# Sato–Tate theorem for families and low-lying zeros of automorphic \(L\)-functions

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## 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 low-lying 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).

### Mathematics Subject Classification

11F55 11F67 11F70 11F72 11F75 14L15 20G30 22E30 22E35## 1 Introduction

The non-trivial 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 non-archimedean 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 quasi-split; 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.

*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 \(k\rightarrow \infty \),

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 \(\widehat{\mu }^{\mathrm {ST}}\).

- (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.

### 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 non-trivial 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^{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 low-lying 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 low-lying zeros.

More precisely [60] predicts that the low-lying 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.^{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**

- (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 non-trivial 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 low-lying 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 non-trivial 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.

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 6-A 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)

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 non-canonical 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**

- (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**

- (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 \(\mathcal {G}(N)={{\mathrm{USp}}}(2N)\)].

^{3}The kernel of the limiting point process when \(\mathcal {G}={{\mathrm{U}}}\) is given by the Dyson sine kernel

### 1.6 Low-lying 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 self-dual 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 low-lying 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 low-lying 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.

**Theorem 1.5**

- (i)
- there is a limiting \(1\)-level density for the low-lying 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 Arthur-Selberg 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 \(\infty \).

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 continue with Theorem 1.5. The proof relies heavily on Theorem 1.3. The connection between the two statements might not be immediately apparent.

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 \(\mathrm{GL}(d)\).

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 Ramanujan-Petersson for \(\mathrm{GL}(d,\mathbb {A})\).

The integer \(i\geqslant 1\) in Corollary 6.9 is an upper-bound 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).

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 non-split 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 so-called 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 low-lying 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

### 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 \({}^L G^{\mathrm {ur}}:=\widehat{G}\rtimes W_F^{\mathrm {ur}}\).

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. \(\square \)

**Lemma 2.2**

*Proof*

### 2.3 Truncated unramified Hecke algebras

**Lemma 2.3**

- (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*

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**

*Proof*

- (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 \(|\lambda |_{{\mathcal {B}}}\leqslant \kappa \).

### 2.4 The case of \(\mathrm{GL}_d\)

**Lemma 2.5**

### 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 \(\eta ^*:{\mathcal {H}}^{\mathrm {ur}}(G)\rightarrow {\mathcal {H}}^{\mathrm {ur}}(H)\).

**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*

### 2.6 Partial Satake transform

**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 \(\lambda \).\(\square \)

### 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**

- (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 \(G={{\mathrm{Sp}}}_{2n}\), \(n\geqslant 1\))

*Example 2.11*

(When \(G={{\mathrm{SO}}}_{2n}\), \(n\geqslant 2\))

*Example 2.12*

(When \(G={{\mathrm{SO}}}_{2n+1}\))

### 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*

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*

An elementary matrix computation shows the bound below, which will be used several times.

**Lemma 2.15**

*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*

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 \(\alpha _\gamma \).

*Remark 2.20*

Suppose that \(F\) runs over the completions of a number field \(\mathbf {F}\) at non-archimedean 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 well-defined 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 sub-spectrum. 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 \(G^{\wedge ,\mathrm {ur},\text {temp}}\). Harish-Chandra’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

- (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 \((\widehat{G}\rtimes \mathrm{Fr})_{\mathrm{ss-conj}}\); 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 \(\theta :{\mathcal {H}}^{\mathrm {ur}}(G)\rightarrow \mathbb {C}\).

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

- \(\mathrm{(i)}_t\)
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 \(\widehat{A}_c\).

*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 \(\widehat{T}_{nc}/(\mathrm{Fr}-\mathrm{id})\widehat{T}_{nc}\rightarrow \widehat{A}_{nc}\)).

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**

*Proof*

*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 L-functions

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 set-up notation. More detailed discussions may be found in [32, 55, 75], [86, §2] and [52, §5].

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

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

\(Z=Z(G)\) 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

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 Arthur-Selberg 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 quasi-character 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 square-integrable 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

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 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 non-cuspidal 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 Piatetski-Shapiro 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.

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**

*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 non-tempered 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 non-trivial 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 \(p(s)=(s-it)(1-it-s)\)).

^{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 \(j\).

**Proposition 4.4**

^{5}

*Proof*

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 real-valued 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 real-valued as well for all \(r\in (-\infty ,\infty )\).

## 5 Sato–Tate equidistribution

### 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 \((\widehat{K},\widehat{T}_c)\).

**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**

*Proof*

### 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.

**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^{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**

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**

*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

\(F\) 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 non-semisimple 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 \(M\).

**Lemma 6.1**

*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 \({\mathcal {S}}^G(\phi )= {\mathcal {S}}^M({\mathcal {S}}^G_M \phi )=\phi _T\).

### 6.2 Gross’s motives

**Proposition 6.3**

- (i)
\(\mathrm{Mot}_{G,d}\) is self-dual for each \(d\geqslant 1\).

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

- (iii)
\(\sum _{d\geqslant 1} (2d-1)\dim \mathrm{Mot}_{G,d} = \dim G\).

- (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 \(\mathrm{Gal}(E/F)\).

**Lemma 6.4**

*Proof*

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**

*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*

**Lemma 6.8**

*Proof*

**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 \(i=[E:F](1+e_{F/\mathbb {Q}_p} \log _p [E:F])-1\).

### 6.4 Stable discrete series characters

\(G\) 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 \(G(\mathbb {R})\), \(D_\infty ^G(\gamma )=1\).]

*regular*\(\gamma \in T(\mathbb {R})\), let us define (cf. [3, (4.4)])

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*

**Lemma 6.11**

*Proof*

### 6.5 Euler–Poincaré functions

\(\chi _{\mathrm{EP}}(\pi \otimes \xi )=0\) 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

**Proposition 6.12**

**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**

- (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 non-trivial 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 \(\mathrm{Gal}(E/F)\),where \(d_{E/F}=\mathbb {N}_{F/\mathbb {Q}}({\mathfrak D}_{E{/}F})\) is the relative discriminant of \(E{/}F\); recall that \(d_E= d_F^{[E:F]} d_{E/F}\).$$\begin{aligned} d_{E/F}^{A_1} \leqslant \mathbb {N}_{F/\mathbb {Q}}(\mathfrak {f}_\rho ) \leqslant d_{E/F}^{1/\dim (\rho )}, \end{aligned}$$

*Proof*

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 discriminant-conductor 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 non-trivial 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**

- (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),$$\begin{aligned} |L(\mathrm{Mot}_G)|\leqslant c_1 \prod _{v\in \mathrm{Ram}(G)} q_v^{A_{20}}. \end{aligned}$$

*Proof*

**Corollary 6.17**

*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 well-known 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**

*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**

- (i)
\(s(r)=1\);

- (ii)
\(r\) is self-dual 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 anti-linear map \(j:V\rightarrow V\) which satisfies \(j^2=1\).

- (vi)
\(r\) is self-dual 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).

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**

- (i)
\(s(r)=0\);

- (ii)
\(r\) is not self-dual;

- (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**

- (i)
\(s(r)=-1\);

- (ii)
\(r\) is self-dual and cannot be defined over \(\mathbb {R}\).

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

- (iv)
\(r\) is self-dual 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

\(G\) 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)].

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**

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.

*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 2-1*. 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 2-2*. When \(\gamma \) is non-central and non-elliptic.

*Step 2-3*. When \(\gamma \) is noncentral and elliptic in \(G\).

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 2-2) to find \(a_G,b_G,e_G\geqslant 0\) which satisfy the conditions described at the ends of Step 2-2 and Step 2-3. We are done with the proof of Proposition 7.1.

### 7.2 A global consequence

\(G\) 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.

**Theorem 7.3**

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.

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

parabolic subgroups \(P_v\) of \(G\) containing \(B_v\)

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 group-theoretic 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 2-3 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.

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 (non-canonical choice of) a Weyl-group 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**

*Proof of Lemma 7.7*

- 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 \((x'_0)^{-1} n^{-1} \delta n x'_0\in K'\lambda '(\varpi ') K'\). SoLet \(w\) 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$$\begin{aligned} \delta ^{-1}n^{-1} \delta n\in \delta ^{-1} K'\lambda '(\varpi ') K' \subset (K'\lambda _\delta ^{-1}(\varpi ') K')(K'\lambda '(\varpi ') K'). \end{aligned}$$We have \(\langle \alpha ,\lambda '\rangle \leqslant c_\mathbf {G}\Vert \lambda '\Vert \). Note also that$$\begin{aligned} \langle \alpha ,\lambda _0\rangle \leqslant \left\langle \alpha ,w\lambda _\delta ^{-1}+\lambda '\right\rangle ,\quad \alpha \in \Phi ^+(G,A'). \end{aligned}$$by Lemma 7.12 since a conjugate of \(\delta \) belongs to \(K'\lambda '(\varpi ')K'\). This implies 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)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)]$$\begin{aligned} \left\langle \alpha ,w\lambda _\delta ^{-1}\right\rangle = v'(w\alpha ^{-1}(\delta ))\leqslant m_{\Xi ^{\mathrm{spl}}}\Vert \lambda '\Vert . \end{aligned}$$for unique \(X_{\alpha _1},\ldots ,X_{\alpha _{|\Phi ^+|}}\in F'\).$$\begin{aligned} n=x_{\alpha _1}(X_{\alpha _1})\ldots x_{\alpha _{|\Phi ^+|}}(X_{\alpha _{|\Phi ^+|}}) \end{aligned}$$(7.17)
- 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 computewhere 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).$$\begin{aligned} \delta ^{-1}n^{-1}\delta n= & {} \delta ^{-1} \left( \prod _{i=|\Phi ^+|}^{1} x_{\alpha _i}(-X_{\alpha _i})\right) \delta \prod _{i=1}^{|\Phi ^+|} x_{\alpha _i}(X_{\alpha _i})\nonumber \\= & {} \left( \prod _{i=|\Phi ^+|}^{1} x_{\alpha _i}(-\alpha ^{-1}_i(\delta ) X_{\alpha _i}) \right) \prod _{i=1}^{|\Phi ^+|} x_{\alpha _i}(X_{\alpha _i}) \nonumber \\ {}= & {} \prod _{i=1}^{|\Phi ^+|} x_{\alpha _i}\left( (1-\alpha ^{-1}_i(\delta ))X_{\alpha _i} + P_{\alpha _i} \right) \end{aligned}$$(7.18)
^{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 \(\delta ^{-1}n^{-1}\delta n\in K'\lambda _0(\varpi ')K'\) thatFor \(1\leqslant i\leqslant |\Phi ^+|\), put$$\begin{aligned} v'\left( (1-\alpha ^{-1}_i(\delta ))X_{\alpha _i} + P_{\alpha _i}\right) \geqslant -m_{\mathbf {G}} n_\mathbf {G}(\lambda _0). \end{aligned}$$(7.19)Obviously \(0\leqslant {\mathcal {M}}_1\leqslant {\mathcal {M}}_2\leqslant \cdots \leqslant {\mathcal {M}}_{|\Phi ^+|}\). We claim that for every \(i\geqslant 1\),$$\begin{aligned} {\mathcal {M}}_i:= & {} \sum _{j=1}^i \left( Y^{i-j} (|v'(1-\alpha _j^{-1}(\delta ))|+ m_{\mathbf {G}}n_{\mathbf {G}}(\lambda _0))\right) \nonumber \\&+\sum _{j=1}^{i-1}Y^j m_{\Xi ^{\mathrm{spl}}} e_{F'/F}\kappa . \end{aligned}$$(7.20)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),$$\begin{aligned} v'(X_{\alpha _i}) \geqslant -{\mathcal {M}}_i. \end{aligned}$$(7.21)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$$\begin{aligned} v'(X_{\alpha _i})+v'\left( 1-\alpha ^{-1}_i(\delta )\right) \geqslant \min \left( -m_{\mathbf {G}} n_\mathbf {G}(\lambda _0),v'(P_{\alpha _i})\right) . \end{aligned}$$where the inequality follows from (7.16), (7.14), the induction hypothesis, and the fact that \(0\leqslant {\mathcal {M}}_{j}\leqslant {\mathcal {M}}_{i-1}\). Hence$$\begin{aligned}&v'\left( \alpha _j^{-1}(\delta )^{k_1}X_{\alpha _j}^{k_2}\right) =k_1v'\left( \alpha _j^{-1}(\delta )\right) +k_2v'(X_{\alpha _j})\nonumber \\&\quad \geqslant -Ym_{\Xi ^\mathrm{spl}}e_{F'/F}\kappa -Y{\mathcal {M}}_{i-1}, \end{aligned}$$Now$$\begin{aligned} v'(P_{\alpha _i})\geqslant -Ym_{\Xi ^\mathrm{spl}}e_{F'/F}\kappa -Y{\mathcal {M}}_{i-1}. \end{aligned}$$as desired. Now that the claim is verified, we have a fortiori$$\begin{aligned} v'(X_{\alpha _i})\geqslant & {} \min \left( -m_{\mathbf {G}}n_\mathbf {G}(\lambda _0),v'(P_{\alpha _i})\right) -v'\left( 1-\alpha ^{-1}_i(\delta )\right) \\\geqslant & {} -m_{\mathbf {G}}n_\mathbf {G}(\lambda _0) -Ym_{\Xi ^\mathrm{spl}}e_{F'/F}\kappa -Y{\mathcal {M}}_{i-1}\\&\quad -\left| v'\left( 1-\alpha ^{-1}_i(\delta )\right) \right| =-{\mathcal {M}}_i, \end{aligned}$$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.$$\begin{aligned} v'(X_{\alpha _i})\geqslant -{\mathcal {M}}_{|\Phi ^+|}, \quad \forall 1\leqslant i\leqslant |\Phi ^+| . \end{aligned}$$(7.22)$$\begin{aligned} {\mathcal {M}}_{|\Phi ^+|}\leqslant & {} Y^{|\Phi ^+|} \sum _{\alpha \in \Phi ^+} \Big ( |v'(1-\alpha ^{-1}(\delta )| + (m_\mathbf {G}m_{\Xi ^{\mathrm{spl}}}+m_\mathbf {G}c_\mathbf {G}\nonumber \\&\quad + m_{\Xi ^{\mathrm{spl}}})e_{F'/F}\kappa \Big ). \end{aligned}$$(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[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$$\begin{aligned} 1\leqslant \sum _{\alpha \in \Delta ^+} (-a^0_\alpha ) \langle \beta ,\alpha ^\vee \rangle \leqslant C,\quad \forall \beta \in \Delta ^+. \end{aligned}$$In fact (7.24) implies that the left inequality holds for all \(\beta \in \Phi ^+\). Hence$$\begin{aligned} {\mathcal {M}}_{|\Phi ^+|}\leqslant -v(\beta (a))\leqslant C\cdot {\mathcal {M}}_{|\Phi ^+|}, \quad \forall \beta \in \Delta ^+. \end{aligned}$$(7.24)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 \(a^{-1}na\in K'\).$$\begin{aligned}&a^{-1}na=\prod _{i=1}^{|\Phi ^+|} x_{\alpha _i}\left( \alpha _i(a)^{-1} X_{\alpha _{i}}\right) \\&\in \prod _{i=1}^{|\Phi ^+|} U_{\alpha _i,v(X_{\alpha _i})-v(\alpha _i(a))}\subset \prod _{i=1}^{|\Phi ^+|} U_{\alpha _i,{\mathcal {M}}_{|\Phi ^+|} +v(X_{\alpha _i})}. \end{aligned}$$
- 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 haveOn the other hand, for any \(\overline{x}'\in \mathrm{Apt}(A'(F'))\) and any positive simple coroot \(\alpha ^{\vee }\), we have$$\begin{aligned}&d(\overline{x}', \mathrm{Apt}(A'(F')))\leqslant d(n\overline{x}'_0, a\overline{x}'_0)= d(n\overline{x}'_0, na\overline{x}'_0)\nonumber \\&\quad = d(\overline{x}'_0, a\overline{x}'_0). \end{aligned}$$(7.25)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$$\begin{aligned} d(\overline{x}', \alpha ^\vee (\varpi ')^{-1} \overline{x}')\leqslant l_{\max }(\Phi ). \end{aligned}$$(7.26)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).\(\square \)$$\begin{aligned} d(\overline{x}'_0, a\overline{x}'_0)\leqslant l_{\max }(\Phi )\cdot C\cdot {\mathcal {M}}_{|\Phi ^+|}\cdot |\Delta ^+|. \end{aligned}$$(7.27)

**Lemma 7.8**

\(X_F(\gamma ,\lambda )~\subset ~\mathrm {Ball}(\overline{x}_1,{\mathcal {M}}(\gamma ,\kappa )).\)

*Proof*

**Lemma 7.9**

*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 group-theoretic 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.

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

\(Z_2:=Z'_2(d_G+2r_G+1)\).

**Corollary 7.11**

\(|O^{G(F)}_{\gamma }({\mathbf {1}}_{K\lambda (\varpi )K}, \mu _G,\mu ^{\mathrm{EP}}_{I_\gamma })|\!\leqslant \! q_v^{r_G(d_G+1)} q_v^{1+Z_1\kappa } D^G(\gamma )^{-Z_2}\).

### 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**

*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 \((\mathbf {B},\mathbf {T})\).

We thank Kottwitz for explaining the proof of the following lemma.

**Lemma 7.13**

*Proof*

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 \(V_{\geqslant i}/V_{\geqslant i+1}\).

**Corollary 7.14**

*Proof*

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 set-theoretic 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

\(M_0\) 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 \({\mathcal {H}}'\subset C^\infty _c(G(F_S))\), definewhere the union is taken over \(\phi _S\in {\mathcal {H}}'\).$$\begin{aligned} \mathrm{supp}\,{\mathcal {H}}'=\cup \, \mathrm{supp}\,\phi _S \end{aligned}$$
\(q_{S}:=\prod _{v\in S} q_v\) where \(q_v\) is the cardinality of the residue field at \(v\). (Convention: \(q_S=1\) if \(S=\emptyset \).)

It is worth stressing that this article treats a reductive group \(G\) without any hypothesis on \(G\) being split (or quasi-split). 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**

\({\mathfrak G}\times _{\mathcal {O}_F} \mathcal {O}_F[\frac{1}{\mathrm{Ram}(G)}]\) 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 scheme-theoretic 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 Bruhat-Tits 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 Bruhat-Tits 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 finite-type \(\mathcal {O}_F\)-schemes to that of triples \((X,\{\mathfrak {X}(v)\}_{v\in S}, \{f_v\}_{v\in S})\) where \(X\) is a finite-type \(\mathcal {O}_F[1/S]\)-scheme, \(\mathfrak {X}(v)\) is a finite-type \(\mathcal {O}_{(v)}\)-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 \(\Xi :G\hookrightarrow GL_m\). \(\square \)

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.

### 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 quasi-split 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**

*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 \(\det (\Xi (\gamma )-(1-X))=X^m+a_{m-1}(\gamma )X^{m-1}+\cdots + a_0(\gamma )\), where \(a_i(\gamma )\in F\) for \(0\leqslant i\leqslant m-1\). 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.

### 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**

*Proof*

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*

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**

*Remark 8.8*

*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.

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.

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.

**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 \(\alpha (U_{H,S_1})=U_{S_1}\).

^{8}) Set

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 quasi-split), we may assume that \(H\) and \(G\) are quasi-split over \(F\) by replacing \(H\) and \(G\) with their quasi-split inner forms.

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.

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 \(\Gamma (v)'\).

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 \(\gamma \).

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**

*Proof*

## 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 low-lying 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 (complex-valued) 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 \(\mathrm{n{\text {-}}ind}^G_M(\sigma )\).

**Proposition 9.1**

*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.

**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*

*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

**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 square-integrable 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.

(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$$\begin{aligned} G_\infty \times _\mathbb {R}\mathbb {C}=({\mathrm {Res}}_{F/\mathbb {Q}} G)\times _\mathbb {Q}\mathbb {C}=\prod _{v|\infty } G\times _{F,v} \mathbb {C}. \end{aligned}$$

*Remark 9.9*

*Example 9.10*

*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*

*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 pseudo-coefficient 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

*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.

^{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

*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**

- (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 \(S_1\).

*Remark 9.17*

*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*

The first assertion of the theorem follows immediately from (9.9). Henceforth we may assume that \({\fancyscript{L}}_c(M_0)\backslash \{G\}\ne \emptyset \).

### 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**

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 \(G(F_{S_1})\),

*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 \(U^{S,\infty }=\prod _{v\notin S\cup S_\infty } K_v\).

\(|1-\alpha (\gamma )|_{S_0}\leqslant \delta _{S_0}\). (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 \(S_1\).)

### 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**

*Proof*

*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) non-tempered 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**

*Proof*

### 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**

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

\(\xi \) has regular highest weight.

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

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

\(q_{v_j}^{N} \mathbb {N}({\mathfrak n}_k)^{-1}\rightarrow 0\) for all \(N>0\).

*Proof*

**Theorem 9.27**

*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**

*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**

*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.

*Remark 9.32*

It would be desirable to improve Theorems 9.16 and 9.19 similarly by prescribing conditions at \(S_0\) in terms of \(\widehat{f}_{S_0}\) rather than the less general \(\widehat{\phi }_{S_0}\). Unfortunately the argument proving Corollary 9.31 does not carry over. For instance in the case of Theorem 9.16, one should know in addition that the multiplicative constant implicit in \(O(q_{S_1}^{A_{\mathrm {lv}}+B_{\mathrm {lv}} \kappa } \mathbb {N}({\mathfrak n}_k)^{-C_{\mathrm {lv}}})\) is bounded as a sequence of \(\widehat{\phi }_{S_0}\) approaches \(\widehat{f}_{S_0}\).

## 10 Langlands functoriality

Let \(r:{}^L G\rightarrow \mathrm{GL}_d(\mathbb {C})\) be a representation of \({}^L G\). Let \(\pi \in {\mathcal {AR}}_{\mathrm{disc},\chi }(G)\) be such that with \(\pi _v \in \Pi _{\mathrm{disc}}(\xi _v^\vee )\) for each \(v|\infty \) (recall the notation from Sects. 6.4 and 9.2). The Langlands correspondence for \(G(F_v)\) [71] associates an \(L\)-parameter \(\varphi _{\xi ^\vee _v}:W_\mathbb {R}\rightarrow {}^L G\) to the \(L\)-packet \(\Pi _{\mathrm{disc}}(\xi _v^\vee )\), cf. Sect. 6.4.

The following asserts the existence of the functorial lift of \(\pi \) under \(r\) as predicted by the Langlands functoriality principle.

**Hypothesis 10.1**

- (i)
\(\Pi \) is isobaric,

- (ii)
\(\Pi _v=r_*(\pi _v)\) [defined in (2.9)] when \(G\), \(r\) and \(\pi \) are unramified at \(v\),

- (iii)
\(\Pi _v\) corresponds to \(r\varphi _{\xi ^\vee _v}\) via the Langlands correspondence for \(\mathrm{GL}_d(F_v)\) for all \(v|\infty \).

If \(\Pi \) as above exists then it is uniquely determined by (i) and (ii) thanks to the strong multiplicity one theorem. Moreover

**Lemma 10.2**

Hypothesis 10.1 (iii) implies that \(\Pi _v\) is tempered for all \(v|\infty \).

*Proof*

Recall the following general fact from [71, §3, (vi)]: let \(\varphi \) be an \(L\)-parameter for a real reductive group and \(\Pi (\varphi )\) its corresponding \(L\)-packet. Then \(\varphi \) has relatively compact image if and only if \(\Pi (\varphi )\) contains a tempered representation if and only if \(\Pi (\varphi )\) contains only tempered representations. In our case this implies that \(\varphi _{\xi ^\vee _v}\) has relatively compact image for every \(v|\infty \), and the continuity of \(r\) shows that the image of \(r\varphi _{\xi ^\vee _v}\) is also relatively compact. The lemma follows. \(\square \)

**Lemma 10.3**

*Proof*

## 11 Statistics of low-lying zeros

As explained in the introduction an application of the quantitative Plancherel Theorems 9.16 and 9.19 is to the study the distribution of the low-lying zeros of families of \(L\)-functions \(\Lambda (s,\Pi )\). The purpose of this section is to state the main results and make our working hypothesis precise.

### 11.1 The random matrix models

**Proposition 11.1**

### 11.2 The \(1\)-level density of low-lying zeros

Since \(\Phi \) decays rapidly at infinity, the zeros \(\gamma _j(\Pi )\) of \(\Lambda (s,\Pi )\) that contribute to the sum are within \(O(1/\log C(\mathcal {F}_k))\) distance of the central point. Therefore the sum over \(j\) only captures a few zeros for each \(\Pi \). The average over the family \(\Pi \in \mathfrak {F}_k\) is essential to have a meaningful statistical quantity.

### 11.3 Properties of families of \(L\)-functions

By definition [see (9.4)], if \(\pi \in \mathcal {F}_k\) then \(\pi _\infty \) has the same infinitesimal character as \(\xi _k^{\vee }\), i.e. \(\pi \in \Pi _{disc}(\xi _k)\). If \(\Pi \in \mathfrak {F}_k\) then \(\Pi _\infty \) corresponds to the composition \(r\circ \phi _{\xi _k}\) via the Langlands correspondence for \(\mathrm{GL}_d(F_\infty )\) [This is Hypothesis 10.1 (iii)]. In particular \(\Pi _\infty \) is uniquely determined by \(\xi _k\) and \(r\). It is identical for all \(\Pi \in \mathfrak {F}_k\).

It is shown in Lemma 10.2 that \(\Pi _\infty \) is tempered. Therefore Proposition 4.1 applies and the bounds towards Ramanujan (4.6) are satisfied for all \(\Pi \in \mathfrak {F}_k\).

### 11.4 Occurrence of poles

We make the following hypothesis concerning poles of \(L\)-functions in our families.

**Hypothesis 11.2**

The hypothesis is natural because it is related to the functoriality Hypothesis 10.1 in many ways. Of course it would be difficult to define the event that “\(L(s,\Pi ) \text { has a pole}\)” without assuming Hypothesis 10.1. Also when Functoriality is known unconditionally it is usually possible to establish the Hypothesis 11.2 unconditionally as well. We shall return to this question in a subsequent article.

### 11.5 Analytic conductors

As in [53] we define an analytic conductor \(C(\mathfrak {F}_k)\) associated to the family. The significance of \(C(\mathfrak {F}_k)\) is that each \(\Pi \in \mathfrak {F}_k\) have an analytic conductor \(C(\Pi )\) comparable to \(C(\mathfrak {F}_k)\). The hypothesis in this subsection will ensure that \(\log |\mathfrak {F}_k| \asymp \log C(\mathfrak {F}_k)\). We distinguish between families in the weight and level aspect.

#### 11.5.1 Weight aspect

For families in the weight aspect we set \(C(\mathfrak {F}_k)\) to be the analytic conductor \(C(\Pi _\infty )\) of the archimedean factor \(\Pi _\infty \) (recall that \(\Pi _\infty \) is the same for all \(\Pi \in \mathfrak {F}_k\)). Then \(C(\Pi )\asymp C(\mathfrak {F}_k)\) for all \(\Pi \in \mathfrak {F}_k\).

From Corollary 9.25 we have that \(\left| \mathfrak {F}_k\right| \asymp \dim \xi _k\) as \(k\rightarrow \infty \). It remains to relate the quantities \(C(\mathfrak {F}_k)\), \(\dim \xi _k\) and \(m(\xi _k)\), which is achieved in (11.6) and (11.7) below.

**Lemma 11.3**

Let \(v|\infty \). Let \(\xi _v\) be an irreducible finite dimensional algebraic representation of \(G(F_v)\). Then \(m(\xi _v)^{\left| \Phi ^+\right| }\ll \dim \xi _v \ll M(\xi _v)^{\left| \Phi ^+\right| }\). Also \(M(\xi _v)\ll \dim \xi _v\).

*Proof*

This follows from Lemma 6.10. Recall the definition of \(m(\xi _v)\) in Sect. 6.4 and \(M(\xi _v)\) in Sect. 10. \(\square \)

#### 11.5.2 Level aspect

**Hypothesis 11.4**

### 11.6 Main result

We may now state our main results on low-lying zeros of the family \(\mathfrak {F}=r_*\mathcal {F}\). The following is a precise version of Theorem 1.5 from the introduction [compare with (11.2)].

**Theorem 11.5**

## 12 Proof of Theorem 11.5

The method of proof of the asymptotic distribution of the \(1\)-level density of low-lying zeros of families of \(L\)-functions has appeared at many places in the literature and is by now relatively standard. However we must justify the details carefully as families of \(L\)-functions haven’t been studied in such a general setting before. The advantage of working in that degree of generality is that we can isolate the essential mechanisms and arithmetic ingredients involved.

In order to keep the analysis concise we have introduced some technical improvements which can be helpful in other contexts: we use non-trivial bounds towards Ramanujan in a systematic way to handle ramified places; we clarify that it is not necessary to assume that the representation be self-dual or any other symmetry property to carry out the analysis; most importantly we exploit the properties of the Plancherel measure when estimating Satake parameters. Previous articles on the subject rely in a way or another on explicit Hecke relations which made the proof indirect and lengthy, although manageable for groups of low rank.

### 12.1 Notation

The proof of the main theorems will follow by a fine estimation of \(\widehat{\mathcal {L}}_{k,v}(y)\) as \(k\rightarrow \infty \). The uniformity in both the places \(v\in \mathcal {V}_F\) and the parameter \(y\in \mathbb {R}\) will play an important role. Typically \(q_v\) will be as large as \(C(\mathfrak {F}_k)^{O(\delta )}\) and \(y\) will be of size proportional to \(\log C(\mathfrak {F}_k)\).

- (i)
the archimedean places \(S_\infty \), the contribution of which is evaluated in Sect. 12.5;

- (ii)
a fixed set \(S_0\) of non-archimedean places. These may be thought of as the “ramified places”. Their contribution is negligible as shown in Sect. 12.7;

- (iii)
the set \(\left\{ v\mid \mathfrak {n}_k\right\} \) of places that divide the level. These play a role only when the level varies and we show in Sect. 12.10 that their contribution is negligible. We use the convention that for families in the weight aspect this set of places is empty;

- (iv)the generic places \(S_{\text {gen}}\) which is the complement in \(\mathcal {V}_F\) of the above three sets of places. This set will actually be decomposed in two parts:$$\begin{aligned} S_{\text {gen}} = S_{\text {main}} \sqcup S_{\text {cut}}, \end{aligned}$$
- (v)
where the set \(S_{\text {cut}}\) is infinite and consists of those non-archimedean places \(v\in S_\text {gen}\) such that \(\tfrac{\log q_v}{2\pi }\) is large enough to be outside of the support of \(\Psi \) [see (12.18) below for the exact definition of \(S_\text {cut}\)]. Then the pairing in (12.2) vanishes;

- (vi)the remaining set \(S_\text {main}\) is finite (but growing as \(k \rightarrow \infty \)). It will produce the main contribution of (12.2). For all places \(v\in S_\text {main}\), each of \(G\), \(r\) and \(\pi \) is unramified over \(F_v\). Using the notation of Sect. 5 we split \(S_\text {main}\) further as the disjoint union of$$\begin{aligned} S_\text {main}\cap \mathcal {V}_F(\theta ),\ \theta \in \fancyscript{C}(\Gamma _1). \end{aligned}$$

### 12.2 Outline

For non-archimedean places \(v\in S_\text {main}\) we study in Sect. 12.6 various moments of Satake parameters. The quantity \(\widehat{\mathcal {L}}_{\text {pl},v}\) in (12.11) below will be the analogue of (12.1) where the average over automorphic representations \(\Pi \in \mathfrak {F}_k\) gets replaced by an average of \(\Pi _v\) against the unramified Plancherel measure. Our Plancherel equidistribution theorems for families (Theorems 9.16 and 9.19) imply that \(\widehat{\mathcal {L}}_{k,v}\) is asymptotic to \(\widehat{\mathcal {L}}_{\text {pl},v}\) as \(k\rightarrow \infty \).

It is essential that our equidistribution theorems are quantitative in a strong polynomial sense. Details on handling the remainder terms are given in Sects. 12.8–12.10.

The overall conclusion of the below analysis is that the limit of (12.4) as \(k\rightarrow \infty \) is equal^{10} to \(-\frac{s(r)}{2}\Phi (0)\), where \(s(r)\) is the Frobenius–Schur indicator of \(r\). In the derivation of the one-level density there is an additional term \(\widehat{\Phi }(0)\) which easily comes from the explicit formula and the contribution of the archimedean terms. Thereby we finish the proof of Theorem 11.5.

### 12.3 Explicit formula

^{11}

### 12.4 Contribution of the poles

### 12.5 Archimedean places

In this subsection we handle the archimedean places \(v\in S_\infty \). Recall from Lemma 10.2 that \(\Pi _{\infty }\) is tempered. In fact we shall only need here a bound towards Ramanujan \(0<\theta <\frac{1}{2}\) as in Sect. 4.2.

**Lemma 12.1**

*Proof*

*Remark 12.2*

**Corollary 12.3**

*Proof*

### 12.6 Moments of Satake parameters

The supports of both measures \(\widehat{\mathcal {L}}_{k,v}\) and \(\widehat{\mathcal {L}}_{\text {pl},v}\) are contained in the discrete set \(\frac{\log q_v}{2\pi }\mathbb {N}_{\geqslant 1}\). If \(q_v\) is large enough this is disjoint from the support of \(\Psi \) and thus all sums over places \(v\in \mathcal {V}_F\) considered below shall be finitely supported.

### 12.7 General upper-bounds

**Proposition 12.4**

- (i)For all \(v\in \mathcal {V}^{\infty }_F\) and all continuous function \(\Psi \),$$\begin{aligned} \left\langle \widehat{\mathcal {L}}_{k,v},\Psi \right\rangle \ll q_v^{\theta -\frac{1}{2}}\log q_v \left||\Psi \right||_{\infty }. \end{aligned}$$
- (ii)For all \(v\in S_{\text {gen}}\) and all continuous function \(\Psi \),$$\begin{aligned} \left\langle \widehat{\mathcal {L}}_{\text {pl},v},\Psi \right\rangle \ll q_v^{-\frac{1}{2}}\log q_v \left||\Psi \right||_{\infty }. \end{aligned}$$

*Proof*

- (i)Inserting the above upper bound into (12.10) we haveBecause \(0<\theta <\frac{1}{2}\), the conclusion easily follows.$$\begin{aligned} \left\langle \widehat{\mathcal {L}}_{k,v},\Psi \right\rangle \ll \log q_v\sum _{\nu \geqslant 1} q_v^{\nu (\theta -1/2)}\left| \Psi \left( \frac{\nu }{2\pi }\log q_v\right) \right| . \end{aligned}$$
- (ii)The Plancherel measure \(\widehat{\mu }^{\text {pl},\mathrm {ur}}\) has total mass one and is supported on the tempered spectrum \(\widehat{G}(F_v)^{\wedge ,\mathrm {ur},\text {temp}}\) (see Sect. 3.2). We deduce similarly that for every \(\nu \geqslant 1\),Indeed the image of any unramified \(L\)-parameter \(r\circ \varphi :W_{F_v}^{\mathrm {ur}} \rightarrow \mathrm{GL}_d(\mathbb {C})\) is bounded and all Frobenius eigenvalues have therefore absolute value one. \(\square \)$$\begin{aligned} \left| \beta _{\text {pl},v}^{(\nu )}\right| \leqslant d \end{aligned}$$(12.13)

### 12.8 Plancherel equidistribution

We are in position to apply the Plancherel equidistribution theorem for families established in Sect. 9. We shall derive uniform asymptotics as \(k\rightarrow \infty \) for \(\beta ^{(\nu )}_{v}(\mathfrak {F}_k)\).

**Proposition 12.5**

*Proof*

For families in the weight aspect the assumptions in Theorem 9.19 are always satisfied. This yields the main term in (12.14) with the error term \(O(q_{S_1}^{A_{\mathrm {wt}}+B_{\mathrm {wt}}\kappa } m(\xi _k)^{-C_{\mathrm {wt}}})\). By the estimate (11.7) we may choose \(C_5:=C_{\mathrm {wt}}/C_2\) to conclude the proof of (12.14). \(\square \)

### 12.9 Main term

We deduce from Proposition 12.5 the following estimate for \(\widehat{\mathcal {L}}_{k,v}\).

**Proposition 12.6**

*Proof*

### 12.10 Handling remainder terms

For archimedean places \(v\in S_\infty \) we encountered in Sect. 12.5 the remainder term \(O(||\widehat{\Psi } ||_1)\). Because \(\log C(\mathfrak {F}_k)\rightarrow \infty \), this remainder term is negligible for \(D_v(\mathfrak {F}_k,\Phi )\) as \(k\rightarrow \infty \).

### 12.11 Sum over primes

It remains to estimate the above terms (12.22) which consist of sums over the places \(v\in S_\text {main}\). We shall use the prime number theorem and the Cebotarev equidistribution theorem which we now proceed to recall, following e.g. [79, Chap. 7]. Let \(E/F\) be a finite Galois extension with Galois group \(\Gamma =\mathrm{Gal}(E/F)\). For all conjugacy class \(\theta \in \fancyscript{C}(\Gamma )\), recall that \(\mathcal {V}_F(\theta )\) consists of those unramified places \(v\in \mathcal {V}_F^\infty \) such that \(\mathrm{Fr}_v\in \theta \).

**Proposition 12.7**

Note that if we replace \(\log q_v\) by \(-\log q_v\), the same estimate holds with the integral on the right-hand side ranging from \(-\infty \) to \(0\). We shall use this observation below when adding the contribution of \(\overline{D_{v}}(\mathfrak {F}_k,\Phi )\) which will then produce produce the integral \(\int \nolimits _{-\infty }^{\infty } \widehat{\Phi }(y)dy = \Phi (0)\).

### 12.12 Computing the moments \(M^{(1)}\) and \(M^{(2)}\)

**Lemma 12.8**

The restriction \(r|_{\widehat{G}}\) does not contain the trivial representation.

*Proof*

If there were a non-zero vector in \(\mathbb {C}^d\) invariant by \(r(\widehat{G})\) then all its translates by \(\mathrm{Gal}(\bar{F}/F)\) would still be invariant because \(\widehat{G}\) is a normal subgroup of \({}^LG\). Because \(r\) is irreducible these translates generate \(\mathbb {C}^d\) and thus the restriction \(r|_{\widehat{G}}\) would be trivial ^{12} which yields a contradiction. For an extension of this argument see e.g. [93, Prop. 24, § I.8.1]. \(\square \)

Since \(v\in S_\text {main}\), the group \(G\) is unramified over \(F_v\) and the restriction \(r|_{\widehat{G}\rtimes W_{F_v}}\) is an unramified \(L\)-morphism which factors through \(\widehat{G} \rtimes W^{\mathrm {ur}}_{F_v}\). Note that this restriction might reducible in general.

**Proposition 12.9**

*Proof*

We decompose the restriction of \(r\) to \(\widehat{G} \rtimes W^{\mathrm {ur}}_{F_v}\) into a direct sum of irreducible \(\oplus _i r_i\). By Lemma 12.8 each \(r_i|_{\widehat{G}}\) does not contain the trivial representation. In particular each \(r_i\) does not factor through \(W_{F_v}^{\mathrm {ur}}\).

For the second moment \(M^{(2)}\) we shall need a more refined estimate. Recall the finite extension \(F_1/F\) from Sect. 5. We also choose a finite extension \(F_2/F_1\) such that \(r\) factors through \(\widehat{G} \rtimes \mathrm{Gal}(F_2/F)\). Let \(\Gamma _2:=\mathrm{Gal}(F_2/F)\) and denote by \(\fancyscript{C}(\Gamma _2)\) the set of conjugacy classes in \(\Gamma _2\).

**Proposition 12.10**

- (i)
- For all \(\theta \in \fancyscript{C}(\Gamma _2)\) there is an algebraic integer \(s(r,\theta )\) such that uniformly for all \(v\in S_\text {main}\),Here \([\mathrm{Fr}_v]\in \fancyscript{C}(\Gamma _2)\) is the conjugacy class of \(\mathrm{Fr}_v\) in \(\Gamma _2\).$$\begin{aligned} \beta ^{(2)}_{\mathrm {pl},v}=s(r,[\mathrm{Fr}_v]) + O\left( q_v^{-1}\right) . \end{aligned}$$(12.25)
- (ii)
- The following identity holdswhere \(s(r)\in \{-1,0,1\}\) is the Frobenius-Schur indicator of \(r\).$$\begin{aligned} s(r)= \sum _{\theta \in \fancyscript{C}(\Gamma _2)} \frac{\left| \theta \right| }{\left| \Gamma _2\right| } s(r,\theta ) \end{aligned}$$

*Proof*

(i) We proceed in way similar to the proof of Proposition 12.9 above. We shall give an explicit formula (12.26) for \(s(r,\theta )\).

We decompose \({{\mathrm{Sym}}}^2 r=\oplus \rho ^+_i\) (resp. \(\bigwedge ^2 r=\oplus \rho ^-_i\)) into a direct sum of irreducible representation of \(\widehat{G} \rtimes \mathrm{Gal}(E/F)\). Then we can decompose for each \(i\) the restriction \(\rho ^+_i|_{\widehat{G}\rtimes W^{\mathrm {ur}}_{F_v}}=\oplus _j \rho ^+_{ij}\) as a direct sum of irreducible representations of \(\widehat{G}\rtimes W^{\mathrm {ur}}_{F_v}\). Similarly we let \(\rho ^-_i|_{\widehat{G}\rtimes W^{\mathrm {ur}}_{F_v}}=\oplus _j \rho ^-_{ij}\).

In the second case, \(i\) is such that \(\rho ^+_i\) does factor through \(\mathrm{Gal}(E/F)\). Then for all \(j\), \(\rho ^+_{ij}\) factors through \(W_F^{\mathrm {ur}}\) (in particular it is \(1\)-dimensional). We have that \(\widehat{\Phi }^+_{ij}(1)=\rho ^+_{ij}(\mathrm{Fr}_v)\). By linearity we deduce that \(\sum _{j} \phi ^+_{ij}(1)={{\mathrm{tr}}}\rho ^+_i(\mathrm{Fr}_v)\). This is an algebraic integer which depends only on the conjugacy class of \(\mathrm{Fr}_v\) in \(\Gamma _2\).

### 12.13 Conclusion

We now gather all the estimates and conclude the proof of Theorem 11.5. The explicit formula (12.5) expresses \(D(\mathfrak {F}_k,\Phi )\) as the sum of four terms. The term \(D_{\text {pol}}(\mathfrak {F}_k,\Phi )\) goes to zero as \(k\rightarrow \infty \) as consequence of Hypothesis 11.2, see Sect. 12.4.

We now turn to the non-archimedean contribution. The places \(v\in S_0\) and \(v\mid \mathfrak {n}_k\) are negligible thanks to (12.16) and (12.17), respectively.

It remains the non-archimedean places \(v\in S_\text {gen}=S_\text {main}\sqcup S_\text {cut}\). The contribution from \(v\in S_\text {cut}\) is zero because the support of \(\widehat{\Phi }\) is included in \((-\delta ,\delta )\), see (12.18).

For each \(v\in S_\text {main}\) we apply Proposition 12.6. The sum over \(v\in S_\text {main}\) of the remainder terms is shown to be negligible in (12.19) and (12.20). For the main term the estimate (12.21) shows that the contribution of the higher moments is negligible. It remains the two terms \(M^{(1)}\) and \(M^{(2)}\) as defined in (12.22).

## Footnotes

- 1.
Sarnak and the authors gave a more refined and updated framework in [89] while our paper was under review.

- 2.
In this paper we do not distinguish in the orthogonal ensemble between the \(\mathrm {O}\), \({{\mathrm{SO}}}(odd)\) and \({{\mathrm{SO}}}(even)\) symmetries. We will return to this question in a subsequent work.

- 3.
For other values of \(\beta \ne 1,2,4\), the limiting statistics attached to (1.4) has been determined recently by Valkó–Virág in terms of the Brownian carousel.

- 4.
One should be aware that Theorem 5.8 in [52] does not apply directly to our setting because it is valid under certain further assumptions on \(\Pi \) such as \(\mu _i(\Pi _v)\) being real for archimedean places \(v\).

- 5.
Note however that it is never allowed to switch the sum and integration symbols in (4.9). This is because the \(L\)-function is evaluated at the center of the critical strip in which the Euler product does not converge absolutely.

- 6.
If \(\pi \) is not cuspidal then the hypothesis is never supposed to be satisfied.

- 7.
For instance if \(P_{\alpha _i}=\alpha _i^{-1}(\delta )^2 X_{\alpha _i}^4 + \alpha _i^{-1}(\delta )^3 X_{\alpha _i}^3\) then \(Y=4\).

- 8.
Choose \(U^{S_1}_Z\) to be any open compact subgroup. Then \(U_{Z,S_1}U_{Z}^{S_1}Z(F)\) has a finite index in \(Z(\mathbb {A})^1\) by compactness. Then enlarge \(U^{S_1}_Z\) without breaking compactness such that the equality holds.

- 9.
Without the hypothesis that the center is trivial, one should work with fixed central character and apply the trace formula in such a setting. Then our results and arguments in the weight aspect should remain valid without change.

- 10.
A quick explanation for the minus sign is as follows. A local \(L\)-factor is of the form \((1-\alpha q^{-s})^{-1}\) with three minus signs thus its logarithmic derivative is \(-\log q \sum \nolimits _{\nu \geqslant 1}\alpha ^{\nu }q^{-\nu s}\) with one minus sign.

- 11.
Note that the exponential in (12.1) is \(e^{2i\pi xy}\) with a plus sign.

- 12.
In the sense that \(r|_{\widehat{G}}\) would be a direct sum of trivial representations. In the sequel we use this slight abuse of notation when saying that a representation is “trivial”.

## Notes

### Acknowledgments

We would like to thank Jim Arthur, Joseph Bernstein, Laurent Clozel, Julia Gordon, Nicholas Katz, Emmanuel Kowalski, Erez Lapid, Jasmin Matz, Philippe Michel, Peter Sarnak, Kannan Soundararajan and Akshay Venkatesh for helpful discussions and comments. We would like to express our gratitude to Robert Kottwitz and Bao Châu Ngô for helpful discussions regarding Sect. 7, especially about the possibility of a geometric approach. We appreciate Brian Conrad for explaining us about the integral models for reductive groups. We thank the referee for a very careful reading. Most of this work took place during the AY2010-2011 at the Institute for Advanced Study and some of the results have been presented there in March. We thank the audience for their helpful comments and the IAS for providing excellent working conditions. S. W. S. acknowledges support from the National Science Foundation during his stay at the IAS under Agreement No. DMS-0635607 and thanks Massachusetts Institute of Technology and Korea Institute for Advanced Study for providing a very amiable work environment. N. T. is supported by a Grant #209849 from the Simons Foundation. We gratefully acknowledge BIRS, and the organizers of the workshop on L-packets, where this appendix was conceived. J. G. is deeply grateful to Sug-Woo Shin, Nicolas Templier, Loren Spice, Tasho Statev-Kaletha, William Casselman, and Gopal Prasad for helpful conversations. R.C. was supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007–2013) with ERC Grant Agreement No. 615722 MOTMELSUM and by the Labex CEMPI (ANR-11-LABX-0007-01); J. G. was supported by NSERC; I. H. was supported by the SFB 878 of the Deutsche Forschungsgemeinschaft.