Freydoon Shahidi

A proof of Langlands' conjecture on Plancherel measures; Complementary series for p-adic groups

analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at least when the group is quasi-split and the inducing representation is generic. Assuming two natural conjectures in harmonic analysis of p-adic groups, we also prove the validity of the conjecture in general (Theorem 9.5). As our second result we obtain all the complementary series and special representations of quasi-split p-adic groups coming from rank-one parabolic subgroups and generic supercuspidal represen

On Certain L-Functions

We report briefly on an endoscopic classification of representations by focusing on one aspect of the problem, the question of embedded Hecke eigenvalues. 1. The problem for G By “eigenvalue”, we mean the family of unramified Hecke eigenvalues of an automorphic representation. The question is whether there are any eigenvalues for the discrete spectrum that are also eigenvalues for the continuous spectrum. The answer for classical groups has to be part of any general classification of their automorphic representations. The continuous spectrum is to be understood narrowly in the sense of the spectral theorem. It corresponds to representations in which the continuous induction parameter is unitary. For example, the trivial one-dimensional automorphic representation of the group SL(2) does not represent an embedded eigenvalue. This is because it corresponds to a value of the one-dimensional induction parameter at a nonunitary point in the complex domain. For general linear groups, the absence of embedded eigenvalues has been known for some time. It is a consequence of the classification of Jacquet-Shalika [JS] and Moeglin-Waldspurger [MW]. For other classical groups, the problem leads to interesting combinatorial questions related to the endoscopic comparison of trace formulas. We shall consider the case that G is a (simple) quasisplit symplectic or special orthogonal group over a number field F . Suppose for example that G is split and of rank n. The continuous spectrum of maximal dimension is then parametrized by n-tuples of (unitary) idele class characters. Is there any n-tuple whose unramified Hecke eigenvalue family matches that of an automorphic representation π in the discrete spectrum of G? The answer is no if π is required to have a global Whittaker model. This follows from the work of Cogdell, Kim, Piatetskii-Shapiro and Shahidi 2010 Mathematics Subject Classification. Primary 22E55, 22E50; Secondary 20G35, 58C40.

Cuspidality of symmetric powers with applications

The purpose of this paper is to prove that the symmetric fourth power of a cusp form on GL(2), whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the RamanujanPetersson and Sato-Tate conjectures. In particular, we establish the bound q 1/9 v for

Functorial products for GL~2 x GL~3 and the symmetric cube for GL~2

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on GL2 × GL3 as automorphic forms on GL6, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on GL2. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain L-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Sections 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for GL2. As noted below, many other applications follow. To be precise, let π1 and π2 be two automorphic cuspidal representations of GL2(AF ) and GL3(AF ), respectively, where AF is the ring of adèles of a number field F . Write π1 = ⊗vπ1v and π2 = ⊗vπ2v. For each v, finite or otherwise, let π1v ⊠ π2v be the irreducible admissible representation of GL6(Fv), attached to (π1v, π2v) through the local Langlands correspondence by Harris-Taylor [HT], Henniart [He], and Langlands [La4]. We point out that, if φiv, i = 1, 2, are the twoand the three-dimensional representations of Deligne-Weil group, parametrizing πiv, respectively, then π1v⊠π2v is attached to the six-dimensional representation φ1v ⊗ φ2v . Let π1 ⊠ π2 = ⊗v(π1v ⊠ π2v). Our first result (Theorem 5.1) is:

Twisted endoscopy and reducibility of induced representations for

1. Introduction. This is the first in a series of papers in which we study the reducibility of representations induced from discrete series representations of the Levi factors ofmaximal parabolic subgroups of p-adic groups on the unitary axis. The problem is equivalent to determining certain local Langlands L-functions which are of arithmetic significance by themselves. One of the main aims of this paper is to interpret our results in the direction of the parametrization problem by means of the theory of twisted endoscopy for which our method seems to be very suitable. When the representation is supercuspidal, we also study the reducibility of the induced representations off the unitary axis. Let F be a p-adic field of characteristic zero. Fix a positive integer n > 1 and let G be either of the groups SP2n, S02n or S02n+l. In all three cases there is a conjugacy class of maximal parabolic subgroups which has GL,, as its Levi factor. Let P MN be the standard parabolic subgroup ofG in this conjugacy class. Then M GL,,. Let tr be a discrete series representation of M GLn(F) and, given s C, let I(s, tr) be the representation of G G(F) induced from tr (R) Idet( )ls. Let I(tr) I(0, tr).

p

Introduction. The purpose of this paper is to prove the equality of certain local coefficients of arithmetic significance which were attached to representations of quasi-split real reductive algebraic groups in [27] with their corresponding Artin factors attached by local class field theory [21]. As a consequence, we establish an identity satisfied by certain normalized intertwining operators. It seems to be useful in applications of the trace formula [1, 29]. More precisely, let G be the group of real points of a quasi-split reductive algebraic group over R. Let A be the set of simple roots defined by a fixed minimal parabolic subgroup Po = M0A0U of G. Fix 6 C A, and let P = P9 be the corresponding standard parabolic subgroup of G and write P = MAN for its Langlands decomposition. Fix a nondegenerate character x °f U i ^ (a, H{o)) be an irreducible admissible x-gic Banach (in particular x~generic unitary) representation of M (cf. Section 1). Given v G a£, the complex dual of the Lie algebra of A, let I(v,o,0) be the continuously (quasi-unitarily, if o is unitary) induced representation IndP^Ga® e v 9 and let V(v,o,0) be its space (Section 0). Then F ^ a , ^ = Vfoo^O). Now, let W be the Weyl group of Ao in G. Choose w G W such that w(0) C A. Let N~ be the unipotent group opposite to N. Define N

-adic groups

= U C\ wN~w~\ where w is a representative of w in G. For/ G F(^,a,^)00, define

Local coefficients as Artin factors for real groups

The purpose of this paper is to prove the boundedness in vertical strips of finite width for all the L-functions that appear in constant terms of Eisenstein series, under a certain natural assumption on local normalized intertwining operators. As a corollary, we prove the boundedness for a number of important L-functions, among them the symmetic cube and triple product L-functions attached to cusp forms on GL(2), as well as several Rankin-Selberg product L-functions, where the local assumption was alredy proved to be valid in each case. Under the same assumption, the finiteness of poles on all of C for all of these L-functions is also proved, a necessary step in the proof of boundedness in vertical strips.

Boundedness of Automorphic L-Functions in Vertical Strips

We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose L-groups have classical derived groups. The important transfer from GSp4 to GL4 follows from our result as a special case.

Generic transfer for general spin groups

Abstract In this paper the authors generalize a result of Vogan on irreducibility of standard modules for generic representations from real groups to p -adic ones whenever the inducing data is supercuspidal. They also prove injectivity for standard modules in this case. As applications, the authors prove a number of results relating the poles of intertwining operators and points of reducibility of induced representations to the poles of L -functions defined by the second author, modulo a conjecture on them whose validity for classical groups is also verified here. A result on certain real groups with applications in liftings of automorphic forms from classical groups to general linear groups via L -functions is also proved.