ON HOLOMORPHY OF LOCAL LANGLANDS L-FUNCTIONS

A Thesis

Submitted to the Faculty

of

Purdue University

by

Mahdi Asgari

In Partial Fulfillment of the

Requirements for the Degree

of

Doctor of Philosophy

August 2000 ii

To Narges iii

ACKNOWLEDGMENTS

First and foremost, I would like to thank my advisor, Professor Freydoon Shahidi, for all his help and great patience throughout these years. His encouragements were especially precious during hard times in my work. He set an example of utmost professionalism for me to learn and follow. I will always be grateful to him. I also thank members of my committee, Professors Donu Arapura, Luchezar Avramov, David Goldberg, and Joseph Lipman. In particular, I would like to ac- knowledge all the help I have received from D. Goldberg throughout this work. Thanks are due to D. Ban, C. Jantzen, A. Roche, R. Schmidt, J. K. Yu, and Y. Zhang for many helpful discussions at different stages of this work. Many other friends, both in the Mathematics Department and outside, have made my stay at Purdue more pleasant. I thank them for their friendship. Last but not the least, my indebtedness to my wife, Narges, goes beyond words. This work would not have been possible without her love and many sacrifices. My deepest gratitude goes to my parents for giving me the opportunities they never had, and to my sister, Shidrokh, and my brother, Mohsen, for making my life so much joyful. iv

TABLE OF CONTENTS

Page ABSTRACT ...... v 1 Introduction ...... 1 2 Preliminaries ...... 4 2.1 Notation ...... 4 2.2 Review of Structure Theory ...... 4 2.3 Definition of local L-functions ...... 5 2.4 Langlands-Shahidi Method ...... 7 2.5 Examples ...... 10 2.6 A Conjecture and Known Results ...... 14 3 Local L-functions for Spin Groups ...... 17 3.1 Structure Theory for Spin Groups ...... 17 3.2 Discrete Series Representations of GSpin Groups ...... 22 3.3 Dual Groups and the Adjoint Representation ...... 27

4 L-functions for Groups of Type F4 ...... 29 4.1 Levi Subgroups ...... 29 4.2 Holomorphy of Local L-functions ...... 33 LIST OF REFERENCES ...... 42 v

ABSTRACT

Asgari, Mahdi, Ph.D., Purdue University, August, 2000. On Holomorphy of Local Langlands L-functions. Major Professor: Freydoon Shahidi.

This work studies local Langlands L-functions defined via the Langlands-Shahidi method which associates a complex function with a generic representation of a Levi subgroup in a quasi-split reductive p-adic group. Defining these functions via exis- tence of certain γ-factors, essentially quotients of these L-functions, Shahidi set forth a conjecture on their holomorphy in a half plane if the representation is tempered. He subsequently proved this conjecture for classical groups. The goal here is to prove this conjecture for Levi subgroups of the so-called spin groups, the double coverings of special orthogonal groups. We also verify the conjecture for Levi subgroups of

exceptional groups of type F4. H. Kim has recently proved the conjecture for E-type exceptional groups except for a few cases in which the representation theory of the Levi subgroup is not known well enough. The conjecture for an exceptional group of type G2 is already well-known. 1

1. Introduction

The use of automorphic representations has helped prove many powerful results in number theory. For this, one needs to continually study the representation theory of reductive groups over both local and global fields. The philosophy is that anything that is done for multiplicative group of a number field (cf. Tate’s thesis [Tat50]) can also be done for any reductive group. Let M be a connected reductive linear algebraic group defined over a number field, k, with A its ring of adeles. Let π = π be an automorphic form on M = M(A ). k ⊗v v k For each place v of k, M can be naturally considered to be defined over kv, the completion of k at the v. This group is unramified for almost all places v, i.e., it splits over an unramified Galois extension of Fv with Galois group Γv. To this data is attached a group, LM = LM 0 ⋊ Γ, called the L-group, whose connected component M = LM 0 is a complex linear Lie group. Also, for almost all non-archimedean places c v,πv is an irreducibe admissible representation of M(kv) which is unramified, i.e., its representation space has a vector fixed by M( ), where is the ring of integers of Ov Ov the local field kv. The Satake isomorphism attaches a unique semisimple conjugacy class A in M to each such πv. c If r is a (finite dimensional) complex analytic representation of M, one defines the c local Langlands L-function by

L(s,π , r) = det(I r(A)q−s)−1, s C, v − ∈ where q is the cardinality of the residue class field of Fv. However, it remains to define such local L-functions for the remaining places and these are expected to satisfy some global properties. More precisely, Langlands introduced the following: 2

Conjecture. Let S be a finite set of places which includes all the archimedean ones and for v S, all the defining data in the above is unramified. If L (s,π,r) denotes 6∈ S the product of local Langlands L-functions over v S, then one can complete this 6∈ product by defining the L-functions for v S, each being the inverse of a polynomial ∈ −s in qv when v is a finite place, such that the completed Euler product extends to a meromorphic function of s on C with a finite number of poles and a functional equation. The L-functions at the archimedean places are Artin L-functions defined by local Langlands correspondence (cf.[Lan71]).

There have been two different methods suggested in order to define these L- fucntions directly. The first method is usually called the Rankin-Selberg method and uses integral representations. The other is the Langlands-Shahidi method. Our focus in this work is the second method.

The idea of this method is to use Eisenstein series ([Lan71],[Sha81],[Sha88], and [Sha90c]). In particular, one uses intertwining operators to define the γ-factors, which are essentially quotients of local L-functions. These factors are defined in terms of the harmonic analysis on the group. One hence obtains L(s, σ, r)(cf. Chapter 2). Here, σ denotes an irreducible admissible representation of M = M(F ), where F is a local field of characteristic zero. In this method one assumes M to be the Levi component of a parabolic subgroup P = MN in a (quasi-split) connected reductive group G with N the unipotent radical of P. Also, assume that σ is generic, i.e., it has a Whittaker model. (One could define the L-functions for any irreducible admissible tempered σ to be the same as that of the unique generic one that conjecturally exists in its L-packet. This conjecture predicts the existence of a generic representation in certain finite sets of irreducible admissible representations all of which are supposed to have the same L-function.) The representation r is one of the L-group whose restriction to the connected component, M, of the L-group is assumed to be an irreducible constituent c of the adjoint representation of M on the Lie algebra of the dual of N = N(F ). Such c r’s turn out to cover most of the interesting L-functions studied so far.

Defining these L-functions, Shahidi in [Sha90c] set forth the following 3

Conjecture. If σ is tempered, then L(s, σ, r) is holomorphic for (s) > 0. ℜ This conjecture was subsequently proved in a series of papers for the cases where G is a general linear, symplectic, special orthogonal, or unitary group. Given that it is enough to prove the cojecture for G simple, it remained to prove the conjecture for the so-called spin groups, simply-connected double coverings of special orthogonal groups, and exceptional groups.

The conjecture is well-known for an exceptional group of type G2 (cf. Page 284 of [Sha90b]). In this work, we prove it for spin groups as well as an exceptional group of type F4. Recently, H. Kim has proved the conjecture, along with many other results, for exceptional groups of types E6,E7, and E8, except for a few cases where the Levi component involves either a spin group or another exceptional group ([Kim00]). The general idea of the proof both in our work and in Kim’s is based on construc- tion of discrete series representations out of supercuspidal ones ([CS98]). The latter has been worked on by many people including I. Bernstein and A. Zelevinsky ([BZ77] and [Zel80]) for general linear groups and D. Ban, C. Jantzen, G. Mui´c, and M. Tadi´c for other classical groups (cf. [Ban99], [Jan93a], [Jan93b], [Mui98], [Tad94], [Tad98], and [Tad99]). Such results have important applications in the theory of automorphic forms. In fact, in a recent important work, H. Kim and F. Shahidi have used some of our results in order to handle some local problems in their long-awaited result on the existence of symmetric cubes cusp forms on GL2 (cf. [KS00]). 4

2. Preliminaries

2.1 Notation

Throughout this work we let F be a non-archimedean local field of characteristic zero. Such fields are completions of a number field, a finite extension of Q, with respect to one of its prime ideals. Let be the ring of integers in F and denote O P its unique maximal ideal. Then / is a finite field whose cardinality we will denote O P by q (cf. [Ser79] as a general reference).

2.2 Review of Structure Theory

The theory of L-functions that we will review is available in more generality than we present here. We can take any quasi-split connected reductive group to work with. Instead we choose to work with a split simple group. This will avoid a few technicalities while we are not imposing any real restriction since our new results in this work address the so-called Spin groups and exceptional groups of type F4, both of which are split groups. This will also simplify the definition of L-group. In fact, we will only deal with the connected component of what is usually called the L-group which involves Galois group or, more precisely, the Weil-Deligne group (cf. [Tat79] and [Bor79]). We refer to the connected component of the L-group as the complex dual or simply its dual.

Let G be a split connected reductive linear algebraic group defined over F. Fix a Borel subgroup B of G and write B = TU where U is the unipotent radical of B and T is a fixed maximal torus in G. Let ∆ be the corresponding system of simple roots of G with respect to T. Let P be a maximal standard parabolic subgroup in G with Levi decomposition P = MN, (N U) and T M. Then M = M , where ⊃ ⊂ θ θ =∆ α for a unique simple root α ∆. \ { } ∈ 5

Let a = Hom(X(M)F , R) be the real Lie algebra of connected component of the

center of M, where X(M)F denotes the F -rational characters of M. We define the dual of a and its complexification as follows.

∗ a := X(M) Z R, F ⊗ ∗ ∗ aC := a C. ⊗

∗ ∗ Recall that since M is assumed to be maximal, aC/zC is one-dimensional and we will shortly choose an identification of it with C. Here z denotes the Lie algebra of the split torus in the center of G.

Let ρP be half of the sum of positive roots of G whose root vectors generate N and define α to be e 1 ∗ ρP a , ρP , α∨ ∈ h i where , is the Z-pairing between the roots and coroots of G. We now identify h· ·i ∗ s C with sα˜ aC. Also, define a homomorphism ∈ ∈ H : M a P −→ via

qhχ,HP (·)i = χ( ) , χ X(M) . | · |F ∈ F We also have

qhsα,H˜ P (·)i = α˜( ) s , | · |F whereα ˜( ) is the value ofα ˜ on an element of M and s C. · ∈ 2.3 Definition of local L-functions A connected reductive group G as in the above is characterized by its root datum, a quadruple (X,R,X∨,R∨), where X and X∨ denote the root lattice and coroot lattice of G equipped with a Z-pairing

, : X X∨ Z h· ·i × → 6

while R and R∨ are the set of roots and coroots. The root datum obtained by interchanging the roots and coroots, (X∨,R∨,X,R), determines another connected reductive group called the dual of G (cf. [Spr98],[Bor91],[Hum75]). When considered as a group over complex numbers, we refer to it as the complex dual of G and denote it by G. It is the connected component of the L-group of G. b Now consider M, the complex dual of M which can be realized as a Levi subgroup c in G. We now define the local Langlands L-function. b Definition 2.3.1. Let σ be an unramified representation of M = M(F ), (i.e., hav- ing an M( )-fixed vector). The Satake isomorphism then associates a semisimple O conjugacy class, Aσ, in M (which we will normally identify with a representative Aσ c in T , the maximal torus of M ). For a finite-dimensional complex representation r b c of M define the local Langlands L-function as follows. c L(s, σ, r) := det(I r(A )q−s)−1, s C. − σ ∈ Clearly, in the above definition, we did not have to assume M to be a Levi sub- group in another group. In fact, the definition is usually presented in that generality. However, we will need this assumption for generalization of this definition via the Langlands-Shahidi method (cf. 2.4). That method uses specific r’s which turn out to include many important examples of L-functions studied so far. To describe them let P be a standard parabolic in G b b having M as its Levi factor and write P = MN (cf. [Bor79]). Define n to be the Lie c b c b algebra of N on which M acts via the adjoint action, r. b b c Proposition 2.3.2. There exists a positive integer m such that m n = V M i b i=1 ∨ where V , spanned by root vectors X ∨ n with α,˜ β = i, is an irreducible subrep- i β ∈ h i m resentation of r = ri. b Li=1 Proof. This is due to Langlands. Proposition 4.1. of [Sha88] gives an algebraic proof. 7

Hence, one gets the L-functions L(s, σ, r ) for each 1 i m. i ≤ ≤ 2.4 Langlands-Shahidi Method

In the definition we gave for local L-function we had to assume that the represen- tation σ is unramified in order to be able to use the Satake isomorphism. How do we

define L(s, σ, ri) if σ is ramified? Two methods have been suggested, each of which has its own advantages and power in proving new results. One is the Rankin-Selberg method which uses integral representations. The second method is the Langlands- Shahidi method which assumes σ is a an irreducible admissible generic representation of a Levi subgroup in another reductive group as above. We should mention that these L-functions are the same no matter what method one uses to define them since they are to satisfy certain properties that uniquely determine them.

Shahidi studied intertwining operators and local coefficients in a series of papers which are the basis on which these L-functions are defined. These in turn lead to the definition of γ-factors which we recall below. We should remark that these results cover both global and local (archimedean and non-archimedean) cases. Combining local and global methods is in fact one of the aspects of this method that contributes to its strength in proving powerful new results. However, since in this work we are only concerned with non-archimedean local places, we present the part of existence of γ-factors that we will be using.

We should also remark that throughout this work a multiplicative character χ of U = U(F ) is fixed which one needs in the definition of a generic representation (i.e., having Whittaker functions with respect to χ or being χ-generic). We do this by

fixing an additive character ψF of F on which the γ-factors as well as ǫ-factors, to be introduced later, depend. The L-functions, however, do not depend on this additive character. We also assume that χ and ψF are compatible in the sense of Section 3 of [Sha90c]. For simplicity, we may drop these throughout this work.

Let M, G and r be as in Sections 2.2 and 2.3. Let σ be an irreducible admissible χ-generic representation of M = M(F ). Shahidi proved in Theorem 3.5. of [Sha90c] 8

that there exist m complex functions

γi(s,σ,r,ψF )= γ(s, σ, ri),

each a rational function in q−s, satisfying certain local and global properties. Below we recall some of these properties that will be helpful to us. We should remark that the above theorem in its complete generality is the main result of [Sha90c] which also establishes uniqueness of these γ-factors.

Proposition 2.4.1. For each 1 i m we have, ≤ ≤ γ(s, σ, r ,ψ )γ(1 s, σ,˜ r , ψ¯ ) = 1. i F − i F

Moreover, γ(s, σ, r˜i)= γ(s, σ,˜ ri).

Definition 2.4.2. Let P C[X] with P (0) = 1 be such that P (q−s) is the σ,i ∈ σ,i σ,i normalized numerator of γ(s, σ, ri). If σ is tempered, define the local Langlands L- −s −1 function L(s, σ, ri) to be the inverse of this polynomial, Pσ,i(q ) . One then defines

L(s, σ, ri) for arbitrary irreducible admissible generic σ via Langlands classification (cf. Page 308 of [Sha90c]).

−s Proposition 2.4.3. There exists a monomial in q , denoted by ǫ(s, σ, ri,ψF ), such that

γ(s, σ, r ,ψ )= ǫ(s, σ, r ,ψ )L(1 s, σ, r˜)/L(s, σ, r ). i F i F − i i Remark 2.4.4. Note that the L-functions on the right hand side of the above Propo-

sition do not depend on ψF while the ǫ-factor does. Also, by arguments on page 287 of [Sha90b] and Proposition 7.8 of [Sha90c], we know that if σ is unitary supercuspidal, then L(s, σ, r) is, up to a monomial is q−s, equal to L( s, σ, r˜). − Another important property of the γ-factors is their multiplicativity which is es-

sential in our work. Let M = Mθ be a maximal standard Levi subgroup in G and σ be an irreducible admissible χ-generic representation of M(F ). Assume P′ = M′ N′ · ′ ′ to be a parabolic subgroup inside M and M = M ′ with θ θ. Also assume that θ ⊂ 9

σ Ind σ′ 1 where σ′ is an irreducible admissible generic representation of M ′. Let ⊂ P ′↑M ⊗ −1 w = wℓ,∆wℓ,θ be the longest element of the Weyl group of G modulo that of M. Set θ = θ′ and w′ = w. There exists a simple root α ∆ such that w′ (α ) < 0. Define 1 1 1 ∈ 1 1 Ω = θ α . Now M contains M as a maximal Levi. Let w = w w−1 and 1 1 ∪ { 1} Ω1 θ1 1 ℓ,Ω1 ℓ,θ1 ′ ′ −1 w2 = w1w1 . Let θ2 = w1(θ1) and continue this process to define successive θj’s and Ω ’s. There exists n such that w′ = 1, i.e., w = w w w . j n n−1 n−2 ··· 1

Proposition 2.4.5. (Shahidi) If we set w = w w for 2 j n 1 and j j−1 ··· 1 ≤ ≤ − ′ w1 = 1, then wj(σ ) is a representation of Mθj (F ) and we have

γ(s, σ, r ,ψ )= γ(s, w (σ′),R ,ψ ). i F Y j ij F j∈Si

Each r , 1 i m, is an irreducible constituent of the adjoint action of the L-group i ≤ ≤ of M on the Lie algebra of the L-group of N and R , 1 j n 1, is the same j ≤ ≤ − action for Mθj in MΩj . Every irreducible component of the restriction of ri to the ′ L-group of M is equivalent, under wj, to an irreducible constituent Rij of some Rj for a unique j. The set Si consists of all such j for a given i.

To explain the sets Si more clearly, note that in the above product, i is fixed and out of (possibly) several j’s we take the appropriate constituent of appropriate Rj’s on the right hand side for the product corresponding to the fixed i on the left hand

side. All these constituent of all the j’s would then cover all the ri’s, i = 1, 2,...,m. This means, for a fixed j, if we choose all the appropriate constituents of the i’s to whose products R contributed, then we recover R . In other words, R = R with j j j ⊕ ij the sum over all i’s such that j S . Examples of such products with detail will occur ∈ i later on when we deal with an exceptional group of type F4.

Remark 2.4.6. Local coefficients, via which γ-factors are defined, only depend on the derived group of a reductive group. This implies that we only need to study the γ-factors and L-functions for semisimple groups. In fact, it is enough to prove the holomorphy for these L-functions for Levi subgroups of simple groups. 10

2.5 Examples

Another property of γ-factors that we will need is that twisting the representation on the Levi by a certain character translates into a shift in the complex parameter s. More precisely, we have the following.

Proposition 2.5.1. Let σ0 be an irreducible admissible representation of M and let σ = σ qhs0α,HP (·)i, where H is as in 2.2. Then L(s, σ, r)= L(s + s , σ , r). 0 ⊗ e P 0 0

This property turns out to be essential in proving that the L-functions for discrete series (and tempered) representations, which are obtained as products of those for supercuspidal representations, have their possible poles for (s) > 0 cancelled among ℜ different terms, i.e., they are holomorphic. In the cases we are going to consider, the group G is simply-connected. (Example 2.5.6 considers a case where G is not simply-connected.) For the remainder, assume G to be simple and simply-connected. Then every t T can be written uniquely as ∈

t = β∨(t ), t GL Y β β ∈ 1 β∈∆

∨ - where β : GL1 T is the coroot attached to β (cf. Page 44, Corollary (a) to Lemma 28 of [Ste67]). Define the map

- ξα : T GL1

via ξα(t)= tα. Note that ker ξ = T M , where M is the derived group of M. To see this α ∩ D D observe that if

t = β∨(t ) Y β β∈∆

with t = 1, then t M since β∨(t ) M for β = α. α ∈ D β ∈ D 6

Proposition 2.5.2. With notation as in the above, we have α = ξα on T. e 11

Proof. Let ∆ = α , α ,...,α be a system of simple roots for G and α = α . 0 { 1 2 n} r Write ∆ α as a disjoint union 0 \ { r} ∆ ∆ ∆ 1 ∪ 2 ∪···∪ s

where each ∆ℓ is an irreducible subsystem whose Dynkin diagram is a connected

component of the Dynkin diagram of ∆0 after removing αr. Let ρ0 be half of the sum

of positive roots in ∆0 and ρℓ be that of ∆ℓ. Every positive root of G with a root

vector in N is a positive root generated by ∆0 but not by any of ∆ℓ’s. Therefore, s ρP = ρ ρ . 0 − X ℓ ℓ=1 n ∨ Let t = α (tj) be an element in the maximal split torus T in G. Write Qj=1 j s t = α∨(t ) T r r Y ℓ ℓ=1 where

T = α∨(t ). ℓ Y j j αj ∈∆ℓ Notice that ρ (T ) = 1 if k = ℓ. To see this note that ℓ k 6 hρ ,α∨i ρ (T )= t ℓ j ℓ k Y j αj ∈∆k and β, α∨ = 0 for β ∆ , k = 1, 2,...,s. h j i ∈ k Now

ρ0(t) ρP(t) = s ℓ=1 ρℓ(t) Q s ∨ ρ0( ℓ=1 Tℓ) ρ0(αr (tr)) = s Q∨ · s s ℓ=1 ρℓ(αr (tr)) ℓ=1 k=1 ρℓ(Tk) Qs · Q ∨ Q ℓ=1 ρ0(Tℓ) ρ0(αr (tr)) = Qs s ∨ . ρℓ(Tℓ) · ρℓ(α (tr)) Qℓ=1 Qℓ=1 r

We claim that ρℓ(Tℓ)= ρ0(Tℓ) for ℓ = 1, 2,...,s. To see this observe that

hρ ,α∨i ρ (T )= t ℓ j ℓ ℓ Y j αj ∈∆ℓ 12

and

∨ hρ0,α i ρ (T )= t j . 0 ℓ Y j αj ∈∆ℓ

It is enough to show that for α ∆ , we have j ∈ ℓ

ρ , α∨ = ρ , α∨ = 1. h ℓ j i h 0 j i

To see this note that if ρ is half of the sum of positive roots in any irreducible root system, in particular, ∆0 and ∆ℓ, then for any simple root α we have,

ρ, α∨ = ω, α∨ = ω, α∨ = 1, h i hX i Xh i ω ω where ω runs over the set of fundamental weights, since all ω, α∨ are 0 except for h i one which is 1. Therefore,

∨ ρ0(αr (tr)) P ρ (t) = s ∨ ℓ=1 ρℓ(αr (tr)) Q ∨ hρ0,αr i tr = ∨ s hρℓ,αr i ℓ=1 tr Q s ∨ = thρ0−Pℓ=1 ρℓ,αr i

∨ hρP,αr i = tr .

This implies that αr(t)= tr which completes the proof. f Recall that α defined in Section 2.2 is a character on T, the maximal split torus in G which extendse to a character of M. We can also extend ξ, defined before 2.5.2, to a character of M as follows. Write M = A M where A T is the split torus in the center of M. For m = ad with a A · D ⊂ ∈ and d M , define ξ(m) = ξ(a). To see this is well-defined, let m = a d = a d ∈ D 1 1 2 2 with a ,a A and d ,d M . Then, a−1a = d d−1 A M . This implies that 1 2 ∈ 1 2 ∈ D 2 1 2 1 ∈ ∩ D −1 ξ(a2 a1)=1or ξ(a1)= ξ(a2). 13

Moreover, note that we now have, ker ξ = MD and the following sequence is exact.

- - ξ - - 1 MD M GL1 1

Corollary 2.5.3. Let σ0 be an irreducible generic representation of M = M(F ) and

s0 σ = σ ξ( ) for some s C. Then, we have 0 ⊗ · F 0 ∈

L(s, σ, r)= L(s + s0, σ0, r) where r denotes an irreducible constituent of the adjoint action of the L-group of M on the Lie algebra of the L-group of N.

Proof. Clear from 2.5.2 and 2.5.1.

Example 2.5.4. Let G be SLn+1 and

g M =   : g GL , a−1 = det(g) GL  ∈ n  ≃ n  a    be the maximal Levi component in G corresponding to θ =∆ α where \ { n} ∆= α , α ,...,α { 1 2 n} is the fundamental system of simple roots for G with αi(t)= ai/ai+1 for

t = diag(a ,a ,...,a ) T. 1 2 n+1 ∈ In this case, 1 α = (α + 2α + + nα ). n n + 1 1 2 ··· n f n+1 Note that ai = 1. Then, Qi=1 n i 1 α (t) = ( (a /a ) ) n+1 n Y i i+1 f i=1 n+1 1 = ((a a a ) ) n+1 1 2 ··· n

= det(diag(a1,a2,...,an))

Therefore, ξ(g) = det(g) on GLn. 14

Example 2.5.5. Let G be Sp with ∆ = α , α where α is the long root. Let 4 { 1 2} 2 θ = α . Then { 2}  a     M = Mθ =  g  : g SL2, a GL1 GL1 SL2.   ∈ ∈ ≃ ×  a−1       With t = diag(a ,a ,a−1,a−1) T we have, 1 2 2 1 ∈

 a1/a2 : i = 1 αi(t)=  2 a2 : i = 2  and 1 α (t) = (α + α )(t) 1 1 2 2 f = (a /a ) a 1 2 · 2

= a1.

Therefore, ξ is the projection on the first component on GL SL and ker ξ = 1 × 2 1 SL as expected. { } × 2

Example 2.5.6. Assume G = SO2n+1 or SO2n and let M = GLn be the Siegel Levi in G. Note that G is not simply-connected. Denote the rational character defined by

the determinant on M by ξ. Again we have ker ξ = MD. Computing α˜ explicitly, we see that in fact ξ =α ˜ but ξ =α ˜2. This means that twisting the representation on 6 s GL (F ) by det( ) would result in a shift by 2s in the L-function. n · F

Notice that if G = Sp2n and M = GLn is the Siegel Levi inside G, then twisting as above would translate in a shift by s by our Proposition. This explains the appearance of s and 2s in the above cases discussed on Page 7 of [Sha92].

2.6 A Conjecture and Known Results

The following is Conjecture 7.1. of [Sha90c].

Conjecture 2.6.1. Assume that σ is a tempered representation. Then for 1 i m ≤ ≤

L(s, σ, ri) 15

is holomorphic for (s) > 0. ℜ Remark 2.6.2. It is enough to prove the conjecture for σ in discrete series. The L-functions for tempered σ are then a product of those for discrete series σ by 2.4.5.

This conjecture is a theorem in the following cases (cf. [Sha90c]).

Proposition 2.6.3. If m = 1, then L(s, σ, r ) is holomorphic for (s) > 0. 1 ℜ Proposition 2.6.4. If m = 2 and

L(s, σ, r )= (1 a q−s)−1, 2 Y − j j

possibly an empty product with all a C of absolute value one, then L(s, σ, r ) is j ∈ 1 holomorphic for (s) > 0. The assumption of the Proposition holds in particular when ℜ r2 is one-dimensional.

Proposition 2.6.5. Assume σ to be unitary supercuspidal. Then L(s, σ, ri) is holo- morphic for (s) > 0, 1 i m. In fact each L(s, σ, r ) is a product (possibly empty) ℜ ≤ ≤ i as in Proposition 2.6.4 with a = 1. | j| The conjecture for Levi subgroups of classical groups has been proved in various papers ([Sha83], [Sha90b], [Sha92] [CS98]). It is also well-known for Levi subgroups

of an exceptional group of type G2. To establish the conjecture in general, one needs to prove it for Levi subgroups of exceptional groups of type F4,E6,E7,E8 as well as the so-called spin groups, the double coverings of special orthogonal groups which are not classical. (For us, classical would mean a group that is defined via fixed points of a bilinear form. Some consider classical to be any non-exceptional group which would include spin groups.) In the next two chapters, we establish this conjecture for Levi subgroups of spin groups and Levi subgroups of an exceptional group of type F4. H. Kim has recently proved the conjecture for Levi subgroups of E-type exceptional groups except for the following four Levi subgroups: Cases E 3,E 3, and E 4 7 − 8 − 8 − of [Sha88] as well as Case (xxviii) of [Lan71] (D E ) (cf.[Kim00]). The above four 7 ⊂ 8 Levis all involve spin group or exceptional group of type E6 where we do not have 16 enough information on discrete series representations (e.g., how to obtain them from supercuspidals). 17

3. Local L-functions for Spin Groups

3.1 Structure Theory for Spin Groups

To study the L-functions for Spin groups we need to look at parabolic induction which requires one to know the structure of Levi factors in these groups. These factors turn out to be rather complicated to describe.

Example 3.1.1. The group Spin5 is simply-connected of type B2. Note that this group is the same as Sp4 if we interchange the roles of long and short roots. This means the Siegel Levi of Spin is the non-Siegel Levi of Sp which is GL Sp or GL Spin 5 4 1 × 2 1 × 3 while the non-Siegel Levi of Spin5 is GL2 which is not what we would expect if Spin5 were to have the same type of Levi subgroups as classical groups.

As a more complicated example, one could look at the Siegel Levi in Spin2n+1 when n = 2k is even. As in [Sha97] one can verify that the Levi subgroup is isomorphic to

GL SL 1 × n , (λ,λ I ) : λk = 1 { · n } while one would expect to get GLn if the Levi subgroups were to have the same structure as in the case of classical groups. (What we get in this case doubly covers GLn.)

It turns out that the Levi subgroups in Spin groups are not always a product of general linear groups and another Spin of smaller rank and one can not describe them in a straight forward way. A few examples such as above, especially in the case of non-maximal Levis, soon reveals this fact. To remedy this we will instead look at another class of groups, namely, GSpin groups, which have Spin groups as their derived groups. These groups turn out to have simpler Levi factors. Since local factors depend only on the derived group of the group in question, the L-functions are the same for both class of groups. 18

Let Spinm be the split simple simply-connected algebraic group of type Bn if m = 2n + 1 and of type Dn if m = 2n. This group is a double covering, as algebraic groups, of the group SOm.

Fix a maximal torus T contained in a fixed Borel subgroup B in Spinm. This corresponds to a choice of a system of simple roots ∆ = α , α ,...,α . We denote { 1 2 n} ∨ ∨ ∨ ∨ - the associated coroots by ∆ = α1 , α2 ,...,αn . We then have αi : T GL1 { } ∨ and α∨ : GL - T such that α α∨(x) = xhαi,αj i where , : X X∨ - Z i 1 i ◦ j h· ·i × is the Z-pairing between characters and cocharacters.

Proposition 3.1.2. (a) Each t T can be written uniquely as ∈ n t = α∨(t ), t GL . Y j j j ∈ 1 j=1

(b) The center of the group consists of t as above such that

n hα ,α∨i t i j = 1 Y j j=1

for 1 i n. ≤ ≤

Proof. Both statements are parts of Lemma 28 on Page 43 of [Ste67]. Note that part (a) is true whenever the group is simply connected while part (b) is true in general.

Corollary 3.1.3. The center of Spinm is equal to

1,c Z/2Z { } ≃ if m = 2n + 1, where c = α∨( 1). If m = 2n, then it is equal to n −

1,c,z,cz Z/2Z Z/2Z { } ≃ ×

provided that n is even and

1,z,c,z3 Z/4Z  ≃ 19

otherwise, where

c = α∨ ( 1)α∨( 1) n−1 − n − and

n−2 z = α∨(( 1)j) α∨ ( 1) Y j − · n−1 − j=1 if n is even and

n−2 z = α∨(( 1)j) α∨ (√ 1)α∨(√ 1) Y j − · n−1 − n − j=1 if it is odd. Observe that c = z2.

Proof. By part (a) of the previous proposition t is in the center if and only if

n t ker α . ∈ \ i i=1 Now assume m = 2n + 1. If t ker α , then ∈ i

2 −1  t1 t2 = 1 if i = 1 n  ∨  t−1 t2 t−1 = 1 if 2 i n 2 hαi,αj i  i−1 i i+1 αi(t)= tj =  ≤ ≤ − Y  −1 2 −2 j=1 tn−2 tn−1 tn = 1 if i = n 1  −  t−1 t2 = 1 if i = n  n−1 n  Hence t is in the center if and only if

t t t t t2 1 = 2 = = n−2 = n−1 = n = 1. 1 t1 ··· tn−3 tn−2 tn−1

Therefore, t = 1 and all other t ’s are equal to 1. n ± j Now assume m = 2n. Again, t is in the center if and only if

t1 t2 tn−2 tn−1tn 2 2 = = = = and tn−1 = tn = tn−2. 1 t1 ··· tn−3 tn−2

Solving these equations simultaneously will yield the corollary.

We can now define the group GSpinm. 20

Definition 3.1.4. For m 3 define ≥ GL Spin G = 1 × m (1, 1), ( 1,c) { − }

where c is defined as in 3.1.3. We will call this group GSpinm. Write A for the image of (λ, 1) λ GL and H for the image of (1,g) g Spin in G. Then, G = A H { | ∈ 1} { | ∈ m} · with S = A H = 1,c . Clearly, the derived group of GSpin is Spin . ∩ { } m m We also define GSpin0 and GSpin1 to be GL1. (One could also define GSpin2 to be GL2 although this group will not come up in our discussions.)

We now give an alternative description of GSpinm in terms of root datum which will be helpful later on.

Proposition 3.1.5. The reductive group GSpinm can be given by the root datum (X,R,X∨,R∨) where

X = Ze Ze Ze 0 ⊕ 1 ⊕···⊕ n

and

X∨ = Ze∗ Ze∗ Ze∗ 0 ⊕ 1 ⊕···⊕ n

equipped with the standard Z-pairing while R and R∨ are defined as follows.

(a) If m = 2n + 1, then

R = α = e e , α = e e ,...,α = e e , α = e , { 1 1 − 2 2 2 − 3 n−1 n−1 − n n n} R∨ = α∨ = e∗ e∗, α∨ = e∗ e∗,...,α∨ = e∗ e∗ , α∨ = 2e∗ e∗ . { 1 1 − 2 2 2 − 3 n−1 n−1 − n n n − 0}

(b) If m = 2n, then

R = α = e e ,...,α = e e , α = e + e , { 1 1 − 2 n−1 n−1 − n n n−1 n} R∨ = α∨ = e∗ e∗,...,α∨ = e∗ e∗ , α∨ = e∗ + e∗ e∗ . { 1 1 − 2 n−1 n−1 − n n n1 n − 0} 21

Proof. We give the proof for the case of m = 2n + 1. The other case is similar. We compute the root datum of GSpin2n+1 from that of Spin2n+1 and verify that it can be written as above. Start with the character lattice of Spin2n+1. Since Spin2n+1 is simply-connected, this lattice is the same as the weight lattice of the Lie algebra so2n+1 which is well-known. Choose f1, f2,...,fn (cf.[FH91]) such that the Z-span of f , f ,...,f , (f + + f )/2 is the character lattice of Spin . The character 1 2 n 1 ··· n 2n+1 lattice of GL Spin can then be written in the same way as the Z-span of 1 × 2n+1 f , f , f ,...,f , (f + + f )/2. Now the characters of GSpin are those that 0 1 2 n 1 ··· n 2n+1 have trivial value at the element ( 1,c). Note that f ( 1,c) = 1 for 1 i n and − i − ≤ ≤ (f +(f + + f )/2)( 1,c) = 1. In fact, the character lattice of GSpin can be 0 1 ··· n − 2n+1 written as the Z-span of f , f ,...,f , f +(f + + f )/2. Using the Z-pairing of 1 2 n 0 1 ··· n the root datum, one can also compute the cocharacter lattice which turns out to be spanned by f ∗ and f ∗ +(f ∗/2), 1 i n. Now for 1 i n set e = f and set 0 i 0 ≤ ≤ ≤ ≤ i i e = f +(f + + f )/2. Also set e∗ = f ∗ +(f ∗)/2 and e∗ = f ∗. Notice that the 0 0 1 ··· n i i 0 0 0 roots and coroots which were primarily given in terms of fi’s can now be written in ∗ terms of ei’s as in the proposition. For example, the coroot 2fn can be written as 2(e∗ f ∗/2) = 2e∗ f ∗ = 2e∗ e∗. We chose to write them in terms of e ’s since they n − 0 n − 0 n − 0 i become more similar to the way root datum of other classical groups, e.g., GSp2n are written (cf.3.1.7).

Remark 3.1.6. The fact that (f + +f )/2 lies in the character lattice of Spin 1 ··· n 2n+1 is what is keeping this group from having Levi subgroups similar to those of classi- cal groups. On the other hand, there is no such element in the character lattice of

GSpin2n+1.

Remark 3.1.7. The root datum of GSpin2n+1 is the dual root datum to the one for

the group GSp2n while that of GSpin2n is dual to root datum of GSO2n (cf. [Tad94]

for the root datum of GSp2n, for example).

We are now ready to compute the Levi subgroups of G. 22

Theorem 3.1.8. Let L be a Levi subgroup in G. Then, L is isomorphic to

GL GL GL GSpin ′ n1 × n2 ×···× nb × m

with 2(n + n + + n )+ m′ = m and m′ = 2. 1 2 ··· b 6

Remark 3.1.9. Note that if m′ = 2, then m is automatically even and G is a group

of type Dm/2 in which case leaving any of the last two roots out will produce isomorphic Levi subgroups. Since we are only interested in L-functions, we will not deal with the Levi corresponding to the simple root that is one to the last in our ordering. This will allow us to avoid some minor changes that would be otherwise necessary later on when we study irreducibility of induced representations in this case (cf. [Ban99]).

Proof. By 3.1.7 we know that the dual of GSpinm is GSp2n if m = 2n +1 and GSO2n if m = 2n. Now notice that the Levi subgroups in Spin are just duals of those of GSp or GSO which we know are products of general linear groups and another smaller rank group of the same type. This will complete the proof.

Remark 3.1.10. The above theorem shows that the groups GSpin have rather simple Levi components. This is by no means the case for Spin groups and that is why we choose to study the L-functions using the former (cf. 3.1.1). This is similar to the

situation of Levi subgroups of SLn versus GLn.

3.2 Discrete Series Representations of GSpin Groups

In this section we describe the parabolically induced representations on GSpin(F ) that contain discrete series subrepresentations. We apply methods similar to the ones used by M. Tadi´cand D. Ban. Their method is based on Casselman’s square integrability criteria for which we need some preparation.

Let ∆ be the set of simple roots of G = GSpinm with respect to some maximal torus T. For any θ ∆, define ⊂

A =( ker α)0. θ ∩α∈θ 23

Standard Levi components in G are centralizers in G of these Aθ’s, hence in one-one correspondence with the subsets of ∆. We denote such Levi subgroups by Mθ and a

standard parabolic containing them by Pθ. We also define

A− = a A : α(a) 1, α ∆ θ . θ { ∈ θ | | ≤ ∀ ∈ \ } Moreover, two subsets θ and Ω of ∆ are said to be associate if there is an element w in the Weyl group of G with respect to T such that w(θ) = Ω.

For an irreducible admissible representation σ of M = M(F ), we let iG,M (σ) denote the normalized parabolically induced representation from σ 1 on the parabolic ⊗ subgroup P = MN to G = G(F ). Similarly, if π is an irreducible representation of G,

then we let rM,G(π) denote the Jacquet module of π with respect to P. (cf. [Cas] for more detailed explanation of these notations.) We can now state Casselman’s square integrability criteria.

Theorem 3.2.1. Let Mθ be the Levi component of a parabolic subgroup Pθ = MθNθ

in G = GSpinm. Assume that σ is a supercuspidal representation of M = Mθ(F ). An irreducible admissible representation π ֒ i (σ) is square integrable if and only if → G,M

(a) π restricted to A∆ is unitary.

(b) for every Ω ∆ associate to θ and every central character χ of r (π) we ⊂ MΩ,G have χ(a) < 1 for all a A− A ( )A . | | ∈ Ω \ ∅ O ∆ In the above, denotes the ring of integers of our local field F. O Proof. This is a version of Theorem 6.5.1 of [Cas].

We now give another version of the above that is more suitable to use for our purposes.

Proposition 3.2.2. Define

β = (1,..., 1, 0,..., 0) Rn. i ∈ i times | {z } Let π ֒ i (σ) be an irreducible smooth representation of GSpin (F ) and assume → G,M m that σ = ρ ρ ρ τ, where ρ is a supercuspidal representations of GL (F ) 1 ⊗ 2 ⊗···⊗ k ⊗ i ni 24

and τ is one of GSpinm′ (F ), is such that the corresponding Levi is minimal among the ones with r (π) = 0. Write each ρ as ρ = νe(ρi)ρu with e(ρ ) R and ρu M,G 6 i i i i ∈ i unitarizable. Define

e∗(σ)=(e(ρ1),...,e(ρ1),...,e(ρk),...,e(ρk), 0,..., 0). ′ n1 times n times m times | {z } | k {z } | {z } If π is square integrable. then

(e∗(σ), βn1 ) > 0,

(e∗(σ), βn1+n2 ) > 0, . .

(e∗(σ), βn1+···+nk ) > 0.

Conversely, if the above inequalities hold for all such partitions and all such σ’s, then π is square integrable.

Proof. The proof of this statement is straight forward from 3.2.1.

We will need the following result of Zelevinsky.

Proposition 3.2.3. (Zelevinsky) Let πi,i = 1, 2 be irreducible supercuspidal repre- sentations of GL (F ). If π π ν±1, then π π is irreducible. ni 1 6≃ 2 1 × 2

We now state the main statements of this section. Let G denote GSpinm(F ) and M denote the F -points of the standard Levi corresponding to the partition m = 2(n +n + +n )+m′. Assume that ρ is a supercuspidal representation of GL (F ) 1 2 ··· k i ni and τ is one of GSpinm′ (F ) and let ρ be the representation on M defined by

ρ = ρ ρ ρ τ. 1 ⊗ 2 ⊗···⊗ k ⊗

Also, set

ρ ρ ρ ⋊ τ = i (ρ). 1 × 2 ×···× k G,M

Write ρ = νei ρu with e R and ρu unitary. i i i ∈ i 25

Theorem 3.2.4. If ρ ρ ρ ⋊ τ has a discrete series subrepresentation, 1 × 2 ×···× k then ρu ρu for i = 1, 2,...,k. i ≃ i e Theorem 3.2.5. If ρ ρ ρ ⋊ τ has a discrete series subrepresentation, 1 × 2 ×···× k then 2e Z for i = 1, 2,...,k. i ∈

Proof. We use the notation and techniques used in [Tad98],[Tad94],[Tad92], and [Ban99]. Let π be the discrete series subrepresentation in the above theorems. Then

ρ is a quotient of the Jacquet module rM,G(π) by Frobenius reciprocity. Fix i 1, 2,...,k and set 0 ∈ { }

Y 0 = i 1,...,k α Z such that ρ ναρ , i0 { ∈ { }|∃ ∈ i0 ≃ i} Y 1 = i 1,...,k α Z such that ρ ναρ , i0 { ∈ { }|∃ ∈ i0 ≃ i} f Y = Y 0 Y 1, i0 i0 ∪ i0 Y c = 1,...,k Y . i0 { } \ i0

First assume ρu ρ˜u . For j , j′ Y 0, j , j′ Y 1, and j Y c we have all the i0 6≃ i0 0 0 ∈ i0 1 1 ∈ i0 c ∈ i0 following relations (cf. 3.2.3).

ρ ρ˜ ′ ρ˜ ′ ρ , ρ ρ˜ ′ ρ˜ ′ ρ , j0 × j0 ≃ j0 × j0 j1 × j1 ≃ j1 × j1 ρ ρ ρ ρ , ρ˜ ρ˜ ρ˜ ρ˜ , j0 × j1 ≃ j1 × j0 j1 × j1 ≃ j1 × j0 ρ ρ ρ ρ , ρ˜ ρ ρ ρ˜ , j0 × jc ≃ jc × j0 j0 × jc ≃ jc × j0 ρ ρ ρ ρ , ρ˜ ρ ρ ρ˜ . j1 × jc ≃ jc × j1 j1 × jc ≃ jc × j1

If m′ = 0 or 1, then 6

ρ ⋊ τ ρ˜ ⋊ τ, ρ ⋊ τ ρ˜ ⋊ τ j0 ≃ j0 j1 ≃ j1

and if m′ = 0 or 1, then

ρ ⋊ 1 ρ˜ ⋊ 1, ρ ⋊ 1 ρ˜ ⋊ 1. j0 ≃ j0 j1 ≃ j1 26

Write

Y 0 = a ,...,a , i0 { 1 k0 } Y 1 = b , . . . , b , i0 { 1 k1 } Y c = d ,...,d , i0 { 1 kc } with ai

ρ ρ ⋊ τ 1 ×···× k

ρa1 ρa ρd1 ρd ρb1 ρb ⋊ τ ≃ ×···× k0 × ×···× kc × ×···× k1

ρa1 ρa ρd1 ρd ρb1 ρb − ρ˜b ⋊ τ ≃ ×···× k0 × ×···× kc × ×···× k1 1 × k1

ρa1 ρa ρd1 ρd ρ˜b ρb1 ρb − ⋊ τ ≃ ×···× k0 × ×···× kc × k1 × ×···× k1 1 . . . ≃

ρa1 ρa ρ˜b . . . ρ˜b1 ρd1 ρd ⋊ τ. ≃ ×···× k0 × k1 × × ×···× kc In the same way, we get

ρ1 ρk ⋊ τ ρb1 ρb ρ˜a . . . ρ˜a1 ρd1 ρd ⋊ τ. ×···× ≃ ×···× k1 × k0 × × ×···× kc We also get similar relations in the case of m′ = 0 or 1 with τ in the above replaced by 1. Now Frobenius reciprocity implies that the representations

′ ρ = ρa1 ρa ρ˜b . . . ρ˜b1 ρd1 ρd ⋊ τ ⊗···⊗ k0 ⊗ k1 ⊗ ⊗ ⊗···⊗ kc and

′′ ρ = ρb1 ρb ρ˜a . . . ρ˜a1 ρd1 ρd ⋊ τ ⊗···⊗ k1 ⊗ k0 ⊗ ⊗ ⊗···⊗ kc are the quotients of the corresponding Jacquet modules. Consider the representation

′ ρa1 ρa ρb1 ρb of GLu(F ) for some integer u. Now we have (βu,e∗(ρ )) = ×···× k0 × ×···× k1 (β ,e (ρ′′)) which is a contradiction with Proposition 3.2.2 since not both of them − u ∗ could be strictly positive.

The second theorem is proved similarly. Note that since ρi is in particular a α tempered representation of GLni (F ) it is generic which implies that if ν ⋊ τ reduces, then 2α Z. One will need this to finish the proof. ∈ 27

3.3 Dual Groups and the Adjoint Representation

Another ingredient we will need before we can study the L-functions is the adjoint action r.

Proposition 3.3.1. The Langlands dual group of the group GSpinm is isomorphic to GSp if m = 2n + 1, and GSO if m = 2n and n = 1. 2n 2n 6 Proof. Use root datum (cf. Remark 3.1.7).

Corollary 3.3.2. The complex dual of a Levi subgroup of the form described in 3.1.8 is isomorphic to

GL (C) GL (C) GL (C) GSp (C) k1 × k2 ×···× kb × 2r if n = 2r + 1 and

GL (C) GL (C) GL (C) GSO (C) k1 × k2 ×···× kb × 2r if n = 2r.

We can now describe the representation r.

Proposition 3.3.3. Let M be a maximal Levi subgroup in GSpin of the form GL m k × GSpin , where m = 2k + n and n = 2. Then the adjoint action of the complex dual of n 6 this Levi on the Lie algebra of the unipotent radical of a parabolic containing this dual Levi may be written as follows. Denote the standard representations of the groups

1 2 GLk(C), GSp2l(C), and GSO2l(C) by ρk,R2l, and R2l, respectively and let µ be the multiplicative character defining GSp2l or GSO2l in each case.

(a) Assume n = 2l + 1 and l 1. Then, r = r r and r = ρ R1 while ≥ 1 ⊕ 2 1 k ⊗ 2l f r = Sym2ρ µ−1. If l = 0, then r = r and r = Sym2ρ µ−1. In the case, 2 k ⊗ 1 1 k ⊗ l = 1 note that GSpin3 is GL2 and its dual is GSp2(C)= GL2(C) and the results are well-known.

(b) Assume n = 2l and l 2. Then, r = r r and r = ρ R2 while r = ≥ 1 ⊕ 2 1 k ⊗ 2l 2 f 2ρ µ−1. If l = 0, then r = r and r = 2ρ µ−1. Note that we are not ∧ k ⊗ 1 1 ∧ k ⊗ allowing the case l = 1 since it gives the same Levi factor as the case l = 0. 28

We now fix a maximal Levi subgroup M = GL (F ) GSpin (F ). Let π π n × m 1 ⊗ 2 be an irreducible admissible generic representation of M which is in the discrete series. For i = 1, 2, the L-function L(s,π π , r ) is the inverse of the numerator of 1 ⊗ 2 i γ(s,π π , r ). Our final results of this section are the following theorems. 1 ⊗ 2 i Theorem 3.3.4. The local L-function L(s,π π , r ) is holomorphic for (s) > 0. 1 ⊗ 2 1 ℜ Theorem 3.3.5. The local L-function L(s,π π , r ) is holomorphic for (s) > 0. 1 ⊗ 2 2 ℜ

Proof. Note that the L-functions in the second theorem do not depend on π2 and already appear in other classical groups for which we know the result, i.e., the first theorem is valid. For the first theorem, notice that the arguments in Section 4 of [CS98] extend word for word to this case. We remark that the requirement discussed after Corollary 4.2 of [CS98] is our Theorem 3.2.4 and our Theorem 3.2.5 is used in Lemma 4.12 of [CS98]. We remark that our second L-function appears as a single L-function in the

case of Siegel Levi in Spin2n+1 or Spin2n in [Sha97]. 29

4. L-functions for Groups of Type F4

4.1 Levi Subgroups

Let F be a non-archimedean local field of characteristic zero and let G be a split group of type F defined over F. We let ∆ = α , α , α , α be a system of simple 4 { 1 2 3 4} roots for G where α1 and α2 are the long roots while α3 and α4 are the short ones. We first compute the standard Levi subgroups in G. There are four maximal ones corresponding to subsets θ , θ , θ , and θ in ∆ where θ =∆ α . Set M = M . To 1 2 3 4 i \{ i} i θi compute each of these, notice that each Mi being a reductive group, is the product of its derived group and the connected component of its center. Since G is simply connected, the derived group of each Mi is also simply connected, hence easily known given the type of its root system. We only need to find the intersection of the derived group with the connected component of the center of each Levi. For example, in the case of M1 we know that M1D, the derived group of M1, is isomorphic to the simply connected split group of type C3, i.e., it is isomorphic to Sp6. To find the center of

M1, we use the fact that every element in it is in the kernel of the roots generating M . Note that since G is simply-connected, every element h T, the maximal torus 1 ∈ of G, can be written uniquely as

∨ ∨ ∨ ∨ h = α1 (t1)α2 (t2)α3 (t3)α4 (t4),

∨ - where αj : GL1 T is the coroot associated with αj. Now

4 hα ,α∨i α (h)= t i j i Y j j=1 30

where , denotes the Z-pairing between the roots and coroots. Thus, h (M ), h· ·i ∈ Z 1 the center of M1, if and only if we have

−1 2 −2 t1 t2t3 = 1

−1 2 −1 t2 t3t4 = 1

−1 2 t3 t4 = 1.

These hold if and only if we have,

t t2 t t 2 = 3 = 4 = 3 = λ t1 t2 1 t4 for some λ GL . Therefore, A, the connected component of (M ) can be given ∈ 1 Z 1 by

∨ 2 ∨ 3 ∨ 2 ∨ A = α (λ )α (λ )α (λ )α (λ) λ GL1 .  1 2 3 4 | ∈

In A M , we have λ2 = 1 and therefore, ∩ 1D

∨ ∨ 2 A M1 = α (λ)α (λ) λ = 1 . ∩ D  2 4 |

This is the center of M Sp , thus, 1D ≃ 6 GL Sp M 1 × 6 GSp , 1 ≃ 1,a ≃ 6 { }

where a = (a1,a2) with a1 the element of order two in GL1 and a2 the non-trivial element in the center of Sp6. One can similarly compute the other maximal Levi subgroups to conclude

Proposition 4.1.1. With notation as in above, we have

M GSp 1 ≃ 6

and

GL Spin M 1 × 7 GSpin 4 ≃ 1,a ≃ 7 { } 31

with a = (a1,a2) where a1 is the order two element in GL1 and a2 is the non-trivial element in the center of Spin7. The other two Levis, M2 and M3, are isomorphic to two (isomorphic) Levi subgroups of SL , namely, (GL GL ) SL and (GL GL ) 5 2 × 3 ∩ 5 3 × 2 ∩ SL5 (cf. 4.1.3). To see this note that M2 (respectively, M3) and its corresponding Levi in SL have derived groups isomorphic to SL SL (respectively, SL SL ) 5 2 × 3 3 × 2 and the intersection of the connected component of the center of this Levi with its derived group is the same as that of M2 (respectively, M3) since we will have the same computations as in the above.

Remark 4.1.2. Note that M1 and M4 are dual of each other, i.e., one is obtained from the other by interchanging the roots and coroots in its root datum. The same is true for M2 and M3. This is, of course, expected since the dual of G is isomorphic to itself.

Remark 4.1.3. Both M2 and M3 are isomorphic to GL SL SL 1 × 2 × 3 A where A = (ζ , ζ3 I , ζ2 I ) ζ6 = 1 ζ , where ζ is a primitive 6th root of unity { · 2 · 3 | }≃h 6i 6 and I2 and I3 are the identity elements in SL2 and SL3. They can also be written as GL SL 2 × 3 B where B = (ω I ,ω I ) ω3 = 1 . To see this consider the following maps: { · 2 · 3 | } GL SL GL SL SL f - GL SL π - 2 × 3 1 × 2 × 3 2 × 3 B where f(x,g,h)=(x−1g,h) and π is the natural projection. Then,

−1 3 2 ker(π f) = (x,g,h) x g = ω Id2,h = ω I3,ω = 1, (xω) = 1 ◦  | · · 3 2 = (x, xω I2,ω I3) ω = 1, x = ω  · · | 3 2 6 = (x, x I2, x I3) x = 1 = A.  · · | Therefore, π f induces the desired isomorphism. This is the description of M used ◦ 3 in [Sha90a] to prove a theorem on bounds on Fourier coefficients of Maass forms. 32

Using the same type of computations, one can compute other non-maximal stan- dard Levis in G which themselves will be Levi components of the maximal Levis.

Proposition 4.1.4. For θ ∆, let M be the Levi component generated by the roots ⊂ θ in θ. Then we have,

M M GL GL , {α1,α2} ≃ {α3,α4} ≃ 1 × 3

M M M GL GL , {α1,α3} ≃ {α1,α4} ≃ {α2,α4} ≃ 2 × 2

M GL GSp , {α2,α3} ≃ 1 × 4

and

M M M M GL GL GL . {α1} ≃ {α2} ≃ {α3} ≃ {α4} ≃ 1 × 1 × 2

Proof. The proof is similar to the maximal Levi cases in the above. For example, if θ = α , α , then h M if and only if { 3 4} ∈ θ t t 1 1 3 = 4 = = . t2 t3 t4 λ

Thus, the connected component of the center of Mθ can be given by

∨ ∨ 3 ∨ 2 ∨ α (µ)α (λ )α (λ )α (λ) λ,µ GL1 .  1 2 3 4 | ∈

In the intersection with the derived group we have, µ = λ3 = 1. Hence the intersection is given by

α∨(λ2)α∨(λ) λ3 = 1 .  3 4 |

which is the center of the derived group, itself being isomorphic to SL3. Therefore,

GL SL M GL 1 × 3 GL GL . θ ≃ 1 × ζ ≃ 1 × 3 h 3i Other cases are similar. 33

4.2 Holomorphy of Local L-functions In this section we prove the conjecture on holomorphy of local L-functions for

maximal Levi subgroups of the exceptional group of type F4. We start with the Levi

M4.

Theorem 4.2.1. Assume σ to be an irreducible admissible generic representation of

M4 = M4(F ). Let r denote the adjoint action of the L-group of M4 on the Lie algebra

of the L-group of the unipotent radical of the parabolic in G having M4 as its Levi factor. Then, r = r r where r ’s are the irreducible constituents of r of dimensions 1 ⊕ 2 i 14 and 1, respectively. Now if σ is a discrete series representation, then each

L(s, σ, ri)

is holomorphic for (s) > 0. ℜ

Proof. In this case m = 2 and r2 is one-dimensional so the theorem is clear by 2.6.4.

Next, we treat Levi subgroups M2 and M3.

Theorem 4.2.2. Assume σ to be an irreducible admissible generic representation of M2 = M2(F ). Let r denote the adjoint action of the L-group of M2 on the Lie algebra of the L-group of the unipotent radical of the parabolic in G having M2 as its Levi factor. Then, r = r r r r where r ’s are the irreducible constituents 1 ⊕ 2 ⊕ 3 ⊕ 4 i of r of dimensions 6, 9, 2, and 3, respectively. Assume that σ is a discrete series representation, then each

L(s, σ, ri)

is holomorphic for (s) > 0. ℜ Proof. Consider the group SL with the usual simple roots ∆ = β , β , β , β . Then 5 { 1 2 3 4} M2 could be considered as the Levi inside this group associated with the subset β , β , β (cf. 4.1.1). { 1 3 4} 34

Let M = GL GL be the corresponding Levi inside GL . Then we have 2 2 × 3 5 M = M SL and the irreducible representations of M are obtained via restriction 2 2 ∩ 5 2 from M (F ) = GL (F ) GL (F ). Choose σ and σ to be irreducible admissible 2 2 × 3 2 3 representations of GL2(F ) and GL3(F ) such that

σ (σ σ ) . ⊂ 2 ⊗ 3 |M2

We now describe the L-function for each ri in terms of the above data. Notice

that tables in [Sha88] and [Lan71] give the highest weight for each ri from which one can expect the general type of the L-function that arises but we will need more information for our exact statements below as we present in the proof.

2 Lemma 4.2.3. Let ρn denote the standard representation of GLn(C) and let Sym ρn denote the n(n+1)/2-dimensional irreducible representation of this group on the space

of symmetric tensors of rank two. Also let σ2 and σ3 be as in the above and let ω2

and ω3 denote their central characters, respectively. We then have

(a) L(s, σ, r ) = L(s, σ σ ,ρ ρ ) = L(s, σ σ ). This is the Rankin-Selberg 1 2 ⊗ 3 2 ⊗ 3 2 × 3 product of σ2 and σ3. e e e (b) L(s, σ, r )= L(s, σ (ω−1σ ), Sym2ρ ρ ). 2 2 ⊗ 3 3 2 ⊗ 3

2 (c) L(s, σ, r3)= L(s,ω2σ2,ρ2).

−3 (d) L(s, σ, r4)= L(s,ω2 σ˜3,ρ3).

Proof. Due to lack of a simple matrix representation of exceptional groups we make use of computations with roots and coroots to verify the lemma. In fact, we show that if σ corresponds to the semisimple conjugacy class represented by the diagonal element A = diag(a ,a , b , b , b ) in M = (GL (C) GL (C)) SL (C), the complex σ 1 2 1 2 3 2 2 × 3 ∩ 5 c dual group of M2, then the following are true from which the lemma follows.

−1 −1 −1 −1 −1 −1 (a) r1(Aσ) = diag(a1b1 ,a1b2 ,a1b3 ,a2b1 ,a2b2 ,a2b3 ).

2 −1 −1 2 −1 −1 2 −1 −1 −1 −1 −1 −1 −1 −1 (b) r2(Aσ) = diag(a1b1 b2 ,a1b1 b3 ,a1b2 b3 ,a1a2b1 b2 ,a1a2b1 b3 ,a1a2b2 b3 , 2 −1 −1 2 −1 −1 2 −1 −1 a2b1 b2 ,a2b1 b3 ,a2b2 b3 ). 35

3 2 2 3 (c) r3(Aσ) = diag(a1a2,a1a2).

3 3 −1 3 3 −1 3 3 −1 (d) r4(Aσ) = diag(a1a2b1 ,a1a2b2 ,a1a2b3 ).

To show these, take an element tˆ = diag (a ,a , b , b , b ) M SL (C). Let 1 2 1 2 3 ∈ 2 ⊂ 5 c β , β , β , β denote the simple roots of the dual of our exceptional group of type { 1 2 3 4} F4. Since the dual groups, being again of the same type, is simply-connected, we can write tˆ uniquely as

4 tˆ= β∨(t ). Y j j j=1

We identify the simple roots β1, β3, and β4 with the corresponding ones in SL5(C) and solve for tj’s in terms of a1, . . . , b3. On the other hand, using the tables in [Sha88] and [Lan71] as well as those of [Bou68] we can compute the eigenvalues of the action of each ri on every associated Xα∨ in terms of tj’s. Substituting for tj’s will give the results in the lemma.

Going back to the proof of the theorem, since σ is in discrete series, σ2 and σ3 are discrete series representations of GL2(F ) and GL3(F ), respectively. The proof is now clear for r1 since it just gives the Rankin-Selberg product L-function of σ2 andσ ˜3(cf. [Sha84] and [Sha90b]). Also note that the central character of σ is unitary, i.e., a a = 1, which means 2 | 1 2| that r3 gives the standard L-function for σ2 twisted by a unitary character while r4

gives the standard L-function forσ ˜3 again twisted by a unitary character. Hence we know the holomorphy for (s) > 0 for both of these cases. ℜ To prove the theorem for r2 we use the multiplicativity of γ-factors (2.4.5). First

note that if both σ2 and σ3 are unitary supercuspidal, then we know the result by 2.6.5. Otherwise, at least one of σ or σ is induced. First assume σ Ind(µ 1/2 µ −1/2) 2 3 2 ⊂ |·| ⊗ |·| where µ is a unitary character of F × and induction is from GL (F ) GL (F ) to 1 × 1 GL (F ) (cf. [BZ77] and [Zel80]). With notation as in 2.4.5 we have j 1, 2, 3, 4, 5 2 ∈ { } with θ = α , α for all j and Ω = Ω = Ω = α , α , α while Ω = Ω = j { 3 4} 1 3 5 { 2 3 4} 2 4 α , α , α . Also, S = 1, 3, 5 and R = R R ,R = R R and R = { 1 3 4} 2 { } 1 11 ⊕ 21 3 23 ⊕ 43 5 36

R R . Apply 2.4.5 and 2.5.3 to write the γ-factor for r as a product and use 15 ⊕ 25 2 2.4.6 to write these factors in terms of the ones for GL2 and GL3 to get

γ(s, σ, r )= γ(s 1, τ, 2ρ )γ(s,τ, 2ρ )γ(s + 1, τ, 2ρ ) 2 − ∧ 3 ∧ 3 ∧ 3 where τ isσ ˜3 twisted by a unitary character, i.e., µ and ρ3 is the standard represen- tation of GL3(C). By 2.4.3 the right hand side of the above is, up to a monomial in q−s, equal to

L(s,τ, 2ρ ) L(s 1, τ, 2ρ ) L(s 2, τ, 2ρ ) ∧ 3 − ∧ 3 − ∧ 3 L(s + 1, τ, 2ρ ) · L(s,τ, 2ρ ) · L(s 1, τ, 2ρ ) ∧ 3 ∧ 3 − ∧ 3 while the left hand side is, up to a monomial in q−s, equal to

L(1 s, σ,˜ r ) − 2 . L(s, σ, r2) Now holomorphy of L(s,τ, 2ρ ) for (s) > 0 (cf.[Sha92]) and the fact that any term ∧ 3 ℜ that could possibly create a pole for L(s, σ, r2) is cancelled by another term on the right hand side of the above, prove that L(s, σ, r ) is holomorphic for (s) > 0. 2 ℜ × Next assume σ3 to be induced and choose a unitary character χ of F such that σ Ind(χ χ χ −1), where χ is a unitary character of F × and induction 3 ⊂ |·|⊗ ⊗ | · | is from GL (F ) GL (F ) GL (F ) to GL (F ) (cf. [BZ77] and [Zel80]). We argue 1 × 1 × 1 3 similarly. We have j 1, 2,..., 10 and S = 2, 7, 9 (for appropriate choice of ∈ { } 2 { } decomposition in 2.4.5). Also, θ = θ = θ = α while Ω = Ω = Ω = α , α . 2 7 9 { 2} 2 7 9 { 2 3} In this case, R2 = R22,R7 = R27 and R9 = R29. This will lead us, as above, to the the following equality.

γ(s, σ, r )= γ(s 1, π, Sym2ρ )γ(s,π, Sym2ρ )γ(s + 1, π, Sym2ρ ). 2 − 2 2 2

Here, π is just σ2 twisted by a unitary character, i.e., χ and ρ2 is the standard representation of GL2(C). The same argument again proves the theorem.

Theorem 4.2.4. Assume σ to be an irreducible admissible generic representation of

M3 = M3(F ). Let r denote the adjoint action of the L-group of M3 on the Lie algebra of the L-group of the standard unipotent radical of the parabolic in G having M3 as 37 its Levi factor. Then, r = r r r where r ’s are the irreducible constituents of r 1 ⊕ 2 ⊕ 3 i of dimensions 12, 6, and 2, respectively. Now if σ is a discrete series representation, then each

L(s, σ, ri) is holomorphic for (s) > 0. ℜ

Proof. The proof is similar to that of the previous theorem. The Levi subgroup M3 can also be considered as the Levi inside SL5 with the corresponding Levi in GL5 equal to GL GL . Again we will choose σ and σ to be irreducible admissible 3 × 2 2 3 representations of GL2(F ) and GL3(F ) such that

σ (σ σ ) . ⊂ 3 ⊗ 2 |M3

The same arguments as in the previous lemma gives the following description for ri’s.

Lemma 4.2.5. With notation as in the previous lemma, we have

(a) L(s, σ, r )= L(s, σ σ , Sym2ρ ρ ). 1 3 ⊗ 2 3 ⊗ 2 e −1 2 2 (b) L(s, σ, r2)= L(s,ω2 ω3σ3, Sym ρ3). e −1 2 (c) L(s, σ, r3)= L(s,ω2 ω3ω2σ2,ρ2). e Proof. Computations similar to the previous lemma show that if σ corresponds to the semisimple conjugacy class represented by Aσ = diag(a1,a2,a3, b1, b2) in M3 = c (GL (C) GL (C)) SL (C), the complex dual group of M , then the following are 3 × 2 ∩ 5 3 true from which the lemma follows.

2 −1 2 −1 2 −1 −1 −1 −1 2 −1 2 −1 2 −1 (a) r1(Aσ) = diag(a1b1 ,a2b1 ,a3b1 ,a1a2b1 ,a1a3b1 ,a2a3b1 ,a1b2 ,a2b2 ,a3b2 , −1 −1 −1 a1a2b2 ,a1a3b2 ,a2a3b2 ).

2 2 −1 −1 2 2 −1 −1 2 2 −1 −1 2 −1 −1 2 −1 −1 (b) r2(Aσ) = diag(a1a2b1 b2 ,a1a3b1 b2 ,a2a3b1 b2 ,a1a2a3b1 b2 ,a1a2a3b1 b2 , 2 −1 −1 a1a2a3b1 b2 ). 38

2 2 2 −2 −1 2 2 2 −1 −2 (c) r3(Aσ) = diag(a1a2a3b1 b2 ,a1a2a3b1 b2 ).

As before, σ2 and σ3 are discrete series representations of GL2(F ) and GL3(F ) with unitary central characters and we denote their central characters by ω2 and ω3, re- spectively. The proof of the theorem is now clear for r3 since it gives the standard 2 −1 L-function L(s,π,ρ2) where π isσ ˜2 twisted by the unitary character ω3ω2 . It is also ′ 2 ′ clear for r2 since it gives L(s,π , Sym ρ3) where π is nowσ ˜3 twisted by the unitary 2 −1 character ω3ω2 . To establish the theorem for r1 we again use 2.4.5.

If both σ2 and σ3 are unitary supercuspidal, then the theorem is clear by 2.6.5.

Otherwise, first assume that σ2 is induced as in the previous proof. In this case we have, j 1, 2, 3, 4, 5 and S = 1, 5 . Also, R = R and R = R . In fact, all ∈ { } 1 { } 1 11 3 23 Rj’s are irreducible in this case. We then get

γ(s, σ, r )= γ(s 1/2, τ, Sym2ρ )γ(s + 1/2, τ, Sym2ρ ) 1 − 3 3

−1 where τ isσ ˜3 twisted by a unitary character, i.e., µ and ρ3 is the standard rep- resentation of GL3(C). Again by 2.4.3 the right hand side of the above is, up to a monomial in q−s, equal to

L(s 3/2, τ, Sym2ρ ) L(s 1/2, τ, Sym2ρ ) − 3 − 3 L(s 1/2, τ, Sym2ρ ) · L(s + 1/2, τ, Sym2ρ ) − 3 3 while the left hand side is, up to a monomial in q−s, equal to

L(1 s, σ, r ) − 1 . L(s, σ, r1) Holomorphy for (s) > 0 is again clear because of cancellations and holomorphy of ℜ L(s,τ, Sym2ρ ) for (s) > 0 (cf.[Sha92]). 3 ℜ

Finally, we prove the theorem for M1.

Theorem 4.2.6. Assume σ to be an irreducible admissible generic representation of

M1 = M1(F ). Let r denote the adjoint action of the L-group of M1 on the Lie algebra

of the L-group of the unipotent radical of the parabolic in G having M1 as its Levi 39

factor. Then, r = r r where r and r are the irreducible constituents of r of 1 ⊕ 2 1 2 dimensions 8 and 7, respectively. Now if σ is a discrete series representation, then each

L(s, σ, ri)

is holomorphic for (s) > 0. ℜ Proof. We showed in 4.1.1 that M GSp . Recall that the dual of M is GSpin (C). 1 ≃ 6 1 7 We now describe the representations r1 and r2.

Lemma 4.2.7. We have

(a) L(s, σ, r1) gives the so-called spin L-functions. The representation r1 is the

eight-dimensional standard representation of GSpin7(C).

(b) L(s, σ, r2)= L(s, τ, R7), where τ is any irreducible constituent of the restriction

of σ to Sp6(F ), the derived group of M1, and R7 denotes the standard represen-

tation of SO7(C), the complex dual of Sp6(F ).

Proof. Similar to our previous lemmas assume that σ corresponds to the semisimple

conjugacy class represented by an element Aσ in M1 = GSpin7(C). Such an element c lies in the maximal torus of our exceptional group of type F4 which is shared by both

GSpin7 as well as GSp6 as Levis in our group. Therefore, Aσ can be given by three

pairs of complex numbers, ak, bk, k = 1, 2, 3 such that a1b1 = a2b2 = a3b3. Then the following hold.

(a) r1(Aσ) = diag(a1a2a3,a1a2b3,a1b2a3,a1b2a3, b1a2a3, b1a2b3, b1b2a3, b1b2b3).

2 2 2 2 2 2 (b) r2(Aσ) = diag(a1,a2,a3, 1, b3, b2, b1).

We now prove the theorem for r1 and r2. Although the argument we are about to give for r1, which makes use of 2.4.5, works for r2 as well, notice that the proof for r2 is already known since L(s, σ, r2) can be obtained in a classical group situation 40

as follows. Consider GL GSp as a standard Levi subgroup in GSp . Consider 1 × 6 8 ∗ the trivial character on GL1(F )= F and σ on GSp6(F ). The adjoint action in this case is seven-dimensional and the L-functions obtained is what we would get for r2.

As for r1 notice that if σ is unitary supercuspidal, then L(s, σ, r1) is holomorphic by 2.6.5. Next assume σ is induced from a smaller Levi. There are seven standard Levi subgroups in M1 as follows (cf.4.1).

L = M = GL GSp 1 {α2,α3} 1 × 4 L = M = GL GSp 2 {α2,α4} 2 × 2 L = M = GL GSp 3 {α3,α4} 3 × 0 L = M = GL GL GSp 4 {α2} 1 × 1 × 2 L = M = GL GL GSp 5 {α3} 1 × 2 × 0 L = M = GL GL GSp 6 {α4} 2 × 1 × 0 L = M = GL GL GL GSp 7 ∅ 1 × 1 × 1 × 0 Since there are not too many Levis, we will look at them one by one. First we will consider the maximal ones, L , L , and L . Let π τ be a discrete series representation 1 2 3 ⊗ of L = L (F ), 1 k 3 such that k k ≤ ≤

σ Ind ((π τ) 1), ⊂ LkNk↑M1 ⊗ ⊗

where Nk is the unipotent radical of the parabolic in M1 containing Lk. For L we get j 1, 2, 3 , θ = α , α , Ω = Ω = α , α , α , and Ω = 1 ∈ { } j { 2 3} 1 3 { 1 2 3} 2 α , α , α . Also, S = 1, 3 with R = R R ,R = R R , and hence { 2 3 4} 1 { } 1 11 ⊕ 21 3 13 ⊕ 23

γ(s, σ, r1)= γ(s,τ1,R4)γ(s,τ2,R4),

−1 where τ1 and τ2 are just τ twisted by unitary characters π and π , respectively and

R4 denotes the standard representation of GSp4(C). The holomprphy is now clear since both factor on the right would give holomorphic L-functions for (s) > 0. ℜ 41

For L we get j 1, 2, 3, 4, 5 , θ = α , α , θ = θ = α , α , θ = θ = 2 ∈ { } 1 { 2 4} 2 5 { 2 4} 3 4 α , α , and Ω = Ω = α , α , α , Ω = Ω = α , α , α , while Ω = α , α , α . { 1 3} 1 5 { 1 2 4} 2 4 { 1 3 4} 3 { 1 2 3} Also, S = 1, 3, 5 with R = R ,R = R R ,R = R , and hence 1 { } 1 11 3 13 ⊕ 23 5 15

γ(s, σ, r )= γ(s,τ ,ρ )γ(s,τ ,ρ )γ(s,π τ) 1 1 2 2 2 ×

−1 where τ1 and τ2 in the first two terms are τ twisted by ωπ and ωπ , respectively where

ωπ is the central character of π. As before, ρ2 denotes the standard representation of GSp2(C)=GL2(C). The last term is just the Rankin-Selberg product of π and τ. Holomorphy of L(s, σ, r ) is again clear for (s) > 0 since the first two terms give 1 ℜ classical L-functions for a discrete series representation τ twisted by ωπ which is a unitary since π is discrete series (cf. [BZ77] and [Zel80]). Finally for L we get j 1, 2, 3, 4, 5 , θ = α , α , and Ω = Ω = Ω = 3 ∈ { } j { 3 4} 1 3 5 α , α , α , while Ω = Ω = α , α , α . Also, S = 1, 2, 4, 5 with R = R ,R = { 1 3 4} 2 4 { 2 3 4} 1 { } 1 11 2 R R ,R = R R ,R = R , and hence 12 ⊕ 22 4 14 ⊕ 24 5 15

−1 γ(s, σ, r1)= γ(s,ωπ,ρ1)γ(s,ωπ ,ρ1)γ(s,π1,ρ3)γ(s,π2,ρ3),

−1 where π1 and π2 are π twisted by the unitary character ωπ and ωπ , respectively.

Holomorphy of L(s, σ, r1) is again clear since in each of the above terms we get a factor of the L-function which is holomorphic for (s) > 0. ℜ If σ is induced from another smaller Levi, we could induce in steps to one of the above Levis first and use our argument above. Note that in the above three cases, we did not assume that the inducing data is necessarily supercupidal but discrete series. This will complete the proof of the theorem. 42

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VITA

Mahdi Asgari was born in Tehran, Iran, on October 22, 1973. He received a B.S. degree in Mathematics from Sharif University of Technology in 1994. He then came to Purdue University where he received a Ph.D. in Mathematics in August, 2000.