take \(U^{S_0,\infty }\) to be the product of \(K_v\) over all finite places \(v\notin S_0\). Then for each \(\pi \in {\mathcal {AR}}_{\mathrm{disc},\chi }(G)\),
if \(\pi ^{S_0,\infty }\) is unramified, \(\pi _{S_0}\in \widehat{U}_{S_0}\) (in which case \(a_{{\mathcal {F}}}(\pi )\ne 0\) if moreover \(\chi _{\mathrm{EP}}(\pi _\infty \otimes \xi )\ne 0\); otherwise \(a_{{\mathcal {F}}}(\pi )=0\)). If the highest weight of \(\xi \) is regular, \(\chi _{\mathrm{EP}}(\pi _\infty \otimes \xi )\ne 0\) exactly when \(\pi _\infty \in \Pi _{\mathrm{disc}}(\xi ^\vee )\), in which case (9.7) simplifies as
Compare this with Example 9.10. (The analogy in the case of modular forms is that \(\pi \) as newforms are counted in the current example whereas oldforms are also counted in Example 9.10.) Finally we observe that since the highest weight of \(\xi \) is regular and \(\pi _\infty \in \Pi _{\mathrm{disc}}(\xi ^\vee )\), the discrete automorphic representation \(\pi \) is automatically cuspidal [107, Thm. 4.3]. In the present example the discrete multiplicity coincides with the cuspidal multiplicity.
Remark 9.12
As the last example shows, the main reason to include \(S_0\) is to prescribe local conditions at finitely many places (namely at \(S_0\)) on automorphic families. For instance one can take \(\widehat{f}_{S_0}=\widehat{\phi }_{S_0}\) where \(\phi _{S_0}\) is a pseudocoefficient of a supercuspidal representation (or a truncation thereof if the center of \(G\) is not anisotropic over \(F_{S_0}\)). Then it allows us to consider a family of \(\pi \) whose \(S_0\)components are a particular supercuspidal representation (or an unramified character twist thereof). By using various \(\widehat{f}_{S_0}\) [which are in general not equal to \(\widehat{\phi }_{S_0}\) for any \(\phi _{S_0}\in C^\infty _c(G(F_{S_0}))\)] one obtains great flexibility in prescribing a local condition as well as imposing weighting factors for a family.
9.3 Families of automorphic representations
Continuing from the previous subsection (in particular keeping Hypothesis 9.8) let us introduce two kinds of families \(\{{\mathcal {F}}_k\}_{k\geqslant 1}\) which will be studied later on. We will measure the size of \(\xi \) in the following way. Let \(T_\infty \) be a maximal torus of \(G_\infty \) over \(\mathbb {R}\). For a \(B\)dominant \(\lambda \in X^*(T_\infty )\), set \(m(\lambda ):=\min _{\alpha \in \Phi ^+} \langle \lambda ,\alpha \rangle \). For \(\xi \) with \(B\)dominant highest weight \(\lambda _{\xi }\), define \(m(\xi ):=m(\lambda _\xi )\).
Let \(\phi _{S_0}\in C^\infty _c(G(F_{S_0}))\). [More generally we will sometimes prescribe a local condition at \(S_0\) by \(\widehat{f}_{S_0}\in {\fancyscript{F}}(G(F_{S_0})^{\wedge })\) rather than \(\phi _{S_0}\).] In the remainder of Sect. 9 we mostly focus on families in the level or weight aspect, respectively described as the following:
Example 9.13
( Level aspect : varying level, fixed weight) Let \({\mathfrak n}_k\subset \mathcal {O}_F\) be a nonzero ideal prime to \(S\) for each \(k\geqslant 1\) such that \(\mathbb {N}({\mathfrak n}_k)=[\mathcal {O}_F:{\mathfrak n}_k]\) tends to \(\infty \) as \(k\rightarrow \infty \). Take
Then \({\mathcal {F}}_k\rightarrow \infty \) as \(k\rightarrow \infty \).
Example 9.14
( Weight aspect : fixed level, varying weight) For our study of weight aspect it is always supposed that \(Z(G)=1\) so that \(A_{G,\infty }=1\) and \(\chi =1\) in order to eliminate the technical problem with central character when weight varies.^{Footnote 9} Let \(\{\xi _k\}_{k\geqslant 1}\) be a sequence of irreducible algebraic representations of \(G_\infty \times _\mathbb {R}\mathbb {C}\) such that \(m(\xi _k)\rightarrow \infty \) as \(k\rightarrow \infty \). Take
Then \({\mathcal {F}}_k\rightarrow \infty \) as \(k\rightarrow \infty \).
Remark 9.15
Sarnak proposed a definition of families of automorphic representations (or automorphic \(L\)functions) in [87]. The above two examples fit in his definition.
9.4 Level aspect
We are in the setting of Example 9.13. Recall that \({\mathrm {Res}}_{F/\mathbb {Q}} G\) is assumed to be cuspidal. Fix \(\Xi :G\hookrightarrow \mathrm{GL}_m\) as in Proposition 8.1 and let \(B_\Xi \) and \(c_\Xi \) be as in (8.2) and Lemma 8.4. Write \({\fancyscript{L}}_c(M_0)\) for the set of \(F\)rational cuspidal Levi subgroups of \(G\) containing the minimal Levi \(M_0\).
Theorem 9.16
Fix \(\phi _{S_0}\in C^\infty _c(G(F_{S_0}))\) and \(\xi \). Let \(S_1\subset \mathcal {V}_F^\infty \) be a subset where \(G\) is unramified. Let \(\phi _{S_1}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{S_1}))^{\leqslant \kappa }\) be such that \(\phi _{S_1}\leqslant 1\) on \(G(F_{S_1})\). If \({\fancyscript{L}}_c(M_0)=\{G\}\) (in particular if \(G\) is abelian) then \(\widehat{\mu }_{{\mathcal {F}}_k,S_1}(\widehat{\phi }_{S_1})=\widehat{\mu }^{\mathrm {pl}}_S(\widehat{\phi }_S)\). Otherwise there exist constants \(A_{\mathrm {lv}}, B_{\mathrm {lv}}>0\) and \(C_{\mathrm {lv}}\geqslant 1\) such that
as \({\mathfrak n},\kappa \in \mathbb {Z}_{\geqslant 1}\), \(S_1\) and \(\phi _{S_1}\) vary subject to the following conditions:
 (i):

\(\mathbb {N}({\mathfrak n}) \geqslant c_\Xi q_{S_1}^{B_\Xi m \kappa }\),
 (ii):

no prime divisors of \({\mathfrak n}\) are contained in \(S_1\).
[The implicit constant in \(O(\cdot )\) is independent of \({\mathfrak n}\), \(\kappa \), \(S_1\) and \(\phi _{S_1}\).]
Remark 9.17
When \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\phi }_{S_0})\ne 0\) (9.8) is equivalent to
.
Remark 9.18
One can choose \(A_{\mathrm {lv}}, B_{\mathrm {lv}}, C_{\mathrm {lv}}\) to be explicit integers. See the proof below. For instance \(C_{\mathrm {lv}}\geqslant n_G\) for \(n_G\) defined in Sect. 1.8.
Proof
Put \(\phi ^{S,\infty }:={\mathbf {1}}_{K^{S,\infty }({\mathfrak n})}\). The right hand side of (9.6) is expanded as in [3, Thm 6.1] as shown by Arthur. Arguing as at the start of the proof of [99, Thm 4.4], we obtain from Lemma 8.4 in view of the imposed lower bound on \(N({\mathfrak n})\) that
where the sum runs over proper cuspidal Levi subgroups of \(G\) containing a fixed minimal \(F\)rational Levi subgroup (see [45, p. 539] for the reason why only cuspidal Levi subgroups contribute) and \(a_M\in \mathbb {C}\) are explicit constants depending only on \(M\) and \(G\). A further explanation of (9.9) needs to be given. Since only semisimple conjugacy classes contribute to Arthur’s trace formula for each \(M\), Lemma 8.4 tells us that any contribution from nonidentity elements vanishes. Note that \(\widehat{\mu }^{\mathrm {pl}}_S(\widehat{\phi }_S)\) comes from the \(M=G\) term on the right hand side.
The first assertion of the theorem follows immediately from (9.9). Henceforth we may assume that \({\fancyscript{L}}_c(M_0)\backslash \{G\}\ne \emptyset \).
Clearly \(\phi _{S_0,M}(1)\) and \(\Phi ^G_M(1,\xi )/\dim \xi \) are constants. It was shown in Lemma 2.14 that \(\phi _{S_1,M}(1)=O(q_{S_1}^{d_G+r_G+b_G\kappa })\) for \(b_G> 0\) in that lemma. We take
We will be done if it is checked that \(\phi ^{S,\infty }_{M}(1)= O(\mathbb {N}({\mathfrak n})^{C_{\mathrm {lv}}})\) for some \(C_{\mathrm {lv}}\geqslant 1\). Let \(P=MN\) be a parabolic subgroup with Levi decomposition where \(M\) is as above. Then
The last equality uses the standard fact about the filtration that \({{\mathrm{vol}}}(N(F_v)_{x,v({\mathfrak n})}) =\varpi _v^{v({\mathfrak n})\dim N} {{\mathrm{vol}}}(N(F_v)_{x,0})\) and the fact (8.1) that \({{\mathrm{vol}}}(N(F_v)_{x,0}) = {{\mathrm{vol}}}(N(F_v)\cap K_v)=1\) when \(G\) is unramified at \(v\). Take
to be the minimum dimension of the unipotent radical of a proper parabolic subgroup of \(G\) with cuspidal Levi part. Then \(\phi ^{S,\infty }_{M}(1)\leqslant \mathbb {N}({\mathfrak n})^{C_{\mathrm {lv}}}\prod _{v\in \mathrm{Ram}(G)} {{\mathrm{vol}}}(K_v\cap N(F_v))\) for every \(M\) in (9.9). \(\square \)
9.5 Weight aspect
We put ourselves in the setting of Example 9.14 and exclude the uninteresting case of \(G=\{1\}\). By the assumption \(Z(G)=\{1\}\), for every \(\gamma \ne 1\in G(F)\) the connected centralizer \(I_\gamma \) has a strictly smaller set of roots so that \(\Phi _{I_\gamma }<\Phi \). Our next task is to prove a similar error bound as in the last subsection.
Theorem 9.19
Fix \(\phi _{S_0}\in C^\infty _c(G(F_{S_0}))\) and \(U^{S,\infty }\subset G(\mathbb {A}^{S,\infty })\). There exist constants \(A_{\mathrm {wt}},B_{\mathrm {wt}}>0\) and \(C_{\mathrm {wt}}\geqslant 1\) satisfying the following: for

any \(\kappa \in \mathbb {Z}_{>0}\),

any finite subset \(S_1\subset \mathcal {V}_F^\infty \) disjoint from \(S_0\) and \(S_{\mathrm{bad}}\) (Sect. 7.2) and

any \(\phi _{S_1}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{S_1}))^{\leqslant \kappa }\) such that \(\phi _{S_1}\leqslant 1\) on \(G(F_{S_1})\),
where the implicit constant in \(O(\cdot )\) is independent of \(\kappa \), \(S_1\) and \(\phi _{S_1}\). (Equivalently, \(\widehat{\mu }^\natural _{{\mathcal {F}},S_1}(\widehat{\phi }_{S_1})\widehat{\mu }^{\mathrm {pl}}_{S_1}(\widehat{\phi }_{S_1}) = O(q_{S_1}^{A_{\mathrm {wt}}+B_{\mathrm {wt}}\kappa } m(\xi )^{C_{\mathrm {wt}}})\) if \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\phi }_{S_0})\ne 0\).)
Remark 9.20
We always assume that \(S_0\) and \(S_1\) are disjoint. So the condition on \(S_1\) is really that it stays away from the finite set \(S_{\mathrm{bad}}\). This enters the proof where a uniform bound on orbital integrals from Sect. 7.2 is applied to the places in \(S_1\).
Remark 9.21
Again \(A_{\mathrm {wt}}, B_{\mathrm {wt}}, C_{\mathrm {wt}}\) can be chosen explicitly as can be seen from the proof below. For instance a choice can be made such that \(C_{\mathrm {wt}}\geqslant n_G\) for \(n_G\) defined in Sect. 1.8.
Proof
We can choose a sufficiently large finite set \(S'_0\supset S_0\cup \mathrm{Ram}(G)\) in the complement of \(S_1\cup S_\infty \) such that \(U^{S,\infty }\) is a finite disjoint union of groups of the form \((\prod _{v\notin S'_0\cup S_1\cup S_\infty } K_v)\times U_{S'_0\backslash S_0}\) for open compact subgroups \(U_{S'_0\backslash S_0}\) of \(G(\mathbb {A}_{F,S'_0\backslash S_0})\). By replacing \(S_0\) with \(S'_0\) (and thus \(S\) with \( S'_0\coprod S_1\)), we reduce the proof to the case where \(U^{S,\infty }=\prod _{v\notin S\cup S_\infty } K_v\).
For an \(F\)rational Levi subgroup \(M\) of \(G\), let \(Y_M\) be as in Proposition 8.7, where \(\kappa \), \(S_0\) and \(S_1\) are as in the theorem. (So the set \(Y_M\) varies as \(\kappa \) and \(S_1\) vary.) Take (9.6) as a starting point. Arthur’s trace formula ([3, Thm 6.1]) and the argument in the proof of [99, Thm 4.11] show (note that our \(Y_M\) contains \(Y_M\) of [99] but could be strictly bigger):
where \(a_{M,\gamma }\) (including \(M=G\)) is given by
Let us work with one cuspidal Levi subgroup \(M\) at a time. Observe that clearly \(\Omega _{I^M_\gamma }/\Omega _{I^M_\gamma ,c}\leqslant \Omega \) and that \(\tau (I^M_\gamma )\) is bounded by a constant depending only on \(G\) in view of (6.3) and Corollary 8.12 or Lemma 8.13.
By Corollary 6.17, there exist constants \(c_2,A_2>0\) such that
It is convenient to define the following finite subset of \(\mathcal {V}_F^\infty \) for each \(\gamma \in Y_M\). We fix a maximal torus \(T^M_\gamma \) in \(M\) over \(\overline{F}\) containing \(\gamma \) and write \(\Phi _{M,\gamma }\) for the set of roots of \(T^M_\gamma \) in \(M\). (A different choice of \(T^M_\gamma \) does not affect the argument.)
(If \(\gamma \) is in the center of \(M(F)\) then \(S_{M,\gamma }=\emptyset \) and \(q_{S_{M,\gamma }}=1\).)
We know that \(O^{M(F_v)}_\gamma ({\mathbf {1}}_{K_{M,v}})=1\) for \(v\notin S\cup S_{M,\gamma }\cup S_\infty \) and that \(S_{M,\gamma }\supset \mathrm{Ram}(I^M_\gamma )\) from [63, Cor 7.3]. According to Lemma 6.2 \(\phi _v={\mathbf {1}}_{K_{v}}\) implies \(\phi _{v,M}={\mathbf {1}}_{K_{M,v}}\). Hence
By Theorem 13.1, there exists a constant \(c(\phi _{S_0,M})>0\) such that
By Theorem 7.3, there exist \(a,b,c,e_G\in \mathbb {R}_{\geqslant 0}\) (independent of \(\gamma \), \(S_1\), \(\kappa \) and \(k\)) such that
[To obtain (9.12) and (9.13), apply Theorem 7.3 to \(v\in S_1\) and \(v\in S_{M,\gamma }\).]
Hence
On the other hand there exist \(\delta _{S_0},\delta _{\infty }\), \(\delta _{S_1}\geqslant 1\) such that for every \(\gamma \in Y_M\) with \(\alpha (\gamma )\ne 1\),

\(1\alpha (\gamma )_{S_0}\leqslant \delta _{S_0}\). (compactness of \(\mathrm{supp}\,\phi _{S_0}\))

\(1\alpha (\gamma )_{\infty }\leqslant \delta _{\infty }\). (compactness of \(U_\infty \))

\(1\alpha (\gamma )_{S_1}\leqslant \delta _{S_1}q_{S_1}^{B_5 \kappa }\). (Lemma 2.18 and Remark 2.20 explains the independence of \(B_1\) of \(S_1\).)
[When \(\alpha (\gamma )=1\), our convention is that \(1\alpha (\gamma )_v=1\) for every \(v\) to be consistent with the first formula of Appendix A.] Hence, together with the product formula for \(1\alpha (\gamma )\),
Set \(\delta :=\delta _{S_0} \delta _{\infty } \delta _{S_1}\). Note that \(1\alpha (\gamma )_v \leqslant 1\) for all \(\alpha \in \Phi _{M,\gamma }\) and all \(v\in S_{M,\gamma }\). If \(\gamma \in Z(M)(F)\) then \(q_{S_{M,\gamma }}=1\). Otherwise for each \(v\in S_{M,\gamma }\), we may choose \(\alpha \in \Phi _{M,\gamma }\) such that \(1\alpha (\gamma )_v\ne 1\). Then \(1\alpha (\gamma )_v \leqslant q_v^{1/w_G s_G}\) (for the same reason as in the proof of Proposition 8.7, Step I) so
In particular the crude bound \(\max _{v\in S_{M,\gamma }} q_v\geqslant 2S_{M,\gamma }\) holds, hence
Notice that the upper bound is independent of \(\gamma \) (and depends only on the fixed data). Keep assuming that \(\gamma \) is not central in \(M\) and that \(\alpha (\gamma )\ne 1\). Again by the product formula \(\prod _{v\in S_1\cup S_{M,\gamma }} 1\alpha (\gamma )_v = \prod _{v\in S_0\cup S_\infty } 1\alpha (\gamma )_v^{1} \geqslant (\delta _{S_0}\delta _\infty )^{1}\), thus
The above holds also when \(\gamma \) is central in \(M\), in which case the left hand side equals 1.
Now (9.14), (9.15), (9.16), and (9.17) imply
Lemma 6.11 gives a bound on the stable discrete series character:
Multiplying (9.11), (9.18) and (9.19) altogether (and noting \(\iota ^M(\gamma )^{1}\leqslant 1\)), the absolute value of the summand for \(\gamma \) in (9.10) (including \(M=G\)) is
All in all, \(\widehat{\mu }_{{\mathcal {F}},S}(\widehat{\phi }_S)\widehat{\mu }^{\mathrm {pl}}_S(\widehat{\phi }_S)\) is
Set (excluding \(\gamma =1\) in the second minimum when \(M=G\))
Note that \(C_{\mathrm {wt}}\) depends only on \(G\). It is automatic that \(\Phi ^+\Phi ^+_{I^M_\gamma }\geqslant 1\) on \(Y_G\backslash \{1\}\) and \(Y_M\) for \(M\in {\fancyscript{L}}_c(M_0)\backslash \{G\}\). The proof is concluded by invoking Corollary 8.10 (applied to \(Y_G\) and \(Y_M\)) with the choice
\(\square \)
9.6 Automorphic Plancherel density theorem
In the situation of either Examples 9.13 or 9.14, let us write \({\mathcal {F}}_k(\phi _{S_0})\) for \({\mathcal {F}}_k\) in order to emphasize the dependence on \(\phi _{S_0}\). Take \(S_1=\emptyset \) so that \(S=S_0\). Then \(\widehat{\mu }_{{\mathcal {F}}_k(\phi _{S}),\emptyset }\) may be viewed as a complex number (as it is a measure on a point). In fact we can consider \({\mathcal {F}}_k(\widehat{f}_{S})\), a family whose local condition at \(S\) is prescribed by \(\widehat{f}_S\in {\fancyscript{F}}(G(F_S)^{\wedge })\), even if \(\widehat{f}_{S}\) does not arise from any \(\phi _{S}\) in \(C^\infty _c(G(F_S))\). Put \(\widehat{\mu }_k(\widehat{f}_{S}):=\widehat{\mu }_{{\mathcal {F}}_k(\widehat{f}_{S}),\emptyset }\in \mathbb {C}\). We recover the automorphic Plancherel density theorem [99, Thms 4.3, 4.7].
Corollary 9.22
Consider families \({\mathcal {F}}_k\) in level or weight aspect as above. In level aspect assume that the highest weight of \(\xi \) is regular. (No assumption is necessary in the weight aspect.) For any \(\widehat{f}_S\in {\fancyscript{F}}(G(F_S)^{\wedge })\),
Proof
Theorems 9.16 and 9.19 tell us that
(Even though there was a condition on \(S_1\), note that there was no condition on \(S_0\) in either theorem.)
We would like to improve (9.20) to allow more general test functions. What needs to be shown [cf. (9.21) below] is that for every \(\epsilon >0\),
Thanks to Proposition 9.1 there exist \(\phi _{S},\psi _{S}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{S}))\) such that \(\widehat{f}_{S}\widehat{\phi }_{S}\leqslant \widehat{\psi }_{S}\) on \(G(F_{S})^{\wedge }\) and \(\widehat{\mu }^{\mathrm {pl}}_{S}(\widehat{\psi }_{S})\leqslant \epsilon \). Then [cf. (9.22) below]
Now \(\widehat{\mu }^{\mathrm {pl}}_{S}(\widehat{f}_{S}\widehat{\phi }_{S})\leqslant \widehat{\mu }^{\mathrm {pl}}_S(\widehat{\psi }_S)\leqslant \epsilon \), and \(\widehat{\mu }_k(\widehat{\phi }_{S}) \widehat{\mu }^{\mathrm {pl}}_{S}(\widehat{\phi }_{S})\leqslant \epsilon \) for \(k\gg 1\) by (9.20). Finally \(\widehat{\mu }_k\) is a positive measure since the highest weight of \(\xi \) is regular (see Example 9.11), and we get
[To see the positivity of \(\widehat{\mu }_k\), notice that \(\widehat{\mu }_k(\widehat{f}_{S}\widehat{\phi }_{S})\) is unraveled via (9.4) and (9.5) as a sum of \((\widehat{f}_{S}\widehat{\phi }_{S})(\pi )\) with coefficients having nonnegative signs. This is because \(\chi _{\mathrm{EP}}(\pi _\infty \otimes \xi )\) is either 0 or \((1)^{q(G)}\) when \(\xi \) has regular highest weight, cf. Sect. 6.5.] According to (9.20), \(\lim _{k\rightarrow \infty }\widehat{\mu }_k(\widehat{\psi }_S)=\widehat{\mu }^{\mathrm {pl}}_S(\widehat{\psi }_S)\leqslant \epsilon \). In particular \(\widehat{\mu }_k(\widehat{f}_{S}\widehat{\phi }_{S})\leqslant 2\epsilon \) for \(k\gg 1\). The proof is complete. \(\square \)
Remark 9.23
If \(G\) is anisotropic modulo center over \(F\) so that the trace formula for compact quotients is available, or if a further local assumption at finite places is imposed so as to avail the simple trace formula, the regularity condition on \(\xi \) can be removed by an argument of De George and Wallach [38] and Clozel [22]. The main point is to show that the contribution of (\(\xi \)cohomological) nontempered representations at \(\infty \) to the trace formula is negligible compared to the contribution of discrete series. Their argument requires some freedom of choice of test functions at \(\infty \), so it breaks down in the general case since one has to deal with new terms in the trace formula which disappear when Euler–Poincaré functions are used at \(\infty \). In other words, it seems necessary to prove analytic estimates on more terms (if not all terms) in the trace formula than we did in order to get rid of the assumption on \(\xi \). (This remark also applies to the same condition on \(\xi \) in Sects. 9.7 and 9.8 for level aspect families.) We may return to this issue in future work.
Remark 9.24
In the case of level aspect families [99, Thm 4.3] assumes that the level subgroups form a chain of decreasing groups whose intersection is the trivial group. The above corollary deals with some new cases as it assumes only that \(\mathbb {N}({\mathfrak n}_k)\rightarrow \infty \).
Corollary 9.25
Keep assuming that \(S_1=\emptyset \). Let \((U_k^{S,\infty },\xi _k)=(K^{S,\infty }({\mathfrak n}_k),\xi )\) or \((U^{S,\infty },\xi _k)\) in Examples 9.13 or 9.14, respectively, but prescribe local conditions at \(S\) by \(\widehat{f}_{S}\) rather than \(\phi _{S}\). Then
Proof
The corollary results from Corollary 9.22 since
\(\square \)
9.7 Application to the Sato–Tate conjecture for families
As an application of Theorems 9.16 and 9.19, we are about to fulfill the promise of Sect. 5.4 by showing that the Satake parameters in the automorphic families \(\{{\mathcal {F}}_k\}\) are equidistributed according to the Sato–Tate measure in a suitable sense (cf. Conjecture 5.9).
The notation and convention of Sect. 5 are retained here. Let \(\theta \in {\fancyscript{C}}(\Gamma _1)\) and \(\widehat{f}\in {\mathcal {F}}(\widehat{T}_{c,\theta }/\Omega _{c,\theta })\). For each \(v\in \mathcal {V}_F(\theta )\), the image of \(\widehat{f}\) in \({\mathcal {F}}(G(F_v)^{\wedge ,\mathrm {ur}})\) via (9.1) will be denoted \(\widehat{f}_v\).
Theorem 9.26
(Level aspect) Pick any \(\theta \in {\fancyscript{C}}(\Gamma _1)\) and let \(\{v_j\}_{j\geqslant 1}\) be a sequence in \(\mathcal {V}_F(\theta )\) such that \(q_{v_j}\rightarrow \infty \) as \(j\rightarrow \infty \). Suppose that

\(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\phi }_{S_0})\ne 0\) and

\(\xi \) has regular highest weight.
Then for every \(\widehat{f}\in {\mathcal {F}}(\widehat{T}_{c,\theta }/\Omega _{c,\theta })\),
where the limit is taken over \((j,k)\) subject to the following conditions:

\(\mathbb {N}({\mathfrak n}_k) q_{v_j}^{B_\Xi m \kappa }\geqslant c_\Xi ^{1}\),

\(v_j\not \mid {\mathfrak n}_k\),

\(q_{v_j}^{N} \mathbb {N}({\mathfrak n}_k)^{1}\rightarrow 0\) for all \(N>0\).
Proof
Fix \(\widehat{f}\). We are done if \(\limsup _{(j,k)\rightarrow \infty } \widehat{\mu }^{\natural }_{{\mathcal {F}}_k,v_j}(\widehat{f}_{v_j}) \widehat{\mu }^{\mathrm {ST}}_{\theta }(\widehat{f})\leqslant 4\epsilon \) for every \(\epsilon >0\). By Proposition 5.3, \(\widehat{\mu }^{\mathrm {pl}}_{v_j}(\widehat{f}_{v_j})\widehat{\mu }^{\mathrm {ST}}_{\theta }(\widehat{f})\leqslant \epsilon \) for sufficiently large \(j\). So it is enough to show that
For every \(j\geqslant 1\), Proposition 9.4 allows us to find \(\phi _{v_j},\psi _{v_j}\in {\mathcal {H}}^{\mathrm {ur}}(G(F_{v_j}))^{\leqslant \kappa }\) such that \(\widehat{f}_{v_j}\widehat{\phi }_{v_j}\leqslant \widehat{\psi }_{v_j}\) on \(G(F_{v_j})^{\wedge }\) and \(\widehat{\mu }^{\mathrm {pl}}_{v_j}(\widehat{\psi }_{v_j})\leqslant \epsilon \).
For each \(j\geqslant 1\),
Since \(\widehat{\mu }^{\mathrm {pl}}_{v_j}\) is a positive measure,
Theorem 9.16 and the assumptions of the theorem imply that for sufficiently large \((j,k)\), \(\widehat{\mu }^{\natural }_{{\mathcal {F}}_k,v_j}(\widehat{\phi }_{v_j}) \widehat{\mu }^{\mathrm {pl}}_{v_j}(\widehat{\phi }_{v_j})\leqslant \epsilon \). So we will be done if for sufficiently large \((j,k)\),
Arguing as in the proof of Corollary 9.22 we deduce the following: when \(\widehat{\mu }^{\natural }_{{\mathcal {F}}_k,v_j}(\widehat{f}_{v_j}\widehat{\phi }_{v_j})\) is unraveled as a sum over \(\pi \) [cf. (9.4) and (9.5)], each summand is \(\widehat{\phi }_{S_0}(\pi _{S_0})(\widehat{f}_{v_j}\widehat{\phi }_{v_j})(\pi _{v_j})\) times a nonnegative real number. (This uses the regularity assumption on \(\xi \). Certainly the absolute value of the sum does not get smaller when every summand is replaced with (something greater than or equal to) its absolute value, i.e.
Now choose \(\phi '_{S_0}\in C^\infty _c(G(F_{S_0}))\) according to Lemma 9.6 so that \(\phi _{S_0}(\pi _{S_0})\leqslant \phi '_{S_0}(\pi _{S_0})\) for every \(\pi _{S_0}\in G(F_{S_0})^{\wedge }\). Then
Theorem 9.16 applied to \(\widehat{\psi }_{v_j}\) and the inequality \(\widehat{\mu }^{\mathrm {pl}}_{v_j}(\widehat{\psi }_{v_j})\leqslant \epsilon \) imply that
This concludes the proof of (9.23), thus also (9.21). \(\square \)
Theorem 9.27
(Weight aspect) Let \(\theta \in {\fancyscript{C}}(\Gamma _1)\) and \(\widehat{\phi }_{S_0}\in C^\infty _c(G(F_{S_0}))\). Suppose that \(\{v_j\}_{j\geqslant 1}\) is a sequence in \(\mathcal {V}_F(\theta )\) such that \(q_{v_j}\rightarrow \infty \) as \(j\rightarrow \infty \) and that \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\phi }_{S_0})\ne 0\). Then for every \(\widehat{f}\in {\mathcal {F}}(\widehat{T}_{c,\theta }/\Omega _{c,\theta })\),
if \(q_{v_j}^{N} m(\xi _k)^{1}\rightarrow 0\) as \(k\rightarrow \infty \) for any integer \(N\geqslant 1\).
Proof
Same as above, except that Theorem 9.19 is used instead of Theorem 9.16. \(\square \)
Remark 9.28
As we have mentioned in Sect. 5.4, Theorems 9.26 and 9.27 indicate that \(\{{\mathcal {F}}_k\}_{\geqslant 1}\) are “general” families of automorphic representations in the sense of Conjecture 5.9.
Corollary 9.29
In the setting of Theorems 9.26 or 9.27, suppose in addition that \({\mathcal {F}}_k\ne 0\) for all \(k\geqslant 1\). Then
Proof
Follows from Corollary 9.25 and the two preceding theorems (cf. Remark 9.9). \(\square \)
Remark 9.30
The assumption that \({\mathcal {F}}_k\ne 0\) is almost automatically satisfied. Corollary 9.25 and the assumption that \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\phi }_{S_0})\ne 0\) imply that \({\mathcal {F}}_k\ne 0\) for any sufficiently large \(k\).
9.8 More general test functions at \(S_0\)
So far we worked primarily with families of Examples 9.13 and 9.14. We wish to extend Theorems 9.26 and 9.27 when the local condition at \(S_0\) is given by \(\widehat{f}_{S_0}\), which may not be of the form \(\widehat{\phi }_{S_0}\) for any \(\phi _{S_0}\in C^\infty _c(G(F_{S_0}))\) (cf. Example 9.11 and Remark 9.12).
Corollary 9.31
Let \(\theta \in {\fancyscript{C}}(\Gamma _1)\) and let \(\{v_j\}_{j\geqslant 1}\) be a sequence of places in \(\mathcal {V}_F(\theta )\) such that \(q_{v_j}\rightarrow \infty \) as \(j\rightarrow \infty \). Consider \(\widehat{\mu }_{{\mathcal {F}}_k,v_j}\) where
satisfying the conditions of Theorems 9.26 or 9.27, respectively. Then
where the limit is taken as in Theorem 9.26 (resp. Theorem 9.27).
Proof
The basic strategy is to reduce to the case of \(\widehat{\phi }\) and \(\widehat{\phi }_{v_j}\) in place of \(\widehat{f}\) and \(\widehat{f}_{v_j}\) via Sauvageot’s density theorem, as in the proof of Theorem 9.26. We can decompose \(\widehat{f}=\widehat{f}^+ +\widehat{f}^\) with \(\widehat{f}^+,\widehat{f}^\in {\mathcal {F}}(\widehat{T}_{c,\theta }/\Omega _{c,\theta })\) such that \(\widehat{f}^+\) and \(\widehat{f}^\) are nonnegative everywhere. The corollary for \(\widehat{f}\) is proved as soon as it is proved for \(\widehat{f}^+\) and \(\widehat{f}^\). Thus we may assume that \(\widehat{f}\geqslant 0\) from now on.
Fix any choice of \(\epsilon >0\). Proposition 9.1 ensures the existence of \(\phi _{S_0},\psi _{S_0}\in C^\infty _c(G(F_{S_0}))\) such that \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\psi }_{S_0})\leqslant \epsilon \) and \(\widehat{f}_{S_0}(\pi _{S_0})\widehat{\phi }_{S_0}(\pi _{S_0})\leqslant \widehat{\psi }_{S_0}(\pi _{S_0})\) for all \(\pi _{S_0}\in G(F_{S_0})^\wedge \). Of course we can guarantee in addition that \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\phi }_{S_0})\ne 0\). Put
Likewise we define \({\mathcal {F}}_k(\widehat{\psi }_{S_0})\) and so on. Then (cf. a similar step in the proof of Theorem 9.26)
The first term on the right side tends to 0 as \((j,k)\rightarrow \infty \) by Theorems 9.26 and 9.27. The last term is bounded by \(\widehat{\mu }^{\mathrm {pl}}_{S_0\cup \{v_j\}}(\widehat{\psi }_{S_0}\widehat{f}_{v_j})\leqslant \epsilon \widehat{\mu }^{\mathrm {pl}}_{v_j}(\widehat{f}_{v_j})\) using the fact that \(\widehat{\mu }^{\mathrm {pl}}_{S_0}\) is a positive measure. In order to bound the second term, recall that we are either in the weight aspect, or in the level aspect with regular highest weight for \(\xi \). Then \(a_{{\mathcal {F}}_k(\widehat{f}_{S_0}\widehat{\phi }_{S_0})}(\pi )\) is a nonnegative multiple of \(\widehat{f}_{S_0}(\pi _{S_0})\widehat{\phi }_{S_0}(\pi _{S_0})\) as in the proof of Theorem 9.26. Thus
the last inequality coming from the bound \(\widehat{\mu }^{\mathrm {pl}}_{S_0}(\widehat{\psi }_{S_0})\leqslant \epsilon \).
Hence we have shown that
By Theorems 9.26 and 9.27 and the fact that \(\lim \limits _{j\rightarrow \infty }\widehat{\mu }^{\mathrm {pl}}_{v_j}(\widehat{f}_{v_j})=\widehat{\mu }^{\mathrm {ST}}_{\theta }(\widehat{f})\), the right hand side is seen to be bounded by \(2\epsilon \widehat{\mu }^{\mathrm {ST}}_{\theta }(\widehat{f})\). As we are free to choose \(\epsilon >0\), we deduce that
\(\square \)
Remark 9.32
It would be desirable to improve Theorems 9.16 and 9.19 similarly by prescribing conditions at \(S_0\) in terms o