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

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 acknowledge 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.

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 F_{4} . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Levi Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Holomorphy of Local L-functions . . . . . . . . . . . . . . . . . . . . 33

LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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 existence 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 F_{4}. 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 G_{2} is already well-known.

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_{k} its ring of adeles. Let π = ⊗_{v}π_{v} be an automorphic form on M = M(𝔸_{k}). For each place v of k, M can be naturally considered to be defined over k_{v}, 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 F_{v} with Galois group Γ_{v}. To this data is attached a group, ^{L}M = ^{L}M^{0} ⋊ Γ, called the L-group, whose connected component M̂ = ^{L}M^{0} is a complex linear Lie group. Also, for almost all non-archimedean places v, π_{v} is an irreducibe admissible representation of M(k_{v}) which is unramified, i.e., its representation space has a vector fixed by M(O_{v}), where O_{v} is the ring of integers of the local field k_{v}. The Satake isomorphism attaches a unique semisimple conjugacy class A in M̂ to each such π_{v}.

If r is a (finite dimensional) complex analytic representation of M̂, one defines the local Langlands L-function by

L(s, π_{v}, r) = det(I − r(A)q^{−s})^{−1}, s ∈ ℂ,

where q is the cardinality of the residue class field of F_{v}.

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:

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}(s, π, r) denotes the product of local Langlands L-functions over v ∉ S, then one can complete this product by defining the L-functions for v ∈ S, each being the inverse of a polynomial in q_{v}^{−s} 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 Lfucntions 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, M̂, of the L-group is assumed to be an irreducible constituent of the adjoint representation of M̂ on the Lie algebra of the dual of N = N(F). Such 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 Conjecture. If σ is tempered, then L(s, σ, r) is holomorphic for ℜ(s) > 0.

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 G_{2} (cf. Page 284 of [Sha90b]). In this work, we prove it for spin groups as well as an exceptional group of type F_{4}. Recently, H. Kim has proved the conjecture, along with many other results, for exceptional groups of types E_{6}, E_{7}, and E_{8}, 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 construction 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. Muić, and M. Tadić 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 GL_{2} (cf. [KS00]).

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 O be the ring of integers in F and P denote its unique maximal ideal. Then O/P is a finite field whose cardinality we will denote 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 F_{4}, 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 Δ 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_{θ}, where θ = Δ \ {α} for a unique simple root α ∈ Δ.

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)_{F} ⊗_{ℤ} ℝ,
𝔞_{ℂ}^{∗} := 𝔞^{∗} ⊗ ℂ.

Recall that since M is assumed to be maximal, 𝔞_{ℂ}^{∗}/𝔷_{ℂ}^{∗} 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 ρ_{P} be half of the sum of positive roots of G whose root vectors generate N and define α̃ to be

\frac{1}{⟨ρ_{P} , α^{∨}⟩}ρ_{P} ∈ 𝔞^{∗},

where ⟨· , ·⟩ is the ℤ-pairing between the roots and coroots of G. We now identify s ∈ ℂ with sα̃ ∈ 𝔞_{ℂ}^{∗}. Also, define a homomorphism

H_{P} : M ⟶ 𝔞

via

q^{⟨χ,H_{P} (·)⟩} = |χ(·)|_{F} , χ ∈ X(M)_{F} .

We also have

q^{⟨sα̃,H_{P} (·)⟩} = |α̃(·)|_{F}^{s} ,

where α̃(·) is the value of α̃ 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

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 Ĝ. It is the connected component of the L-group of G.

Now consider M̂, the complex dual of M which can be realized as a Levi subgroup in Ĝ. We now define the local Langlands L-function. Definition 2.3.1. Let σ be an unramified representation of M = M(F), (i.e., having an M(O)-fixed vector). The Satake isomorphism then associates a semisimple conjugacy class, A_{σ}, in M̂ (which we will normally identify with a representative A_{σ} in T̂, the maximal torus of M̂ ). For a finite-dimensional complex representation r of M̂ define the local Langlands L-function as follows.

L(s, σ, r) := det(I − r(A_{σ})q^{−s})^{−1}, s ∈ ℂ.

Clearly, in the above definition, we did not have to assume M to be a Levi subgroup 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 P̂ be a standard parabolic in Ĝ having M̂ as its Levi factor and write P̂ = M̂N̂ (cf. [Bor79]). Define n̂ to be the Lie algebra of N̂ on which M̂ acts via the adjoint action, r. Proposition 2.3.2. There exists a positive integer m such that

n̂ = ⨁_{i=1}^{m} V_{i}

where V_{i}, spanned by root vectors X_{β^{∨}} ∈ n̂ with ⟨α̃, β^{∨}⟩ = i, is an irreducible subrepresentation of r = ⨁_{i=1}^{m} r_{i}. Proof. This is due to Langlands. Proposition 4.1. of [Sha88] gives an algebraic proof.

Hence, one gets the L-functions L(s, σ, r_{i}) for each 1 ≤ i ≤ m.

2.4 Langlands-Shahidi Method

In the definition we gave for local L-function we had to assume that the representation σ is unramified in order to be able to use the Satake isomorphism. How do we define L(s, σ, r_{i}) 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 ψ_{F} of F on which the γ-factors as well as ϵ-factors, to be introduced later, depend. The L-functions, however, do not depend on this additive character. We also assume that χ and ψ_{F} 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] that there exist m complex functions

γ_{i}(s, σ, r, ψ_{F}) = γ(s, σ, r_{i}),

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_{i}, ψ_{F})γ(1 − s, σ̃, r_{i}, ψ̄_{F}) = 1.

Moreover, γ(s, σ, r̃_{i}) = γ(s, σ̃, r_{i}).

Definition 2.4.2. Let P_{σ,i} ∈ ℂ[X] with P_{σ,i}(0) = 1 be such that P_{σ,i}(q^{−s}) is the normalized numerator of γ(s, σ, r_{i}). If σ is tempered, define the local Langlands Lfunction L(s, σ, r_{i}) to be the inverse of this polynomial, P_{σ,i}(q^{−s})^{−1}. One then defines L(s, σ, r_{i}) for arbitrary irreducible admissible generic σ via Langlands classification (cf. Page 308 of [Sha90c]).

Proposition 2.4.3. There exists a monomial in q^{−s}, denoted by ϵ(s, σ, r_{i}, ψ_{F}), such that

γ(s, σ, r_{i}, ψ_{F}) = ϵ(s, σ, r_{i}, ψ_{F})L(1 − s, σ, r̃_{i})/L(s, σ, r_{i}).

Remark 2.4.4. Note that the L-functions on the right hand side of the above Proposition do not depend on ψ_{F} while the ϵ-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, r̃).

Another important property of the γ-factors is their multiplicativity which is essential in our work. Let M = M_{θ} 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_{θ′} with θ′ ⊂ θ. Also assume that σ ⊂ Ind_{P′↑M}σ′ ⊗ 1 where σ′ is an irreducible admissible generic representation of M′. Let w = w_{ℓ,Δ}w_{ℓ,θ}^{−1} be the longest element of the Weyl group of G modulo that of M. Set θ_{1} = θ′ and w′_{1} = w. There exists a simple root α_{1} ∈ Δ such that w′_{1}(α_{1}) < 0. Define Ω_{1} = θ_{1} ∪ {α_{1}}. Now M_{Ω_{1}} contains M_{θ_{1}} as a maximal Levi. Let w_{1} = w_{ℓ,Ω_{1}}w_{ℓ,θ_{1}}^{−1} and w′_{2} = w′_{1}w_{1}^{−1}. Let θ_{2} = w_{1}(θ_{1}) and continue this process to define successive θ_{j}’s and Ω_{j}’s. There exists n such that w′_{n} = 1, i.e., w = w_{n−1}w_{n−2} ⋯ w_{1}.

Proposition 2.4.5. (Shahidi) If we set \overline{w}_{j} = w_{j−1} ⋯ w_{1} for 2 ≤ j ≤ n − 1 and \overline{w}_{1} = 1, then \overline{w}_{j}(σ′) is a representation of M_{θ_{j}}(F) and we have

γ(s, σ, r_{i}, ψ_{F}) = ∏_{j∈S_{i}} γ(s, \overline{w}_{j}(σ′), R_{ij}, ψ_{F}).

Each r_{i}, 1 ≤ i ≤ m, is an irreducible constituent of the adjoint action of the L-group of M on the Lie algebra of the L-group of N and R_{j}, 1 ≤ j ≤ n − 1, is the same action for M_{θ_{j}} in M_{Ω_{j}}. Every irreducible component of the restriction of r_{i} to the L-group of M′ is equivalent, under \overline{w}_{j}, to an irreducible constituent R_{ij} of some R_{j} for a unique j. The set S_{i} consists of all such j for a given i.

To explain the sets S_{i} 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 R_{j}’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 r_{i}’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_{j} contributed, then we recover R_{j}. In other words, R_{j} = ⊕R_{ij} with the sum over all i’s such that j ∈ S_{i}. Examples of such products with detail will occur later on when we deal with an exceptional group of type F_{4}.

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.

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 σ_{0} be an irreducible admissible representation of M and let σ = σ_{0} ⊗ q^{⟨s_{0}α̃,H_{P} (·)⟩}, where H_{P} is as in 2.2. Then L(s, σ, r) = L(s + s_{0}, σ_{0}, r).

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 ℜ(s) > 0 cancelled among 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∈Δ} β^{∨}(t_{β}), t_{β} ∈ GL_{1}

where β^{∨} : GL_{1} \overline{}T is the coroot attached to β (cf. Page 44, Corollary (a) to Lemma 28 of [Ste67]). Define the map

ξ_{α} : T \overline{}GL_{1}

via ξ_{α}(t) = t_{α}.

Note that ker ξ_{α} = T ∩ M_{D}, where M_{D} is the derived group of M. To see this observe that if

t = ∏_{β∈Δ} β^{∨}(t_{β})

with t_{α} = 1, then t ∈ M_{D} since β^{∨}(t_{β}) ∈ M_{D} for β ≠ α.

Proposition 2.5.2. With notation as in the above, we have α̃ = ξ_{α} on T.

Proof. Let Δ_{0} = {α_{1}, α_{2}, …, α_{n}} be a system of simple roots for G and α = α_{r}. Write Δ_{0} \ {α_{r}} as a disjoint union

Δ_{1} ∪ Δ_{2} ∪ ⋯ ∪ Δ_{s}

where each Δ_{ℓ} is an irreducible subsystem whose Dynkin diagram is a connected component of the Dynkin diagram of Δ_{0} after removing α_{r}. Let ρ_{0} be half of the sum of positive roots in Δ_{0} and ρ_{ℓ} be that of Δ_{ℓ}. Every positive root of G with a root vector in N is a positive root generated by Δ_{0} but not by any of Δ_{ℓ}’s. Therefore,

ρ_{P} = ρ_{0} − ∑_{ℓ=1}^{s} ρ_{ℓ}.

Let t = ∏_{j=1}^{n} α_{j}^{∨}(t_{j}) be an element in the maximal split torus T in G. Write

t = α_{r}^{∨}(t_{r}) ∏_{ℓ=1}^{s} T_{ℓ}

where

T_{ℓ} = ∏_{α_{j}∈Δ_{ℓ}} α_{j}^{∨}(t_{j}).

Notice that ρ_{ℓ}(T_{k}) = 1 if k ≠ ℓ. To see this note that

ρ_{ℓ}(T_{k}) = ∏_{α_{j}∈Δ_{k}} t_{j}^{⟨ρ_{ℓ},α_{j}^{∨}⟩}

and ⟨β, α_{j}^{∨}⟩ = 0 for β ∈ Δ_{k}, k = 1, 2, …, s.

Now

ρ_{P}(t) = \frac{ρ_{0}(t)}{∏_{ℓ=1}^{s} ρ_{ℓ}(t)}
= \frac{ρ_{0}(∏_{ℓ=1}^{s} T_{ℓ}) · ρ_{0}(α_{r}^{∨}(t_{r}))}{∏_{ℓ=1}^{s} ρ_{ℓ}(α_{r}^{∨}(t_{r})) · ∏_{ℓ=1}^{s} ∏_{k=1}^{s} ρ_{ℓ}(T_{k})}
= \frac{∏_{ℓ=1}^{s} ρ_{0}(T_{ℓ})}{∏_{ℓ=1}^{s} ρ_{ℓ}(T_{ℓ})} · \frac{ρ_{0}(α_{r}^{∨}(t_{r}))}{∏_{ℓ=1}^{s} ρ_{ℓ}(α_{r}^{∨}(t_{r}))}.

We claim that ρ_{ℓ}(T_{ℓ}) = ρ_{0}(T_{ℓ}) for ℓ = 1, 2, …, s. To see this observe that

ρ_{ℓ}(T_{ℓ}) = ∏_{α_{j}∈Δ_{ℓ}} t_{j}^{⟨ρ_{ℓ},α_{j}^{∨}⟩}

and

ρ_{0}(T_{ℓ}) = ∏_{α_{j}∈Δ_{ℓ}} t_{j}^{⟨ρ_{0},α_{j}^{∨}⟩}.

It is enough to show that for α_{j} ∈ Δ_{ℓ}, we have

⟨ρ_{ℓ}, α_{j}^{∨}⟩ = ⟨ρ_{0}, α_{j}^{∨}⟩ = 1.

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

⟨ρ, α^{∨}⟩ = ⟨∑_{ω} ω, α^{∨}⟩ = ∑_{ω}⟨ω, α^{∨}⟩ = 1,

where ω runs over the set of fundamental weights, since all ⟨ω, α^{∨}⟩ are 0 except for one which is 1.

Therefore,

ρ_{P}(t) = \frac{ρ_{0}(α_{r}^{∨}(t_{r}))}{∏_{ℓ=1}^{s} ρ_{ℓ}(α_{r}^{∨}(t_{r}))}
= \frac{t_{r}^{⟨ρ_{0},α_{r}^{∨}⟩}}{∏_{ℓ=1}^{s} t_{r}^{⟨ρ_{ℓ},α_{r}^{∨}⟩}}
= t^{⟨ρ_{0}−∑_{ℓ=1}^{s} ρ_{ℓ},α_{r}^{∨}⟩}
= t_{r}^{⟨ρ_{P},α_{r}^{∨}⟩}.

This implies that α̃_{r}(t) = t_{r} which completes the proof.

Recall that α̃ defined in Section 2.2 is a character on T, the maximal split torus in G which extends 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_{D} where A ⊂ T is the split torus in the center of M. For m = ad with a ∈ A and d ∈ M_{D}, define ξ(m) = ξ(a). To see this is well-defined, let m = a_{1}d_{1} = a_{2}d_{2} with a_{1}, a_{2} ∈ A and d_{1}, d_{2} ∈ M_{D}. Then, a_{2}^{−1}a_{1} = d_{2}d_{1}^{−1} ∈ A ∩ M_{D}. This implies that ξ(a_{2}^{−1}a_{1}) = 1 or ξ(a_{1}) = ξ(a_{2}).

Moreover, note that we now have, ker ξ = M_{D} and the following sequence is exact.

1 \overline{}M_{D}  \overline{}M \overline{}^{ξ} GL_{1}  \overline{}1

Corollary 2.5.3. Let σ_{0} be an irreducible generic representation of M = M(F) and σ = σ_{0} ⊗ |ξ(·)|_{F}^{s_{0}} for some s_{0} ∈ ℂ. Then, we have

L(s, σ, r) = L(s + s_{0}, σ_{0}, 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 SL_{n+1} and

M = \{( \begin{matrix}g \\ & a \end{matrix} ) : g ∈ GL_{n}, a^{−1} = det(g)\} ≃ GL_{n}

be the maximal Levi component in G corresponding to θ = Δ \ {α_{n}} where

Δ = {α_{1}, α_{2}, …, α_{n}}

is the fundamental system of simple roots for G with α_{i}(t) = a_{i}/a_{i+1} for

t = diag(a_{1}, a_{2}, …, a_{n+1}) ∈ T.

In this case,

α̃_{n} = \frac{1}{n + 1}(α_{1} + 2α_{2} + ⋯ + nα_{n}).

Note that ∏_{i=1}^{n+1} a_{i} = 1. Then,

α̃_{n}(t) = (∏_{i=1}^{n}(a_{i}/a_{i+1})^{i})^{\frac{1}{n+1}}
= ((a_{1}a_{2} ⋯ a_{n})^{n+1})^{\frac{1}{n+1}}
= det(diag(a_{1}, a_{2}, …, a_{n}))

Therefore, ξ(g) = det(g) on GL_{n}.

Example 2.5.5. Let G be Sp_{4} with Δ = {α_{1}, α_{2}} where α_{2} is the long root. Let θ = {α_{2}}. Then

M = M_{θ} = \{( \begin{matrix}a \\ & g \\ & & a^{−1} \end{matrix} ) : g ∈ SL_{2}, a ∈ GL_{1}\} ≃ GL_{1} × SL_{2}.

With t = diag(a_{1}, a_{2}, a_{2}^{−1}, a_{1}^{−1}) ∈ T we have,

α_{i}(t) = \{ \begin{matrix}a_{1}/a_{2} : & i = 1 \\ a_{2}^{2} : & i = 2 \end{matrix}

and

α̃_{1}(t) = (α_{1} + \frac{1}{2}α_{2})(t)
= (a_{1}/a_{2}) · a_{2}
= a_{1}.

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

Example 2.5.6. Assume G = SO_{2n+1} or SO_{2n} and let M = GL_{n} 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 = M_{D}. Computing α̃ explicitly, we see that in fact ξ ≠ α̃ but ξ = α̃^{2}. This means that twisting the representation on GL_{n}(F) by | det(·)|_{F}^{s} would result in a shift by 2s in the L-function.

Notice that if G = Sp_{2n} and M = GL_{n} 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, r_{i})

is holomorphic for ℜ(s) > 0.

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_{1}) is holomorphic for ℜ(s) > 0.

Proposition 2.6.4. If m = 2 and

L(s, σ, r_{2}) = ∏_{j}(1 − a_{j}q^{−s})^{−1},

possibly an empty product with all a_{j} ∈ ℂ of absolute value one, then L(s, σ, r_{1}) is holomorphic for ℜ(s) > 0. The assumption of the Proposition holds in particular when r_{2} is one-dimensional.

Proposition 2.6.5. Assume σ to be unitary supercuspidal. Then L(s, σ, r_{i}) is holomorphic for ℜ(s) > 0, 1 ≤ i ≤ m. In fact each L(s, σ, r_{i}) is a product (possibly empty) as in Proposition 2.6.4 with |a_{j}| = 1.

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 G_{2}. To establish the conjecture in general, one needs to prove it for Levi subgroups of exceptional groups of type F_{4}, E_{6}, E_{7}, E_{8} 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 F_{4}.

H. Kim has recently proved the conjecture for Levi subgroups of E-type exceptional groups except for the following four Levi subgroups: Cases E_{7} − 3, E_{8} − 3, and E_{8} − 4 of [Sha88] as well as Case (xxviii) of [Lan71] (D_{7} ⊂ E_{8}) (cf.[Kim00]). The above four Levis all involve spin group or exceptional group of type E_{6} where we do not have enough information on discrete series representations (e.g., how to obtain them from supercuspidals).

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 Spin_{5} is simply-connected of type B_{2}. Note that this group is the same as Sp_{4} if we interchange the roles of long and short roots. This means the Siegel Levi of Spin_{5} is the non-Siegel Levi of Sp_{4} which is GL_{1} × Sp_{2} or GL_{1} × Spin_{3} while the non-Siegel Levi of Spin_{5} is GL_{2} which is not what we would expect if Spin_{5} 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 Spin_{2n+1} when n = 2k is even. As in [Sha97] one can verify that the Levi subgroup is isomorphic to

\frac{GL_{1} × SL_{n}}{\{(λ, λ · I_{n}) : λ^{k} = 1\}},

while one would expect to get GL_{n} 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 GL_{n}.)

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.

Let Spin_{m} be the split simple simply-connected algebraic group of type B_{n} if m = 2n + 1 and of type D_{n} if m = 2n. This group is a double covering, as algebraic groups, of the group SO_{m}.

Fix a maximal torus T contained in a fixed Borel subgroup B in Spin_{m}. This corresponds to a choice of a system of simple roots Δ = {α_{1}, α_{2}, …, α_{n}}. We denote the associated coroots by Δ^{∨} = \{α_{1}^{∨}, α_{2}^{∨}, …, α_{n}^{∨}\}. We then have α_{i} : T \overline{}GL_{1} and α_{i}^{∨} : GL_{1} \overline{}T such that α_{i} ∘ α_{j}^{∨}(x) = x^{⟨α_{i},α_{j}^{∨}⟩} where ⟨·, ·⟩ : X × X^{∨} \overline{}ℤ is the ℤ-pairing between characters and cocharacters.

Proposition 3.1.2. (a) Each t ∈ T can be written uniquely as

t = ∏_{j=1}^{n} α_{j}^{∨}(t_{j}), t_{j} ∈ GL_{1}.

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

∏_{j=1}^{n} t_{j}^{⟨α_{i},α_{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 Spin_{m} is equal to

{1, c} ≃ ℤ/2ℤ

if m = 2n + 1, where c = α_{n}^{∨}(−1). If m = 2n, then it is equal to

{1, c, z, cz} ≃ ℤ/2ℤ × ℤ/2ℤ

provided that n is even and

{1, z, c, z^{3}} ≃ ℤ/4ℤ

otherwise, where

c = α_{n−1}^{∨}(−1)α_{n}^{∨}(−1)

and

z = ∏_{j=1}^{n−2} α_{j}^{∨}((−1)^{j}) · α_{n−1}^{∨}(−1)

if n is even and

z = ∏_{j=1}^{n−2} α_{j}^{∨}((−1)^{j}) · α_{n−1}^{∨}(\sqrt{−1})α_{n}^{∨}(\sqrt{−1})

if it is odd. Observe that c = z^{2}. Proof. By part (a) of the previous proposition t is in the center if and only if

t ∈ ⋂_{i=1}^{n} ker α_{i}.

Now assume m = 2n + 1. If t ∈ ker α_{i}, then

α_{i}(t) = ∏_{j=1}^{n} t_{j}^{⟨α_{i},α_{j}^{∨}⟩} = \{ \begin{matrix}t_{1}^{2} t_{2}^{−1} = 1 & if i = 1 \\ t_{i−1}^{−1} t_{i}^{2} t_{i+1}^{−1} = 1 & if 2 ≤ i ≤ n − 2 \\ t_{n−2}^{−1} t_{n−1}^{2} t_{n}^{−2} = 1 & if i = n − 1 \\ t_{n−1}^{−1} t_{n}^{2} = 1 & if i = n \end{matrix}

Hence t is in the center if and only if

\frac{t_{1}}{1} = \frac{t_{2}}{t_{1}} = ⋯ = \frac{t_{n−2}}{t_{n−3}} = \frac{t_{n−1}}{t_{n−2}} = \frac{t_{n}^{2}}{t_{n−1}} = 1.

Therefore, t_{n} = ±1 and all other t_{j}’s are equal to 1.

Now assume m = 2n. Again, t is in the center if and only if

\frac{t_{1}}{1} = \frac{t_{2}}{t_{1}} = ⋯ = \frac{t_{n−2}}{t_{n−3}} = \frac{t_{n−1}t_{n}}{t_{n−2}} and t_{n−1}^{2} = t_{n}^{2} = t_{n−2}.

Solving these equations simultaneously will yield the corollary.

We can now define the group GSpin_{m}.

Definition 3.1.4. For m ≥ 3 define

G = \frac{GL_{1} × Spin_{m}}{\{(1, 1), (−1, c)\}}

where c is defined as in 3.1.3. We will call this group GSpin_{m}. Write A for the image of {(λ, 1)|λ ∈ GL_{1}} and H for the image of {(1, g)|g ∈ Spin_{m}} in G. Then, G = A·H with S = A ∩ H = {1, c}. Clearly, the derived group of GSpin_{m} is Spin_{m}.

We also define GSpin_{0} and GSpin_{1} to be GL_{1}. (One could also define GSpin_{2} to be GL_{2} although this group will not come up in our discussions.)

We now give an alternative description of GSpin_{m} in terms of root datum which will be helpful later on.

Proposition 3.1.5. The reductive group GSpin_{m} can be given by the root datum (X, R, X^{∨}, R^{∨}) where

X = ℤe_{0} ⊕ ℤe_{1} ⊕ ⋯ ⊕ ℤe_{n}

and

X^{∨} = ℤe_{0}^{∗} ⊕ ℤe_{1}^{∗} ⊕ ⋯ ⊕ ℤe_{n}^{∗}

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

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

R = {α_{1} = e_{1} − e_{2}, α_{2} = e_{2} − e_{3}, …, α_{n−1} = e_{n−1} − e_{n}, α_{n} = e_{n}},
R^{∨} = {α_{1}^{∨} = e_{1}^{∗} − e_{2}^{∗}, α_{2}^{∨} = e_{2}^{∗} − e_{3}^{∗}, …, α_{n−1}^{∨} = e_{n−1}^{∗} − e_{n}^{∗}, α_{n}^{∨} = 2e_{n}^{∗} − e_{0}^{∗}}.

(b) If m = 2n, then

R = {α_{1} = e_{1} − e_{2}, …, α_{n−1} = e_{n−1} − e_{n}, α_{n} = e_{n−1} + e_{n}},
R^{∨} = {α_{1}^{∨} = e_{1}^{∗} − e_{2}^{∗}, …, α_{n−1}^{∨} = e_{n−1}^{∗} − e_{n}^{∗}, α_{n}^{∨} = e_{n_{1}}^{∗} + e_{n}^{∗} − e_{0}^{∗}}.

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

Remark 3.1.6. The fact that (f_{1}+⋯+f_{n})/2 lies in the character lattice of Spin_{2n+1} is what is keeping this group from having Levi subgroups similar to those of classical groups. On the other hand, there is no such element in the character lattice of GSpin_{2n+1}.

Remark 3.1.7. The root datum of GSpin_{2n+1} is the dual root datum to the one for the group GSp_{2n} while that of GSpin_{2n} is dual to root datum of GSO_{2n} (cf. [Tad94] for the root datum of GSp_{2n}, for example).

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

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

GL_{n_{1}} × GL_{n_{2}} × ⋯ × GL_{n_{b}} × GSpin_{m′}

with 2(n_{1} + n_{2} + ⋯ + n_{b}) + m′ = m and m′ ≠ 2.

Remark 3.1.9. Note that if m′ = 2, then m is automatically even and G is a group of type D_{m/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 GSpin_{m} is GSp_{2n} if m = 2n + 1 and GSO_{2n} 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 SL_{n} versus GL_{n}.

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. Tadić and D. Ban. Their method is based on Casselman’s square integrability criteria for which we need some preparation.

Let Δ be the set of simple roots of G = GSpin_{m} with respect to some maximal torus T. For any θ ⊂ Δ, define

A_{θ} = (∩_{α∈θ} ker α)^{0}.

Standard Levi components in G are centralizers in G of these A_{θ}’s, hence in one-one correspondence with the subsets of Δ. We denote such Levi subgroups by M_{θ} and a standard parabolic containing them by P_{θ}. We also define

A_{θ}^{−} = {a ∈ A_{θ} : |α(a)| ≤ 1, ∀α ∈ Δ \ θ}.

Moreover, two subsets θ and Ω of Δ 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 i_{G,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 r_{M,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 M_{θ} be the Levi component of a parabolic subgroup P_{θ} = M_{θ}N_{θ} in G = GSpin_{m}. Assume that σ is a supercuspidal representation of M = M_{θ}(F). An irreducible admissible representation π ↪ i_{G,M}(σ) is square integrable if and only if

(a) π restricted to A_{Δ} is unitary.

(b) for every Ω ⊂ Δ associate to θ and every central character χ of r_{M_{Ω},G}(π) we have |χ(a)| < 1 for all a ∈ A_{Ω}^{−} \ A_{∅}(O)A_{Δ}.

In the above, O denotes the ring of integers of our local field F.

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_{i} = (\underbrace{1, …, 1}_{i times}, 0, …, 0) ∈ ℝ^{n}.

Let π ↪ i_{G,M}(σ) be an irreducible smooth representation of GSpin_{m}(F) and assume that σ = ρ_{1} ⊗ ρ_{2} ⊗ ⋯ ⊗ ρ_{k} ⊗ τ, where ρ_{i} is a supercuspidal representations of GL_{n_{i}}(F) and τ is one of GSpin_{m′}(F), is such that the corresponding Levi is minimal among the ones with r_{M,G}(π) ≠ 0. Write each ρ_{i} as ρ_{i} = ν^{e(ρ_{i})}ρ_{i}^{u} with e(ρ_{i}) ∈ ℝ and ρ_{i}^{u} unitarizable. Define

e_{∗}(σ) = (\underbrace{e(ρ_{1}), …, e(ρ_{1})}_{n_{1} times}, …, \underbrace{e(ρ_{k}), …, e(ρ_{k})}_{n_{k} times}, \underbrace{0, …, 0}_{m′ times}).

If π is square integrable. then

(e_{∗}(σ), β_{n_{1}}) > 0,
(e_{∗}(σ), β_{n_{1}+n_{2}}) > 0,
⋮
(e_{∗}(σ), β_{n_{1}+···+n_{k}}) > 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 π_{i}, i = 1, 2 be irreducible supercuspidal representations of GL_{n_{i}}(F). If π_{1} ≄ π_{2}ν^{±1}, then π_{1} × π_{2} is irreducible.

We now state the main statements of this section. Let G denote GSpin_{m}(F) and M denote the F -points of the standard Levi corresponding to the partition m = 2(n_{1}+n_{2}+⋯+n_{k})+m′. Assume that ρ_{i} is a supercuspidal representation of GL_{n_{i}}(F) and τ is one of GSpin_{m′}(F) and let ρ be the representation on M defined by

ρ = ρ_{1} ⊗ ρ_{2} ⊗ ⋯ ⊗ ρ_{k} ⊗ τ.

Also, set

ρ_{1} × ρ_{2} × ⋯ × ρ_{k} ⋊ τ = i_{G,M}(ρ).

Write ρ_{i} = ν^{e_{i}}ρ_{i}^{u} with e_{i} ∈ ℝ and ρ_{i}^{u} unitary.

Theorem 3.2.4. If ρ_{1} × ρ_{2} × ⋯ × ρ_{k} ⋊ τ has a discrete series subrepresentation, then ρ_{i}^{u} ≃ ρ̃_{i}^{u} for i = 1, 2, …, k.

Theorem 3.2.5. If ρ_{1} × ρ_{2} × ⋯ × ρ_{k} ⋊ τ has a discrete series subrepresentation, then 2e_{i} ∈ ℤ for i = 1, 2, …, k.

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 r_{M,G}(π) by Frobenius reciprocity.

Fix i_{0} ∈ {1, 2, …, k} and set

Y_{i_{0}}^{0} = {i ∈ {1, …, k}|∃α ∈ ℤ such that ρ_{i_{0}} ≃ ν^{α}ρ_{i}},
Y_{i_{0}}^{1} = {i ∈ {1, …, k}|∃α ∈ ℤ such that ρ̃_{i_{0}} ≃ ν^{α}ρ_{i}},
Y_{i_{0}} = Y_{i_{0}}^{0} ∪ Y_{i_{0}}^{1},
Y_{i_{0}}^{c} = {1, …, k} \ Y_{i_{0}}.

First assume ρ_{i_{0}}^{u} ≄ ρ̃_{i_{0}}^{u}. For j_{0}, j′_{0} ∈ Y_{i_{0}}^{0}, j_{1}, j′_{1} ∈ Y_{i_{0}}^{1}, and j_{c} ∈ Y_{i_{0}}^{c} we have all the following relations (cf. 3.2.3).

ρ_{j_{0}} × ρ̃_{j′_{0}} ≃ ρ̃_{j′_{0}} × ρ_{j_{0}}, ρ_{j_{1}} × ρ̃_{j′_{1}} ≃ ρ̃_{j′_{1}} × ρ_{j_{1}},
ρ_{j_{0}} × ρ_{j_{1}} ≃ ρ_{j_{1}} × ρ_{j_{0}}, ρ̃_{j_{1}} × ρ̃_{j_{1}} ≃ ρ̃_{j_{1}} × ρ̃_{j_{0}},
ρ_{j_{0}} × ρ_{j_{c}} ≃ ρ_{j_{c}} × ρ_{j_{0}}, ρ̃_{j_{0}} × ρ_{j_{c}} ≃ ρ_{j_{c}} × ρ̃_{j_{0}},
ρ_{j_{1}} × ρ_{j_{c}} ≃ ρ_{j_{c}} × ρ_{j_{1}}, ρ̃_{j_{1}} × ρ_{j_{c}} ≃ ρ_{j_{c}} × ρ̃_{j_{1}}.

If m′ ≠ 0 or 1, then

ρ_{j_{0}} ⋊ τ ≃ ρ̃_{j_{0}} ⋊ τ, ρ_{j_{1}} ⋊ τ ≃ ρ̃_{j_{1}} ⋊ τ

and if m′ = 0 or 1, then

ρ_{j_{0}} ⋊ 1 ≃ ρ̃_{j_{0}} ⋊ 1, ρ_{j_{1}} ⋊ 1 ≃ ρ̃_{j_{1}} ⋊ 1.

Write

Y_{i_{0}}^{0} = {a_{1}, …, a_{k_{0}}},
Y_{i_{0}}^{1} = {b_{1}, …, b_{k_{1}}},
Y_{i_{0}}^{c} = {d_{1}, …, d_{k_{c}}},

with a_{i} < a_{j}, b_{i} < b_{j}, and d_{i} < d_{j} for i < j. We use the above relations to get

ρ_{1} × ⋯ × ρ_{k}⋊ τ
≃ ρ_{a_{1}} × ⋯ × ρ_{a_{k_{0}}} × ρ_{d_{1}} × ⋯ × ρ_{d_{kc}} × ρ_{b_{1}} × ⋯ × ρ_{b_{k_{1}}} ⋊ τ
≃ ρ_{a_{1}} × ⋯ × ρ_{a_{k_{0}}} × ρ_{d_{1}} × ⋯ × ρ_{d_{kc}} × ρ_{b_{1}} × ⋯ × ρ_{b_{k_{1}−1}} × ρ̃_{b_{k_{1}}} ⋊ τ
≃ ρ_{a_{1}} × ⋯ × ρ_{a_{k_{0}}} × ρ_{d_{1}} × ⋯ × ρ_{d_{kc}} × ρ̃_{b_{k_{1}}} × ρ_{b_{1}} × ⋯ × ρ_{b_{k_{1}−1}} ⋊ τ
≃ …
≃ ρ_{a_{1}} × ⋯ × ρ_{a_{k_{0}}} × ρ̃_{b_{k_{1}}} × … ρ̃_{b_{1}} × ρ_{d_{1}} × ⋯ × ρ_{d_{kc}} ⋊ τ.

In the same way, we get

ρ_{1} × ⋯ × ρ_{k} ⋊ τ ≃ ρ_{b_{1}} × ⋯ × ρ_{b_{k_{1}}} × ρ̃_{a_{k_{0}}} × … ρ̃_{a_{1}} × ρ_{d_{1}} × ⋯ × ρ_{d_{kc}} ⋊ τ.

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′ = ρ_{a_{1}} ⊗ ⋯ ⊗ ρ_{a_{k_{0}}} ⊗ ρ̃_{b_{k_{1}}} ⊗ … ρ̃_{b_{1}} ⊗ ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{kc}} ⋊ τ

and

ρ′′ = ρ_{b_{1}} ⊗ ⋯ ⊗ ρ_{b_{k_{1}}} ⊗ ρ̃_{a_{k_{0}}} ⊗ … ρ̃_{a_{1}} ⊗ ρ_{d_{1}} ⊗ ⋯ ⊗ ρ_{d_{kc}} ⋊ τ

are the quotients of the corresponding Jacquet modules. Consider the representation ρ_{a_{1}}×⋯×ρ_{a_{k_{0}}} ×ρ_{b_{1}}×⋯×ρ_{b_{k_{1}}} of GL_{u}(F) for some integer u. Now we have (β_{u}, e_{∗}(ρ′)) = −(β_{u}, e_{∗}(ρ′′)) which is a contradiction with Proposition 3.2.2 since not both of them could be strictly positive.

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

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 GSpin_{m} is isomorphic to GSp_{2n} if m = 2n + 1, and GSO_{2n} if m = 2n and n ≠ 1. 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_{k_{1}}(ℂ) × GL_{k_{2}}(ℂ) × ⋯ × GL_{k_{b}}(ℂ) × GSp_{2r}(ℂ)

if n = 2r + 1 and

GL_{k_{1}}(ℂ) × GL_{k_{2}}(ℂ) × ⋯ × GL_{k_{b}}(ℂ) × GSO_{2r}(ℂ)

if n = 2r. We can now describe the representation r. Proposition 3.3.3. Let M be a maximal Levi subgroup in GSpin_{m} of the form GL_{k}× GSpin_{n}, where m = 2k + n and n ≠ 2. Then the adjoint action of the complex dual of 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 GL_{k}(ℂ), GSp_{2l}(ℂ), and GSO_{2l}(ℂ) by ρ_{k}, R_{2l}^{1}, and R_{2l}^{2}, respectively and let μ be the multiplicative character defining GSp_{2l} or GSO_{2l} in each case.

(a) Assume n = 2l + 1 and l ≥ 1. Then, r = r_{1} ⊕ r_{2} and r_{1} = ρ_{k} ⊗ R̃_{2l}^{1} while r_{2} = Sym^{2}ρ_{k} ⊗ μ^{−1}. If l = 0, then r = r_{1} and r_{1} = Sym^{2}ρ_{k} ⊗ μ^{−1}. In the case, l = 1 note that GSpin_{3} is GL_{2} and its dual is GSp_{2}(ℂ) = GL_{2}(ℂ) and the results are well-known.

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

We now fix a maximal Levi subgroup M = GL_{n}(F) × GSpin_{m}(F). Let π_{1} ⊗ π_{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_{1} ⊗ π_{2}, r_{i}) is the inverse of the numerator of γ(s, π_{1} ⊗ π_{2}, r_{i}). Our final results of this section are the following theorems.

Theorem 3.3.4. The local L-function L(s, π_{1} ⊗ π_{2}, r_{1}) is holomorphic for ℜ(s) > 0.

Theorem 3.3.5. The local L-function L(s, π_{1} ⊗ π_{2}, r_{2}) is holomorphic for ℜ(s) > 0.

Proof. Note that the L-functions in the second theorem do not depend on π_{2} 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 Spin_{2n+1} or Spin_{2n} in [Sha97].

4. L-functions for Groups of Type F_{4}

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_{4} defined over F. We let Δ = {α_{1}, α_{2}, α_{3}, α_{4}} be a system of simple roots for G where α_{1} and α_{2} are the long roots while α_{3} and α_{4} are the short ones. We first compute the standard Levi subgroups in G. There are four maximal ones corresponding to subsets θ_{1}, θ_{2}, θ_{3}, and θ_{4} in Δ where θ_{i} = Δ\{α_{i}}. Set M_{i} = M_{θ_{i}}. To compute each of these, notice that each M_{i} 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 M_{i} 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 M_{1} we know that M_{1}_{D}, the derived group of M_{1}, is isomorphic to the simply connected split group of type C_{3}, i.e., it is isomorphic to Sp_{6}. To find the center of M_{1}, we use the fact that every element in it is in the kernel of the roots generating M_{1}. Note that since G is simply-connected, every element h ∈ T, the maximal torus of G, can be written uniquely as

h = α_{1}^{∨}(t_{1})α_{2}^{∨}(t_{2})α_{3}^{∨}(t_{3})α_{4}^{∨}(t_{4}),

where α_{j}^{∨} : GL_{1} \overline{}T is the coroot associated with α_{j}. Now

α_{i}(h) = ∏_{j=1}^{4} t_{j}^{⟨α_{i},α_{j}^{∨}⟩}

where ⟨· , ·⟩ denotes the ℤ-pairing between the roots and coroots. Thus, h ∈ Z(M_{1}), the center of M_{1}, if and only if we have

t_{1}^{−1}t_{2}^{2}t_{3}^{−2} = 1
t_{2}^{−1}t_{3}^{2}t_{4}^{−1} = 1
t_{3}^{−1}t_{4}^{2} = 1.

These hold if and only if we have,

\frac{t_{2}}{t_{1}} = \frac{t_{3}^{2}}{t_{2}} = \frac{t_{4}}{1} = \frac{t_{3}}{t_{4}} = λ

for some λ ∈ GL_{1}. Therefore, A, the connected component of Z(M_{1}) can be given by

A = {α_{1}^{∨}(λ^{2})α_{2}^{∨}(λ^{3})α_{3}^{∨}(λ^{2})α_{4}^{∨}(λ) | λ ∈ GL_{1}} .

In A ∩ M_{1}_{D}, we have λ^{2} = 1 and therefore,

A ∩ M_{1}_{D} = {α_{2}^{∨}(λ)α_{4}^{∨}(λ) | λ^{2} = 1} .

This is the center of M_{1}_{D} ≃ Sp_{6}, thus,

M_{1} ≃ \frac{GL_{1} × Sp_{6}}{\{1, a\}} ≃ GSp_{6},

where a = (a_{1}, a_{2}) with a_{1} the element of order two in GL_{1} and a_{2} the non-trivial element in the center of Sp_{6}.

One can similarly compute the other maximal Levi subgroups to conclude

Proposition 4.1.1. With notation as in above, we have

M_{1} ≃ GSp_{6}

and

M_{4} ≃ \frac{GL_{1} × Spin_{7}}{\{1, a\}} ≃ GSpin_{7}

with a = (a_{1}, a_{2}) where a_{1} is the order two element in GL_{1} and a_{2} is the non-trivial element in the center of Spin_{7}. The other two Levis, M_{2} and M_{3}, are isomorphic to two (isomorphic) Levi subgroups of SL_{5}, namely, (GL_{2}×GL_{3})∩SL_{5} and (GL_{3}×GL_{2})∩ SL_{5} (cf. 4.1.3). To see this note that M_{2} (respectively, M_{3}) and its corresponding Levi in SL_{5} have derived groups isomorphic to SL_{2} × SL_{3} (respectively, SL_{3} × SL_{2}) and the intersection of the connected component of the center of this Levi with its derived group is the same as that of M_{2} (respectively, M_{3}) since we will have the same computations as in the above.

Remark 4.1.2. Note that M_{1} and M_{4} 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 M_{2} and M_{3}. This is, of course, expected since the dual of G is isomorphic to itself.

Remark 4.1.3. Both M_{2} and M_{3} are isomorphic to

\frac{GL_{1} × SL_{2} × SL_{3}}{A}

where A = {(ζ , ζ^{3}·I_{2} , ζ^{2}·I_{3}) | ζ^{6} = 1} ≃ ⟨ζ_{6}⟩, where ζ_{6} is a primitive 6th root of unity and I_{2} and I_{3} are the identity elements in SL_{2} and SL_{3}. They can also be written as

\frac{GL_{2} × SL_{3}}{B}

where B = {(ω · I_{2}, ω · I_{3}) | ω^{3} = 1}. To see this consider the following maps:

GL_{1} × SL_{2} × SL_{3}  \overline{}^{f} GL_{2} × SL_{3}  \overline{}^{π} \frac{GL_{2} × SL_{3}}{B}

where f(x, g, h) = (x^{−1}g, h) and π is the natural projection. Then,

ker(π ∘ f) = {(x, g, h) | x^{−1}g = ω · Id_{2}, h = ω · I_{3}, ω^{3} = 1, (xω)^{2} = 1}
= {(x, xω · I_{2}, ω · I_{3}) | ω^{3} = 1, x^{2} = ω}
= {(x, x^{3} · I_{2}, x^{2} · I_{3}) | x^{6} = 1} = A.

Therefore, π ∘ f induces the desired isomorphism. This is the description of M_{3} used in [Sha90a] to prove a theorem on bounds on Fourier coefficients of Maass forms.

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

Proposition 4.1.4. For θ ⊂ Δ, let M_{θ} be the Levi component generated by the roots in θ. Then we have,

M_{{α_{1},α_{2}}} ≃ M_{{α_{3},α_{4}}} ≃ GL_{1} × GL_{3},
M_{{α_{1},α_{3}}} ≃ M_{{α_{1},α_{4}}} ≃ M_{{α_{2},α_{4}}} ≃ GL_{2} × GL_{2},
M_{{α_{2},α_{3}}} ≃ GL_{1} × GSp_{4},

and

M_{{α_{1}}} ≃ M_{{α_{2}}} ≃ M_{{α_{3}}} ≃ M_{{α_{4}}} ≃ GL_{1} × GL_{1} × GL_{2}.

Proof. The proof is similar to the maximal Levi cases in the above. For example, if θ = {α_{3}, α_{4}}, then h ∈ M_{θ} if and only if

\frac{t_{3}}{t_{2}} = \frac{t_{4}}{t_{3}} = \frac{1}{t_{4}} = \frac{1}{λ}.

Thus, the connected component of the center of M_{θ} can be given by

{α_{1}^{∨}(μ)α_{2}^{∨}(λ^{3})α_{3}^{∨}(λ^{2})α_{4}^{∨}(λ) | λ, μ ∈ GL_{1}} .

In the intersection with the derived group we have, μ = λ^{3} = 1. Hence the intersection is given by

{α_{3}^{∨}(λ^{2})α_{4}^{∨}(λ) | λ^{3} = 1} .

which is the center of the derived group, itself being isomorphic to SL_{3}. Therefore,

M_{θ} ≃ GL_{1} × \frac{GL_{1} × SL_{3}}{⟨ζ_{3}⟩} ≃ GL_{1} × GL_{3}.

Other cases are similar.

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 F_{4}. We start with the Levi M_{4}.

Theorem 4.2.1. Assume σ to be an irreducible admissible generic representation of M_{4} = M_{4}(F). Let r denote the adjoint action of the L-group of M_{4} on the Lie algebra of the L-group of the unipotent radical of the parabolic in G having M_{4} as its Levi factor. Then, r = r_{1} ⊕r_{2} where r_{i}’s are the irreducible constituents of r of dimensions 14 and 1, respectively. Now if σ is a discrete series representation, then each

L(s, σ, r_{i})

is holomorphic for ℜ(s) > 0.

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

Next, we treat Levi subgroups M_{2} and M_{3}.

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

L(s, σ, r_{i})

is holomorphic for ℜ(s) > 0.

Proof. Consider the group SL_{5} with the usual simple roots Δ = {β_{1}, β_{2}, β_{3}, β_{4}}. Then M_{2} could be considered as the Levi inside this group associated with the subset {β_{1}, β_{3}, β_{4}} (cf. 4.1.1).

Let \overline{M}_{2} = GL_{2} × GL_{3} be the corresponding Levi inside GL_{5}. Then we have M_{2} = \overline{M}_{2} ∩SL_{5} and the irreducible representations of M_{2} are obtained via restriction from \overline{M}_{2}(F) = GL_{2}(F) × GL_{3}(F). Choose σ_{2} and σ_{3} to be irreducible admissible representations of GL_{2}(F) and GL_{3}(F) such that

σ ⊂ (σ_{2} ⊗ σ_{3})|_{M_{2}}.

We now describe the L-function for each r_{i} in terms of the above data. Notice that tables in [Sha88] and [Lan71] give the highest weight for each r_{i} 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. Lemma 4.2.3. Let ρ_{n} denote the standard representation of GL_{n}(ℂ) and let Sym^{2}ρ_{n} denote the n(n+1)/2-dimensional irreducible representation of this group on the space of symmetric tensors of rank two. Also let σ_{2} and σ_{3} be as in the above and let ω_{2} and ω_{3} denote their central characters, respectively. We then have

(a) L(s, σ, r_{1}) = L(s, σ_{2} ⊗ σ̃_{3}, ρ_{2} ⊗ ρ_{3}) = L(s, σ_{2} × σ̃_{3}). This is the Rankin-Selberg product of σ_{2} and σ̃_{3}.

(b) L(s, σ, r_{2}) = L(s, σ_{2} ⊗ (ω_{3}^{−1}σ_{3}), Sym^{2}ρ_{2} ⊗ ρ_{3}).

(c) L(s, σ, r_{3}) = L(s, ω_{2}^{2}σ_{2}, ρ_{2}).

(d) L(s, σ, r_{4}) = L(s, ω_{2}^{−3}σ̃_{3}, ρ_{3}). 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_{σ} = diag(a_{1}, a_{2}, b_{1}, b_{2}, b_{3}) in M̂_{2} = (GL_{2}(ℂ)×GL_{3}(ℂ))∩SL_{5}(ℂ), the complex dual group of M_{2}, then the following are true from which the lemma follows.

(a) r_{1}(A_{σ}) = diag(a_{1}b_{1}^{−1}, a_{1}b_{2}^{−1}, a_{1}b_{3}^{−1}, a_{2}b_{1}^{−1}, a_{2}b_{2}^{−1}, a_{2}b_{3}^{−1}).

(b) r_{2}(A_{σ}) = diag(a_{1}^{2}b_{1}^{−1}b_{2}^{−1}, a_{1}^{2}b_{1}^{−1}b_{3}^{−1}, a_{1}^{2}b_{2}^{−1}b_{3}^{−1}, a_{1}a_{2}b_{1}^{−1}b_{2}^{−1}, a_{1}a_{2}b_{1}^{−1}b_{3}^{−1}, a_{1}a_{2}b_{2}^{−1}b_{3}^{−1}, a_{2}^{2}b_{1}^{−1}b_{2}^{−1}, a_{2}^{2}b_{1}^{−1}b_{3}^{−1}, a_{2}^{2}b_{2}^{−1}b_{3}^{−1}).

(c) r_{3}(A_{σ}) = diag(a_{1}^{3}a_{2}^{2}, a_{1}^{2}a_{2}^{3}).

(d) r_{4}(A_{σ}) = diag(a_{1}^{3}a_{2}^{3}b_{1}^{−1}, a_{1}^{3}a_{2}^{3}b_{2}^{−1}, a_{1}^{3}a_{2}^{3}b_{3}^{−1}).

To show these, take an element t̂ = diag (a_{1}, a_{2}, b_{1}, b_{2}, b_{3}) ∈ M̂_{2} ⊂ SL_{5}(ℂ). Let {β_{1}, β_{2}, β_{3}, β_{4}} denote the simple roots of the dual of our exceptional group of type F_{4}. Since the dual groups, being again of the same type, is simply-connected, we can write t̂ uniquely as

t̂= ∏_{j=1}^{4} β_{j}^{∨}(t_{j}).

We identify the simple roots β_{1}, β_{3}, and β_{4} with the corresponding ones in SL_{5}(ℂ) and solve for t_{j}’s in terms of a_{1}, …, b_{3}. 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 r_{i} on every associated X_{α^{∨}} in terms of t_{j}’s. Substituting for t_{j}’s will give the results in the lemma.

Going back to the proof of the theorem, since σ is in discrete series, σ_{2} and σ_{3} are discrete series representations of GL_{2}(F) and GL_{3}(F), respectively. The proof is now clear for r_{1} since it just gives the Rankin-Selberg product L-function of σ_{2} and σ̃_{3}(cf.

[Sha84] and [Sha90b]).

Also note that the central character of σ_{2} is unitary, i.e., |a_{1}a_{2}| = 1, which means that r_{3} gives the standard L-function for σ_{2} twisted by a unitary character while r_{4} gives the standard L-function for σ̃_{3} again twisted by a unitary character. Hence we know the holomorphy for ℜ(s) > 0 for both of these cases.

To prove the theorem for r_{2} we use the multiplicativity of γ-factors (2.4.5). First note that if both σ_{2} and σ_{3} are unitary supercuspidal, then we know the result by 2.6.5. Otherwise, at least one of σ_{2} or σ_{3} is induced. First assume σ_{2} ⊂ Ind(μ|·|^{1/2}⊗μ|·|^{−1/2}) where μ is a unitary character of F^{×} and induction is from GL_{1}(F) × GL_{1}(F) to GL_{2}(F) (cf. [BZ77] and [Zel80]). With notation as in 2.4.5 we have j ∈ {1, 2, 3, 4, 5} with θ_{j} = {α_{3}, α_{4}} for all j and Ω_{1} = Ω_{3} = Ω_{5} = {α_{2}, α_{3}, α_{4}} while Ω_{2} = Ω_{4} = {α_{1}, α_{3}, α_{4}}. Also, S_{2} = {1, 3, 5} and R_{1} = R_{11} ⊕ R_{21}, R_{3} = R_{23} ⊕ R_{43} and R_{5} = R_{15} ⊕ R_{25}. Apply 2.4.5 and 2.5.3 to write the γ-factor for r_{2} as a product and use 2.4.6 to write these factors in terms of the ones for GL_{2} and GL_{3} to get

γ(s, σ, r_{2}) = γ(s − 1, τ, ∧^{2}ρ_{3})γ(s, τ, ∧^{2}ρ_{3})γ(s + 1, τ, ∧^{2}ρ_{3})

where τ is σ̃_{3} twisted by a unitary character, i.e., μ and ρ_{3} is the standard representation of GL_{3}(ℂ). By 2.4.3 the right hand side of the above is, up to a monomial in q^{−s}, equal to

\frac{L(s, τ, ∧^{2}ρ_{3})}{L(s + 1, τ, ∧^{2}ρ_{3})} · \frac{L(s − 1, τ, ∧^{2}ρ_{3})}{L(s, τ, ∧^{2}ρ_{3})} · \frac{L(s − 2, τ, ∧^{2}ρ_{3})}{L(s − 1, τ, ∧^{2}ρ_{3})}

while the left hand side is, up to a monomial in q^{−s}, equal to

\frac{L(1 − s, σ̃, r_{2})}{L(s, σ, r_{2})}.

Now holomorphy of L(s, τ, ∧^{2}ρ_{3}) for ℜ(s) > 0 (cf.[Sha92]) and the fact that any term that could possibly create a pole for L(s, σ, r_{2}) is cancelled by another term on the right hand side of the above, prove that L(s, σ, r_{2}) is holomorphic for ℜ(s) > 0.

Next assume σ_{3} to be induced and choose a unitary character χ of F^{×} such that σ_{3} ⊂ Ind(χ| · | ⊗ χ ⊗ χ| · |^{−1}), where χ is a unitary character of F^{×} and induction is from GL_{1}(F) × GL_{1}(F) × GL_{1}(F) to GL_{3}(F) (cf. [BZ77] and [Zel80]). We argue similarly. We have j ∈ {1, 2, …, 10} and S_{2} = {2, 7, 9} (for appropriate choice of decomposition in 2.4.5). Also, θ_{2} = θ_{7} = θ_{9} = {α_{2}} while Ω_{2} = Ω_{7} = Ω_{9} = {α_{2}, α_{3}}. In this case, R_{2} = R_{22}, R_{7} = R_{27} and R_{9} = R_{29}. This will lead us, as above, to the the following equality.

γ(s, σ, r_{2}) = γ(s − 1, π, Sym^{2}ρ_{2})γ(s, π, Sym^{2}ρ_{2})γ(s + 1, π, Sym^{2}ρ_{2}).

Here, π is just σ_{2} twisted by a unitary character, i.e., χ and ρ_{2} is the standard representation of GL_{2}(ℂ). The same argument again proves the theorem.

Theorem 4.2.4. Assume σ to be an irreducible admissible generic representation of M_{3} = M_{3}(F). Let r denote the adjoint action of the L-group of M_{3} on the Lie algebra of the L-group of the standard unipotent radical of the parabolic in G having M_{3} as its Levi factor. Then, r = r_{1} ⊕ r_{2} ⊕ r_{3} where r_{i}’s are the irreducible constituents of r of dimensions 12, 6, and 2, respectively. Now if σ is a discrete series representation, then each

L(s, σ, r_{i})

is holomorphic for ℜ(s) > 0.

Proof. The proof is similar to that of the previous theorem. The Levi subgroup M_{3} can also be considered as the Levi inside SL_{5} with the corresponding Levi in GL_{5} equal to GL_{3} × GL_{2}. Again we will choose σ_{2} and σ_{3} to be irreducible admissible representations of GL_{2}(F) and GL_{3}(F) such that

σ ⊂ (σ_{3} ⊗ σ_{2})|_{M_{3}}.

The same arguments as in the previous lemma gives the following description for r_{i}’s. Lemma 4.2.5. With notation as in the previous lemma, we have

(a) L(s, σ, r_{1}) = L(s, σ_{3} ⊗ σ̃_{2}, Sym^{2}ρ_{3} ⊗ ρ_{2}).

(b) L(s, σ, r_{2}) = L(s, ω_{2}^{−1}ω_{3}^{2}σ̃_{3}, Sym^{2}ρ_{3}).

(c) L(s, σ, r_{3}) = L(s, ω_{2}^{−1}ω_{3}^{2}ω_{2}σ̃_{2}, ρ_{2}).

Proof. Computations similar to the previous lemma show that if σ corresponds to the semisimple conjugacy class represented by A_{σ} = diag(a_{1}, a_{2}, a_{3}, b_{1}, b_{2}) in M̂_{3} = (GL_{3}(ℂ) × GL_{2}(ℂ)) ∩ SL_{5}(ℂ), the complex dual group of M_{3}, then the following are true from which the lemma follows.

(a) r_{1}(A_{σ}) = diag(a_{1}^{2}b_{1}^{−1}, a_{2}^{2}b_{1}^{−1}, a_{3}^{2}b_{1}^{−1}, a_{1}a_{2}b_{1}^{−1}, a_{1}a_{3}b_{1}^{−1}, a_{2}a_{3}b_{1}^{−1}, a_{1}^{2}b_{2}^{−1}, a_{2}^{2}b_{2}^{−1}, a_{3}^{2}b_{2}^{−1}, a_{1}a_{2}b_{2}^{−1}, a_{1}a_{3}b_{2}^{−1}, a_{2}a_{3}b_{2}^{−1}).

(b) r_{2}(A_{σ}) = diag(a_{1}^{2}a_{2}^{2}b_{1}^{−1}b_{2}^{−1}, a_{1}^{2}a_{3}^{2}b_{1}^{−1}b_{2}^{−1}, a_{2}^{2}a_{3}^{2}b_{1}^{−1}b_{2}^{−1}, a_{1}^{2}a_{2}a_{3}b_{1}^{−1}b_{2}^{−1}, a_{1}a_{2}^{2}a_{3}b_{1}^{−1}b_{2}^{−1}, a_{1}a_{2}a_{3}^{2}b_{1}^{−1}b_{2}^{−1}).

(c) r_{3}(A_{σ}) = diag(a_{1}^{2}a_{2}^{2}a_{3}^{2}b_{1}^{−2}b_{2}^{−1}, a_{1}^{2}a_{2}^{2}a_{3}^{2}b_{1}^{−1}b_{2}^{−2}).

As before, σ_{2} and σ_{3} are discrete series representations of GL_{2}(F) and GL_{3}(F) with unitary central characters and we denote their central characters by ω_{2} and ω_{3}, respectively. The proof of the theorem is now clear for r_{3} since it gives the standard L-function L(s, π, ρ_{2}) where π is σ̃_{2} twisted by the unitary character ω_{3}^{2}ω_{2}^{−1}. It is also clear for r_{2} since it gives L(s, π′, Sym^{2}ρ_{3}) where π′ is now σ̃_{3} twisted by the unitary character ω_{3}^{2}ω_{2}^{−1}. To establish the theorem for r_{1} we again use 2.4.5.

If both σ_{2} and σ_{3} are unitary supercuspidal, then the theorem is clear by 2.6.5. Otherwise, first assume that σ_{2} is induced as in the previous proof. In this case we have, j ∈ {1, 2, 3, 4, 5} and S_{1} = {1, 5}. Also, R_{1} = R_{11} and R_{3} = R_{23}. In fact, all R_{j}’s are irreducible in this case. We then get

γ(s, σ, r_{1}) = γ(s − 1/2, τ, Sym^{2}ρ_{3})γ(s + 1/2, τ, Sym^{2}ρ_{3})

where τ is σ̃_{3} twisted by a unitary character, i.e., μ^{−1} and ρ_{3} is the standard representation of GL_{3}(ℂ). Again by 2.4.3 the right hand side of the above is, up to a monomial in q^{−s}, equal to

\frac{L(s − 3/2, τ, Sym^{2}ρ_{3})}{L(s − 1/2, τ, Sym^{2}ρ_{3})} · \frac{L(s − 1/2, τ, Sym^{2}ρ_{3})}{L(s + 1/2, τ, Sym^{2}ρ_{3})}

while the left hand side is, up to a monomial in q^{−s}, equal to

\frac{L(1 − s, σ, r_{1})}{L(s, σ, r_{1})}.

Holomorphy for ℜ(s) > 0 is again clear because of cancellations and holomorphy of L(s, τ, Sym^{2}ρ_{3}) for ℜ(s) > 0 (cf.[Sha92]).

Finally, we prove the theorem for M_{1}.

Theorem 4.2.6. Assume σ to be an irreducible admissible generic representation of M_{1} = M_{1}(F). Let r denote the adjoint action of the L-group of M_{1} on the Lie algebra of the L-group of the unipotent radical of the parabolic in G having M_{1} as its Levi factor. Then, r = r_{1} ⊕ r_{2} where r_{1} and r_{2} are the irreducible constituents of r of dimensions 8 and 7, respectively. Now if σ is a discrete series representation, then each

L(s, σ, r_{i})

is holomorphic for ℜ(s) > 0.

Proof. We showed in 4.1.1 that M_{1} ≃ GSp_{6}. Recall that the dual of M_{1} is GSpin_{7}(ℂ). We now describe the representations r_{1} and r_{2}. Lemma 4.2.7. We have

(a) L(s, σ, r_{1}) gives the so-called spin L-functions. The representation r_{1} is the eight-dimensional standard representation of GSpin_{7}(ℂ).

(b) L(s, σ, r_{2}) = L(s, τ, R_{7}), where τ is any irreducible constituent of the restriction of σ to Sp_{6}(F), the derived group of M_{1}, and R_{7} denotes the standard representation of SO_{7}(ℂ), the complex dual of Sp_{6}(F).

Proof. Similar to our previous lemmas assume that σ corresponds to the semisimple conjugacy class represented by an element A_{σ} in M̂_{1} = GSpin_{7}(ℂ). Such an element lies in the maximal torus of our exceptional group of type F_{4} which is shared by both GSpin_{7} as well as GSp_{6} as Levis in our group. Therefore, A_{σ} can be given by three pairs of complex numbers, a_{k}, b_{k}, k = 1, 2, 3 such that a_{1}b_{1} = a_{2}b_{2} = a_{3}b_{3}. Then the following hold.

(a) r_{1}(A_{σ}) = diag(a_{1}a_{2}a_{3}, a_{1}a_{2}b_{3}, a_{1}b_{2}a_{3}, a_{1}b_{2}a_{3}, b_{1}a_{2}a_{3}, b_{1}a_{2}b_{3}, b_{1}b_{2}a_{3}, b_{1}b_{2}b_{3}).

(b) r_{2}(A_{σ}) = diag(a_{1}^{2}, a_{2}^{2}, a_{3}^{2}, 1, b_{3}^{2}, b_{2}^{2}, b_{1}^{2}).

We now prove the theorem for r_{1} and r_{2}. Although the argument we are about to give for r_{1}, which makes use of 2.4.5, works for r_{2} as well, notice that the proof for r_{2} is already known since L(s, σ, r_{2}) can be obtained in a classical group situation as follows. Consider GL_{1} × GSp_{6} as a standard Levi subgroup in GSp_{8}. Consider the trivial character on GL_{1}(F) = F^{∗} and σ on GSp_{6}(F). The adjoint action in this case is seven-dimensional and the L-functions obtained is what we would get for r_{2}. As for r_{1} notice that if σ is unitary supercuspidal, then L(s, σ, r_{1}) is holomorphic by 2.6.5. Next assume σ is induced from a smaller Levi. There are seven standard Levi subgroups in M_{1} as follows (cf.4.1).

L_{1} = M_{{α_{2},α_{3}}} = GL_{1} × GSp_{4}

L_{2} = M_{{α_{2},α_{4}}} = GL_{2} × GSp_{2}

L_{3} = M_{{α_{3},α_{4}}} = GL_{3} × GSp_{0}

L_{4} = M_{{α_{2}}} = GL_{1} × GL_{1} × GSp_{2}

L_{5} = M_{{α_{3}}} = GL_{1} × GL_{2} × GSp_{0}

L_{6} = M_{{α_{4}}} = GL_{2} × GL_{1} × GSp_{0}

L_{7} = M_{∅} = GL_{1} × GL_{1} × GL_{1} × GSp_{0}

Since there are not too many Levis, we will look at them one by one. First we will consider the maximal ones, L_{1}, L_{2}, and L_{3}. Let π⊗τ be a discrete series representation of L_{k} = L_{k}(F), 1 ≤ k ≤ 3 such that

σ ⊂ Ind_{L_{k}N_{k}↑M_{1}}((π ⊗ τ) ⊗ 1),

where N_{k} is the unipotent radical of the parabolic in M_{1} containing L_{k}.

For L_{1} we get j ∈ {1, 2, 3}, θ_{j} = {α_{2}, α_{3}}, Ω_{1} = Ω_{3} = {α_{1}, α_{2}, α_{3}}, and Ω_{2} = {α_{2}, α_{3}, α_{4}}. Also, S_{1} = {1, 3} with R_{1} = R_{11} ⊕ R_{21}, R_{3} = R_{13} ⊕ R_{23}, and hence

γ(s, σ, r_{1}) = γ(s, τ_{1}, R_{4})γ(s, τ_{2}, R_{4}),

where τ_{1} and τ_{2} are just τ twisted by unitary characters π and π^{−1}, respectively and R_{4} denotes the standard representation of GSp_{4}(ℂ). The holomprphy is now clear since both factor on the right would give holomorphic L-functions for ℜ(s) > 0.

For L_{2} we get j ∈ {1, 2, 3, 4, 5}, θ_{1} = {α_{2}, α_{4}}, θ_{2} = θ_{5} = {α_{2}, α_{4}}, θ_{3} = θ_{4} = {α_{1}, α_{3}}, and Ω_{1} = Ω_{5} = {α_{1}, α_{2}, α_{4}}, Ω_{2} = Ω_{4} = {α_{1}, α_{3}, α_{4}}, while Ω_{3} = {α_{1}, α_{2}, α_{3}}. Also, S_{1} = {1, 3, 5} with R_{1} = R_{11}, R_{3} = R_{13} ⊕ R_{23}, R_{5} = R_{15}, and hence

γ(s, σ, r_{1}) = γ(s, τ_{1}, ρ_{2})γ(s, τ_{2}, ρ_{2})γ(s, π × τ)

where τ_{1} and τ_{2} in the first two terms are τ twisted by ω_{π} and ω_{π}^{−1}, respectively where ω_{π} is the central character of π. As before, ρ_{2} denotes the standard representation of GSp_{2}(ℂ) = GL_{2}(ℂ). The last term is just the Rankin-Selberg product of π and τ. Holomorphy of L(s, σ, r_{1}) is again clear for ℜ(s) > 0 since the first two terms give classical L-functions for a discrete series representation τ twisted by ω_{π} which is a unitary since π is discrete series (cf. [BZ77] and [Zel80]).

Finally for L_{3} we get j ∈ {1, 2, 3, 4, 5}, θ_{j} = {α_{3}, α_{4}}, and Ω_{1} = Ω_{3} = Ω_{5} = {α_{1}, α_{3}, α_{4}}, while Ω_{2} = Ω_{4} = {α_{2}, α_{3}, α_{4}}. Also, S_{1} = {1, 2, 4, 5} with R_{1} = R_{11}, R_{2} = R_{12} ⊕ R_{22}, R_{4} = R_{14} ⊕ R_{24}, R_{5} = R_{15}, and hence

γ(s, σ, r_{1}) = γ(s, ω_{π}, ρ_{1})γ(s, ω_{π}^{−1}, ρ_{1})γ(s, π_{1}, ρ_{3})γ(s, π_{2}, ρ_{3}),

where π_{1} and π_{2} are π twisted by the unitary character ω_{π} and ω_{π}^{−1}, respectively. Holomorphy of L(s, σ, r_{1}) is again clear since in each of the above terms we get a factor of the L-function which is holomorphic for ℜ(s) > 0.

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.

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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.