Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Let G=GL2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\mathrm{GL}_{2n}$$\end{document} over a totally real number field F and n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}. Let Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document} be a cuspidal automorphic representation of G(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(\mathbb {A})$$\end{document}, which is cohomological and a functorial lift from SO(2n+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2n+1)$$\end{document}. The latter condition can be equivalently reformulated that the exterior square L-function of Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document} has a pole at s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1$$\end{document}. In this paper, we prove a rationality result for the residue of the exterior square L-function at s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1$$\end{document} and also for the holomorphic value of the symmetric square L-function at s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1$$\end{document} attached to Π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi $$\end{document}. As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gauß sums of Hecke characters.


General background
Let F be an algebraic number field and let be a cuspidal automorphic representation of GL 2 (A F ). Rationality results for special values of the associate automorphic Lfunction L(s, ) have been studied by several authors over the last decades. For the scope of this paper, we would like to mention Manin and Shimura, who were the first to study special values of L(s, ) in the particular case, when F is totally real, i.e., when comes from a Hilbert modular form, cf. [27] and [33], and Kurchanov, who treated the case of a CM-field F in a series of papers, cf. [24,25]. Shortly later, Harder published some articles, see [15,16], in which he described a general approach to such rationality results. In [15], Harder considered the case of an arbitrary number field F, while in [16], he extended the methods of the above authors to some automorphic representations, which do not necessarily come from cusp forms (for F imaginary quadratic). The case of GL 2 over a general number field F has also been considered independently by Hida, cf. [18] and later on also by Shimura, see [34].
It took some time until extensions of these results to general linear groups GL n of higher rank n were available. Important achievements include Ash-Ginzburg, [1], Kazhdan-Mazur-Schmidt [22], and Mahnkopf [26].
Guided by the above methods, meanwhile, there is a growing number of results that have been proved about the rationality of special values of certain automorphic Lfunctions attached to GL n . As a selection of examples, relevant to the present paper, we refer to Raghuram [28,29], Harder-Raghuram [17], Grobner-Harris [11]; Grobner-Raghuram [14], Grobner-Harris-Lapid [12] and Balasubramanyam-Raghuram [2].
In all of these references, the corresponding rationality result is obtained by writing the special L-value at hand as an algebraic multiple of a certain period invariant. 1 This period is defined by comparison of a rational structure on a cohomology space, attached to the given automorphic representation , with a rational structure on a model-space of (the finite part of) , such as a Whittaker model or a Shalika model. (The word "rational structure" here refers to a subspace of the vector space, carrying the action of , which is essentially defined over the field of rationality of and at the same time stable under the group action.) While the first rational structure on the cohomology space is purely of geometric nature and has its origin in the cohomology of arithmetic groups (or better: the cohomology of arithmetically defined locally symmetric spaces), the latter rational structure is defined by reference to the uniqueness of the given modelspace.
In this paper, we continue the above considerations. But while most of the aforementioned papers deal with special values of the Rankin-Selberg L-function (by some variation or the other), the principal L-function, or the Asai L-functions, here we would like to study the algebraicity of the exterior square L-function and the symmetric square L-function, attached to a cuspidal automorphic representation of the general linear group.

The main results of this paper
To put ourselves in medias res, let F be a totally real number field and let G = GL N /F, N = 2n with n ≥ 2. The restrictions on F and the index of the general linear groups under consideration are owed to the inevitable, as it will become clear below. Indeed, let be a cuspidal automorphic representation of G(A) and let W ψ ( ) be its ψ-Whittaker model. As we want to exploit the results of Bump-Friedberg [6], we shall assume that the partial exterior square L-function L S (s, , 2 ) of has a pole at s = 1. (Here, S is a finite set of places of F, containing all archimedean ones, such that for a place v / ∈ S, the local components v and ψ v are unramified.) In particular, this forces N = 2n to be even, see [20,Theorem 2], and furthermore to be self-dual, ∼ = ∨ , and to have trivial central character.
Our first main result gives a rationality statement for the residue Res s=1 (L S (s, , 2 )) of the exterior square L-function. More precisely, we obtain the following result: Theorem 1.1 Let F be a totally real number field and G = GL 2n /F, n ≥ 2. Let be a unitary cuspidal automorphic representation of G(A), which is cohomological with respect to an irreducible, self-contragredient, algebraic, finite-dimensional representation E μ of G ∞ . Assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function L S (s, , 2 ) has a pole at s = 1. Then, for every σ ∈ Aut(C), there is a non-trivial period p t ( σ ), defined by a comparison of a given rational structure on the Whittaker model of σ f and a rational structure on a realization of σ f in cohomology in top degree t, and a non-trivial archimedean period p t ( σ ∞ ), such that In particular, where "∼ Q( f ) " means up to multiplication of the right hand side by an element in the number field Q( f ).
This is proved in details in Sect. 7.4, see Theorem 7.4. For a precise definition of the periods p t ( σ ) and p t ( σ ∞ ), as well as for a complete list of choices which enter their involved definitions, we refer to Proposition 4.3 and Remark 4.4, respectively (7.2) and Remark 7.3. The non-vanishing of the archimedean period p t ( σ ∞ ) is shown-building on a result of Sun-in our Theorem 7.1. The number field Q( f ) in the theorem is (by Strong Multiplicity One) the aforementioned field of rationality of the cuspidal automorphic representation . See Sects. 2.5 and 3.3.
The key result, which we use, in order to derive the above theorem, is a certain integral-representation, obtained by Bump-Friedberg [6], of the residue Res s=1 (L S (s, , 2 )) of the exterior square L-function in terms of integrating over a cycle Z (A)H (F)\H (A). Here, Z is the centre of G and H = GL n × GL n , suitably embedded into G, cf. 2.2.
More precisely, if one combines the three main results of [6], then, under the assumptions made in the theorem, one obtains the following equality, shown in our Theorem 6.1: Here, is a certain global Schwartz-Bruhat function on A n , chosen with care in Sect. 6.1, and c n ·ˆ (0) is the (non-zero) residue at s = 1 of an Eisenstein series attached to a section f s = ⊗ v f v,s , which is defined by . See Sects. 6.1 and 6.4 for the precise definitions of the terms appearing in (1.2). What one should observe is that the value of the partial L-function L S (n, 1) of the trivial character of A at n appears in the formula.
In order for the pole of L S (s, , 2 ) at s = 1 not to cancel with the pole of L S (n, 1) at n = 1, we assumed n ≥ 2, which explains the corresponding assumption in Theorem 1.1 (resp. Theorem 7.4). (As for the case of n = 1, L S (s, , 2 ) = L S (s, 1), the analogue of Theorem 1.1 would boil down to a rationality result for the central critical value of the L-function of unitary cusp forms of GL 2 (A), which is known, e.g., by Harder [15]. Therefore, considering only n ≥ 2 is not a serious restriction.) Observe that the top degree t, mentioned in Theorem 1.1, where has non-trivial cohomology, equals the dimension of the locally symmetric spaces, which are associated to the cycle Z (A)H (F)\H (A), cf. Sect. 5.2. (Here, we necessarily have to use that F is totally real, which explains the last obstruction, set in the beginning.) As a consequence, we may use the de Rham isomorphism. Together with (1.2) and Matringe's equivariance-result (Theorem A) in the Appendix, this finally gives Theorem 1.1.
We point out that, if satisfies the assumptions made in the theorem, then automatically satisfies the assumptions made in Grobner-Raghuram [14]. Hence, the non-zero periods ω 0 ( f ) and ω( ∞ ) constructed in loc. cit. are well-defined. See our Sect. 7.5 below for details. If we define the non-zero, top-degree Whittaker-Shalika periods, where "∼ Q( f ) " means up to multiplication of the right hand side by an element in the number field Q( f ).
In order to obtain our second main theorem on the symmetric square L-function, we need a version of one of the main results of Grobner-Harris-Lapid [12] and Balasubramanyam-Raghuram [2], which is tailored to our present situation at hand. This is achieved in Sect. 8, applying [2] to our particular case. The aforementioned result reads as follows: Theorem 1.3 Let be a self-dual, unitary, cuspidal automorphic representation of G(A) (with trivial central character), which is cohomological with respect to an irreducible, self-contragredient, algebraic, finite-dimensional representation E μ of G ∞ . Then, for every σ ∈ Aut(C), In particular, where "∼ Q( f ) × " means up to multiplication by a non-trivial element in the number field Q( f ).
In the statement of the latter theorem, p t ( ) is the top-degree period defined above, while p b ( ) is defined analogously, but using the lowest degree b, where has nontrivial cohomology. The non-vanishing archimedean period p( ∞ ) is defined in (7.1). We refer to Sects. 8.2 and 8.3 for precise assertions and definitions concerning these periods, in particular Remarks 8.1 and 8.4. The second main theorem of this paper finally deals with the value of the symmetric square L-function at s = 1. Recall that we have L S (s, × ) = L S (s, , Sym 2 ) · L S (s, , 2 ).
As by assumption L S (s, , 2 ) carries the (simple) pole of L S (s, × ) at s = 1, the symmetric square L-function L S (s, , Sym 2 ) is holomorphic and non-vanishing at s = 1. Our second main theorem hence follows by combining Theorem 1.1 (resp. Theorem 7.4) with Theorem 1.3 (resp. Theorem 8.5). We obtain, see Theorem 9.2, Theorem 1.4 Let be a unitary cuspidal automorphic representation of G(A), as in the statement of Theorem 1.1. Then, for every σ ∈ Aut(C), In particular, where "∼ Q( f ) " means up to multiplication of L S (1, , Sym 2 ) by an element in the number field Q( f ).
Similar to before, we may define bottom-degree Whittaker-Shalika periods. Set Then, we have the following corollary, see Corollary 9.4, in which we may get once more rid of the L-factor L( 1 2 , f ), if it is non-zero. Corollary 1.4 Let be as in the statement of Theorem 1.4 (resp. Theorem 9.2). If where "∼ Q( f ) × " means up to multiplication of L S (1, , Sym 2 ) by a non-zero element in the number field Q( f ).
On a final note, we may also derive a theorem for quotients of symmetric square Lfunctions, which is independent of all periods appearing in this paper. We hope that this application of Theorem 1.4-our third main result-serves as an interesting example of the strength of the relation provided by Theorem 1.4 between the symmetric square L-function and our a priori only abstract Whittaker period p b ( ). More precisely, we obtain (cf. Theorem 10.1) Theorem 1.5 Let be a cuspidal automorphic representation of G(A) and let χ 1 and χ 2 be two Hecke characters of finite order, such that ⊗ χ i , i = 1, 2, both satisfy the conditions of Corollary 1.4. If χ 1 and χ 2 have moreover the same infinity-type, i.e., χ 1,∞ = χ 2,∞ , then, where "∼ Q( f ,χ 1, f ,χ 2, f ) × " means up to multiplication by a non-zero element in the composition of number fields Q( f ), Q(χ 1, f ) and Q(χ 2, f ).
It shall be noted that, whereas the quantities on the left hand side of the above equation all depend crucially on , the right hand side is not only independent of the all periods considered in this paper, but completely independent of .

Number fields
In this paper, F denotes a totally real number field of degree d = [F : Q] with ring of integers O. For a place v, let F v be the topological completion of F at v. Let S ∞ be the set of archimedean places of F. If v / ∈ S ∞ , we let O v be the local ring of integers of F v with unique maximal ideal ℘ v . Moreover, A denotes the ring of adèles of F and A f its finite part. We use the local and global normalized absolute values, the first being denoted by | · | v , the latter by · . The fact that F has no complex place is crucial, see Sect. 5.2. Once and for all, we fix a non-trivial additive character ψ : F\A → C × as in [14, §2.7].

Algebraic groups and real Lie groups
Throughout this paper G denotes GL 2n /F, n ≥ 2, the general linear group over F. Although much of the paper works also for GL N with N arbitrary (e.g., the Diagram 5.2), it will be crucial for the main result that N = 2n is even (because only then, the exterior square L-function may have a pole, [20, Theorem 2, p. 224]) and that n ≥ 2 (because the ζ -function attached to F has a pole at n, if n = 1). Let H be GL n × GL n over F. We identify H with a subgroup of G, defined as the image of the homomorphism J : GL n × GL n → GL 2n , where The center of G/F is denoted Z /F. We write G ∞ := R F/Q (G)(R) (resp., where R F/Q stands for Weil's restriction of scalars. Lie algebras of real Lie groups are denoted by the same letter, but in lower case gothics. At an archimedean place v ∈ S ∞ we let K v be a maximal compact subgroup of the real Lie group G( As before, we write K • H,v for the connected component of the identity and we let K H,∞ := v∈S ∞ K H,v and K • H,∞ := v∈S ∞ K • H,v . Let A G be the multiplicative group of positive real numbers R + , being diagonally embedded into the center Z ∞ of G ∞ . It is a direct complement of the group G(A) (1) := {g ∈ G(A)| det(g) = 1} in G(A). According to our conventions, the Lie algebra of the real Lie group A G is denoted a G . Furthermore, we let m G := g ∞ /a G , m H := h ∞ /a G and s := z ∞ /a G . Observe that these spaces are Lie subalgebras of g ∞ .

Cohomology of locally symmetric spaces
Define the orbifolds A representation E μ as in Sect. 2.3 defines a locally constant sheaf E μ on S G , whose espace étalé is G(A) 1 /K • ∞ × G(F) E μ (with the discrete topology on E μ ). Along the proper map J :S H → S G , which is induced by J , Sect. 2.2, we also obtain a sheaf onS H , which we will again denote by be the corresponding space of sheaf cohomology with compact support. This is an Corollary 2.13]. Observe that the proper map J from above gives rise to a non-trivial

Complex automorphisms and rational structures
For σ ∈ Aut(C), let us define the σ -twist σ ν of an (abstract) representation ν of G(A f ) (resp., G(F v ), v / ∈ S ∞ ) on a complex vector space W , following Waldspurger [36], I.1: If W is a C-vector space with a σ -linear isomorphism φ : W → W then we set This definition is independent of φ and W up to equivalence of representations, whence we may always take W := W ⊗ σ C, i.e., the abelian group W endowed with the scalar multiplication λ · σ w := σ −1 (λ)w.
Furthermore, if ν ∞ = v∈S ∞ ν v is an (abstract) representation of the real Lie group G ∞ , we let interpreting v ∈ S ∞ as an embedding of fields v : F → R. Combining these two definitions, we may define the σ -twist on a global representation We recall also the definition of the rationality field of a representation from [36], I.1. If ν is any of the representations considered above, then let S(ν) be the group of all automorphisms σ ∈ Aut(C) such that σ ν ∼ = ν. Then the rationality field Q(ν) of ν is defined as the fixed-field of S(ν) within C, i.e., We say that a representation ν on a C-vector space W is defined over a subfield F ⊂ C, if there is a F-vector subspace W F ⊂ W , stable under the given action, such that the canonical map W F ⊗ F C → W is an isomorphism. The following lemma is due to Clozel, [7, p. 122 and p. 128]. (See also [13,Lemma 7.1].) Lemma 2.6 Let E μ be an irreducible, algebraic representation as in Sect. 2.3. As a representation of the diagonally embedded group G(F) → G ∞ , σ E μ is isomorphic to the abstract representation E μ ⊗ σ C. Moreover, as a representation of G(F), E μ is defined over Q(E μ ).
We fix once and for all a Q(E μ )-structure on E μ as a representation of G(F). Clearly, this also fixes a Q(E μ )-structure on E μ as a representation of H (F). As a consequence, the G( [7, p. 123]. Moreover, this also pins down natural σ -linear, equivariant isomorphisms for all σ ∈ Aut(C), cf. [7, p. 128]. The following lemma is obvious.

Lemma 2.7
For all σ ∈ Aut(C) the following diagram commutes, 3 Facts and conventions for cuspidal automorphic representations

Cohomological cusp forms
In this paper, we let be an irreducible unitary cuspidal automorphic representation of G(A) with trivial central character. Furthermore, we assume that is self-dual, i.e., ∼ = ∨ . This is no loss of generality, as the main result will only hold for such cuspidal representations. (Compare this to Proposition 3.5 below.) Recall that has a (unique) Whittaker model (with respect to ψ). We write is canonically determined by the uniqueness of local Whittaker models. We will furthermore assume that is cohomological: By this we understand that there is an irreducible, algebraic representation E μ of G ∞ , as in Sect. 2.3, such that the archimedean component ∞ of has non-vanishing for some degree q.
Then the following assertions are equivalent: Proof This follows combining the following well-known results on relative Lie algebra cohomology : [5], I. 1.3 (the Künneth rule), I. 5.1, I. Theorem 5.3 (Wigner's lemma) and II. Proposition 3.1 (all cochains are closed and harmonic).
As a consequence, the archimedean component ∞ of a cuspidal automorphic representation , as above, is cohomological in our sense, if and only if ∞ has nonvanishing (g ∞ , K • ∞ )-cohomology or equivalently, non-vanishing (g ∞ , (Z ∞ K ∞ ) • )cohomology with respect to the same algebraic, self-dual coefficient module E μ (although the degrees and dimensions of non-trivial cohomology spaces may change).
Observe furthermore, that (for all degrees q and characters This is well-known and follows from [4, §5].

Rational structures
We have the following result:

Proposition 3.3 Let be a cuspidal automorphic representation of G(A), which is cohomological with respect to E μ . Then, the σ -twisted representation σ is also cuspidal automorphic and it is cohomological with respect to
Proof This is essentially due to Clozel [7]. In order to derive the above result from [7], observe that ∞ is "regular algebraic" in Clozel's sense, if and only if it is cohomological in our sense: This follows using Lemma 3.2 and [13, Theorem 6.3].
Hence, σ f is the non-archimedean part of a cuspidal automorphic representation, which is cohomological with respect to σ E μ by [7,Theorem 3.13]. By uniqueness, see e.g. [13, 5.5], the archimedean part of this cuspidal automorphic representation is isomorphic to σ ∞ as defined above. By [7,Proposition 3 [13,Corollary 8.7].). Finally, it is an implicit consequence of [7, Theorem 3.13] and its proof that Q( f ) is a number field containing Q(E μ ). For a detailed exposition of the latter assertion, we refer to [13,Theorem 8. 1] and the proof of [13,Corollary 8.7].

Lifts from SO(2n + 1)
We resume the assumptions made on from Sect. 3.1. As a last part of notation for , let us introduce S = S( , ψ), which is a (sufficiently large) finite set of places of F, containing S ∞ and such that outside S, both and ψ are unramified.

Proposition 3.5 Let be a cuspidal automorphic representation of G(A) as in Sect.
3.1 above. Then the following assertions are equivalent: 1. The partial exterior square L-function, L S (s, , 2 ), has a pole at s = 1, 2.
is the lift of an irreducible unitary generic cuspidal automorphic representation of the split special orthogonal group SO(2n + 1) in the sense of [8, §1].
Proof With our assumptions on this is [8,Theorem 7.1].
This result is recalled for convenience, as it provides an alternative description of what it means that the exterior square L-function of has a pole at s = 1. We will have to make this assumption later, in order to obtain our main theorems. See, Theorems 6.1, 7.4, 9.2, and 10.1. It is not referred to until Sect. 6.4. In any case, the above result is accompanied by

The map W σ
Recall the unique Whittaker model , having 1 as its last entry, which conjugates ψ v to σ • ψ v . (Observe that t σ,v does not depend on ψ v ). See [26, 3.3] and [30, 3.2]. This provides us a σ -linear intertwining operator for all σ ∈ Aut(C). In particular, we get a Q( v ) structure on W ψ v ( v ) by taking the subspace of Aut(C/Q( v ))-invariant vectors. By the same procedure, we obtain

The map F t
This choice of a character of the component group is forced upon us by the proof of Theorem 7.1 and so we restrict our attention from now on to it.
, chosen in Definition 3.4 above and recall the canonical Q( f )-rational structure on the Whittaker model W ψ f ( f ) of f just fixed in Sect. 4.1 above. As it has been mentioned briefly in the introduction, our top-degree Whittaker period-in abbreviated symbol p t ( )will be determined by the comparison of these two rational structures. Hence, in order to actually compare them, we have to specify a concrete comparison isomorphism It is the purpose of this section to explain this choice carefully, as it is all crucial for the definition of our periods. Our choices will be guided by the ideas in [15, p. 79 The first data we will fix once and for all consists of is a generator of the one-dimensional space . (We may and will also assume that {X j } is the extension of a given ordered basis {Y j } of m H /k H,∞ along our embedding J : H → G. This assumption, however, will only be important later on. See, e.g., Sects. 5.2 and 7.1.)

By [5, II. Proposition 3.1] and the uniqueness of the archimedean Whittaker model and its canonical decomposition into local factors
For the sake of readability we suppress its various dependencies, listed in Choice 4.2 above, in its notation.
Next recall (e.g. from Sect. 2.5) that σ ∈ Aut(C) acts on objects at infinity, which are parameterized by S ∞ , by permuting the archimedean places. Having given a generator Finally, this entails the description of the desired "comparison isomorphism" mentioned at the beginning of this subsection, i.e., of a fixed choice of isomorphism of It is important to observe that we did not have to decompose the global map W ψ computing the ψ-Fourier coefficient, hence there are no hidden ambiguities in this definition: A complete set of dependencies of our comparison isomorphism F t is hence given by Choice 4.2. In light of Proposition 3.3 and our discussion above, we obtain isomorphisms F t σ for all σ ∈ Aut(C) with the same precise set of dependencies.

The map H σ,t μ
As a last ingredient in this section, we define a σ -linear, To that end, recall the embedding q from (3.2) and the σ -linear isomorphism H σ,q μ from (2.1). Observe that Im(H σ,t μ • t ) =Im( t σ ). Indeed, by multiplicity one and strong multiplicity one for the discrete automorphic spectrum of G(A), the as desired. (Shortly speaking, this amounts to say that the restriction

Top-degree Whittaker periods
Recall the maps There is the following result: Proof This is essentially due to the uniqueness of essential vectors for v , v / ∈ S ∞ : Otherwise put, the proof of Proposition/Definition 3.3 in Raghuram-Shahidi [30] goes through word for word in our (slightly different) situation at hand.

Remark 4.4 A lot of choices have been made in order to
give the definition of our topdegree Whittaker periods, while (almost) none of them is reflected explicitly in our choice of notation " p t ( )". So, for the sake of precision, we would like to summarize comprehensively at one place on which data, i.e., fixed chosen ingredients, p t ( ) actually depends: 1. , ψ and the cohomological degree t. 2. The fixed concrete choices of a Q( f )-rational structure on the canonical Whit- The (Whittaker) periods p t ( ) defined by Proposition 4.3 are the analogues of the (Shalika) periods ω ( f ) defined in Grobner-Raghuram DefinitionProposition 4.2.1. The idea behind the construction of p t ( ) (as of ω ( f )), however, goes back to [15,26,30].

The map T μ
will be denoted by the same letter Tσ μ .

The de-Rham-isomorphism R
Then it is easy to see thatS H is homeomorphic to the projective limit The normalized maps, R K f := vol dh f (K f ) · R K f form a system of compatible maps with respect to the pull-backs, given by the coveringsS

In summary: a rational diagram
In the following proposition, we abbreviate

An integral representation of the residue of the exterior square L-function
In this section, we will recapitulate some results from Jacquet-Shalika [21] and Bump-Friedberg [6].

Eisenstein series and a result of Jacquet-Shalika
We resume the notation and assumptions made in the previous sections. In addition, for any integer m ≥ 2, we will now fix once and for all a Schwartz-Bruhat function is the modulus character of the standard parabolic subgroup P of GL m , with Levi subgroup L = GL m−1 × GL 1 . Clearly, the analogous assertion holds for the local components f v,s . There is the following result due to Jacquet-Shalika [ Here, c m is a certain non-zero complex number.

Measures
When dealing with rationality results of special values of L-functions, the choice of measures is all-important. In this section, we specify our choices of measures, which will be guided by the explicit choices made in Bump-Friedberg [6]. Recall the group H = GL n × GL n , Sect. 2.2. We will use the notation (g, g ), to specify an element of H (A) (and use analogous notation locally). A measure of H (A) will be the product of a measures dg and dg as chosen above for m = n of the two isomorphic copies of GL n (A) inside H (A). As Z ⊂ H , also the volume vol dg×dg (Z (F)\Z (A)/A G ) is well-defined and finite.

A result of Bump-Friedberg
Let U n be the group of upper triangular matrices in GL n , having 1 on the diagonal and let Z n be the centre of GL n . Recall the finite set of places S = S( , ψ) from Sect. 3.4. By assumption, outside S, both and ψ are unramified (and ψ normalised). Let

It factors over all places of
Recall the value L S (n, 1) of the partial L-function of the trivial character 1 of A × at n. Since we assumed that n ≥ 2, this number is well-defined and non-zero. The following result is crucial for us: since by assumption L S (s, , 2 ) carries the (simple) pole of the above expression.

Consequences for the σ -twisted case
Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1 and assume that the partial exterior square L-function, L S (s, , 2 ), has a pole at s = 1. Then by Proposition 3.6, σ satisfies the same conditions. Hence, we see that once satisfies the assumptions made in Theorem 6.1, then automatically also σ satisfies them, i.e., Theorem 6.1 holds for the whole Aut(C)-orbit of .
As we are going to use this in the proof of the main results, let us render this more and normalized such that ξ v (id v ) = 1. Given σ ∈Aut(C), let σ ξ ∈ W ψ ( σ ) be the σ -twisted Whittaker function, cf. Sect. 4.1 (the action of σ on the archimedean part of ξ being by permutations as in Sect. 4.2), and let σ ϕ := (W ψ ) −1 ( σ ξ) ∈ σ be the corresponding cuspidal automorphic form. Recall our Schwartz-Bruhat function ∈ S (A n ) from Sect. 6.1, with m = n now. We define the constant This is done purely for cosmetic reasons, as it will become clear below (see the proof of Theorem 7.4). By Sect. 6.1, c n ( , σ ) is non-zero. Let σ ∈ S (A n ) be the Schwartz-Bruhat function which is defined as follows: and an associated Eisenstein series E( σ f s , σ ). Clearly, E( σ f s , σ ) satisfies the assertions of Lemma 6.2, with being replaced by σ .
In summary, with this notation, saying that Theorem 6.1 holds for the whole Aut(C)orbit of , amounts to the equation

Archimedean considerations
The integral representation of the exterior square L-function in Theorem 6.1 allows us to combine the results of Sects. 5 and 6. Before we derive out first main result, we need a non-vanishing theorem, which is an application of Sun's main result in [35].
As a last ingredient, before we can state the aforementioned non-vanishing theorem, we need the following lemma:

are a holomorphic multiple (in s ) of the local archimedean L-function L(s , v ).
Proof This follows combining Theorem 6.1 with [9], Proposition 2.3 and Proposition 3.1 loc. cit. . 1 ) is well-defined. Indeed, using [23], Theorem 2 and Theorem 3 loc. cit., it is easy to see that

It follows that the factor
where h(s ) is holomorphic and non-vanishing for all s ∈ C. Since μ v,k ≥ 0 for all 1 ≤ k ≤ n, by the self-duality hypotheses, cf. Sect. 2.3, L(s , v ) is holomorphic at Here, both numbers L(n, 1 ∞ ) = v∈S ∞ L(n, 1 v ) = π −dn/2 ( n 2 ) d and L S (n, 1) are non-zero. We claim that Sun's aforementioned result now implies the following Proof As a first step and in order to be able to apply Sun's result ( [35], Theorem C), we reduce the problem of showing that c t ( ∞ ) is non-zero to showing that a similarly defined number, d t ( v ) is non-zero. This latter number will only depend on one archimedean place v ∈ S ∞ , whence we find ourselves back in the setting of [35].
To this end, observe that there is a projection where c ∞ := z ∞ ⊕ k ∞ and r = t − d + 1. By reasons of degree, L t induces an isomorphism of (one-dimensional) vector spaces As z ∞ ⊂ h ∞ , and as moreover r Hence, L t and L t factor over the injection where u(i) is the uniquely defined complex number, such that the restriction of L r (X * i ) . Therefore, we may finish the proof by showing that d t ( v ) is non-zero for all v ∈ S ∞ and we are in the situation considered by Sun [35].
Let v ∈ S ∞ be an arbitrary archimedean place. For sake of simplicity, we drop the subscript "v" now everywhere, so, e.g., and analogously for all other local archimedean objects. The local integrals Z (ξ, f 1 ) define a non-zero homomorphism This follows from [6], Theorem 2 and Lemma 7.2. Hence, if we let χ := 1 × 1 be the trivial character of H , then Z (., f 1 ) can be taken as the map ϕ χ in Sun's Theorem C [35]. Next, recall T μ ∈ Hom H (C) (E μ ⊗ C) from Sect. 5.1. If we set w 1 := 0 =: w 2 , then we may take T μ to be the non-zero homomorphism ϕ w 1 ,w 2 from [35, Theorem C]. Hence, loc. cit. , Theorem C, asserts that the map (Here, j 2n is Sun's notation for the embedding h/c H → g/c.) By the onedimensionality of the latter cohomology space, it is hence non-zero on L r ([W ψ ( )] t ). But, then, D computes Hence, reintroducing the subscript "v", and recalling that L(n, 1 v ) = π −n/2 (n/2) = 0, the number d t ( v ) is non-zero for all archimedean places, whence so is c t ( ∞ ).

Definition of the archimedean top-degree period
As a consequence of Proposition 3.6, we may hence define the archimedean periods for all σ ∈Aut(C).

Remark 7.3
Analogously to the case of the top-degree Whittaker period p t ( ), which we dealt with in Remark 4.4, (almost) none of the various choices entering the definition of our archimedean top-degree period p t ( ∞ ) may be found in its notation. For the sake of precision, we would like to summarize at this place on which data, i.e., fixed chosen ingredients, p t ( ∞ ) actually depends: 1. ∞ , ψ ∞ , the cohomological degree t, as well as the archimedean measures dg v and dg v chosen for all v ∈ S ∞ in Sect. 6.3 2. The fixed generator . We remark further that this generator depends itself precisely on the data fixed in Choice 4.2 3. The concrete choice of an intertwining operator It is hence clear that the archimedean top-degree period p t ( ∞ ) depends exclusively on data, which is associated with objects at archimedean places (which explains its name); and that its definition and existence is independent of the definition and proof of existence of our global Whittaker periods p t ( ) from Sect. 4.4.

Rationality of the residue of the exterior square L-function at s = 1
This is our first main theorem. For the precise definitions of p t ( ) and p t ( ∞ ), a comprehensive list of their individual dependencies as well as for their mutual independence, we refer to Sects. 4.4, 7.3, Remarks 4.4 and 7.3 Theorem 7.4 Let F be a totally real number field and G = GL 2n /F, n ≥ 2. Let be a unitary cuspidal automorphic representation of G(A) (self-dual and with trivial central character), which is cohomological with respect to an irreducible, selfcontragredient, algebraic, finite-dimensional representation E μ of G ∞ . Assume that satisfies the equivalent conditions of Proposition 3.5, i.e., the partial exterior square L-function L S (s, , 2 ) has a pole at s = 1. Then, for every σ ∈ Aut(C), In particular, where "∼ Q( f ) " means up to multiplication of the right hand side by an element in the number field Q( f ).
Proof Let be as in the statement of the theorem. We consider the commutative diagram (5.2) in Proposition 5.1: Let be the composition of the upper horizontal arrows, and analogously, let σ be the composition of the lower horizontal arrows. Let In order to prove the theorem, we make both sides of this equation explicit. To that end, be the corresponding cuspidal automorphic form. Recall our Schwartz-Bruhat function ∈ S (A n ) (resp. σ ∈ S (A n )) from Sect. 6.1 (resp. Sect. 6.5), with m = n now. Inserting these functions into Theorem 6.1 [likewise, also into (6.2)] and recalling the definition of our archimedean periods p t ( ∞ ) and p t ( σ ∞ ) from (7.2) shows that Eq. (7.6), induced by our Diagram (5.2), may be rewritten as (Recall that was assumed to have trivial central character.) Invoking our cosmetically tuned choice for σ ∈ S (A n ) from Sect. 6.5, and observing that L(n, Since σ (L( 1 2 , v )) = L( 1 2 , σ v ) = 0 for all v ∈ S\S ∞ , cf. [29,Proposition 3.17], and recalling once more that S = S( , ψ) = S( σ , ψ), we may rewrite this by would have no pole at s = 1 (Here we let ξ v = ξ 0 v,i,α at an archimedean place.). However, reading the proof of Theorem 6.1 backwards, respectively, by [6, Theorem 1 and Theorem 3], the latter expression equals Z (ξ, f s ) L(n, 1) as meromorphic functions in s. By [6, Theorem 1] and our assumption that is a functorial lift from SO(2n + 1), cf. Proposition 3.5, the integral Z (ξ, f s ) has a pole at s = 1, whereas L(n, 1) does not by the assumption that n ≥ 2. Hence, we arrived at a contradiction.
We may therefore finish the proof of the first assertion of Theorem 7.4 by showing that Observing that by a simple change of variable and by our specific choice of The last assertion of the theorem follows by strong multiplicity one for cuspidal automorphic representations of G(A).

Whittaker-Shalika periods and the exterior square L-function
Theorem 7.4 above is accompanied by the following corollary. Recall the non-zero Shalika periods ω ( f ) from Grobner-Raghuram [14]: These were defined by comparing a Q( f )-rational structure on a Shalika model of f and a Q( f )-rational For details, we refer to [14, Definition/Proposition 4.2.1]. Observe that ω 0 ( f ) is well-defined, if we assume that satisfies the assumptions made in the statement of Theorem 7.4: Indeed, as these assumptions include that the partial exterior square L-function L S (s, , 2 ) has a pole at s = 1, has a (1, ψ)-Shalika model by [14, Theorem 3.1.1]. (The extremely careful reader may also recall Lemma 3.2 at this place.) Moreover, by the same reasoning, also the archimedean Shalika period ω( ∞ ) = ω( ∞ , 0) from [14, Theorem 6.6.2] is well-defined (and non-zero). A complete list of all choices, which enter the definition of these Shalika periods ω 0 ( f ) and ω( ∞ ), can be extracted (similar to our considerations leading to Remarks 4.4 and 7.3 above) from [14, Definition/Proposition 4.2.1 and Theorem 6.6.2], where they have been constructed in details. We do not provide such a list here, for the reason that neither ω 0 ( f ) nor ω( ∞ ) appear in the statement of the main theorems (but only in some corollaries).
Define the Whittaker-Shalika periods Obviously the left hand side of (7.3) is uninteresting, if L( 1 2 , f ) = 0. Hence, we allow ourselves to make the strong assumption that L( 1 2 , f ) is non-zero in order to derive the following result:  [12] (but with the totally imaginary field E from [12] being replaced by the totally real field F as a groundfield), respectively Theorem 3.3.11 from [2] (but with the L-value L(1, Ad 0 , π) from [2] being replaced by the residue of L S (s, × ∨ ) at s = 1). For the reason of these close analogies we allow ourselves to be rather brief, when it comes to details. Nevertheless, we think it is worthwhile writing down the following, already for reasons of notation, and in order to give precise statements of results in what follows.

exactly as in Definition 3.4) and we may fix once and for all a generator
in complete analogy to Sect. 4.2, replacing the degree of cohomology t by b in Choice 4.2. Observe that here we exchanged the non-trivial additive character ψ by its inverse ψ −1 and (for notational clearness only), also the index α by β.
Moreover, in light of Proposition 3.6, for all σ ∈Aut(C), we obtain non-trivial Whittaker periods p b ( σ ), unique up to multiplication by elements in Q( σ f ) × , such that commutes. This is the analogue of Proposition 4.3, whose proof goes through word for word in the current situation, i.e., for cohomology in degree b instead of t. See [30, Proposition/Definition 3.3]. In the above diagram, , this map being well-defined following by the same argument as in Sect. 4.3. We leave it to the reader to fill in the remaining details.

Another archimedean period
Recall the Schwartz-Bruhat function = ⊗ v v ∈ S (A 2n ) from Sect. 6.1 with m = 2n in this case. Let U 2n be the subgroup of upper triangular matrices in G = GL 2n , whose diagonal entries are all equal to 1.
) be a local Whittaker function, which is SO(2n)-finite from the right. For such Whittaker functions, the local zeta-integrals Furthermore, by assumption E μ ∼ = E ∨ μ . So, the canonical pairing E μ × E ∨ μ → C induces a pairing E μ × E μ → C, which we will denote by e α , e β := e ∨ β (e α ). As a last ingredient, recall our generators 4.2 and 8.2. Similar to Sect. 7.1, we let s(i, j) be the unique complex number, such that X * i ∧ X * j = s(i, j) · X 1 ∧ · · · ∧ X t+b . Putting things together, consider Then there is the following theorem, which follows from Proposition 5.0.3 in [2].
Proof We may adapt the argument given at the beginning of the proof of Theorem 7.1, to see that the non-vanishing of c( ∞ ) may be reduced to showing the non-vanishing of a similarly defined number d( v ), which only depends on one given archimedean place v ∈ S ∞ . Indeed, there is a projection where we wrote again c ∞ := z ∞ ⊕ k ∞ . By reasons of degrees of cohomology, M b induces an isomorphism of (one-dimensional) vector spaces Whence, at the cost of re-scaling M b (X j ) by the non-trivial factor in 0 s * = R, we may and will assume that M b (X j ) ∈ b (g ∞ /c ∞ ) * . Recall the projection L t = L r ⊗ L d−1 and the isomorphism L t = L r ⊗ L d−1 from the proof of 7.1. 2 Moreover, observe that there is an isomorphism which we factor similarly to L t as N t+b (X 1 ∧ · · · ∧ X t+b ) = N r +b (X 1 ∧ · · · ∧ X t+b ) ⊗ N d−1 (X 1 ∧ · · · ∧ X t+b ), where N r +b (X 1 ∧ · · · ∧ X t+b ) ∈ r +b (g ∞ /c ∞ ) * and N d−1 (X 1 ∧ · · · ∧ X t+b ) ∈ d−1 s * . It hence follows that the number c( ∞ ) is a non-trivial multiple of where u(i, j) is the uniquely defined complex number, such that L r (X . Therefore, we may finish the proof by showing that d( v ) is non-zero for all v ∈ S ∞ . This is the reduction to a single archimedean place v ∈ S ∞ , mentioned at the beginning of the proof. The result hence follows by [2,Proposition 5.0.3].
In view of the latter non-vanishing result and Proposition 3.6, we may define for all σ ∈ Aut(C).

Remark 8.4
Analogously to Remark 7.3, let us recollect at one place the various choices which enter the definition of our archimedean period p( ∞ ), since (almost) none of them appear in its notation: 1. ∞ , ψ ∞ , the cohomological degrees b and t, as well as the archimedean measures dg v chosen for all v ∈ S ∞ in Sect. 6.3. It is hence clear that existence and definition of p( ∞ ) is independent of the other periods considered so far in this paper, p t ( ), p b ( ) and p t ( ∞ ).

Rationality of the residue of the Rankin-Selberg L-function at s = 1
Having set up our additional notation above, we obtain the main result of this section.
In particular, where "∼ Q( f ) × " means up to multiplication by a non-trivial element in the number field Q( f ).
Proof The first assertion follows from Theorem 3.3.11 of [2]. The second assertion of Theorem 8.5 follows from the first one, applying strong multiplicity one for the cuspidal automorphic spectrum of G(A) and recalling that Res s=1 (L S (s, × )) is non-zero. In fact, Res s=1 (L S (s, × )) = 0 is well-known and is a consequence of Theorem 8.2 together with [21] (5), p. 550 and Proposition (2.3) in loc. cit..

Definition of the archimedean bottom-degree period
Let be a cuspidal automorphic representation of G(A) as in Sect. 3.1 and σ ∈ Aut(C). Recall the archimedean periods p t ( σ ∞ ) from (7.2) and p( σ ∞ ) from (7.1). We define our bottom-degree, archimedean period by  (s, , 2 ) has a pole at s = 1. Then, for every σ ∈ Aut(C), In particular, where "∼ Q( f ) " means up to multiplication of L S (1, , Sym 2 ) by an element in the number field Q( f ).
Since L S (1, , Sym 2 ) is non-zero (cf. [32, Theorem 5.1]), the first assertion of the theorem follows from Theorems 7.4 and 8.5. The second assertion is now again a consequence of strong multiplicity one for the cuspidal automorphic spectrum of G(A).

Whittaker-Shalika periods and the symmetric square L-function
As in the case of the exterior square L-function, we obtain a corollary of our second main theorem, Theorem 9.2, using the main results of our paper [14]. Recall the non-zero Shalika periods ω ( f ) and ω( ∞ ) = ω( ∞ , 0) from Sect. 7.5 above, respectively from [14, Definition/Proposition 4.2.1 and Theorem 6.6.2], therein, their existence being guaranteed as in Sect. 7.5. Define the Whittaker-Shalika periods Analogously to the situation considered in Sect. 7.5 above, the right hand side of (9.3) is uninteresting if L( 1 2 , f ) = 0. Hence, we allow ourselves to make the strong assumption that L( 1 2 , f ) is non-zero in order to obtain the following result. Corollary 9.4 Let be as in the statement of Theorem 9.2. If L( 1 2 , f ) is non-zero, then 10 Applications for quotients of symmetric square L-functions

Gauß sums of algebraic Hecke characters
It is the purpose of this section to provide a result, independent of the all the periods mentioned above for certain quotients of symmetric square L-functions.
To that end, let χ be a Hecke character of finite order. We define the Gauß sum of its finite part χ f , following Weil [37,VII,Sect. 7]: Let c χ stand for the conductor ideal of χ f and let y = (y v ) v / ∈S ∞ ∈ A × f be chosen such that ord v (y v ) = −ord v (c χ ) − ord v (D F ). Here, D F stands for the absolute different of F, that is, Recall our fixed non-trivial additive character ψ : F\A → C × from Sect. 2.1. The Gauß sum of χ f with respect to y and ψ is now defined as For almost all v, we have G (χ v , ψ v , y v ) = 1, and for all v we have G (χ v , ψ v , y v ) = 0. (See, for example, Godement [10,Eq. 1.22].) Note that, unlike in [37], we do not normalize the Gauß sum to make it have absolute value one. For the sake of easing notation and readability we suppress its dependence on ψ and y, and denote G (χ f , ψ f , y) simply by G (χ f ).

An application of Theorem 9.2
Theorem 10.1 Let F be a totally real number field and G = GL 2n /F, n ≥ 2. Let be any cuspidal automorphic representation of G(A) and let χ 1 and χ 2 be two Hecke characters of finite order, such that ⊗ χ i , i = 1, 2, both satisfy the conditions of Corollary 9.4. If χ 1 and χ 2 have moreover the same infinity-type, i.e., χ 1,∞ = χ 2,∞ , then, where "∼ Q( f ,χ 1, f ,χ 2, f ) × " means up to multiplication by a non-zero element in the composition of number fields Q( f ), Q(χ 1, f ) and Q(χ 2, f ).
It shall be noted that, whereas the quantities on the left hand side of the above equation all depend crucially on , the right hand side is not only independent of the all periods considered in this paper, but completely independent of .
We remind the reader that since both ⊗ χ i , i = 1, 2, satisfy the assumptions of Theorem 9.2, all periods appearing in their definition are well-defined and non-zero, cf. Sect. 7.5.
Since it follows directly from the definition of rationality fields that Q( f ⊗ χ 1, f )Q( f ⊗ χ 2, f ) = Q( f , χ 1, f , χ 2, f ), our Corollary 9.4 (or, alternatively, Theorem 9.2 together with [14, Theorem 7.1.2]) implies that Moreover, the infinity-types of χ 1 and χ 2 are equal by assumption, which implies that ∞ ⊗ χ 1,∞ and ∞ ⊗ χ 2,∞ are not only isomorphic, but literally identical. As a consequence, the contribution of all archimedean periods above cancels out, and we are left with (We remark aside that in order to see this cancellation it would also have been enough to know that the χ i are of finite order, since then ∞ ⊗ χ i,∞ ∼ = ∞ , for i = 1, 2, see [14, 5.3]. However, the equality ∞ ⊗ χ 1,∞ = ∞ ⊗ χ 2,∞ makes the cancellation even more obvious.) It is exactly the main result of [30,Theorem 4.1.], that-if χ 1 and χ 2 have the same infinity-type, which we assume-one has the relation where ∈ K ∞ /K • ∞ is the same for both i = 1, 2, because we assumed that χ 1 and χ 2 have the same infinity-type. Inserting the relations (10.3) and (10.4) into (10.2), we obtain This shows the claim. Suppose that χ is ramified, i.e., non trivial on O × . Then T (q −s , χ, m, φ) = A(q −s , χ, m, φ).
We denote by P n the mirabolic subgroup of G n = G L(n, F), and by A n the diagonal torus of G n , which is contained in the standard Borel B n with unipotent radical N n . For k ∈ {1, . . . , n − 1}, the group G k embeds naturally in G n , so the center Z k of G k embeds in A n , and A n = Z 1 · · · Z n (direct product). The following result follows from Proposition 2.2 of [19]. We fix a non-trivial additive character ψ of F. If z i belongs to Z i ⊂ A n , we set t (z i ) to be the element of F * such that z i = diag(t (z i ), I n−i ) Proposition B Let π be an irreducible generic representation of G n , and ξ ∈ W ψ (π ). For each k ∈ {1, . . . , n − 1}, there exists a finite set I k , a string of characters (c i k ) i k ∈I k of F * , non-negative integers (m ξ i k ) i k ∈I k , and functions (φ ξ i k ) i k ∈I k such that ξ(z 1 · · · z n−1 ) = n−1 k=1 i k ∈I k n−1 k=1 c i k (t (z k )) v(t (z k )) m ξ i k φ ξ i k (t (z k )).
(The characters c ik , which we allow to be equal, depend only on π .) We denote by w n the element of the symmetric group S n naturally embedded in G n , defined by 1 2 · · · m − 1 m m + 1 m + 2 · · · 2m − 1 2m 1 3 · · · 2m − 3 2m − 1 2 4 · · · 2m − 2 2m when n = 2m is even, and by 1 2 · · · m − 1 m m + 1 m + 2 · · · 2m 2m + 1 1 3 · · · 2m − 3 2m − 1 2m + 1 2 · · · 2m − 2 2m when n = 2m + 1 is odd. We denote by L n the standard Levi subgroup of G n which is G (n+1)/2 × G n/2 embedded by the map (g 1 , g 2 ) → diag(g 1 , g 2 ). We denote by H n the group L w n n = w −1 n L n w n , by J (g 1 , g 2 ) the matrix w −1 n diag(g 1 , g 2 )w n of H n (with diag(g 1 , g 2 ) ∈ L n ). Let r be a positive integer. Thanks to the Iwasawa decomposition G r = N r · A r · G r (O), if χ is an unramified character of A r , then the mapχ : n · a · k → χ(a) is well defined on G r . For example, if δ r is the modulus character of the maximal parabolic subgroup of type (r − 1, 1) restricted to A r , we have a mapδ r on G r . Similarly, if λ : z 1 · · · z r ∈ A r → |t (z 1 ) · · · t (z r −1 )|, the mapλ is also defined on G r , and left invariant under Z r .  (J (b, a)) δ m (a) δ (a) −1 where ξ j (g) = ξ(g J (x j , y j )). We identify A m × A m −1 with A n−1 by  (a, b) → J (b, a), and set χ the character of A n−1 defined by  J (b, a) → δ m (a)δ (a) −1 δ (b) −1 . The previous integral becomes We set χ k to be the restriction of χ to Z k . It takes values in q Z ⊂ Q. If we now apply the second proposition of this appendix, we obtain that is the sum for k between 1 and n − 1, i ∈ I k , and j ∈ {1, . . . , } of This implies, according to the first proposition of this appendix, that σ (Z (ξ, q −s )) is the sum for k between 1 and n − 1, i ∈ I k , and j ∈ {1, . . . , } of This means that σ (Z (ξ, q −s )) is equal to Z (σ • ξ, σ (q −s )).