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ON HOLOMORPHY of LOCAL LANGLANDS L-FUNCTIONS a Thesis Submitted to the Faculty of Purdue University by Mahdi Asgari in Partial F

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ON HOLOMORPHY of LOCAL LANGLANDS L-FUNCTIONS a Thesis Submitted to the Faculty of Purdue University by Mahdi Asgari in Partial F 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.
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