Notes on L-Functions for Gln

Home , Automorphic form, Basic Number Theory, Global field, Whittaker model

Notes on L-functions for GLn

J.W. Cogdell ∗

Oklahoma State University, Stillwater, USA

Lectures given at the School on Automorphic Forms on GL(n) Trieste, 31 July - 18 August 2000

LNS0821003

∗[email protected] Abstract The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the ζ-function. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL2(Z) by Hecke and his school. Much of our current theory is a direct outgrowth of Hecke’s. L-functions of automorphic representations were first developed by Jacquet and Langlands for GL2. Their approach followed Hecke combined with the local-global techniques of Tate’s thesis. The theory for GLn was then developed along the same lines in a long series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Shalika. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come from a modular form, the so-called Converse Theorem of Hecke. In the context of automorphic representations, the Converse Theorem for GL2 was developed by Jacquet and Langlands, extended and significantly strengthened to GL3 by Jacquet, Piatetski-Shapiro, and Shalika, and then extended to GLn. What we have attempted to present here is a synopsis of this work and in doing so present the paradigm for the analysis of automorphic L-functions via integral representations. We begin with the Fourier expansion of a cusp form and results on Whittaker models since these are essential for defining Eulerian integrals. We then develop integral representations for L-functions for GLn × GLm which have nice analytic properties (meromorphic continuation, finite order of growth, functional equations) and have Eulerian factorization into products of local integrals. We next turn to the local theory of L-functions for GLn, in both the archimedean and non-archimedean local contexts, which comes out of the Euler factors of the global integrals. We finally combine the global Eulerian integrals with the definition and analysis of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties. We next turn to the various Converse Theorems for GLn and their applications to Langlands liftings. Contents

Introduction 79

1 Fourier Expansions and Multiplicity One Theorems 80 1.1 FourierExpansions...... 81 1.2 Whittaker Models and the Multiplicity One Theorem . . . . . 86 1.3 Kirillov Models and the Strong Multiplicity One Theorem . . 89

2 Eulerian Integrals for GLn 91 2.1 Eulerian Integrals for GL2 ...... 92 2.2 Eulerian Integrals for GLn × GLm with mEisenstein series ...... 101 2.3.2 The global integrals ...... 103

3 Local L-functions 106 3.1 The Non-archimedean Local Factors ...... 106 3.1.1 The local L-function ...... 107 3.1.2 The local functional equation ...... 111 3.1.3 The unramiﬁed calculation ...... 115 3.1.4 The supercuspidal calculation ...... 118 3.1.5 Remarks on the general calculation ...... 118 3.1.6 Multiplicativity and stability of γ–factors ...... 119 3.2 The Archimedean Local Factors ...... 119

4 Global L-functions 124 4.1 The Basic Analytic Properties ...... 125 4.2 Poles of L-functions ...... 128 4.3 Strong Multiplicity One ...... 128 4.4 Non-vanishingResults ...... 130 4.5 The Generalized Ramanujan Conjecture (GRC) ...... 131 4.6 The Generalized Riemann Hypothesis (GRH) ...... 134

5 Converse Theorems 134 5.1 TheResults...... 135 5.2 Inverting the Integral Representation ...... 138 5.3 RemarksontheProofs...... 141 5.3.1 Theorem5.1 ...... 141 5.3.2 Theorem5.2 ...... 142 5.3.3 Theorem5.3 ...... 145 5.3.4 Theorem5.4 ...... 146 5.4 Converse Theorems and Liftings ...... 148 5.5 SomeLiftings ...... 151

References 153 L-functions for GLn 79

Introduction

The purpose of these notes is to develop the analytic theory of L-functions for cuspidal automorphic representations of GLn over a global field. There are two approaches to L-functions of GLn: via integral representations or through analysis of Fourier coefficients of Eisenstein series. In these notes we develop the theory via integral representations. The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the ζ- function [53]. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL2(Z) by Hecke and his school [25]. Much of our current theory is a direct outgrowth of Hecke’s. L-functions of automorphic representations were first developed by Jacquet and Langlands for GL2 [21, 28, 30]. Their approach followed Hecke combined with the local-global techniques of Tate’s thesis [64]. The theory for GLn was then developed along the same lines in a long series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Sha- lika [31–38, 47, 48, 62]. In addition to associating an L-function to an automorphic form, Hecke also gave a criterion for a Dirichlet series to come from a modular form, the so-called Converse Theorem of Hecke [26]. In the context of automorphic representations, the Converse Theorem for GL2 was developed by Jacquet and Langlands [30], extended and significantly strengthened to GL3 by Jacquet, Piatetski-Shapiro, and Shalika [31], and then extended to GLn [7,9]. What we have attempted to present here is a synopsis of this work and in doing so present the paradigm for the analysis of automorphic L-functions via integral representations. Section 1 deals with the Fourier expansion of automorphic forms on GLn and the related Multiplicity One and Strong Multiplicity One theorems. Section 2 then develops the theory of Eulerian integrals for GLn. In Section 3 we turn to the local theory of L-functions for GLn, in both the archimedean and non-archimedean local contexts, which comes out of the Euler factors of the global integrals. In Section 4 we finally combine the global Eulerian integrals with the definition and analysis of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties. In Section 5 we turn to the various Converse Theorems for GLn. We have tried to keep the tone of the notes informal for the most part. We have tried to provide complete proofs where feasible, at least sketches of 80 J.W. Cogdell most major results, and references for technical facts. There is another body of work on integral representations of L-functions for GLn which developed out of the classical work on zeta functions of algebras. This is the theory of principal L-functions for GLn as developed by Godement and Jacquet [22,28]. This approach is related to the one pursued here, but we have not attempted to present it here. The other approach to these L-functions is via the Fourier coefficients of Eisenstein series. This approach also has a classical history. In the context of automorphic representations, and in a broader context than GLn, this approach was originally laid out by Langlands [43] but then most fruitfully pursued by Shahidi. Some of the major papers of Shahidi on this subject are [55–61]. In particular, in [58] he shows that the two approaches give the same L-functions for GLn. We will not pursue this approach in these notes. For a balanced presentation of all three methods we recommend the book of Gelbart and Shahidi [16]. They treat not only L-functions for GLn but L-functions of automorphic representations of other groups as well. We have not discussed the arithmetic theory of automorphic representations and L-functions. For the connections with motives, we recommend the surveys of Clozel [5] and Ramakrishnan [50].

1 Fourier Expansions and Multiplicity One Theo- rems

In this section we let k denote a global ﬁeld, A, its ring of adeles, and ψ will denote a continuous additive character of A which is trivial on k. For the basics on adeles, characters, etc. we refer the reader to Weil [68] or the book of Gelfand, Graev, and Piatetski-Shapiro [18]. We begin with a cuspidal automorphic representation (π, Vπ) of GLn(A). For us, automorphic forms are assumed to be smooth (of uniform moderate growth) but not necessarily K∞–ﬁnite at the archimedean places. This is most suitable for the analytic theory. For simplicity, we assume the central character ωπ of π is unitary. Then Vπ is the space of smooth vectors in an irreducible unitary representation of GLn(A). We will always use cuspidal in this sense: the smooth vectors in an irreducible unitary cuspidal automorphic representation. (Any other smooth cuspidal representation π of GLn(A) is necessarily of the form π = π◦ ⊗ | det |t with π◦ unitary and t real, so there is really no loss of generality in the unitarity assumption. It merely provides L-functions for GLn 81

us with a convenient normalization.) By a cusp form on GLn(A) we will mean a function lying in a cuspidal representation. By a cuspidal function we will simply mean a smooth function ϕ on GLn(k)\ GLn(A) satisfying

U(k)\ U(A) ϕ(ug) du ≡ 0 for every unipotent radical U of standard parabolic subgroups of GL . R n The basic references for this section are the papers of Piatetski-Shapiro [47,48] and Shalika [62].

1.1 Fourier Expansions

Let ϕ(g) ∈ Vπ be a cusp form in the space of π. For arithmetic applications, and particularly for the theory of L-functions, we will need the Fourier expansion of ϕ(g). If f(τ) is a holomorphic cusp form on the upper half plane H, say with respect to SL2(Z), then f is invariant under integral translations, f(τ +1) = f(τ) and thus has a Fourier expansion of the form

∞ 2πinτ f(τ)= ane . n=1 X

If ϕ(g) is a smooth cusp form on GL2(A) then the translations correspond 1 x to the maximal unipotent subgroup N = n = and ϕ(ng)= ϕ(g) 2 0 1 for n ∈ N2(k). So, if ψ is any continuous character of k\A we can deﬁne the ψ-Fourier coeﬃcient or ψ-Whittaker function by

1 x −1 Wϕ,ψ(g)= ϕ g ψ (x) dx. A 0 1 Zk\ We have the corresponding Fourier expansion

ϕ(g)= Wϕ,ψ(g). Xψ (Actually from abelian Fourier theory, one has

1 x ϕ g = W (g)ψ(x) 0 1 ϕ,ψ Xψ as a periodic function of x ∈ A. Now set x = 0.) 82 J.W. Cogdell

If we ﬁx a single non-trivial character ψ of k\A, then by standard duality theory [18,68] the additive characters of the compact group k\A are isomorphic to k via the map γ ∈ k 7→ ψγ where ψγ is the character ψγ (x)= ψ(γx). γ Now, an elementary calculation shows that W (g) = W g ϕ,ψγ ϕ.ψ 1 if γ 6= 0. If we set Wϕ = Wϕ,ψ for our ﬁxed ψ, then the Fourier expansion of ϕ becomes γ ϕ(g)= Wϕ,ψ0 (g)+ Wϕ g . × 1 γX∈k Since ϕ is cuspidal 1 x Wϕ,ψ0 (g)= ϕ g dx ≡ 0 A 0 1 Zk\ and the Fourier expansion for a cusp form ϕ becomes simply γ ϕ(g)= Wϕ g . × 1 γX∈k We will need a similar expansion for cusp forms ϕ on GLn(A). The translations still correspond to the maximal unipotent subgroup

1 x1,2 ∗ .   1 ..  Nn = n = .. ..  ,   . .       1 xn−1,n    0 1        but now this is non-abelian. This difficulty was solved independently by Piatetski-Shapiro [47] and Shalika [62]. We fix our non-trivial continuous character ψ of k\A as above. Extend it to a character of Nn by setting ψ(n)= ψ(x1,2 + · · · + xn−1,n) and define the associated Fourier coefficient or Whittaker function by

−1 Wϕ(g)= Wϕ,ψ(g)= ϕ(ng)ψ (n) dn. A ZNn(k)\ Nn( ) Since ϕ is continuous and the integration is over a compact set this integral is absolutely convergent, uniformly on compact sets. The Fourier expansion takes the following form. L-functions for GLn 83

Theorem 1.1 Let ϕ ∈ Vπ be a cusp form on GLn(A) and Wϕ its associated ψ-Whittaker function. Then γ ϕ(g)= W g ϕ 1 γ∈Nn−1(kX)\GLn−1(k) with convergence absolute and uniform on compact subsets.

The proof of this fact is an induction. It utilizes the mirabolic subgroup Pn of GLn which seems to be ubiquitous in the study of automorphic forms on GLn. Abstractly, a mirabolic subgroup of GLn is simply the stabilizer n of a non-zero vector in (either) standard representation of GLn on k . We n denote by Pn the stabilizer of the row vector en = (0,..., 0, 1) ∈ k . So h y P = p = h ∈ GL ,y ∈ kn−1 ≃ GL ⋉ Y n 1 n−1 n−1 n where I y Y = y = n−1 y ∈ kn−1 ≃ kn−1. n 1

Simply by restriction of functions, a cusp form on GLn(A) restricts to a smooth cuspidal function on Pn(A) which remains left invariant under Pn(k). (A smooth function ϕ on Pn(A) which is left invariant under Pn(k) is called cuspidal if U(k)\ U(A) ϕ(up) du ≡ 0 for every standard unipotent subgroup U ⊂ P .) Since P ⊃ N we may deﬁne a Whittaker function attached to a n R n n cuspidal function ϕ on Pn(A) by the same integral as on GLn(A), namely

−1 Wϕ(p)= ϕ(np)ψ (n) dn. A ZNn(k)\ Nn( ) We will prove by induction that for a cuspidal function ϕ on Pn(A) we have γ 0 ϕ(p)= W p ϕ 0 1 γ∈Nn−1(kX)\ GLn−1(k) with convergence absolute and uniform on compact subsets. The function on Yn(A) given by y 7→ ϕ(yp) is invariant under Yn(k) since Yn(k) ⊂ Pn(k) and ϕ is automorphic on Pn(A). Hence by standard n−1 abelian Fourier analysis for Yn ≃ k we have as before

ϕ(p)= ϕλ(p) \ λ∈(knX−1\An−1) 84 J.W. Cogdell where −1 ϕλ(p)= ϕ(yp)λ (y) dy. A ZYn(k)\ Yn( ) Now, by duality theory [68], (kn−\1\An−1) ≃ kn−1. In fact, if we let h , i n−1 n−1 denote the pairing k × k → k by hx,yi = xiyi we have P ϕ(p)= ϕx(p) n−1 x∈Xk where now we write

−1 ϕx(p)= ϕ(yp)ψ (hx,yi) dy. n−1 An−1 Zk \ n−1 n−1 t GLn−1(k) acts on k with two orbits: {0} and k −{0} = GLn−1(k)· t t en−1 where en−1 = (0,..., 0, 1). The stabilizer of en−1 in GLn−1(k) is Pn−1. Therefore, we may write

t ϕ(p)= ϕ0(p)+ ϕγ· en−1 (p). t γ∈GLn−1X(k)/ Pn−1(k)

Since ϕ(p) is cuspidal and Yn is a standard unipotent subgroup of GLn, we see that

ϕ0(p)= ϕ(yp) dy ≡ 0. A ZYn(k)\ Yn( ) On the other hand an elementary calculation as before gives

tγ 0 ϕ t (p)= ϕt p . γ· en−1 en−1 0 1 Hence we have γ 0 ϕ(p)= ϕt p en−1 0 1 γ∈Pn−1(kX)\ GLn−1(k) and the convergence is still absolute and uniform on compact subsets. Note that if n = 2 this is exactly the fact we used previously for GL2. This then begins our induction. Next, let us write the above in a form more suitable for induction. Structurally, we have Pn = GLn−1 ⋉ Yn and Nn = Nn−1 ⋉ Yn. Therefore, Nn−1 \ GLn−1 ≃ Nn \ Pn. Furthermore, if we let Pn−1 = Pn−1 ⋉ Yn ⊂ Pn,

e L-functions for GLn 85

t then Pn−1 \ GLn−1 ≃ Pn−1\ Pn. Next, note that the function ϕ en−1 (p) sat- n−1 isﬁes, for y ∈ Yn(A) ≃ A , e

t t ϕ en−1 (yp)= ψ(yn−1)ϕ en−1 (p).

t Since ψ is trivial on k we see that ϕ en−1 (p) is left invariant under Yn(k). Hence

γ 0 ϕ(p)= ϕt p = ϕt (δp). en−1 0 1 en−1 γ∈Pn−1(k)\ GLn−1(k) − X δ∈Pn 1X(k)\ Pn(k) e

To proceed with the induction, ﬁx p ∈ Pn(A) and consider the function ′ ′ ′ ′ ϕ (p )= ϕp(p ) on Pn−1(A) given by

′ ′ ′ p 0 ϕ (p )= ϕt p . en−1 0 1

′ ′ ϕ is a smooth function on Pn−1(A) since ϕ was smooth. One checks that ϕ is left invariant by Pn−1(k) and cuspidal on Pn−1(A). Then we may apply our inductive assumption to conclude that

′ ′ ′ γ 0 ′ ϕ (p )= Wϕ′ p ′ 0 1 γ ∈Nn−X2 \ GLn−2 ′ ′ = Wϕ′ (δ p ). ′ δ ∈NnX−1 \ Pn−1

If we substitute this into the expansion for ϕ(p) we see

t ϕ(p)= ϕ en−1 (δp) δ∈Pn−1X(k)\ Pn(k) e ′ = ϕδp(1) δ∈Pn−1X(k)\ Pn(k) e ′ = W ′ (δ ). ϕδp ′ δ ∈N −1 \ P −1 δ∈Pn−1X(k)\ Pn(k) nX n e 86 J.W. Cogdell

Now, as before, Nn−1 \ Pn−1 ≃ Nn \Pn−1 and Nn ≃ Nn−1 ⋉ Yn−1. Thus

′ ′ ′ ′ ′ −e1 ′ ′ Wϕ (δ )= ϕδp(n δ )ψ (n ) dn δp A ZNn−1(k)\ Nn−1( ) ′ ′ −1 −1 ′ ′ = ϕ(yn δ δp)ψ (yn−1)ψ (n ) dy dn A A ZNn−1(k)\ Nn−1( ) ZYn(k)\ Yn( ) = ϕ(nδ′δp)ψ−1(n) dn A ZNn(k)\ Nn( ) ′ = Wϕ(δ δp) and so

′ ϕ(p)= Wϕ(δ δp) ′ δ∈Pn−1X(k)\ Pn(k) δ ∈NXn \Pn−1 e e = Wϕ(δp) δ∈Nn(Xk)\ Pn(k) γ 0 = W p ϕ 0 1 γ∈Nn−1(kX)\ GLn−1(k) which was what we wanted. To obtain the Fourier expansion on GLn from this, if ϕ is a cusp form on GLn(A), then for g ∈ Ω a compact subset the functions ϕg(p) = ϕ(pg) form a compact family of cuspidal functions on Pn(A). So we have

γ 0 ϕ (1) = W g ϕg 0 1 γ∈Nn−1(kX)\ GLn−1(k) with convergence absolute and uniform. Hence

γ 0 ϕ(g)= W g ϕ 0 1 γ∈Nn−1(kX)\ GLn−1(k) again with absolute convergence, uniform for g ∈ Ω.

1.2 Whittaker Models and the Multiplicity One Theorem

Consider now the functions Wϕ appearing in the Fourier expansion of a cusp form ϕ. These are all smooth functions W (g) on GLn(A) which satisfy L-functions for GLn 87

W (ng) = ψ(n)W (g) for n ∈ Nn(A). If we let W(π, ψ) = {Wϕ | ϕ ∈ Vπ} then GLn(A) acts on this space by right translation and the map ϕ 7→ Wϕ intertwines Vπ with W(π, ψ). W(π, ψ) is called the Whittaker model of π. The notion of a Whittaker model of a representation makes perfect sense over a local field or even a finite field. Much insight can be gained by pursuing these ideas over a finite field [20,49], but that would take us too far afield. So let kv be a local field (a completion of k for example [18,68]) and let (πv, Vπv ) be an irreducible admissible smooth representation of GLn(kv). Fix a non- trivial continuous additive character ψv of kv. Let W(ψv) be the space of all smooth functions W (g) on GLn(kv) satisfying W (ng) = ψ(n)W (g) for all n ∈ Nk(kv), that is, the space of all smooth Whittaker functions on GLn(kv) with respect to ψv. This is also the space of the smooth induced representation IndGLn (ψ ). GL (k ) acts on this by right translation. If we Nv v n v have a non-trivial continuous intertwining Vπv → W(ψv) we will denote its image by W(πv, ψv) and call it a Whittaker model of πv.

Whittaker models for a representation (πv, Vπv ) are equivalent to continuous Whittaker functionals on Vπv , that is, continuous functionals Λv satisfying Λv(πv(n)ξv)= ψv(n)Λv(ξv) for all n ∈ Nn(kv). To obtain a Whittaker functional from a model, set Λv(ξv) = Wξv (e), and to obtain a model from a functional, set Wξv (g) = Λv(πv(g)ξv). This is a form of Frobenius reci- procity, which in this context is the isomorphism between HomNn (Vπv , Cψv ) and Hom (V , IndGLn (ψ )) constructed above. GLn πv Nn v The fundamental theorem on the existence and uniqueness of Whittaker functionals and models is the following.

Theorem 1.2 Let (πv, Vπv ) be a smooth irreducible admissible representation of GLn(kv). Let ψv be a non-trivial continuous additive character of kv.

Then the space of continuous ψv–Whittaker functionals on Vπv is at most one dimensional. That is, Whittaker models, if they exist, are unique.

This was first proven for non-archimedean fields by Gelfand and Kazh- dan [19] and their results were later extended to archimedean local fields by Shalika [62]. The method of proof is roughly the following. It is enough GLn to show that W(πv) = IndNn (ψv) is multiplicity free, i.e., any irreducible representation of GLn(kv) occurs in W(ψv) with multiplicity at most one. This in turn is a consequence of the commutativity of the endomorphism algebra End(Ind(ψv)). Any intertwining map from Ind(ψv) to itself is given by convolution with a so-called Bessel distribution, that is, a distribution B on GLn(kv) satisfying B(n1gn2) = ψv(n1)B(g)ψv(n2) for n1,n2 ∈ Nn(kv). 88 J.W. Cogdell

Such quasi-invariant distributions are analyzed via Bruhat theory. By the Bruhat decomposition for GLn, the double cosets Nn \ GLn / Nn are param- eterized by the monomial matrices. The only double cosets that can sup- port Bessel distributions are associated to permutation matrices of the form I . rk .. and the resulting distributions are then stable under the   Ir1   1 . involution g 7→ gσ = w tg w with w = .. the long Weyl n n n   1 element of GLn. Thus for the convolution of Bessel distributions we have σ σ σ B1 ∗B2 = (B1 ∗B2) = B2 ∗B1 = B2 ∗B1. Hence the algebra of intertwining Bessel distributions is commutative and hence W(ψv) is multiplicity free.

A smooth irreducible admissible representation (πv, Vπv ) of GLn(kv) which possesses a Whittaker model is called generic or non-degenerate. Gelfand and Kazhdan in addition show that πv is generic iﬀ its contragredient πv is generic, in fact that π ≃ πι where ι is the outer automorphism gι = tg−1, and −1 in this case the Whittaker model for πv can be obtained as W(πv, ψv e) = t −1 {W (g)= W (wn g e) | W ∈W(π, ψv)}. As a consequence of the local uniquenesse of the Whittaker mode el we can concludef a global uniqueness. If (π, Vπ) is an irreducible smooth admissible representation of GLn(A) then π factors as a restricted tensor product of ′ local representations π ≃ ⊗ πv taken over all places v of k [14, 18]. Conse-

.quently we have a continuous embedding Vπv ֒→ Vπ for each local component Hence any Whittaker functional Λ on Vπ determines a family of local Whit- ′ taker functionals Λv on each Vπv and conversely such that Λ = ⊗ Λv. Hence global uniqueness follows from the local uniqueness. Moreover, once we ﬁx ′ the isomorphism of Vπ with ⊗ Vπv and deﬁne global and local Whittaker functions via Λ and the corresponding family Λv we have a factorization of global Whittaker functions

Wξ(g)= Wξv (gv) v Y ′ for ξ ∈ Vπ which are factorizable in the sense that ξ = ⊗ ξv corresponds to a pure tensor. As we will see, this factorization, which is a direct consequence of the uniqueness of the Whittaker model, plays a most important role in the development of Eulerian integrals for GLn. Now let us see what this means for our cuspidal representations (π, Vπ) of GLn(A). We have seen that for any smooth cusp form ϕ ∈ Vπ we have L-functions for GLn 89 the Fourier expansion

γ ϕ(g)= W g . ϕ 1 γ∈Nn−1(kX)\GLn−1(k) We can thus conclude that W(π, ψ) 6= 0 and that π is (globally) generic with Whittaker functional

−1 Λ(ϕ)= Wϕ(e)= ϕ(ng)ψ (n) dn. Z Thus ϕ is completely determined by its associated Whittaker function Wϕ. From the uniqueness of the global Whittaker model we can derive the Mul- tiplicity One Theorem of Piatetski-Shapiro [48] and Shalika [62].

Theorem (Multiplicity One) Let (π, Vπ) be an irreducible smooth admissible representation of GLn(A). Then the multiplicity of π in the space of cusp forms on GLn(A) is at most one.

Proof: Suppose that π has two realizations (π1, Vπ1 ) and (π2, Vπ2 ) in the space of cusp forms on GLn(A). Let ϕi ∈ Vπi be the two cusp forms associated to the vector ξ ∈ Vπ. Then we have two nonzero Whittaker functionals on

Vπ, namely Λi(ξ) = Wϕi (e). By the uniqueness of Whittaker models, there is a nonzero constant c such that Λ1 = cΛ2. But then we have Wϕ1 (g) =

Λ1(π(g)ξ)= cΛ2(π(g)ξ)= cWϕ2 (g) for all g ∈ GLn(A). Thus

γ ϕ (g)= W g 1 ϕ1 1 γ∈Nn−1(kX)\GLn−1(k) γ = c W g = cϕ (g). ϕ2 1 2 γ∈Nn−1(kX)\GLn−1(k)

But then Vπ1 and Vπ2 have a non-trivial intersection. Since they are irreducible representations, they must then coincide. 2

1.3 Kirillov Models and the Strong Multiplicity One Theo- rem The Multiplicity One Theorem can be signiﬁcantly strengthened. The Strong Multiplicity One Theorem is the following. 90 J.W. Cogdell

Theorem (Strong Multiplicity One) Let (π1, Vπ1 ) and (π2, Vπ2 ) be two cuspidal representations of GLn(A). Suppose there is a ﬁnite set of places S such that for all v∈ / S we have π1,v ≃ π2,v. Then π1 = π2.

There are two proofs of this theorem. One is based on the theory of L- functions and we will come to it in Section 4. The original proof of Piatetski- Shapiro [48] is based on the Kirillov model of a local generic representation.

Let kv be a local ﬁeld and let (πv, Vπv ) be an irreducible admissible smooth generic representation of GLn(kv), such as a local component of a cuspidal representation π. Since πv is generic it has its Whittaker model W(πv, ψv). Each Whittaker function W ∈ W(πv, ψv), since it is a function on GLn(kv), can be restricted to the mirabolic subgroup Pn(kv). A fundamental result of Bernstein and Zelevinsky in the non-archimedean case [1] and Jacquet and Shalika in the archimedean case [36] says that the map

ξv 7→ Wξv |Pn(kv) is injective. Hence the representation has a realization on a space of functions on Pn(kv). This is the Kirillov model

K(πv, ψv)= {W (p)|W ∈W(πv, ψv)}.

Pn(kv) acts naturally by right translation on K(πv, ψv) and the action of all of GLn(kv) can be obtained by transport of structure. But for now, it is the structure of K(πv, ψv) as a representation of Pn(kv) which is of interest.

For kv a non-archimedean ﬁeld, let (τv, Vτv ) be the compactly induced representation τ = indPn(kv ) (ψ ). Then Bernstein and Zelevinsky have v Nn(kv ) v analyzed the representations of Pn(kv) and shown that whenever πv is an irreducible admissible generic representation of GLn(kv) then K(πv, ψv) contains Vτv as a Pn(kv) sub-representation and if πv is supercuspidal then

K(πv, ψv)= Vτv [1].

For kv archimedean, we then let (τv, Vτv ) be the smooth vectors in the irreducible smooth unitarily induced representation IndPn(kv) (ψ ). Then Nn(kv) v Jacquet and Shalika have shown that as long as πv is an irreducible admissible smooth unitary representation of GLn(kv) then in fact K(πv, ψv)= Vτv as representations of Pn(kv) [36, Remark 3.15]. Therefore, for a given place v the local Kirillov models of any two irreducible admissible generic smooth unitary representations have a certain

Pn(kv)-submodule in common, namely Vτv . Let us now return to Piatetski-Shapiro’s proof of the Strong Multiplicity One Theorem [48]. L-functions for GLn 91

Proof: We begin with our cuspidal representations π1 and π2. Since π1 and π2 are irreducible, it suffices to find a cusp form ϕ ∈ Vπ1 ∩ Vπ2 . If we let Bn denote the Borel subgroup of upper triangular matrices in GLn, then Bn(k)\ Bn(A) is dense in GLn(k)\ GLn(A) and so it suffices to find two cusp forms ϕi ∈ Vπi which agree on Bn(A). But Bn ⊂ Pn Zn with Zn the center. If we let ωi be the central character of πi then by assumption ω1,v = ω2,v for v∈ / S and the weak approximation theorem then implies ω1 = ω2. So it suffices to find two ϕi which agree on Pn(A). But as in the proof of the Multiplicity One Theorem, via the Fourier expansion, to show that ϕ1(p)= ϕ2(p) for p ∈ Pn(A) it suffices to show that Wϕ1 (p)= Wϕ2 (p).

Since we can take each Wϕi to be of the form v Wϕi,v this then reduces to a question about the local Kirillov models. For v∈ / S we have by as- Q sumption that K(π1,v, ψv) = K(π2,v, ψv) and for v ∈ S we have seen that

Vτv ⊂ K(π1,v, ψv) ∩ K(π2,v, ψv). So we can construct a common Whit- taker function in the restriction of W(πi, ψ) to Pn(A). This completes the proof. 2

2 Eulerian Integrals for GLn

Let f(τ) again be a holomorphic cusp form of weight k on H for the full modular group with Fourier expansion

2πinτ f(τ)= ane . X Then Hecke [25] associated to f an L-function

−s L(s,f)= ann X and analyzed its analytic properties, namely continuation, order of growth, and functional equation, by writing it as the Mellin transform of f

∞ Λ(s,f) = (2π)−sΓ(s)L(s,f)= f(iy)ysd×y. Z0 An application of the modular transformation law for f(τ) under the transformation τ 7→ −1/τ gives the functional equation

Λ(s,f) = (−1)k/2Λ(k − s,f). 92 J.W. Cogdell

Moreover, if f was an eigenfunction of all Hecke operators then L(s,f) had an Euler product expansion

−s k−1−2s −1 L(s,f)= (1 − app + p ) . p Y We will present a similar theory for cuspidal automorphic representations (π, Vπ) of GLn(A). For applications to functoriality via the Converse Theo- rem (see Lecture 5) we will need not only the standard L-functions L(s,π) ′ ′ but the twisted L-functions L(s,π × π ) for (π , Vπ′ ) a cuspidal automorphic representation of GLm(A) for m

2.1 Eulerian Integrals for GL2 Let us ﬁrst consider the L-functions for cuspidal automorphic representations (π, Vπ) of GL2(A) with twists by an idele class character χ, or what is the same, a (cuspidal) automorphic representation of GL1(A), as in Jacquet- Langlands [30]. Following Jacquet and Langlands, who were following Hecke, for each ϕ ∈ Vπ we consider the integral

a I(s; ϕ, χ)= ϕ χ(a)|a|s−1/2 d×a. × A× 1 Zk \ × Since a cusp form on GL2(A) is rapidly decreasing upon restriction to A as in the integral, it follows that the integral is absolutely convergent for all s, uniformly for Re(s) in an interval. Thus I(s; ϕ, χ) is an entire function of s, bounded in any vertical strip a ≤ Re(s) ≤ b. Moreover, if we let ϕ(g) = t −1 t −1 ϕ( g ) = ϕ(wn g ) then ϕ ∈ Vπ and the simple change of variables a 7→ a−1 in the integral shows that eache integral satisﬁes a functional equatione of the form e I(s; ϕ, χ)= I(1 − s; ϕ, χ−1). So these integrals individually enjoy rather nice analytic properties. e L-functions for GLn 93

If we replace ϕ by its Fourier expansion from Lecture 1 and unfold, we ﬁnd

γa s−1/2 × I(s; ϕ, χ)= Wϕ χ(a)|a| d a × × k \A × 1 Z γX∈k a s−1/2 × = Wϕ χ(a)|a| d a A× 1 Z where we have used the fact that the function χ(a)|a|s−1/2 is invariant under k×. By standard gauge estimates on Whittaker functions [31] this converges for Re(s) >> 0 after the unfolding. As we have seen in Lecture 1, if Wϕ ∈ ′ W(π, ψ) corresponds to a decomposable vector ϕ ∈ Vπ ≃ ⊗ Vπv then the Whittaker function factors into a product of local Whittaker functions

Wϕ(g)= Wϕv (gv). v Y Since the character χ and the adelic absolute value factor into local components and the domain of integration A× also factors we ﬁnd that our global integral naturally factors into a product of local integrals

a s−1/2 × av s−1/2 × Wϕ χ(a)|a| d a = Wϕv χv(av)|av| d av, A× 1 × 1 v kv Z Y Z with the inﬁnite product still convergent for Re(s) >> 0, or

I(s; ϕ, χ)= Ψv(s; Wϕv ,χv) v Y with the obvious deﬁnition of the local integrals

av s−1/2 × Ψv(s; Wϕ ,χv)= Wϕ χv(av)|av| d av. v × v 1 Zkv Thus each of our global integrals is Eulerian. In this way, to π and χ we have associated a family of global Eulerian integrals with nice analytic properties as well as for each place v a family of local integrals convergent for Re(s) >> 0. 94 J.W. Cogdell

2.2 Eulerian Integrals for GLn × GLm with m < n ′ Now let (π, Vπ) be a cuspidal representation of GLn(A) and (π , Vπ′ ) a cus- ′ pidal representation of GLm(A) with m

h ϕ ϕ′(h)| det(h)|s−(n−m)/2 dh. A In−m ZGLm(k)\ GLm( ) This family of integrals would have all the nice analytic properties as before (entire functions of ﬁnite order satisfying a functional equation), but they would not be Eulerian except in the case m = n − 1, which proceeds exactly as in the GL2 case. The problem is that the restriction of the form ϕ to GLm is too brutal to allow a nice unfolding when the Fourier expansion of ϕ is inserted. Instead we will introduce projection operators from cusp forms on GLn(A) to cuspidal functions on on Pm+1(A) which are given by part of the unipotent integration through which the Whittaker function is deﬁned.

2.2.1 The projection operator

In GLn, let Yn,m be the unipotent radical of the standard parabolic subgroup attached to the partition (m + 1, 1,..., 1). If ψ is our standard additive character of k\A, then ψ deﬁnes a character of Yn,m(A) trivial on Yn,m(k) since Yn,m ⊂ Nn. The group Yn,m is normalized by GLm+1 ⊂ GLn and the mirabolic subgroup Pm+1 ⊂ GLm+1 is the stabilizer in GLm+1 of the character ψ.

Deﬁnition If ϕ(g) is a cusp form on GLn(A) deﬁne the projection operator n Pm from cusp forms on GLn(A) to cuspidal functions on Pm+1(A) by

n−m−1 n −“ 2 ” p −1 Pmϕ(p)= | det(p)| ϕ y ψ (y) dy A In−m−1 ZYn,m(k)\ Yn,m( ) for p ∈ Pm+1(A).

As the integration is over a compact domain, the integral is absolutely convergent. We ﬁrst analyze the behavior on Pm+1(A).

n Lemma The function Pmϕ(p) is a cuspidal function on Pm+1(A). L-functions for GLn 95

Proof: Let us let ϕ′(p) denote the non-normalized projection, i.e., for p ∈ Pm+1(A) set n−m−1 ′ “ 2 ” n ϕ (p)= | det(p)| Pmϕ(p). It suﬃces to show this function is cuspidal. Since ϕ(g) was a smooth function ′ ′ on GLn(A), ϕ (p) will remain smooth on Pm+1(A). To see that ϕ (p) is automorphic, let γ ∈ Pm+1(k). Then γ 0 p 0 ϕ′(γp)= ϕ y ψ−1(y) dy. A 0 1 0 1 ZYn,m(k)\ Yn,m( )

Since γ ∈ Pm+1(k) and Pm+1 normalizes Yn,m and stabilizes ψ we may make −1 γ 0 γ 0 the change of variable y 7→ y in this integral to obtain 0 1 0 1 γ 0 p 0 ϕ′(γp)= ϕ y ψ−1(y) dy. A 0 1 0 1 ZYn,m(k)\ Yn,m( )

Since ϕ(g) is automorphic on GLn(A) it is left invariant under GLn(k) and ′ ′ ′ we ﬁnd that ϕ (γp)= ϕ (p) so that ϕ is indeed automorphic on Pm+1(A). ′ We next need to see that ϕ is cuspidal on Pm+1(A). To this end, let U ⊂ Pm+1 be the standard unipotent subgroup associated to the partition (n1,...,nr) of m + 1. Then we must compute the integral

ϕ′(up) du. A ZU(k)\ U( ) Inserting the deﬁnition of ϕ′ we ﬁnd

ϕ′(up) du A ZU(k)\ U( ) u 0 p 0 = ϕ y ψ−1(y) dy du. A A 0 1 0 1 ZU(k)\ U( ) ZYn,m(k)\ Yn,m( ) ′ The group U = U ⋉ Yn,m is the standard unipotent subgroup of GLn associated to the partition (n1,...,nr, 1,..., 1) of n. We may decompose this group in a second manner. If we let U′′ be the standard unipotent subgroup of GLn associated to the partition (n1,...,nr,n−m−1) of n and let Nn−m−1 be the subgroup of GLn obtained by embedding Nn−m−1 into GLn by e I 0 n 7→ m+1 0 n 96 J.W. Cogdell

′ ′′ ′ then U = Nn−m−1 ⋉ U . If we extend the character ψ of Ym,n to U by ′ ′′ making it trivial on U, then in the decomposition U = Nn−m−1 ⋉ U , ψ is dependente only on the Nn−m−1 component and there it is the standard character ψ on Nn−m−1. Hence we may rearrange the integratione to give e ϕ′(up) du A ZU(k)\ U( ) 1 0 p 0 = ϕ u′′ du′′ψ−1(n) dy. A ′′ ′′ A 0 n 0 1 ZNn−m−1(k)\ Nn−m−1( ) ZU (k)\ U ( ) ′′ But since ϕ is cuspidal on GLn and U is a standard unipotent subgroup of GLn then 1 0 p 0 ϕ u′′ du′′ ≡ 0 ′′ ′′ A 0 n 0 1 ZU (k)\ U ( ) from which it follows that

ϕ′(up) du ≡ 0 A ZU(k)\ U( ) ′ so that ϕ is a cuspidal function on Pm+1(A). 2

From Lecture 1, we know that cuspidal functions on Pm+1(A) have a Fourier expansion summed over Nm(k)\ GLm(A). Applying this expansion to our projected cusp form on GLn(A) we are led to the following result. h Lemma Let ϕ be a cusp form on GL (A). Then for h ∈ GL (A), Pn ϕ n m m 1 has the Fourier expansion n−m−1 n h −“ 2 ” γ 0 h Pmϕ = | det(h)| Wϕ 1 0 In−m In−m γ∈Nm(kX)\ GLm(k) with convergence absolute and uniform on compact subsets.

Proof: Once again let n−m−1 ′ “ 2 ” n ϕ (p)= | det(p)| Pmϕ(p) ′ with p ∈ Pm+1(A). Since we have veriﬁed that ϕ (p) is a cuspidal function on Pm+1(A) we know that it has a Fourier expansion of the form

′ γ 0 ϕ (p)= W ′ p ϕ 0 1 γ∈Nm(kX)\ GLm(k) L-functions for GLn 97 where ′ −1 Wϕ′ (p)= ϕ (np)ψ (n) dn. A ZNm+1(k)\ Nm+1( ) n To obtain our expansion for Pmϕ we need to express the right-hand side in terms of ϕ rather than ϕ′. We have

′ ′ −1 ′ ′ Wϕ′ (p)= ϕ (n p)ψ (n ) dn A ZNm+1(k)\ Nm+1( ) n′p 0 = ϕ y g · A A 0 1 ZNm+1(k)\ Nm+1( ) ZYn,m(k)\ Yn,m( ) · ψ−1(y) dy ψ−1(n′) dn′.

It is elementary to see that the maximal unipotent subgroup Nn of GLn can ′ ′ be factored as Nn = Nm+1 ⋉ Yn,m and if we write n = n y with n ∈ Nm+1 ′ and y ∈ Yn,m then ψ(n) = ψ(n )ψ(y). Hence the above integral may be written as

p 0 −1 p 0 Wϕ′ (p)= ϕ n ψ (n) dn = Wϕ . A 0 In−m−1 0 In−m−1 ZNn(k)\ Nn( ) Substituting this expression into the above we ﬁnd that h Pn ϕ m 1 n−m−1 −“ 2 ” γ 0 h = | det(h)| Wϕ 0 In−m In−m γ∈Nm(kX)\ GLm(k) and the convergence is absolute and uniform for h in compact subsets of GLm(A). 2

2.2.2 The global integrals We now have the prerequisites for writing down a family of Eulerian integrals for cusp forms ϕ on GLn twisted by automorphic forms on GLm for m < ′ n. Let ϕ ∈ Vπ be a cusp form on GLn(A) and ϕ ∈ Vπ′ a cusp form on ′ GLm(A). (Actually, we could take ϕ to be an arbitrary automorphic form on GLm(A).) Consider the integrals

′ n h 0 ′ s−1/2 I(s; ϕ, ϕ )= Pmϕ ϕ (h)| det(h)| dh. A 0 1 ZGLm(k)\ GLm( ) 98 J.W. Cogdell

The integral I(s; ϕ, ϕ′) is absolutely convergent for all values of the complex parameter s, uniformly in compact subsets, since the cusp forms are rapidly decreasing. Hence it is entire and bounded in any vertical strip as before. Let us now investigate the Eulerian properties of these integrals. We ﬁrst n replace Pmϕ by its Fourier expansion.

′ n h 0 ′ s−1/2 I(s; ϕ, ϕ )= Pmϕ ϕ (h)| det(h)| dh A 0 In−m ZGLm(k)\ GLm( ) γ 0 h 0 = Wϕ · GLm(k)\ GLm(A) 0 In−m 0 In−m Z γ∈Nm(kX)\ GLm(k) · ϕ′(h)| det(h)|s−(n−m)/2 dh.

′ Since ϕ (h) is automorphic on GLm(A) and | det(γ)| = 1 for γ ∈ GLm(k) we may interchange the order of summation and integration for Re(s) >> 0 and then recombine to obtain

′ h 0 ′ s−(n−m)/2 I(s; ϕ, ϕ )= Wϕ ϕ (h)| det(h)| dh. A 0 In−m ZNm(k)\ GLm( ) This integral is absolutely convergent for Re(s) >> 0 by the gauge estimates of [31, Section 13] and this justiﬁes the interchange. Let us now integrate ﬁrst over Nm(k)\ Nm(A). Recall that for n ∈ Nm(A) ⊂ Nn(A) we have Wϕ(ng)= ψ(n)Wϕ(g). Hence we have I(s; ϕ, ϕ′)= n 0 h 0 Wϕ · A A A 0 In−m 0 In−m ZNm( )\ GLm( ) ZNm(k)\ Nm( ) · ϕ′(nh) dn | det(h)|s−(n−m)/2 dh

h 0 ′ = Wϕ ψ(n)ϕ (nh) dn A A 0 In−m A ZNm( )\ GLm( ) ZNm(k)\ Nm( ) · | det(h)|s−(n−m)/2 dh

h 0 ′ s−(n−m)/2 = Wϕ Wϕ′ (h)| det(h)| dh A A 0 In−m ZNm( )\ GLm( ) ′ = Ψ(s; Wϕ,Wϕ′ ) ′ −1 ′ where Wϕ′ (h) is the ψ -Whittaker function on GLm(A) associated to ϕ , i.e., ′ ′ Wϕ′ (h)= ϕ (nh)ψ(n) dn, A ZNm(k)\ Nm( ) L-functions for GLn 99 and we retain absolute convergence for Re(s) >> 0. From this point, the fact that the integrals are Eulerian is a consequence of the uniqueness of the Whittaker model for GLn. Take ϕ a smooth cusp form in a cuspidal representation π of GLn(A). Assume in addition ′ that ϕ is factorizable, i.e., in the decomposition π = ⊗ πv of π into a restricted tensor product of local representations, ϕ = ⊗ϕv is a pure tensor. Then as we have seen there is a choice of local Whittaker models so that ′ Wϕ(g) = Wϕv (gv). Similarly for decomposable ϕ we have the factoriza- ′ ′ tion W ′ (h)= W ′ (hv). ϕ Q ϕv If we substitute these factorizations into our integral expression, then Q since the domain of integration factors Nm(A)\ GLm(A)= Nm(kv)\ GLm(kv) we see that our integral factors into a product of local integrals Q

′ hv 0 ′ ′ ′ Ψ(s; Wϕ,Wϕ )= Wϕv Wϕ (hv) · 0 In−m v v ZNm(kv)\ GLm(kv) Y s−(n−m)/2 ·| det(hv)|v dhv.

If we denote the local integrals by

′ hv 0 ′ ′ ′ Ψv(s; Wϕv ,Wϕ )= Wϕv Wϕ (hv) · v 0 In−m v ZNm(kv )\ GLm(kv) s−(n−m)/2 ·| det(hv)|v dhv, which converges for Re(s) >> 0 by the gauge estimate of [31, Prop. 2.3.6], we see that we now have a family of Eulerian integrals. Now let us return to the question of a functional equation. As in the case of GL2, the functional equation is essentially a consequence of the existence ι t −1 of the outer automorphism g 7→ ι(g) = g = g of GLn. If we deﬁne the action of this automorphism on automorphic forms by setting ϕ(g) = ι ι n n ϕ(g )= ϕ(wng ) and let Pm = ι ◦ Pm ◦ ι then our integrals naturally satisfy the functional equation e e I(s; ϕ, ϕ′)= I(1 − s; ϕ, ϕ′) where e e e

′ n h ′ s−1/2 I(s; ϕ, ϕ )= Pmϕ ϕ (h)| det(h)| dh. A 1 ZGLm(k)\ GLm( ) Wee have established the followinge result. 100 J.W. Cogdell

′ Theorem 2.1 Let ϕ ∈ Vπ be a cusp form on GLn(A) and ϕ ∈ Vπ′ a cusp ′ form on GLm(A) with m> 0.

The integrals occurring on the right-hand side of our functional equation are again Eulerian. One can unfold the definitions to find first that

′ ′ I(1 − s; ϕ, ϕ )= Ψ(1 − s; ρ(wn,m)Wϕ, Wϕ′ ) where the unfoldede globale integrale e is f f h Ψ(s; W, W ′)= W x I dx W ′(h)| det(h)|s−(n−m)/2 dh  n−m−1  ZZ 1 e   with the h integral over Nm(A)\ GLm(A) and the x integral over Mn−m−1,m(A), the space of (n − m − 1) × m matrices, ρ denoting right translation, and 1 I . w the Weyl element w = m with w = .. n,m n,m w n−m   n−m 1 the standard long Weyl element in GLn−m. Also, for W ∈ W(π, ψ) we set ι −1 W (g)= W (wng ) ∈W(π, ψ ). The extra unipotent integration is the rem- n ′ nant of Pm. As before, Ψ(s; W, W ) is absolutely convergent for Re(s) >> 0. ′ ′ Forf ϕ and ϕ factorizablee as before, these integrals Ψ(s; Wϕ,Wϕ′ ) will factor as well.e Hence we have e e ′ ′ Ψ(s; W ,W ′ )= Ψ (s; W ,W ′ ) ϕ ϕ v ϕv ϕv v Y e e where

hv Ψ (s; W ,W ′ )= W x I dx W ′ (h )| det(h )|s−(n−m)/2 dh v v v v  v n−m−1  v v v v v ZZ 1 e   L-functions for GLn 101

where now the hv integral is over Nm(kv)\ GLm(kv) and the xv integral is over the matrix space Mn−m−1,m(kv). Thus, coming back to our functional equation, we ﬁnd that the right-hand side is Eulerian and factors as

′ ′ ′ I(1−s; ϕ, ϕ )= Ψ(1−s; ρ(w )W , W ′ )= Ψ (1−s; ρ(w )W , W ′ ). n,m ϕ ϕ v n,m ϕv ϕv v Y e e e e f f e f f 2.3 Eulerian Integrals for GLn × GLn

The paradigm for integral representations of L-functions for GLn × GLn is not Hecke but rather the classical papers of Rankin [52] and Selberg [54]. These were ﬁrst interpreted in the framework of automorphic representations by Jacquet for GL2 × GL2 [28] and then Jacquet and Shalika in general [36]. ′ Let (π, Vπ) and (π , Vπ′ ) be two cuspidal representations of GLn(A). Let ′ ϕ ∈ Vπ and ϕ ∈ Vπ′ be two cusp forms. The analogue of the construction above would be simply

ϕ(g)ϕ′(g)| det(g)|s dg. A ZGLn(k)\ GLn( ) This integral is essentially the L2-inner product of ϕ and ϕ′ and is not suitable for deﬁning an L-function, although it will occur as a residue of our integral at a pole. Instead, following Rankin and Selberg, we use an integral representation that involves a third function: an Eisenstein series on GLn(A). This family of Eisenstein series is constructed using the mirabolic subgroup once again.

2.3.1 The mirabolic Eisenstein series

To construct our Eisenstein series we return to the observation that Pn \ GLn ≃ kn −{0}. If we let S(An) denote the Schwartz–Bruhat functions on An, then each Φ ∈ S deﬁnes a smooth function on GLn(A), left invariant by Pn(A), by g 7→ Φ((0,..., 0, 1)g) =Φ(eng). Let η be a unitary idele class character. (For our application η will be determined by the central characters of π and π′.) Consider the function

s ns × F (g, Φ; s, η)= | det(g)| Φ(aeng)|a| η(a) d a. A× Z ′ If we let Pn = Zn Pn be the parabolic of GLn associated to the partition h y (n−1, 1) then one checks that for p′ = ∈ P′ (A) with h ∈ GL (A) 0 d n n−1 102 J.W. Cogdell and d ∈ A× we have,

F (p′g, Φ; s, η) = | det(h)|s|d|−(n−1)sη(d)−1F (g, Φ; s, η) s ′ −1 = δ ′ (p )η (d)F (g, Φ; s, η), Pn with the integral absolutely convergent for Re(s) > 1/n, so that if we extend ′ ′ η to a character of Pn by η(p ) = η(d) in the above notation we have that F (g, Φ; s, η) is a smooth section of the normalized induced representation GLn(A) s−1/2 s−1/2 ′ Ind ′ A (δ ′ η). Since the inducing character δ ′ η of P (A) is invari- Pn( ) Pn Pn n ′ ant under Pn(k) we may form Eisenstein series from this family of sections by E(g, Φ; s, η)= F (γg, Φ; s, η). ′ γ∈Pn(kX)\ GLn(k) If we replace F in this sum by its deﬁnition we can rewrite this Eisenstein series as

E(g, Φ; s, η)= | det(g)|s Φ(aξg)|a|nsη(a) d×a × × k \A n Z ξ∈kX−{0} s ′ ns × = | det(g)| ΘΦ(a, g)|a| η(a) d a × A× Zk \ and this ﬁrst expression is convergent absolutely for Re(s) > 1 [36]. The second expression essentially gives the Eisenstein series as the Mellin transform of the Theta series

ΘΦ(a, g)= Φ(aξg), n ξX∈k where in the above we have written

′ ΘΦ(a, g)= Φ(aξg)=ΘΦ(a, g) − Φ(0). n ξ∈kX−{0} This allows us to obtain the analytic properties of the Eisenstein series from the Poisson summation formula for ΘΦ, namely

Φ(ˆ x)= Φ(y)ψ(ytx) dy. A× Z This allows us to write the Eisenstein series as

Proposition 2.1 The Eisenstein series E(g, Φ; s, η) has a meromorphic continuation to all of C with at most simple poles at s = −iσ, 1 − iσ when η is unramiﬁed of the form η(a) = |a|inσ. As a function of g it is smooth of moderate growth and as a function of s it is bounded in vertical strips (away from the possible poles), uniformly for g in compact sets. Moreover, we have the functional equation

E(g, Φ; s, η)= E(gι, Φ;ˆ 1 − s, η−1) where gι = tg−1.

Note that under the center the Eisenstein series transforms by the central character η−1.

2.3.2 The global integrals Now let us return to our Eulerian integrals. Let π and π′ be our irreducible cuspidal representations. Let their central characters be ω and ω′. Set ′ ′ η = ωω . Then for each pair of cusp forms ϕ ∈ Vπ and ϕ ∈ Vπ′ and each Schwartz-Bruhat function Φ ∈ S(An) set

I(s; ϕ, ϕ′, Φ) = ϕ(g)ϕ′(g)E(g, Φ; s, η) dg. A A ZZn( ) GLn(k)\ GLn( ) 104 J.W. Cogdell

Since the two cusp forms are rapidly decreasing on Zn(A) GLn(k)\ GLn(A) and the Eisenstein series is only of moderate growth, we see that the integral converges absolutely for all s away from the poles of the Eisenstein series and is hence meromorphic. It will be bounded in vertical strips away from the poles and satisﬁes the functional equation

I(s; ϕ, ϕ′, Φ) = I(1 − s; ϕ, ϕ′, Φ)ˆ , coming from the functional equation of the Eisensteine e series, where we still ι ι ′ have ϕ(g)= ϕ(g )= ϕ(wng ) ∈ Vπ and similarly for ϕ . These integrals will be entiree unless we have η(a) = ω(a)ω′(a) = |a|inσ is unramiﬁed.e In that case, the residue at s = −iσ wille be

Res I(s; ϕ, ϕ′, Φ) = −cΦ(0) ϕ(g)ϕ′(g)| det(g)|−iσ dg s=−iσ A A A ZZn( ) GLn( )\ GLn( ) and at s = 1 − iσ we can write the residue as

Res I(s; ϕ, ϕ′, Φ) = cΦ(0)ˆ ϕ(g)ϕ′(g)| det(g)|iσ dg. s=1−iσ A A ZZn( ) GLn(k)\ GLn( )

Therefore these residues define GLn(A) invariante pairingse between π and π′ ⊗ | det |−iσ or equivalently between π and π′ ⊗ | det |iσ. Hence a residues can be non-zero only if π ≃ π′ ⊗ | det |iσ and in this case we can find ϕ, ϕ′, and Φ such that indeed the residue doese not vanish.e We have yet to check thate our integrals are Eulerian. To this end we take the integral, replace the Eisenstein series by its definition, and unfold:

I(s; ϕ, ϕ′, Φ) = ϕ(g)ϕ′(g)E(g, Φ; s, η) dg A A ZZn( ) GLn(k)\ GLn( ) = ϕ(g)ϕ′(g)F (g, Φ; s, η) dg A ′ A ZZn( )Pn(k)\ GLn( ) ′ s ns = ϕ(g)ϕ (g)| det(g)| Φ(aeng)|a| η(a) da dg A A A× ZZn( )Pn(k)\ GLn( ) Z ′ s = ϕ(g)ϕ (g)Φ(eng)| det(g)| dg. A ZPn(k)\ GLn( ) We next replace ϕ by its Fourier expansion in the form

ϕ(g)= Wϕ(γg) γ∈Nn(Xk)\ Pn(k) L-functions for GLn 105 and unfold to ﬁnd

′ ′ s I(s; ϕ, ϕ , Φ) = Wϕ(g)ϕ (g)Φ(eng)| det(g)| dg A ZNn(k)\ GLn( ) ′ s = Wϕ(g) ϕ (ng)ψ(n) dn Φ(eng)| det(g)| dg A A A ZNn( )\ GLn( ) ZNn(k)\ Nn( ) ′ s = Wϕ(g)Wϕ′ (g)Φ(eng)| det(g)| dg A A ZNn( )\ GLn( ) ′ = Ψ(s; Wϕ,Wϕ′ , Φ). This expression converges for Re(s) >> 0 by the gauge estimates as before. To continue, we assume that ϕ, ϕ′ and Φ are decomposable tensors under ′ ′ ′ ′ n ′ n the isomorphisms π ≃ ⊗ πv, π ≃ ⊗ πv, and S(A ) ≃ ⊗ S(kv ) so that we ′ ′ have W (g) = W (g ), W ′ (g) = W ′ (g ) and Φ(g) = Φ (g ). ϕ v ϕv v ϕ v ϕv v v v v Then, since the domain of integration also naturally factors we can decom- Q Q Q pose this last integral into an Euler product and now write ′ ′ Ψ(s; W ,W ′ , Φ) = Ψ (s; W ,W ′ , Φ ), ϕ ϕ v ϕv ϕv v v Y where

′ ′ s Ψ (s; W ,W ′ , Φ )= W (g )W ′ (g )Φ (e g )| det(g )| dg , v ϕv ϕv v ϕv v ϕv v v n v v v ZNn(kv)\ GLn(kv) still with convergence for Re(s) >> 0 by the local gauge estimates. Once again we see that the Euler factorization is a direct consequence of the uniqueness of the Whittaker models.

′ Theorem 2.2 Let ϕ ∈ Vπ and ϕ ∈ Vπ′ cusp forms on GLn(A) and let Φ ∈ S(An). Then the family of integrals I(s; ϕ, ϕ′, Φ) deﬁne meromorphic functions of s, bounded in vertical strips away from the poles. The only possible poles are simple and occur iﬀ π ≃ π′ ⊗ | det |iσ with σ real and are then at s = −iσ and s = 1 − iσ with residues as above. They satisfy the functional equation e ′ ′ ˆ I(s; ϕ, ϕ , Φ) = I(1 − s; Wϕ, Wϕ′ , Φ). Moreover, for ϕ, ϕ′, and Φ factorizable we have that the integrals are Eule- f f rian and we have ′ ′ I(s; ϕ, ϕ , Φ) = Ψ (s; W ,W ′ , Φ ) v ϕv ϕv v v Y with convergence absolute and uniform for Re(s) >> 0. 106 J.W. Cogdell

We remark in passing that the right-hand side of the functional equation also unfolds as ′ ˆ ′ ˆ 1−s I(1 − s; ϕ, ϕ , Φ) = Wϕ(g)Wϕ′ (g)Φ(eng)| det(g)| dg A A ZNn( )\ GLn( ) ′ ˆ e e = Ψv(1 − s; Wfϕ, Wϕf′ , Φ) v Y with convergence for Re(s) << 0. f f We note again that if these integrals are not entire, then the residues give us invariant pairings between the cuspidal representations and hence tell us structural facts about the relation between these representations.

3 Local L-functions

′ If (π, Vπ) is a cuspidal representation of GLn(A) and (π , Vπ′ ) is a cuspidal ′ representation of GLm(A) we have associated to the pair (π,π ) a family of Eulerian integrals {I(s; ϕ, ϕ′)} (or {I(s; ϕ, ϕ′, Φ)} if m = n) and through the Euler factorization we have for each place v of k a family of local integrals ′ ′ {Ψv(s; Wv,Wv)} (or {Ψv(s; Wv,Wv, Φv)}) attached to the pair of local com- ′ ponents (πv,πv). In this lecture we would like to attach a local L-function ′ (or local Euler factor) L(s,πv × πv) to such a pair of local representations through the family of local integrals and analyze its basic properties, including the local functional equation. The paradigm for such an analysis of local L-functions is Tate’s thesis [64]. The mechanics of the archimedean and non- archimedean theories are slightly diﬀerent so we will treat them separately, beginning with the non-archimedean theory.

3.1 The Non-archimedean Local Factors For this section we will let k denote a non-archimedean local ﬁeld. We will let o denote the ring of integers of k and p the unique prime ideal of o. Fix a generator ̟ of p. We let q be the residue degree of k, so q = |o/p| = |̟|−1. We ﬁx a non-trivial continuous additive character ψ of k. ′ (π, Vπ) and (π , Vπ′ ) will now be the smooth vectors in irreducible admissible unitary generic representations of GLn(k) and GLm(k) respectively, as is true for local components of cuspidal representations. We will let ω and ω′ denote their central characters. The basic reference for this section is the paper of Jacquet, Piatetski- Shapiro, and Shalika [33]. L-functions for GLn 107

3.1.1 The local L-function

For each pair of Whittaker functions W ∈ W(π, ψ) and W ′ ∈ W(π′, ψ−1) and in the case n = m each Schwartz-Bruhat function Φ ∈ S(kn) we have deﬁned local integrals

h Ψ(s; W, W ′)= W W ′(h)| det(h)|s−(n−m)/2 dh In−m ZNm(k)\ GLm(k) h Ψ(s; W, W ′)= W x I dx  n−m−1  ZNm(k)\ GLm(k) ZMn−m−1,m(k) 1 e W ′(h)| det(h)|s−(n−m)/2 dh in the case m

′ ′ s Ψ(s; W, W , Φ) = W (g)W (g)Φ(eng)| det(g)| dg ZNn(k)\ GLn(k) in the case n = m, both convergent for Re(s) >> 0. To make the notation more convenient for what follows, in the case m

h ′ Ψj(s : W, W ) = W x Ij dx Nm(k)\ GLm(k) M (k)   Z Z j,m In−m−j W′(h)| det(h)|s−(n−m)/2 dh,

′ ′ ′ ′ so that Ψ(s; W, W )=Ψ0(s; W, W ) and Ψ(s; W, W )=Ψn−m−1(s; W, W ), which is still absolutely convergent for Re(s) >> 0. We need to understand what type of functionse of s these local integrals are. To this end, we need to understand the local Whittaker functions. So let W ∈ W(π, ψ). Since W is smooth, there is a compact open subgroup K, of ﬁnite index in the maximal compact subgroup Kn = GLn(o), so that W (gk)= W (g) for all k ∈ K. If we let {ki} be a set of coset representatives of GLn(o)/K, using that W transforms on the left under Nn(k) via ψ and the Iwasawa decomposition on GLn(k) we see that W (g) is completely determined by the values of W (aki)= Wi(a) for a ∈ An(k), the maximal split (diagonal) torus of GLn(k). So it suﬃces to understand a general Whittaker function on the torus. Let αi, i = 1,...,n − 1, denote the standard simple 108 J.W. Cogdell

a1 . roots of GLn, so that if a =  ..  ∈ An(k) then αi(a) = ai/ai+1. a  n By a finite function on An(k) we mean a continuous function whose trans- lates span a finite dimensional vector space [30, 31, Section 2.2]. (For the field k× itself the finite functions are spanned by products of characters and powers of the valuation map.) The fundamental result on the asymptotics of Whittaker functions is then the following [31, Prop. 2.2].

Proposition 3.1 Let π be a generic representation of GLn(k). Then there is a ﬁnite set of ﬁnite functions X(π) = {χi} on An(k), depending only on π, so that for every W ∈ W(π, ψ) there are Schwartz–Bruhat functions n−1 φi ∈ S(k ) such that for all a ∈ An(k) with an = 1 we have

W (a)= χi(a)φi(α1(a),...,αn−1(a)). XX(π) The ﬁnite set of ﬁnite functions X(π) which occur in the asymptotics near 0 of the Whittaker functions come from analyzing the Jacquet module W(π, ψ)/hπ(n)W − W |n ∈ Nni which is naturally an An(k)–module. Note that due to the Schwartz-Bruhat functions, the Whittaker functions vanish whenever any simple root αi(a) becomes large. The gauge estimates alluded to in Section 2 are a consequence of this expansion and the one in Proposition 3.6. Several nice consequences follow from inserting these formulas for W and ′ ′ ′ W into the local integrals Ψj(s; W, W ) or Ψ(s; W, W , Φ) [31,33].

′ ′ Proposition 3.2 The local integrals Ψj(s; W, W ) or Ψ(s; W, W , Φ) satisfy the following properties. 1. Each integral converges for Re(s) >> 0. For π and π′ unitary, as we have assumed, they converge absolutely for Re(s) ≥ 1. For π and π′ tempered, we have absolute convergence for Re(s) > 0.

2. Each integral deﬁnes a rational function in q−s and hence meromor- phically extends to all of C.

3. Each such rational function can be written with a common denominator which depends only on the ﬁnite functions X(π) and X(π′) and hence only on π and π′. L-functions for GLn 109

In deriving these when m

h W x I 6= 0  j  In−m−j−1   implies that x lies in a compact set independent of h ∈ GLm(k) [33]. ′ Let Ij(π,π ) denote the complex linear span of the local integrals Ψj(s; W , W ′) if m

h Ψ s; π W, π′(h)W ′ = | det(h)|−s−j+(n−m)/2Ψ (s; W, W ′) j I j n−m and Ψ(s; π(h)W, π′(h)W ′, ρ(h)Φ) = | det(h)|−sΨ(s; W, W ′, Φ). ′ So by varying h and multiplying by scalars, we see that each Ij(π,π ) and I(π,π′) is closed under multiplication by C[qs,q−s]. Since we have bounded denominators, we can conclude:

′ ′ s −s Proposition 3.3 Each Ij(π,π ) and I(π,π ) is a fractional C[q ,q ]–ideal of C(q−s).

Note that C[qs,q−s] is a principal ideal domain, so that each fractional ideal ′ −s ′ Ij(π,π ) has a single generator, which we call Qj,π,π′ (q ), as does I(π,π ), −s which we call Qπ,π′ (q ). However, we can say more. In the case m

Moreover, since qs and q−s are units in C[qs,q−s] we can always normalize −s −1 −s −1 our generator to be of the form Pj,π,π′ (q ) or Pπ,π′ (q ) where the polynomial P (X) satisﬁes P (0) = 1. Finally, in the case m

′ ′ −1 Corollary There is a ﬁnite collection of Wi ∈W(π, ψ), Wi ∈W(π , ψ ), n and if necessary Φi ∈ S(k ) such that ′ ′ ′ ′ L(s,π × π )= Ψ(s; Wi,Wi ) or L(s,π × π )= Ψ(s; Wi,Wi , Φi). Xi Xi L-functions for GLn 111

For future reference, let us set

Ψ(s; W, W ′) Ψ (s; W, W ′) e(s; W, W ′) = , e (s; W, W ′)= j , L(s,π × π′) j L(s,π × π′) Ψ(s; W, W ′) e˜(s; W, W ′) = , L(s,π × π′) e and Ψ(s; W, W ′, Φ) e(s; W, W ′, Φ) = . L(s,π × π′) Then all of these functions are Laurent polynomials in q±s, i.e., elements of C[qs,q−s]. As such they are entire and bounded in vertical strips. As above, ′ ′ there are choices of Wi, Wi , and if necessary Φi such that e(s; Wi,Wi ) ≡ 1 or e(s; W ,W ′, Φ ) ≡ 1. In particular we have the following result. i i i P P Corollary The functions e(s; W, W ′) and e(s; W, W ′, Φ) are entire functions, bounded in vertical strips, and for each s0 ∈ C there is a choice of W , ′ ′ ′ W , and if necessary Φ such that e(s0; W, W ) 6= 0 or e(s0; W, W , Φ) 6= 0.

3.1.2 The local functional equation

Either by analogy with Tate’s thesis or from the corresponding global statement, we would expect our local integrals to satisfy a local functional equation. From the functional equations for our global integrals, we would expect ′ ′ these to relate the integrals Ψ(s; W, W ) and Ψ(1 − s; ρ(wn,m)W , W ) when m

h Ψ s; π W, π′(h)W ′ = | det(h)|−s+(n−m)/2Ψ(s; W, W ′) I n−m

′ and one checks that Ψ(1 − s; ρ(wn,m)W , W ) has the same quasi-invariance ′ −1 as a bilinear form on W(π, ψ) ×W(π , ψ ). In addition, if we let Yn,m f f 112 J.W. Cogdell denote the unipotent radical of the standard parabolic subgroup associated to the partition (m + 1, 1,..., 1) as before then we have the quasi-invariance

Ψ(s; π(y)W, W ′)= ψ(y)Ψ(s; W, W ′)

′ for all y ∈ Yn,m. One again checks that Ψ(1 − s; ρ(wn,m)W , W ) satisﬁes the same quasi-invariance as a bilinear form on W(π, ψ) ×W(π′, ψ−1). For n = m we have seen that e f f

Ψ(s; π(h)W, π′(h)W ′, ρ(h)Φ) = | det(h)|−sΨ(s; W, W ′, Φ) and it is elementary to check that Ψ(1−s; W , W ′, Φ)ˆ satisﬁes the same quasi- invariance as a trilinear form on W(π, ψ) ×W(π′, ψ−1) × S(kn). Our local functional equations will now follow fromf thef following result [33, Proposi- tions 2.10 and 2.11].

Proposition 3.4 (i) If m

h B π W, π′(h)W ′ = | det(h)|−s+(n−m)/2B (W, W ′) s I s n−m ′ ′ and Bs(π(y)W, W )= ψ(y)Bs(W, W ) for all h ∈ GLm(k) and y ∈ Yn,m(k). (ii) If n = m, then except for a ﬁnite number of exceptional values of −s ′ −1 n q there is a unique trilinear form Ts on W(π, ψ) ×W(π , ψ ) × S(k ) satisfying

′ ′ −s ′ Ts(π(h)W, π (h)W , ρ(h)Φ) = | det(h)| Ts(W, W , Φ) for all h ∈ GLn(k).

Let us say a few words about the proof of this proposition, because it is another application of the analysis of the restriction of representations of GLn to the mirabolic subgroup Pn [33, Sections 2.10 and 2.11]. In the case where m

Theorem 3.2 There is a rational function γ(s,π × π′, ψ) ∈ C(q−s) such that we have

′ ′ n−1 ′ ′ Ψ(1 − s; ρ(wn,m)W , W )= ω (−1) γ(s,π × π , ψ)Ψ(s; W, W ) if m

Ψ(1 − s; W , W ′, Φ)ˆ = ω′(−1)n−1γ(s,π × π′, ψ)Ψ(s; W, W ′, Φ) if m = n for all W f∈Wf(π, ψ), W ′ ∈W(π′, ψ−1), and if necessary all Φ ∈ S(kn).

′ Again, if π = 1 is the trivial representation of GL1(k) we write γ(s,π,ψ)= γ(s,π × 1, ψ). The fact that γ(s,π × π′, ψ) is rational follows from the fact that it is a ratio of local integrals. An equally important local factor, which occurs in the current formu- lations of the local Langlands correspondence [23, 26], is the local ε-factor. 114 J.W. Cogdell

Deﬁnition The local factor ε(s,π × π′, ψ) is deﬁned as the ratio

γ(s,π × π′, ψ)L(s,π × π′) ε(s,π × π′, ψ)= . L(1 − s, π × π′)

With the local ε-factor the local functional equatione e can be written in the form

Ψ(1 − s; ρ(w )W , W ′) Ψ(s; W, W ′) n,m = ω′(−1)n−1ε(s,π×π′, ψ) if m

′ ′ n−1 ′ ε(s,π × π , ψ)= ω (−1) e˜(1 − s; ρ(wn,m)Wi, Wi ) Xi or f f ′ ′ n−1 ′ ˆ ε(s,π × π , ψ)= ω (−1) e(1 − s; Wi, Wi , Φi) Xi and hence ε(s,π × π′, ψ) ∈ C[qs,q−s]. On the otherf f hand, applying the functional equation twice we get

ε(s,π × π′, ψ)ε(1 − s, π × π′, ψ−1) = 1

′ s −s so that ε(s,π × π , ψ) is a unit in C[q ,qe ].e This can be restated as:

Proposition 3.5 ε(s,π × π′, ψ) is a monomial function of the form cq−fs. L-functions for GLn 115

Let us make a few remarks on the meaning of the number f occurring in the ε–factor in the case of a single representation. Assume that ψ is unramified. In this case write ε(s,π,ψ) = ε(0,π,ψ)q−f(π)s. In [34] it is shown that f(π) is a non-negative integer, f(π) = 0 iff π is unramified, that in general the space of vectors in Vπ which is fixed by the compact open subgroup

∗ . f(π) ∗ . f(π) K1(p )= g ∈ GLn(o) g ≡   (mod p )  ∗      0 · · · 0 1         has dimension exactly 1, and that if t

3.1.3 The unramiﬁed calculation

Let us now turn to the calculation of the local L-functions. The first case to consider is that where both π and π′ are unramified. Since they are assumed generic, they are both full induced representations from unramified characters of the Borel subgroup [69]. So let us write π ≃ IndGLn (µ ⊗ Bn 1 ···⊗ µ ) and π′ ≃ IndGLm (µ′ ⊗···⊗ µ′ ) with the µ and µ′ unramified n Bm 1 m i j characters of k×. The Satake parameterization of unramified representations associates to each of these representations the semi-simple conjugacy classes [Aπ] ∈ GLn(C) and [Aπ′ ] ∈ GLm(C) given by

′ µ1(̟) µ1(̟) .. ′ .. Aπ =  .  Aπ =  .  . µ (̟) µ′ (̟)  n   m      (Recall that ̟ is a uniformizing parameter for k, that is, a generator of p.) In the Whittaker models there will be unique normalized K = GL(o)– ′ ′ −1 ﬁxed Whittaker functions, W◦ ∈W(π, ψ) and W◦ ∈W(π , ψ ), normalized ′ by W◦(e) = W◦(e) = 1. Let us concentrate on W◦ for the moment. Since this function is right Kn–invariant and transforms on the left by ψ under Nn we have that its values are completely determined by its values on diagonal 116 J.W. Cogdell matrices of the form ̟j1 J .. ̟ =  .  ̟jn   n   J for J = (j1,...,jn) ∈ Z . There is an explicit formula for W◦(̟ ) in terms of the Satake parameter Aπ due to Shintani [63] for GLn and generalized to arbitrary reductive groups by Casselman and Shalika [4]. + n Let T (n) betheset of n–tuples J = (j1,...,jn) ∈ Z with j1 ≥···≥ jn. Let ρJ be the rational representation of GLn(C) with dominant weight ΛJ deﬁned by t1 .. j1 jn ΛJ  .  = t1 · · · tn . t  n Then the formula of Shintani says that

+ J 0 if J∈ / T (n) W◦(̟ )= δ1/2(̟J ) tr(ρ (A )) if J ∈ T +(n) ( Bn J π under the assumption that ψ is unramified. This is proved by analyzing the recursion relations coming from the action of the unramified Hecke algebra on W◦. ′ J m We have a similar formula for W◦(̟ ) for J ∈ Z . If we use these formulas in our local integrals, we find [36, I, Prop. 2.3] ̟J Ψ(s; W ,W ′)= W W ′(̟J ) ◦ ◦ ◦ I ◦ + n−m J∈T (Xm), jm≥0 · | det(̟J )|s−(n−m)/2δ−1 (̟J ) Bm −|J|s = tr(ρ(J,0)(Aπ)) tr(ρJ (Aπ′ ))q + J∈T (Xm), jm≥0 −|J|s = tr(ρ(J,0)(Aπ) ⊗ ρJ (Aπ′ ))q + J∈T (Xm), jm≥0 m n →where we let |J| = j1+· · ·+jm and we embed Z ֒→ Z by J = (j1, · · · , jm) 7 (J, 0) = (j1, · · · , jm, 0, · · · , 0). We now use the fact from invariant theory that

r tr(ρ(J,0)(Aπ) ⊗ ρJ (Aπ′ )) = tr(S (Aπ ⊗ Aπ′ )), + J∈T (m),X jm≥0, |J|=r L-functions for GLn 117 where Sr(A) is the rth-symmetric power of the matrix A, and ∞ tr(Sr(A))zr = det(I − Az)−1 Xr=0 for any matrix A. Then we quickly arrive at ′ −s −1 ′ −s −1 Ψ(s; W◦,W◦) = det(I − q Aπ ⊗ Aπ′ ) = (1 − µi(̟)µj(̟)q ) Yi,j a standard Euler factor of degree mn. Since the L-function cancels all poles −s of the local integrals, we know at least that det(I − q Aπ ⊗ Aπ′ ) divides L(s,π × π′)−1. Either of the methods discussed below for the general calculation of local factors then shows that in fact these are equal. There is a similar calculation when n = m and Φ=Φ◦ is the characteristic function of the lattice on ⊂ kn. Also, since π unramified implies that its contragredient π is also unramified, with W◦ as its normalized unramified Whittaker function, then from the functional equation we can conclude that in this situation wee have ε(s,π × π′, ψ) ≡ 1.f

Theorem 3.3 If π, π′, and ψ are all unramiﬁed, then ′ ′ −s −1 Ψ(s; W◦,W◦) m

3.1.4 The supercuspidal calculation The other basic case is when both π and π′ are supercuspidal. In this case ′ the restriction of W to Pn or W to Pm lies in the Kirillov model and is hence compactly supported mod N. In the case of m

cΦ(0) W (g)W ′(g)| det(g)|s0 dg ZZn(k)Nn(k)\ GLn(k) which is the Whittaker form of an invariant pairing between π and π′ ⊗ s0 ′ s0 | det | . Thus we must have s0 is pure imaginary and π ≃ π ⊗ | det | for the residue to be nonzero. This condition is also suﬃcient. e Theorem 3.4 If π and π′ are both (unitary) supercuspidal, then L(s,π × π′) ≡ 1 if m

L(s,π × π′)= (1 − αq−s)−1 Y with the product over all α = qs0 with π ≃ π′ ⊗ | det |s0 .

e 3.1.5 Remarks on the general calculation In the other cases, we must rely on the Bernstein–Zelevinsky classiﬁcation of generic representations of GLn(k) [69]. All generic representations can be realized as prescribed constituents of representations parabolically induced from supercuspidals. One can proceed by analyzing the Whittaker functions of induced representations in terms of Whittaker functions of the inducing data as in [33] or by analyzing the poles of the local integrals in terms of quasi invariant pairings of derivatives of π and π′ as in [11] to compute L(s,π ×π′) in terms of L-functions of pairs of supercuspidal representations. We refer you to those papers or [42] for the explicit formulas. L-functions for GLn 119

3.1.6 Multiplicativity and stability of γ–factors To conclude this section, let us mention two results on the γ-factors. One is used in the computations of L-factors in the general case. This is the multiplicativity of γ-factors [33]. The second is the stability of γ-factors [37]. Both of these results are necessary in applications of the Converse Theorem to liftings, which we discuss in Section 5.

Proposition (Multiplicativity of γ-factors) If π = Ind(π1 ⊗ π2), with

πi and irreducible admissible representation of GLri (k), then

′ ′ ′ γ(s,π × π , ψ)= γ(s,π1 × π , ψ)γ(s,π2 × π , ψ)

′ ′ −1 ′ and similarly for π . Moreover L(s,π × π ) divides [L(s,π1 × π )L(s,π2 × π′)]−1.

Proposition (Stability of γ-factors) If π1 and π2 are two irreducible admissible generic representations of GLn(k), having the same central character, then for every sufficiently highly ramified character η of GL1(k) we have γ(s,π1 × η, ψ)= γ(s,π2 × η, ψ) and L(s,π1 × η)= L(s,π2 × η) ≡ 1. More generally, if in addition π′ is an irreducible generic representation of GLm(k) then for all sufficiently highly ramified characters η of GL1(k) we have ′ ′ γ(s, (π1 ⊗ η) × π , ψ)= γ(s, (π2 ⊗ η) × π , ψ) and ′ ′ L(s, (π1 ⊗ η) × π )= L(s, (π2 ⊗ η) × π ) ≡ 1.

3.2 The Archimedean Local Factors We now take k to be an archimedean local ﬁeld, i.e., k = R or C. We take (π, Vπ) to be the space of smooth vectors in an irreducible admissible unitary generic representation of GLn(k) and similarly for the representation 120 J.W. Cogdell

′ (π , Vπ′ ) of GLm(k). We take ψ a non-trivial continuous additive character of k. The treatment of the archimedean local factors parallels that of the non- archimedean in many ways, but there are some significant differences. The major work on these factors is that of Jacquet and Shalika in [38], which we follow for the most part without further reference, and in the archimedean parts of [36]. One significant difference in the development of the archimedean theory is that the local Langlands correspondence was already in place when the theory was developed [45]. The correspondence is very explicit in terms of the usual Langlands classification. Thus to π is associated an n dimensional ′ semi-simple representation τ = τ(π) of the Weil group Wk of k and to π an ′ ′ m-dimensional semi-simple representation τ = τ(π ) of Wk. Then τ(π) ⊗ ′ τ(π ) is an nm dimensional representation of Wk and to this representation of the Weil group is attached Artin-Weil L– and ε–factors [65], denoted L(s,τ ⊗ τ ′) and ε(s,τ ⊗ τ ′, ψ). In essence, Jacquet and Shalika define

L(s,π×π′)= L(s,τ(π)⊗τ(π′)) and ε(s,π×π′, ψ)= ε(s,τ(π)⊗τ(π′), ψ) and then set ε(s,π × π′, ψ)L(1 − s, π × π′) γ(s,π × π′, ψ)= . L(s,π × π′) e e For example, if π is unramiﬁed, and hence of the form π ≃ Ind(µ1 ⊗···⊗ ri µn) with unramiﬁed characters of the form µi(x)= |x| then

n L(s,π)= L(s,τ(π)) = Γv(s + ri) Yi=1 is a standard archimedean Euler factor of degree n, where

−s/2 s π Γ( 2 ) if kv = R Γv(s)= −s . (2(2π) Γ(s) if kv = C They then proceed to show that these functions have the expected relation to the local integrals. Their methods of analyzing the local integrals ′ ′ Ψj(s; W, W ) and Ψ(s; W, W , Φ), deﬁned as in the non-archimedean case for W ∈ W(π, ψ), W ′ ∈ W(π′, ψ−1), and Φ ∈ S(kn), are direct analogues of those used in [33] for the non-archimedean case. Once again, a most important fact is [38, Proposition 2.2] L-functions for GLn 121

Proposition 3.6 Let π be a generic representation of GLn(k). Then there is a ﬁnite set of ﬁnite functions X(π) = {χi} on An(k), depending only on π, so that for every W ∈ W(π, ψ) there are Schwartz functions φi ∈ n−1 S(k × Kn) such that for all a ∈ An(k) with an = 1 we have

W (nak)= ψ(n) χi(a)φi(α1(a),...,αn−1(a), k). XX(π) Now the ﬁnite functions are related to the exponents of the representation π and through the Langlands classiﬁcation to the representation τ(π) of Wk. We retain the same convergence statements as in the non-archimedean case [36, I, Proposition 3.17; II, Proposition 2.6], [38, Proposition 5.3].

′ ′ Proposition 3.7 The integrals Ψj(s; W, W ) and Ψ(s; W, W , Φ) converge absolutely in the half plane Re(s) ≥ 1 under the unitarity assumption and for Re(s) > 0 if π and π′ are tempered.

The meromorphic continuation and “bounded denominator” statement in the case of a non-archimedean local ﬁeld is now replaced by the following. Deﬁne M(π × π′) to be the space of all meromorphic functions φ(s) with the property that if P (s) is a polynomial function such that P (s)L(s,π × π′) is holomorphic in a vertical strip S[a, b]= {s a ≤ Re(s) ≤ b} then P (s)φ(s) is bounded in S[a, b]. Note in particular that if φ ∈ M(π × π′) then the quotient φ(s)L(s,π × π′)−1 is entire.

′ ′ Theorem 3.5 The integrals Ψj(s; W, W ) or Ψ(s; W, W , Φ) extend to meromorphic functions of s which lie in M(π × π′). In particular, the ratios

Ψ (s; W, W ′) Ψ(s; W, W ′, Φ) e (s; W, W ′)= j or e(s; W, W ′, Φ) = j L(s,π × π′) L(s,π × π′) are entire and in fact are bounded in vertical strips.

This statement has more content than just the continuation and “bounded denominator” statements in the non-archimedean case. Since it prescribes the “denominator” to be the L factor L(s,π × π′)−1 it is bound up with the actual computation of the poles of the local integrals. In fact, a sig- niﬁcant part of the paper of Jacquet and Shalika [38] is taken up with the simultaneous proof of this and the local functional equations: 122 J.W. Cogdell

Theorem 3.6 We have the local functional equations

′ ′ n−1 ′ ′ Ψn−m−j−1(1 − s; ρ(wn,m)W , W )= ω (−1) γ(s,π × π , ψ)Ψj(s; W, W ) or f f Ψ(1 − s; W , W ′, Φ)ˆ = ω′(−1)n−1γ(s,π × π′, ψ)Ψ(s; W, W ′, Φ).

The one factf thatf we are missing is the statement of “minimality” of the L-factor. That is, we know that L(s,π × π′) is a standard archimedean Euler factor (i.e., a product of Γ-functions of the standard type) and has the property that the poles of all the local integrals are contained in the poles of the L-factor, even with multiplicity. But we have not established that the L-factor cannot have extraneous poles. In particular, we do know that we can achieve the local L-function as a finite linear combination of local integrals. Towards this end, Jacquet and Shalika enlarge the allowable space of local ′ integrals. Let Λ and Λ be the Whittaker functionals on Vπ and Vπ′ associated with the Whittaker models W(π, ψ) and W(π′, ψ−1). Then Λˆ = Λ ⊗ Λ′ defines a continuous linear functional on the algebraic tensor product Vπ ⊗ Vπ′ which extends continuously to the topological tensor product Vπ⊗π′ = Vπ⊗ˆ Vπ′ , viewed as representations of GLn(k) × GLm(k). Before proceeding, let us make a few remarks on smooth representations. If (π, Vπ) is the space of smooth vectors in an irreducible admissible unitary representation, then the underlying Harish-Chandra module is the space of Kn-finite vectors Vπ,K. Vπ then corresponds to the (Casselman-Wallach) canonical completion of Vπ,K [66]. The category of Harish-Chandra modules is appropriate for the algebraic theory of representations, but it is useful to work in the category of smooth admissible representations for automorphic forms. If in our context we take the underlying Harish-Chandra modules Vπ,K and Vπ′,K then their algebraic tensor product is an admissible Harish- Chandra module for GLn(k) × GLm(k). The associated smooth admissible representation is the canonical completion of this tensor product, which is in fact Vπ⊗π′ , the topological tensor product of the smooth representations π and π′. It is also the space of smooth vectors in the unitary tensor product of the unitary representations associated to π and π′. So this completion is a natural place to work in the category of smooth admissible representations. Now let

′ ′ W(π ⊗ π , ψ)= {W (g, h)= Λ(ˆ π(g) ⊗ π (h)ξ)|ξ ∈ Vπ⊗π′ }. L-functions for GLn 123

Then W(π⊗π′, ψ) contains the algebraic tensor product W(π, ψ)⊗W(π′, ψ−1) and is again equal to the topological tensor product. Now we can extend all our local integrals to the space W(π ⊗ π′, ψ) by setting h Ψ (s; W )= W x I , h dx | det(h)|s−(n−m)/2 dh j  j   ZZ In−m−j    and s Ψ(s; W, Φ) = W (g, g)Φ(eng)| det(g)| dh Z for W ∈W(π ⊗ π′, ψ). Since the local integrals are continuous with respect to the topology on the topological tensor product, all of the above facts remain true, in particular the convergence statements, the local functional equations, and the fact that these integrals extend to functions in M(π×π′). ′ ′ ′ At this point, let Ij(π,π )= {Ψj(s; W )|W ∈W(π ⊗ π )} and let I(π,π ) be the span of the local integrals {Ψ(s; W, Φ)|W ∈W(π⊗π′, ψ), φ ∈ S(kn)}. ′ Once again, in the case m

Theorem 3.7 I(π,π′)= M(π × π′).

As a consequence of this, we draw the following useful Corollary.

Corollary There is a Whittaker function W in W(π ⊗ π′, ψ) such that ′ L(s,π × π )=Ψ(s; W ) if m

′ Proposition 3.8 For any s0 ∈ C there are choices of W ∈W(π, ψ), W ∈ ′ −1 ′ ′ W(π , ψ ) and if necessary Φ such that e(s0; W, W ) 6= 0 or e(s0; W, W , Φ) 6= 0.

4 Global L-functions

Once again, we let k be a global field, A its ring of adeles, and fix a non-trivial continuous additive character ψ = ⊗ψv of A trivial on k. Let (π, Vπ) be a cuspidal representation of GLn(A) (see Section 1 for ′ all the implied assumptions in this terminology) and (π , Vπ′ ) a cuspidal representation of GLm(A). Since they are irreducible we have restricted ′ ′ ′ ′ tensor product decompositions π ≃ ⊗ πv and π ≃ ⊗ πv with (πv, Vπv ) and ′ ′ (πv, Vπv ) irreducible admissible smooth generic unitary representations of ′ ′ ′ ′ GLn(kv) and GLm(kv) [14,18]. Let ω = ⊗ ωv and ω = ⊗ ωv be their central characters. These are both continuous characters of k×\A×. Let S be the finite set of places of k, containing the archimedean places ′ S∞, such that for all v∈ / S we have that πv, πv, and ψv are unramified. ′ For each place v of k we have defined the local factors L(s,πv × πv) and ′ ε(s,πv × πv, ψv). Then we can at least formally define

′ ′ ′ ′ L(s,π × π )= L(s,πv × πv) and ε(s,π × π )= ε(s,πv × πv, ψv). v v Y Y We need to discuss convergence of these products. Let us ﬁrst consider ′ ′ the convergence of L(s,π × π ). For those v∈ / S, so πv, πv, and ψv are ′ −s ′ −1 unramiﬁed, we know that L(s,πv × πv) = det(I − qv Aπv ⊗ Aπv ) and that 1/2 ′ the eigenvalues of Aπv and Aπv are all of absolute value less than qv . Thus the partial (or incomplete) L-function

S ′ ′ −s ′ −1 L (s,π × π )= L(s,πv × πv)= det(I − q Aπv ⊗ Aπv ) vY∈ /S vY∈ /S is absolutely convergent for Re(s) >> 0. Thus the same is true for L(s,π × π′). ′ For the ε–factor, we have seen that ε(s,πv × πv, ψv) ≡ 1 for v∈ / S so that the product is in fact a ﬁnite product and there is no problem with convergence. The fact that ε(s,π × π′) is independent of ψ can either be checked by analyzing how the local ε–factors vary as you vary ψ, as is done in [7, Lemma 2.1], or it will follow from the global functional equation presented below. L-functions for GLn 125

4.1 The Basic Analytic Properties Our ﬁrst goal is to show that these L-functions have nice analytic properties.

Theorem 4.1 The global L–functions L(s,π ×π′) are nice in the sense that

1. L(s,π × π′) has a meromorphic continuation to all of C,

2. the extended function is bounded in vertical strips (away from its poles),

3. they satisfy the functional equation

L(s,π × π′)= ε(s,π × π′)L(1 − s, π × π′).

To do so, we relate the L-functions to the global integrals.e e Let us begin with continuation. In the case m> 0 and for appropriate choices of ϕ and ϕ′ we have

′ ′ I(s; ϕ, ϕ )= Ψv(s; Wϕv ,Wϕv ) v Y

′ S ′ = Ψv(s; Wϕv ,Wϕv ) L (s,π × π ) ! vY∈S ′ Ψv(s; Wϕv ,Wϕv ) ′ = ′ L(s,π × π ) L(s,πv × πv) ! vY∈S

′ ′ = ev(s; Wϕv ,Wϕv ) L(s,π × π ) ! vY∈S

′ ′ We know that each ev(s; Wv,Wv) is entire. Hence L(s,π × π ) has a mero- ′ morphic continuation. If m = n then for appropriate ϕ ∈ Vπ, ϕ ∈ Vπ′ , and Φ ∈ S(An) we again have

′ ′ ′ ′ I(s; ϕ, ϕ , Φ) = ev(s; Wϕv ,Wϕ , Φv) L(s,π × π ). v ! vY∈S ′ ′ Once again, since each ev(s; Wv,Wv, Φv) is entire, L(s,π × π ) has a meromorphic continuation. 126 J.W. Cogdell

Let us next turn to the functional equation. This will follow from the functional equation for the global integrals and the local functional equations. We will consider only the case where m

I(s; ϕ, ϕ′)= I˜(1 − s; ϕ, ϕ′).

Once again we have for appropriate ϕ and ϕ′ e e

′ ′ ′ ′ I(s; ϕ, ϕ )= ev(s; Wϕv ,Wϕ ) L(s,π × π ) v ! vY∈S while on the other side

′ ′ ′ ˜ ′ I(1 − s; ϕ, ϕ )= e˜v(1 − s; ρ(wn,m)Wϕv , Wϕ ) L(1 − s, π × π ). v ! vY∈S f f However, bye thee local functional equations, for each v ∈ S we havee e

Ψ(1 − s; ρ(w )W , W ′ ) e˜ (1 − s; ρ(w )W , W ′ )= n,m v v v n,m v v L(1 − s, π × π′) e f f Ψ(s; W ,W ′ ) f f = ω′ (−1)n−1ε(s,π × π′ , ψ ) v v v e v e v v L(s,π × π′) ′ n−1 ′ ′ = ωv(−1) ε(s,πv × πv, ψv)ev(s,Wv,Wv) Combining these, we have

′ ′ n−1 ′ ′ L(s,π × π )= ωv(−1) ε(s,πv × πv, ψv) L(1 − s, π × π ). ! vY∈S ′ ′ e e Now, for v∈ / S we know that πv is unramiﬁed, so ωv(−1) = 1, and also that ′ ε(s,πv × πv, ψv) ≡ 1. Therefore

′ n−1 ′ ′ n−1 ′ ωv(−1) ε(s,πv × πv, ψv)= ωv(−1) ε(s,πv × πv, ψv) v vY∈S Y = ω′(−1)n−1ε(s,π × π′) = ε(s,π × π′) and we indeed have

L(s,π × π′)= ε(s,π × π′)L(1 − s, π × π′).

e e L-functions for GLn 127

Note that this implies that ε(s,π × π′) is independent of ψ as well.

Let us now turn to the boundedness in vertical strips. For the global integrals I(s; ϕ, ϕ′) or I(s; ϕ, ϕ, Φ) this simply follows from the absolute convergence. For the L-function itself, the paradigm is the following. For every ′ ﬁnite place v ∈ S we know that there is a choice of Wv,i, Wv,i, and Φv,i if necessary such that

′ ′ L(s,πv × πv) = Ψ(s; Wv,i,Wv′i) or ′ X ′ L(s,πv × πv) = Ψ(s; Wv,i,Wv′i, Φv,i). X If m = n − 1 or m = n then by the unpublished work of Jacquet and Shalika mentioned toward the end of Section 3 we know that we have similar statements for v ∈ S∞. Hence if m = n−1 or m = n there are global choices ′ ϕi, ϕi, and if necessary Φi such that

′ ′ ′ ′ L(s,π × π )= I(s; ϕi, ϕi) or L(s,π × π )= I(s; ϕi, ϕi, Φi). X X Then the boundedness in vertical strips for the L-functions follows from that of the global integrals. However, if m

′ ′ L(s,πv × πv)= I(s; Wv) or L(s,πv × πv)= I(s; Wv,i, Φv,i). X To make the above paradigm work for m

4.2 Poles of L-functions Let us determine where the global L-functions can have poles. The poles of the L-functions will be related to the poles of the global integrals. Recall from Section 2 that in the case of m

Theorem 4.2 If m

Corollary L(s,π × π) has simple poles at s = 0 and s = 1. e

4.3 Strong Multiplicitye One Let us return to the Strong Multiplicity One Theorem for cuspidal representations. First, recall the statement:

′ Theorem (Strong Multiplicity One) Let (π, Vπ) and (π , Vπ′ ) be two cuspidal representations of GLn(A). Suppose there is a ﬁnite set of places S ′ ′ such that for all v∈ / S we have πv ≃ πv. Then π = π . L-functions for GLn 129

We will now present Jacquet and Shalika’s proof of this statement via L- functions [36]. First note the following observation, which follows from our analysis of the location of the poles of the L-functions.

′ ′ Observation For π and π cuspidal representations of GLn(A), L(s,π×π ) has a pole at s = 1 iﬀ π ≃ π′. e Thus the L-function gives us an analytic method of testing when two cuspidal representations are isomorphic, and so by the Multiplicity One Theorem, the same. Proof: If we take π and π′ as in the statement of Strong Multiplicity One, ′ we have that πv ≃ πv for v∈ / S and hence S ′ S ′ L (s,π × π)= L(s,πv × πv)= L(s,πv × πv)= L (s,π × π ) vY∈ /S vY∈ /S Since the localeL-factors never vanishe and for unitarye representationse they have no poles in Re(s) ≥ 1 (since the local integrals have no poles in this region) we see that for s = 1 that L(s,π × π′) has a pole at s = 1 iﬀ LS(s,π × π′) does. Hence we have that since L(s,π × π) has a pole at s = 1, so does LS(s,π × π). But LS(s,π × πe) = LS(s,π × π′), so that both LS(s,πe × π′) and then L(s,π × π′) have poles at se= 1. But then the L-function criterion abovee gives that π ≃eπ′. Now applye Multiplicity One. e e 2 In fact, Jacquet and Shalika push this method much further. If π is an irreducible automorphic representation of GLn(A), but not necessarily cuspidal, then it is a theorem of Langlands [44] that there are cuspidal representations, say τ1,...,τr of GLn1 ,..., GLnr with n = n1 +· · ·+nr, such ′ that π is a constituent of Ind(τ1 ⊗···⊗ τr). Similarly, π is a constituent of ′ ′ Ind(τ1 ⊗···⊗ τr′ ). Then the generalized version of the Strong Multiplicity One theorem that Jacquet and Shalika establish in [36] is the following.

Theorem (Generalized Strong Multiplicity One) Given π and π′ irreducible automorphic representations of GLn(A) as above, suppose that ′ there is a ﬁnite set of places S such that πv ≃ πv for all v∈ / S. Then ′ ′ r = r and there is a permutation σ of the set {1,...,r} such that ni = nσ(i) ′ and τi = τσ(i). ′ Note, the cuspidal representations τi and τi need not be unitary in this statement. 130 J.W. Cogdell

4.4 Non-vanishing Results Of interest for questions from analytic number theory, for example questions of equidistribution, are results on the non-vanishing of L-functions. The fundamental non-vanishing result for GLn is the following theorem of Jacquet and Shalika [35] and Shahidi [56,57].

′ Theorem 4.3 Let π and π be cuspidal representations of GLn(A) and ′ GLm(A). Then the L-function L(s,π × π ) is non-vanishing for Re(s) ≥ 1.

The proof of non-vanishing for Re(s) > 1 is in keeping with the spirit of these notes [36, I, Theorem 5.3]. Since the local L-functions are never zero, to establish the non-vanishing of the Euler product for Re(s) > 1 it suﬃces to show that the Euler product is absolutely convergent for Re(s) > 1, and for this it is suﬃcient to work with the incomplete L-function LS(s,π × π′) where S is as at the beginning of this Section. Then we can write

S ′ ′ −s ′ −1 L (s,π × π )= L(s,πv × πv)= det(I − qv Aπv ⊗ Aπv ) vY∈ /S vY∈ /S with absolute convergence for Re(s) >> 0. Recall that an inﬁnite product (1 + an) is absolutely convergent iﬀ the associated series log(1 + a ) is absolutely convergent. n Q Let us write P ′ µv,1 µv,1 A = .. and A ′ = .. . πv  .  πv  .  µ µ′  v,n  v,m     1/2 ′ 1/2 We have seen that |µv,i|

′ ′ −s log L(s,πv × πv) = − log(1 − µv,iµv,jqv ) Xi,j ∞ ′ d ∞ d d (µ µ ) tr(A ) tr(A ′ ) v,i v,j πv πv = ds = ds dqv dqv Xi,j Xd=1 Xd=1 with the sum absolutely convergent for Re(s) >> 0. Then, still for Re(s) >> 0, ∞ d d tr(A ) tr(A ′ ) S ′ πv πv log(L (s,π × π )) = ds . dqv Xv∈ /S Xd=1 L-functions for GLn 131

If we apply this to π′ = π = π we ﬁnd

∞ | tr(Ad )|2 S e πv log(L (s,π × π)) = ds dqv Xv∈ /S Xd=1 which is a Dirichlet series with non-negative coefficients. By Landau’s Lemma this will be absolutely convergent up to the its first pole, which we know is at s = 1. Hence this series, and the associated Euler product L(s,π × π), is absolutely convergent for Re(s) > 1. An application of the Cauchy–Schwatrz inequality then implies thate the series ∞ d d tr(A ) tr(A ′ ) S ′ πv πv log(L (s,π × π )) = ds dqv Xv∈ /S Xd=1 is also absolutely convergent for Re(s) > 1. Thus L(s,π × π′) is absolutely convergent and hence non-vanishing for Re(s) > 1. To obtain the non-vanishing on the line Re(s) = 1 requires the technique of analyzing L-functions via their occurrence in the constant terms and Fourier coefficients of Eisenstein series, which we have not discussed. They can be found in the references [35] and [56,57] mentioned above.

4.5 The Generalized Ramanujan Conjecture (GRC) The current version of the GRC is a conjecture about the structure of cuspidal representations.

Conjecture (GRC) Let π be a (unitary) cuspidal representation of GLn(A) ′ with decomposition π ≃ ⊗ πv. Then the local components πv are tempered representations.

However, it has an interesting interpretation in terms of L-functions which is more in keeping with the origins of the conjecture. If π is cuspidal, then at every ﬁnite place v where πv is unramiﬁed we have associated µv,1 .. a semisimple conjugacy class, say Aπv =  .  so that µ  v,n   n −s −1 −s −1 L(s,πv) = det(I − qv Aπv ) = (1 − µv,iqv ) . Yi=1 132 J.W. Cogdell

If v is an archimedean place where πv is unramiﬁed, then we can similarly write n L(s,π)= Γv(s + µv,i) Yi=1 where −s/2 s π Γ( 2 ) if kv ≃ R Γv(s)= −s . (2(2π) Γ(s) if kv ≃ C Then the statement of the GRC in these terms is

Conjecture (GRC for L-functions) If π is a cuspidal representation of GLn(A) and if v is a place where πv is unramiﬁed, then |µv,i| = 1 for v non-archimedean and Re(µv,i) = 0 for v archimedean.

Note that by including the archimedean places, this conjecture encom- passes not only the classical Ramanujan conjectures but also the various versions of the Selberg eigenvalue conjecture [27]. −1/2 Recall that by the Corollary to Theorem 3.3 we have the bounds qv < 1/2 |µv,i|

−( 1 − 1 ) 1 − 1 2 n2+1 2 n2+1 qv ≤ |µv,i|≤ qv if v is non-archimedean and 1 1 | Re(µ )|≤ − for v archimedean. v,i 2 n2 + 1 Their techniques are global and rely on the theory of Rankin–Selberg L- functions as presented here, a technique of persistence of zeros in families of L-functions, and a positivity argument.

For GL2 there has been much recent progress. The best general estimates I am aware of at present are due to Kim and Shahidi [41], who use the holomorphy of the symmetric ninth power L-function for Re(s) > 1 to obtain

1 1 − 9 9 qv < |µv,i|

Definition A cuspidal representation π of GLn(A) is called weakly Ra- manujan if for every ǫ> 0 there is a constant cǫ > 0 and an infinite sequence of places {vm} with the property that each πvm is unramified and the Satake parameters µvm,i satisfy

−1 −ǫ ǫ cǫ qvm < |µvm,i| < cǫqvm .

For example, if we knew that all cuspidal representations on GLn(A) were weakly Ramanujan, then we would know that under Langlands liftings between general linear groups, the property of occurrence in the spectral decomposition is preserved [8]. For n = 2, 3 our techniques let us show the following.

Proposition 4.1 For n = 2 or n = 3 all cuspidal representations are weakly Ramanujan.

Proof: First, let π be a cuspidal representation or GLn(A). Recall that from the absolute convergence of the Euler product for L(s,π × π) we know d 2 | tr(Aπv )| that the series ds is absolutely convergent for Re(s) > 1, so dqv Xv∈ /S Xd | tr(A )|2 that in particular we have that πv is absolutely convergent for qs v∈ /S v Re(s) > 1. Thus, for a set of placesX of positive density, we have the estimate 2 ǫ −1 | tr(Aπv )| < qv for each ǫ. Since Aπv = Aπv for components of cuspidal −1 representations, we have the same estimate for | tr(Aπv )|. In the case of n = 2 and n = 3, these estimates and the fact that

| det Aπv | = |ωv(̟v)| = 1 give us estimates on the coefficients of the characteristic polynomial for Aπv . For example, if n = 3 and the characteristic 3 2 ǫ/2 polynomial of Aπv is X +aX +bX +c then we know |a| = | tr(Aπv )| 3. Applying this to both Aπv and Aπv we find that for our set primes of positive density above we have the estimate −ǫ ǫ qv < |µvm,i| < qv. Thus we find that for n = 2, 3 cuspidal representations of GLn are weakly Ramanujan. 2 134 J.W. Cogdell

4.6 The Generalized Riemann Hypothesis (GRH) This is one of the most important conjectures in the analytic theory of L- functions. Simply stated, it is

Conjecture (GRH) For any cuspidal representation π, all the zeros of 1 the L-function L(s,π) lie on the line Re(s)= 2 .

Even in the simplest case of n = 1 and π = 1 the trivial representation this reduces to the Riemann hypothesis for the Riemann zeta function! For an interesting survey on these and other conjectures on L-functions and their relation to number theoretic problems, we refer the reader to the survey of Iwaniec and Sarnak [27].

5 Converse Theorems

Let us return first to Hecke. Recall that to a modular form ∞ 2πinτ f(τ)= ane nX−1 for, say, SL2(Z) Hecke attached an L function L(s,f) and they were related via the Mellin transform ∞ Λ(s,f) = (2π)−sΓ(s)L(s,f)= f(iy)ys d×y Z0 and derived the functional equation for L(s,f) from the modular transformation law for f(τ) under the modular transformation law for the transformation τ 7→ −1/τ. In his fundamental paper [24] he inverted this process by taking a Dirichlet series ∞ a D(s)= n ns Xn=1 and assuming that it converged in a half plane, had an entire continuation to a function of finite order, and satisfied the same functional equation as the L-function of a modular form of weight k, then he could actually reconstruct a modular form from D(s) by Mellin inversion

2+i∞ −2πny 1 −s s f(iy)= ane = (2π) Γ(s)D(s)y ds 2πi 2−i∞ Xi Z L-functions for GLn 135 and obtain the modular transformation law for f(τ) under τ 7→ −1/τ from the functional equation for D(s) under s 7→ k − s. This is Hecke’s Converse Theorem. In this Section we will present some analogues of Hecke’s theorem in the context of L-functions for GLn. Surprisingly, the technique is exactly the same as Hecke’s, i.e., inverting the integral representation. This was ﬁrst done in the context of automorphic representation for GL2 by Jacquet and Langlands [30] and then extended and signiﬁcantly strengthened for GL3 by Jacquet, Piatetski-Shapiro, and Shalika [31]. For a more extensive bibliography and history, see [10]. This section is taken mainly from our survey [10]. Further details can be found in [7,9].

5.1 The Results Once again, let k be a global ﬁeld, A its adele ring, and ψ a ﬁxed non-trivial continuous additive character of A which is trivial on k. We will take n ≥ 3 to be an integer. To state these Converse Theorems, we begin with an irreducible admissible representation Π of GLn(A). In keeping with the conventions of these notes, we will assume that Π is unitary and generic, but this is not neces- ′ sary. It has a decomposition Π = ⊗ Πv, where Πv is an irreducible admissible generic representation of GLn(kv). By the local theory of Section 3, to each Πv is associated a local L-function L(s, Πv) and a local ε-factor ε(s, Πv, ψv). Hence formally we can form

L(s, Π) = L(s, Πv) and ε(s, Π, ψ)= ε(s, Πv, ψv). v v Y Y We will always assume the following two things about Π:

1. L(s, Π) converges in some half plane Re(s) >> 0,

2. the central character ωΠ of Π is automorphic, that is, invariant under k×.

Under these assumptions, ε(s, Π, ψ) = ε(s, Π) is independent of our choice of ψ [7]. Our Converse Theorems will involve twists by cuspidal automorphic representations of GLm(A) for certain m. For convenience, let us set A(m) 136 J.W. Cogdell

to be the set of automorphic representations of GLm(A), A0(m) the set of m cuspidal representations of GLm(A), and T (m)= A0(d). d=1 ′ ′ ′ a Let π = ⊗ π v be a cuspidal representation of GLm(A) with m

′ ′ ′ ′ L(s, Π×π )= L(s, Πv×π v) and ε(s, Π×π )= ε(s, Πv×π v, ψv) v v Y Y since again the local factors make sense whether Π is automorphic or not. A consequence of (1) and (2) above and the cuspidality of π′ is that both L(s, Π × π′) and L(s, Π × π′) converge absolutely for Re(s) >> 0, where Π and π′ are the contragredient representations, and that ε(s, Π × π′) is independent of the choicee ofeψ. e We saye that L(s, Π×π′) is nice if it satisﬁes the same analytic properties it would if Π were cuspidal, i.e., 1. L(s, Π × π′) and L(s, Π × π′) have analytic continuations to entire functions of s, e e 2. these entire continuations are bounded in vertical strips of ﬁnite width,

3. they satisfy the standard functional equation

L(s, Π × π′)= ε(s, Π × π′)L(1 − s, Π × π′).

The basic Converse Theorem for GLn is the following.e e

Theorem 5.1 Let Π be an irreducible admissible representation of GLn(A) as above. Suppose that L(s, Π × π′) is nice for all π′ ∈T (n − 1). Then Π is a cuspidal automorphic representation.

In this theorem we twist by the maximal amount and obtain the strongest possible conclusion about Π. The proof of this theorem essentially follows that of Hecke [24] and Weil [67] and Jacquet–Langlands [30]. It is of course valid for n = 2 as well. For applications, it is desirable to twist by as little as possible. There are essentially two ways to restrict the twisting. One is to restrict the rank of the groups that the twisting representations live on. The other is to restrict ramiﬁcation. When we restrict the rank of our twists, we can obtain the following result. L-functions for GLn 137

Theorem 5.2 Let Π be an irreducible admissible representation of GLn(A) as above. Suppose that L(s, Π × π′) is nice for all π′ ∈T (n − 2). Then Π is a cuspidal automorphic representation.

This result is stronger than Theorem 5.1, but its proof is a bit more delicate. The theorem along these lines that is most useful for applications is one in which we also restrict the ramification at a finite number of places. Let us fix a finite set of S of finite places and let T S(m) denote the subset of T (m) consisting of representations that are unramified at all places v ∈ S.

Theorem 5.3 Let Π be an irreducible admissible representation of GLn(A) as above. Let S be a finite set of finite places. Suppose that L(s, Π × π′) is nice for all π′ ∈T S(n − 2). Then Π is quasi-automorphic in the sense that ′ ′ there is an automorphic representation Π such that Πv ≃ Πv for all v∈ / S. Note that as soon as we restrict the ramification of our twisting representations we lose information about Π at those places. In applications we usually choose S to contain the set of finite places v where Πv is ramified. The second way to restrict our twists is to restrict the ramification at all but a finite number of places. Now fix a non-empty finite set of places S which in the case of a number field contains the set S∞ of all archimedean ′ places. Let TS(m) denote the subset consisting of all representations π in T (m) which are unramified for all v∈ / S. Note that we are placing a grave restriction on the ramification of these representations.

Theorem 5.4 Let Π be an irreducible admissible representation of GLn(A) as above. Let S be a non-empty finite set of places, containing S∞, such that ′ the class number of the ring oS of S-integers is one. Suppose that L(s, Π×π ) ′ is nice for all π ∈TS(n − 1). Then Π is quasi-automorphic in the sense that ′ ′ there is an automorphic representation Π such that Πv ≃ Πv for all v ∈ S ′ and all v∈ / S such that both Πv and Πv are unramified. There are several things to note here. First, there is a class number restriction. However, if k = Q then we may take S = S∞ and we have a Converse Theorem with “level 1” twists. As a practical consideration, if we let SΠ be the set of finite places v where Πv is ramified, then for applications we usually take S and SΠ to be disjoint. Once again, we are losing all information at those places v∈ / S where we have restricted the ramification unless Πv was already unramified there. 138 J.W. Cogdell

The proof of Theorem 5.1 essentially follows the lead of Hecke, Weil, and Jacquet–Langlands. It is based on the integral representations of L- functions, Fourier expansions, Mellin inversion, and finally a use of the weak form of Langlands spectral theory. For Theorems 5.2, 5.3, and 5.4, where we have restricted our twists, we must impose certain local conditions to compensate for our limited twists. For Theorem 5.2 and 5.3 there are a finite number of local conditions and for Theorem 5.4 an infinite number of local conditions. We must then work around these by using results on generation of congruence subgroups and either weak or strong approximation.

5.2 Inverting the Integral Representation

Let Π be as above and let ξ ∈ VΠ be a decomposable vector in the space VΠ of Π. Since Π is generic, then ﬁxing local Whittaker models W(Πv, ψv) at all places, compatibly normalized at the unramiﬁed places, we can associate to ξ a non-zero function Wξ(g) = Wξv (gv) on GLn(A) which transforms by the global character ψ under left translation by N (A), i.e., W (ng) = Q n ξ ψ(n)Wξ(g). Since ψ is trivial on rational points, we see that Wξ(g) is left invariant under Nn(k). We would like to use Wξ to construct an embedding of VΠ into the space of (smooth) automorphic forms on GLn(A). The simplest idea is to average Wξ over Nn(k)\ GLn(k), but this will not be convergent. However, if we average over the rational points of the mirabolic P = Pn then the sum

Uξ(g)= Wξ(pg) Nn(kX)\ P(k) is absolutely convergent. For the relevant growth properties of Uξ see [7]. Since Π is assumed to have automorphic central character, we see that Uξ(g) is left invariant under both P(k) and the center Zn(k). Suppose now that we know that L(s, Π × π′) is nice for all π′ ∈ T (m). Then we will hope to obtain the remaining invariance of Uξ from the GLn × GLm functional equation by inverting the integral representation for L(s, Π× ′ π ). With this in mind, let Q = Qm be the mirabolic subgroup of GLn which t stabilizes the standard unit vector em+1, that is, the column vector all of whose entries are 0 except the (m + 1)th, which is 1. Note that if m = n − 1 then Q is nothing more than the opposite mirabolic P=t P−1 to P. If we let L-functions for GLn 139

αm be the permutation matrix in GLn(k) given by

1 α = I m  m  In−m−1   −1 −1 then Qm = αm αn−1Pαn−1αm is a conjugate of P and for any m we have that P(k) and Q(k) generate all of GLn(k). So now set

Vξ(g)= Wξ(αmqg) ′ N (kX)\ Q(k) ′ −1 where N = αm Nn αm ⊂ Q. This sum is again absolutely convergent and is invariant on the left by Q(k) and Z(k). Thus, to embed Π into the space of automorphic forms it suﬃces to show Uξ = Vξ, for the we get invariance of Uξ under all of GLn(k). It is this that we will attempt to do using the integral representations. ′ Now let (π , Vπ′ ) be an irreducible subrepresentation of the space of au- ′ tomorphic forms on GLm(A) and assume ϕ ∈ Vπ′ is also factorizable. Let

′ n h ′ s−1/2 I(s; Uξ, ϕ )= PmUξ ϕ (h)| det(h)| dh. A 1 ZGLm(k)\ GLm( ) This integral is always absolutely convergent for Re(s) >> 0, and for all s if π′ is cuspidal. As with the usual integral representation we have that this unfolds into the Euler product

′ h 0 ′ s−(n−m)/2 I(s; Uξ, ϕ )= Wξ Wϕ′ (h)| det(h)| dh A A 0 In−m ZNm( )\ GLm( ) hv 0 ′ s−(n−m)/2 ′ = Wξv Wϕ (hv)| det(hv)|v dhv 0 In−m v v Nm(kv)\ GLm(kv ) Y Z ′ = Ψ (s; W ,W ′ ). v ξv ϕ v v Y Note that unless π′ is generic, this integral vanishes. Assume ﬁrst that π′ is cuspidal. Then from the local theory of L- ′ functions from Section 3, for almost all ﬁnite places we have Ψv(s; Wξ ,W ′ ) v ϕ v ′ ′ ′ = L(s, Πv ×π v) and for the other places Ψv(s; W ,W ′ )= ev(s; W ,W ′ ) ξv ϕ v ξv ϕ v ′ ′ L(s, Πv ×π v) with the ev(s; W ,W ′ ) entire and bounded in vertical strips. ξv ϕ v ′ ′ So in this case we have I(s; Uξ, ϕ ) = e(s)L(s, Π × π ) with e(s) entire and 140 J.W. Cogdell bounded in vertical strips. Since L(s; Π × π′) is assumed to be nice we may ′ conclude that I(s; Uξ, ϕ ) has an analytic continuation to an entire function which is bounded in vertical strips. When π′ is not cuspidal, it is a subrepresentation of a representation that is induced from (possibly non-unitary) cuspidal representations σi of GLri (A) for ri < m with ri = m and is in fact, if our integral doesn’t vanish, the unique generic constituent of this P induced representation. Then we can make a similar argument using this induced representation and the fact that the L(s, Π × σi) are nice to again ′ ′ ′ ′ conclude that for all π , I(s; Uξ, ϕ )= e(s)L(s, Π × π )= e (s) L(s, Π × σi) is entire and bounded in vertical strips. (See [7] for more details on this Q point.) ′ ′ ′ Similarly, consider I(s; Vξ, ϕ ) for ϕ ∈ Vπ′ with π an irreducible subrepresentation of the space of automorphic forms on GLm(A), still with

′ n h ′ s−1/2 I(s; Vξ, ϕ )= PmVξ ϕ (h)| det(h)| dh. A 1 ZGLm(k)\ GLm( ) Now this integral converges for Re(s) << 0. However, when we unfold, we ﬁnd

′ ′ I(s; V , ϕ )= Ψ (1 − s; ρ(w )W , W ′ ) =e ˜(1 − s)L(1 − s, Π × π′) ξ v n,m ξv ϕ v

Y ′ as above. Thus I(es; Vξ, ϕ ) also hasf an analyticf continuation to ane entiree function of s which is bounded in vertical strips. Now, utilizing the assumed global functional equation for L(s, Π × π′) in ′ ′ the case where π is cuspidal, or for the L(s, Π × σi) in the case π is not cuspidal, as well as the local functional equations at v ∈ S∞ ∪ SΠ ∪ Sπ′ ∪ Sψ as in Section 3 one ﬁnds

′ ′ ′ ′ I(s; Uξ, ϕ )= e(s)L(s, Π × π ) =e ˜(1 − s)L(1 − s, Π × π )= I(s; Vξ, ϕ )

′ ′ for all ϕ in all irreducible subrepresentations π of GLe m(eA), in the sense of analytic continuation. This concludes our use of the L-function. ′ ′ We now rewrite our integrals I(s; Uξ, ϕ ) and I(s; Vξ, ϕ ) as follows. We × a ﬁrst stratify GLm(A). For each a ∈ A let GLm(A) = {g ∈ GLm(A) | a a det(g)= a}. We can, and will, always take GLm(A)=SLm(A)· . Im−1 Let n ′ n h ′ hPmUξ, ϕ ia = PmUξ ϕ (h) dh a A 1 ZSLm(k)\ GLm( ) L-functions for GLn 141

n ′ and similarly for hPmVξ, ϕ ia. These are both absolutely convergent for all a and deﬁne continuous functions of a on k×\A×. We now have that ′ n ′ I(s; Uξ, ϕ ) is the Mellin transform of hPmUξ, ϕ ia,

′ n ′ s−1/2 × I(s; Uξ, ϕ )= hPmUξ, ϕ ia |a| d a, × A× Zk \ ′ similarly for I(s; Vξ, ϕ ), and that these two Mellin transforms are equal in the sense of analytic continuation. By Mellin inversion as in Lemma 11.3.1 n ′ n ′ of Jacquet-Langlands [30], we have that hPmUξ, ϕ ia = hPmVξ, ϕ ia for all a, and in particular for a = 1. Since this is true for all ϕ′ in all irreducible subrepresentations of automorphic forms on GLm(A), then by the weak form n n of Langlands’ spectral theory for SLm we may conclude that PmUξ = PmVξ as functions on Pm+1(A). More speciﬁcally, we have the following result.

Proposition 5.1 Let Π be an irreducible admissible representation of GLn(A) as above. Suppose that L(s, Π × π′) is nice for all π′ ∈T (m). Then for each n n ξ ∈ VΠ we have PmUξ(Im+1)= PmVξ(Im+1).

This proposition is the key common ingredient for all our Converse The- orems.

5.3 Remarks on the Proofs All of our Converse Theorems take Proposition 5.1 as their starting point. Theorem 5.1 follows almost immediately. In Theorems 5.2, 5.3, and 5.4 we must add local conditions to compensate for the fact that we do not have the full family of twists from Theorem 5.1 and then work around them. We will sketch these arguments here. Details for Theorems 5.1 and 5.4 can be found in [7] and for Theorems 5.2 and 5.3 can be found in [9].

5.3.1 Theorem 5.1 Let us ﬁrst look at the proof of Theorem 5.1. So we now assume that Π is as above and that L(s, Π × π′) is nice for all π′ ∈ T (n − 1). Then we n n have that for all ξ ∈ VΠ, Pn−1Uξ(In) = Pn−1Vξ(In). But for m = n − 1 the n projection operator Pn−1 is nothing more than restriction to Pn. Hence we have Uξ(In) = Vξ(In) for all ξ ∈ VΠ. Then for each g ∈ GLn(A), we have Uξ(g) = UΠ(g)ξ(In) = VΠ(g)ξ(In) = Vξ(g). So the map ξ 7→ Uξ(g) gives our embedding of Π into the space of automorphic forms on GLn(A), since now 142 J.W. Cogdell

Uξ is left invariant under P(k), Q(k), and hence all of GLn(k). Since we still have Uξ(g)= Wξ(pg) Nn(kX)\ P(k) we can compute that Uξ is cuspidal along any parabolic subgroup of GLn. Hence Π embeds in the space of cusp forms on GLn(A) as desired.

5.3.2 Theorem 5.2 Next consider Theorem 5.2, so now suppose that n ≥ 3, and that L(s, Π×π′) is nice for all π′ ∈T (n−2). Then from Proposition 5.1 we may conclude that n n Pn−2Uξ(In−1)= Pn−2Vξ(In−1) for all ξ ∈ VΠ. Since the projection operator n n−1 n−1 Pn−2 now involves a non-trivial integration over k \A we can no longer argue as in the proof of Theorem 5.1. To get to that point we will have to impose a local condition on the vector ξ at one place. Before we place our local condition, let us write Fξ = Uξ −Vξ. Then Fξ is n rapidly decreasing as a function on Pn−1. We have Pn−2Fξ(In−1) = 0 and we n−1 would like to have simply that Fξ(In) = 0. Let u = (u1,...,un−1) ∈ A and consider the function I tu f (u)= F n−1 . ξ ξ 1 n−1 n−1 Now fξ(u) is a function on k \A and as such has a Fourier expansion

fξ(u)= fˆξ(α)ψα(u) n−1 α∈Xk t where ψα(u)= ψ(α · u) and

fˆξ(α)= fξ(u)ψ−α(u) du. n−1 An−1 Zk \ n ˆ In this language, the statement Pn−2Fξ(In−1) = 0 becomes fξ(en−1) = 0, where as always, ek is the standard unit vector with 0’s in all places except the kth where there is a 1. Note that Fξ(g) = Uξ(g) − Vξ(g) is left invariant under (Z(k) P(k)) ∩ (Z(k) Q(k)) where Q = Qn−2. This contains the subgroup

In−2 R(k)= r = α′ α α α′ ∈ kn−2, α 6= 0 .   n−1 n n−1  1       L-functions for GLn 143

Using this invariance of Fξ, it is now elementary to compute that, with ˆ ˆ ′ n−1 this notation, fΠ(r)ξ(en−1) = fξ(α) where α = (α , αn−1) ∈ k . Since fˆξ(en−1) = 0 for all ξ, and in particular for Π(r)ξ, we see that for every ξ we have fˆξ(α) = 0 whenever αn−1 6= 0. Thus

ˆ ˆ ′ fξ(u)= fξ(α)ψα(u)= fξ(α , 0)ψ(α′,0)(u). n−1 ′ n−2 α∈Xk α ∈Xk

ˆ ′ Hence fξ(0,..., 0, un−1) = α′∈kn−2 fξ(α , 0) is constant as a function of u . Moreover, this constant is f (e )= F (I ), which we want to be 0. n−1 P ξ n−1 ξ n This is what our local condition will guarantee.

If v is a ﬁnite place of k, let ov denote the ring of integers of kv, and let pv denote the prime ideal of ov. We may assume that we have chosen v so that the local additive character ψv is normalized, i.e., that ψv is trivial on −1 ov and non-trivial on pv . Given an integer nv ≥ 1 we consider the open compact group

nv nv K00,v(pv )= {g = (gi,j) ∈ GLn(ov) |(i) gi,n−1 ∈ pv for 1 ≤ i ≤ n − 2; nv (ii) gn,j ∈ pv for 1 ≤ j ≤ n − 2; 2nv (iii) gn,n−1 ∈ pv }.

(As usual, gi,j represents the entry of g in the i-th row and j-th column.)

Lemma Let v be a ﬁnite place of k as above and let (Πv, VΠv ) be an irreducible admissible generic representation of GLn(kv). Then there is a vector ′ ξv ∈ VΠv and a non-negative integer nv such that

nv ′ ′ 1. for any g ∈ K00,v(pv ) we have Πv(g)ξv = ωΠv (gn,n)ξv

In−2 ′ 2. Πv 1 u ξv du = 0. −1   pv 1 R   The proof of this Lemma is simply an exercise in the Kirillov model of Πv and can be found in [9]. If we now ﬁx such a place v0 and assume that our vector ξ is chosen so 144 J.W. Cogdell

′ that ξv0 = ξv0 , then we have

F (I )= f (e ) = Vol(p−1)−1 f (0,..., 0, u ) du ξ n ξ n−1 v0 − ξ v0 v0 p 1 Z v0 In−2 = Vol(p−1)−1 F 1 u du = 0 v0 − ξ v0 v0 p 1   Z v0 1   for such ξ. ′ Hence we now have Uξ(In) = Vξ(In) for all ξ ∈ VΠ such that ξv0 = ξv0 ′ nv0 v0 v0 at our ﬁxed place. If we let G = K00,v0 (pv0 ) G , where we set G = ′ v6=v0 GLn(kv), then we have that this group preserves the local component ξ′ up to a constant factor so that for g ∈ G′ we have U (g)= U (I )= Qv0 ξ Π(g)ξ n VΠ(g)ξ(In)= Vξ(g). We now use a fact about generation of congruence type subgroups. Let ′ ′ ′ Γ1 = (P(k) Z(k)) ∩ G , Γ2 = (Q(k) Z(k)) ∩ G , and Γ=GLn(k) ∩ G . Then Uξ(g) is left invariant under Γ1 and Vξ(g) is left invariant under Γ2. It is essentially a matrix calculation that together Γ1 and Γ2 generate Γ. So, ′ as a function on G , Uξ(g) = Vξ(g) is left invariant under Γ. So if we let v0 ′ v0 ′ Π = ⊗ Π then the map ξ 7→ U ′ v0 (g) embeds V v0 into A(Γ\ G ), v6=v0 v ξvo ⊗ξ Π the space of automorphic forms on G′ relative to Γ. Now, by weak approx- ′ ′ imation, GLn(A) = GLn(k) · G and Γ = GLn(k) ∩ G , so we can extend v0 Π to an automorphic representation of GLn(A). Let Π0 be an irreducible component of the extended representation. Then Π0 is automorphic and coincides with Π at all places except possible v0.

One now repeats the entire argument using a second place v1 6= v0. Then we have two automorphic representations Π1 and Π0 of GLn(A) which agree at all places except possibly v0 and v1. By the generalized Strong Multiplicity One for GLn we know that Π0 and Π1 are both constituents of the same induced representation Ξ = Ind(σ1 ⊗···⊗ σr) where each σi is a cuspidal representation of some GLmi (A), each mi ≥ 1 and mi = n. We can write each σ = σ◦ ⊗ | det |ti with σ◦ unitary cuspidal and t ∈ R and i i i P i assume t1 ≥···≥ tr. If r > 1, then either m1 ≤ n − 2 or mr ≤ n − 2 (or both). For simplicity assume mr ≤ n − 2. Let S be a ﬁnite set of places containing all archimedean places, v0, v1, SΠ, and Sσi for each i. Taking L-functions for GLn 145

′ π = σr ∈T (n − 2), we have the equality of partial L-functions

S ′ S ′ S ′ eL (s, Π × π )= L (s, Π0 × π )= L (s, Π1 × π ) S ′ S ◦ ◦ = L (s,σi × π )= L (s + ti − tr,σi × σr ). Yi Yi S e Now L (s,σr ×σr) has a pole at s = 1 and all other terms are non-vanishing at s = 1. Hence L(s, Π × π′) has a pole at s = 1 contradicting the fact that ′ L(s, Π × π ) is nice.e If m1 ≤ 2, then we can make a similar argument using L(s, Π × σ1). So in fact we must have r = 1 and Π0 = Π1 = Ξ is cuspidal. Since Π0 agrees with Π at v1 and Π1 agrees with Π at v0 we see that in fact Π =e Π0 = Π1 and Π is indeed cuspidal automorphic.

5.3.3 Theorem 5.3 Now consider Theorem 5.3. Since we have restricted our ramification, we no longer know that L(s, Π × π′) is nice for all π′ ∈T (n − 2) and so Proposition 5.1 above is not immediately applicable. In this case, for each place v ∈ S ′ we fix a vector ξv ∈ VΠv as in the above Lemma. (So we must assume we ′ ′ have chosen ψ so it is unramified at the places in S.) Let ξS = v∈S ξv ∈ Π . Consider now only vectors ξ of the form ξS ⊗ ξ′ with ξS arbitrary S S Q ′ n h in V S and ξ fixed. For these vectors, the functions P U and Π S n−2 ξ 1 h Pn V are unramified at the places v ∈ S, so that the integrals n−2 ξ 1 ′ ′ ′ I(s; Uξ, ϕ ) and I(s; Vξ, ϕ ) vanish unless ϕ (h) is also unramified at those places in S. In particular, if π′ ∈T (n−2) but π′ ∈T/ S(n−2) these integrals ′ will vanish for all ϕ ∈ Vπ′ . So now, for this fixed class of ξ we actually have ′ ′ ′ ′ I(s; Uξ, ϕ ) = I(s; Vξ, ϕ ) for all ϕ ∈ Vπ′ for all π ∈ T (n − 2). Hence, as n n before, Pn−2Uξ(In−1)= Pn−2Vξ(In−1) for all such ξ. Now we proceed as before. Our Fourier expansion argument is a bit more subtle since we have to work around our local conditions, which now have been imposed before this step, but we do obtain that Uξ(g) = Vξ(g) for all ′ nv S g ∈ G = ( v∈S K00,v(pv )) G . The generation of congruence subgroups goes as before. We then use weak approximation as above, but then take for Q Π′ any constituent of the extension of ΠS to an automorphic representation of GLn(A). There is no use of strong multiplicity one nor any further use of the L-function in this case. More details can be found in [9]. 146 J.W. Cogdell

5.3.4 Theorem 5.4 Let us now sketch the proof of Theorem 5.4. We fix a non-empty finite set of places S, containing all archimedean places, such that the ring oS of S- integer has class number one. Recall that we are now twisting by all cuspidal ′ ′ representations π ∈TS(n − 1), that is, π which are unramified at all places v∈ / S. Since we have not twisted by all of T (n − 1) we are not in a position to apply Proposition 5.1. To be able to apply that, we will have to place local conditions at all v∈ / S. We begin by recalling the definition of the conductor of a representation Πv of GLn(kv) and the conductor (or level) of Π itself. Let Kv = GLn(ov) be the standard maximal compact subgroup of GLn(kv). Let pv ⊂ ov be the unique prime ideal of ov and for each integer mv ≥ 0 set ∗ . K (pmv )= g ∈ GL (o ) g ≡  ∗ . (mod pmv ) 0,v v  n v   ∗      0 · · · 0 ∗   mv  mv  mv  and K1,v(pv ) = {g ∈ K0,v(pv ) | gn,n ≡ 1 (mod pv ))}. Note that for 0 0 mv = 0 we have K1,v(pv)=K0,v(pv)=Kv. Then for each local component mv Πv of Π there is a unique integer mv ≥ 0 such that the space of K1,v(pv )– fixed vectors in Πv is exactly one. For almost all v, mv = 0. We take the ideal mv mv pv = f(Πv) as the conductor of Πv. Then the ideal n = f(Π) = v pv ⊂ o ◦ is called the conductor of Π. For each place v we fix a non-zero vector ξv ∈ Πv mv Q which is fixed by K1,v(pv ), which at the unramified places is taken to be ′ the vector with respect to which the restricted tensor product Π= ⊗ Πv is mv ◦ ◦ taken. Note that for g ∈ K0,v(pv ) we have Πv(g)ξv = ωΠv (gn,n)ξv . Now fix a non-empty finite set of places S, containing the archimedean places if there are any. As is standard, we will let GS = v∈S GLn(kv), S S ′ G = v∈ /S GLn(kv), ΠS = ⊗v∈SΠv, Π = ⊗v∈ /SΠv, etc. Then the compact S mv S Q S subring n = v∈ /S pv ⊂ k or the ideal it determines nS = k ∩ kSn ⊂ oS Q S mv is called the S–conductor of Π. Let K1 (n) = v∈ /S K1,v(pv ) and similarly S Q ◦ ◦ S S for K0 (n). Let ξ = ⊗v∈ /Sξv ∈ Π . Then this vector is fixed by K1 (n) and S Q transforms by a character under K0 (n). In particular, since v∈ /S GLn−1(ov) h embeds in KS(n) via h 7→ we see that when we restrictQ ΠS to GL 1 1 n−1 the vector ξ◦ is unramified. Now let us return to the proof of Theorem 5.4 and in particular the version of Proposition 5.1 we can salvage. For every vector ξS ∈ ΠS consider L-functions for GLn 147

◦ ◦ the functions UξS ⊗ξ and VξS⊗ξ . When we restrict these functions to GLn−1 they become unramified for all places v∈ / S. Hence we see that the integrals ◦ ′ ◦ ′ ′ ′ I(s; UξS ⊗ξ , ϕ ) and I(s; VξS ⊗ξ , ϕ ) vanish identically if the function ϕ ∈ Vπ ′ ′ is not unramified for v∈ / S, and in particular if ϕ ∈ Vπ′ for π ∈T (n−1) but ′ ◦ π ∈T/ S(n − 1). Hence, for vectors of the form ξ = ξS ⊗ ξ we do indeed have ◦ ′ ◦ ′ ′ ′ ′ that I(s; UξS ⊗ξ , ϕ )= I(s; VξS ⊗ξ , ϕ ) for all ϕ ∈ Vπ and all π ∈T (n − 1). ◦ ◦ Hence, as in Proposition 5.1 we may conclude that UξS ⊗ξ (In)= VξS ⊗ξ (In) for all ξS ∈ VΠS . Moreover, since ξS was arbitrary in VΠS and the fixed vector ◦ S ◦ ξ transforms by a character of K0 (n) we may conclude that UξS ⊗ξ (g) = ◦ S VξS⊗ξ (g) for all ξS ∈ VΠS and all g ∈ GS K0 (n). ◦ What invariance properties of the function UξS ⊗ξ have we gained from ◦ S our equality with VξS⊗ξ . Let us let Γi(nS) = GLn(k) ∩ GS Ki (n) which we may view naturally as congruence subgroups of GLn(oS ). Now, as a S ◦ function on GS K0 (n), UξS⊗ξ (g) is naturally left invariant under Γ0,P(nS) S ◦ = Z(k) P(k) ∩ GS K0 (n) while VξS ⊗ξ (g) is naturally left invariant under S Γ0,Q(nS) = Z(k) Q(k) ∩ GS K0 (n) where Q = Qn−1. Similarly we set S S Γ1,P(nS) = Z(k) P(k) ∩ GS K1 (n) and Γ1,Q(nS) = Z(k) Q(k) ∩ GS K1 (n). The crucial observation for this Theorem is the following result.

Proposition The congruence subgroup Γi(nS) is generated by Γi,P(nS) and Γi,Q(nS) for i = 0, 1.

This proposition is a consequence of results in the stable algebra of GLn due to Bass which were crucial to the solution of the congruence subgroup problem for SLn by Bass, Milnor, and Serre. This is the reason for the restriction to n ≥ 3 in the statement of Theorem 5.4. From this we get not an embedding of Π into a space of automorphic forms on GLn(A), but rather an embedding of ΠS into a space of classical automorphic forms on GS. To this end, for each ξS ∈ VΠS let us set

◦ S ◦ S ΦξS (gS )= UξS ⊗ξ ((gS , 1 )) = VξS⊗ξ ((gS , 1 ))

for gS ∈ GS. Then ΦξS will be left invariant under Γ1(nS) and transform by a Nebentypus character χS under Γ0(nS) determined by the central char- S acter ωΠS of Π . Furthermore, it will transform by a character ωS = ωΠS under the center Z(kS) of GS. The requisite growth properties are satisﬁed and hence the map ξS 7→ ΦξS deﬁnes an embedding of ΠS into the space A(Γ0(nS)\ GS; ωS,χS) of classical automorphic forms on GS relative to the congruence subgroup Γ0(nS) with Nebentypus χS and central character ωS. 148 J.W. Cogdell

We now need to lift our classical automorphic representation back to an adelic one and hopefully recover the rest of Π. By strong approximation for GLn and our class number assumption we have the isomorphism between the S space of classical automorphic forms A(Γ0(nS)\ GS; ωS,χS) and the K1 (n) invariants in A(GLn(k)\ GLn(A); ω) where ω is the central character of Π. Hence ΠS will generate an automorphic subrepresentation of the space of automorphic forms A(GLn(k)\ GLn(A); ω). To compare this to our original ◦ Π, we must check that, in the space of classical forms, the ΦξS⊗ξ are Hecke eigenforms for a classical Hecke algebra and that their Hecke eigenvalues agree with those from Π. We do this only for those v∈ / S which are unram- iﬁed, where it is a rather standard calculation. As we have not talked about Hecke algebras, we refer the reader to [7] for details. Now if we let Π′ be any irreducible subrepresentation of the represen- ′ tation generated by the image of ΠS in A(GLn(k)\ GLn(A); ω), then Π ′ is automorphic and we have Πv ≃ Πv for all v ∈ S by construction and ′ ′ Πv ≃ Πv for all v∈ / S by the Hecke algebra calculation. Thus we have proven Theorem 5.4.

5.4 Converse Theorems and Liftings

In this section we would like to make some general remarks on how to apply these Converse Theorems to the problem of functorial liftings [3]. In order to apply these theorems, you must be able to control the global properties of the L-function. However, for the most part, the way we have of controlling global L-functions is to associate them to automorphic forms or representations. A minute’s thought will then lead one to the conclusion that the primary application of these results will be to the lifting of automorphic representations from some group H to GLn. Suppose that H is a split classical group, π an automorphic representation of H, and ρ a representation of the L-group of H. Then we should be able to associate an L-function L(s, π, ρ) to this situation [3]. Let us L assume that ρ : H → GLn(C) so that to π should be associated an automorphic representation Π of GLn(A). What should Π be and why should it be automorphic. We can see what Πv should be at almost all places. Since we have the (arithmetic) Langlands (or Langlands-Satake) parameterization of representations for all archimedean places and those ﬁnite places where the representations are unramiﬁed [3], we can use these to associate to πv and the map L-functions for GLn 149

L ρv : Hv → GLn(C) a representation Πv of GLn(kv). If H happensto beGLm then we in principle know how to associate the representation Πv at all places now that the local Langlands conjecture has been solved for GLm [23, 26], but in practice this is still not feasible. For other situations, we do not know what Πv should be at the ramified places. We will return to this difficulty momentarily. But for now, let’s assume we can finesse this local problem ′ and arrive at a representation Π = ⊗ Πv such that L(s, π, ρ) = L(s, Π). Π should then be the Langlands lifting of π to GLn associated to ρ. For simplicity of exposition, let us now assume that ρ is simply the L standard embedding of H into GLn(C) and write L(s, π, ρ) = L(s,π) = L(s, Π). We have our candidate Π for the lift of π to GLn, but how to tell whether Π is automorphic. This is what the Converse Theorem lets us do. But to apply them we must first be able to define and control the twisted L-functions L(s,π × π′) for π′ ∈T with an appropriate twisting set T from one of our Converse Theorems. This is one reason why it is always crucial to define not only the standard L-functions for H, but also the twisted versions. If we know, from the theory of L-functions of H twisted by GLm for appropriate π′, that L(s,π × π′) is nice and L(s,π × π′) = L(s, Π × π′) for twists, then we can use Theorem 5.1 or 5.2 to conclude that Π is cuspidal automorphic or Theorem 5.3 or 5.4 to conclude that Π is quasi-automorphic and at least obtain a weak automorphic lifting Π′ which is verifiably the correct representation at almost all places. At this point this relies on the state of our knowledge of the theory of twisted L-functions for H. Let us return now to the (local) problem of not knowing the appropriate local lifting πv 7→ Πv at the ramified places. We can circumvent this by a combination of global and local means. The global tool is simply the following observation.

Observation Let Π be as in Theorem 5.3 or 5.4. Suppose that η is a ﬁxed (highly ramiﬁed) character of k×\A×. Suppose that L(s, Π × π′) is nice for all π′ ∈T ⊗ η, where T is either of the twisting sets of Theorem 5.3 or 5.4. Then Π is quasi-automorphic as in those theorems.

The only thing to observe, say by looking at the local or global integrals, is that if π′ ∈T then L(s, Π×(π′⊗η)) = L(s, (Π⊗η)×π′) so that applying the Converse Theorem for Π with twisting set T ⊗η is equivalent to applying the Converse Theorem for Π ⊗ η with the twisting set T . So, by either Theorem 5.3 or 5.4, whichever is appropriate, Π ⊗ η is quasi-automorphic and hence Π is as well. 150 J.W. Cogdell

Now, if we begin with π automorphic on H(A), we will take T to be the set of finite places where πv is ramified. For applying Theorem 5.3 we want S = T and for Theorem 5.4 we want S ∩ T = ∅. We will now take η to be highly ramified at all places v ∈ T . So at v ∈ T our twisting representations are all locally of the form (unramified principal series)⊗(highly ramified character). We now need to know the following two local facts about the local theory of L-functions for H.

′ ′ ′ ′ 1. Multiplicativity of γ-factors: If π v = Ind(π 1,v ⊗ π 2,v), with π i,v and

irreducible admissible representation of GLri (kv), then

′ ′ ′ γ(s,πv × π v, ψv)= γ(s,πv × π 1,v, ψv)γ(s,πv × π 2,v, ψv)

′ −1 ′ ′ −1 and L(s,πv × π v) should divide [L(s,πv × π 1,v)L(s,π × π 2,v)] . ′ If πv = Ind(σv ⊗ πv) with σv an irreducible admissible representation ′ ′ of GLr(kv) and πv an irreducible admissible representation of H (kv) ′ ′ with H ⊂ H such that GLr × H is the Levi of a parabolic subgroup of H, then

′ ′ ′ ′ ′ γ(s,πv × π v, ψv)= γ(s,σv × π v, ψv)γ(s,πv × π v, ψv)γ(s, σv × π v, ψv).

2. Stability of γ-factors: If π1,v and π2,v are two irreduciblee admissible representations of H(kv), then for every suﬃciently highly ramiﬁed character ηv of GL1(kv) we have

γ(s,π1,v × ηv, ψv)= γ(s,π2,v × ηv, ψv)

and L(s,π1,v × ηv)= L(s,π2,v × ηv) ≡ 1.

Once again, for these applications it is crucial that the local theory of L-functions is suﬃciently developed to establish these results on the local γ-factors. As we have seen in Section 3, both of these facts are known for GLn. To utilize these local results, what one now does is the following. At the places where πv is ramiﬁed, choose Πv to be arbitrary, except that it should have the same central character as πv. This is both to guarantee that the central character of Π is the same as that of π and hence automorphic and L-functions for GLn 151

to guarantee that the stable forms of the γ–factors for πv and Πv agree. Now ′ form Π = ⊗ Πv. We choose our character η so that at the places v ∈ T we have that the L– and γ–factors for both πv ⊗ ηv and Πv ⊗ ηv are in their stable form and agree. We then twist by T ⊗ η for this ﬁxed character η. If ′ ′ ′ π ∈T⊗η, then for v ∈ T , π v is of the form π v = Ind(µv,1 ⊗···⊗µv,m)⊗ηv × with each µv,i an unramiﬁed character of kv . So at the places v ∈ T we have

′ γ(s,πv × π v)= γ(s,πv × (Ind(µv,1 ⊗···⊗ µv,m) ⊗ ηv))

= γ(s,πv ⊗ (µv,iηv)) (by multiplicativity)

= Y γ(s, Πv ⊗ (µv,iηv)) (by stability)

= γY(s, Πv × (Ind(µv,1 ⊗···⊗ µv,m) ⊗ ηv)) (by multiplicativity) ′ = γ(s, Πv × π v) and similarly for the L-factors. From this it follows that globally we will have L(s,π×π′)= L(s, Π×π′) for all π′ ∈T⊗η and the global functional equation for L(s,π × π′) will yield the global functional equation for L(s, Π × π′). So L(s, Π × π′) is nice and we may proceed as before. We have, in essence, twisted away all information about π and Π at those v ∈ T . The price we pay is that we also lose this information in our conclusion since we only know that Π is quasi-automorphic. In essence, the Converse Theorem ﬁlls in a correct set of data at those places in T to make the resulting global representation automorphic.

5.5 Some Liftings To conclude, let us make a list of some of the liftings that have been ac- complished using these Converse Theorems. Some have used the above trick of multiplicativity and stability of γ–factors to handle the ramiﬁed places. Others, principally those that involve GL2, have adopted a technique of Ra- makrishnan [51] involving a sequence of base changes and descents to get a more complete handle on the ramiﬁed places.

1. The symmetric square lifting from GL2 to GL3 by Gelbart and Jacquet [15].

2. Non-normal cubic base change for GL2 by Jacquet, Piatetski-Shapiro, and Shalika [32]. 152 J.W. Cogdell

3. The tensor product lifting from GL2 × GL2 to GL4 by Ramakrishnan [51].

4. The lifting of generic cusp forms from SO2n+1 to GL2n, with Kim, Piatetski-Shapiro, and Shahidi [6].

5. The tensor product lifting from GL2 × GL3 to GL6 and the symmetric cube lifting from GL2 to GL4 by Kim and Shahidi [40].

6. The exterior square lifting from GL4 to GL6 and the symmetric fourth power lift from GL2 to GL5 by Kim [39].

For the most part, it was Theorem 5.3 that was used in each case, with the exception of (4), where a simpler variant was used requiring twists by T S(n − 1). For the non-normal cubic base change both Theorem 5.3 with n = 3 and Theorem 5.1 with n = 2 were used.

Acknowledgments

Most of what I know about L-functions for GLn I have learned through my years of work with Piatetski-Shapiro. I owe him a great debt of gratitude for all that he has taught me. For several years Piatetski-Shapiro and I have envisioned writing a book on L-functions for GLn [13]. The contents of these notes essentially follows our outline for that book. In particular, the exposition in Sections 1, 2, and parts of 3 and 4 is drawn from drafts for this project. The exposition in Section 5 is drawn from the survey of our work on Converse Theorems [10]. I would also like to thank Piatetski-Shapiro for graciously allowing me to present part of our joint eﬀorts in these notes. I would also like to thank Jacquet for enlightening conversations over the years on his work on L-functions for GLn. L-functions for GLn 153

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