Distinguished non-Archimedean representations

For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation \pi of G. In particular an important question is whether or not the dimension of this space is bounded by one. We cover the known results for the pair (G=R_{E/F}GL(n),H=GL(n)), and then discuss the corresponding SL(n) case. In this paper, we show that (G=R_{E/F}SL(n),H=SL(n)) is a Gelfand pair when n is odd. When $n$ is even, the space of H-invariant forms on \pi can have dimension more than one even when \pi is supercuspidal. The latter work is joint with Dipendra Prasad

Root Numbers of Asai L-Functions

Let E/F be a quadratic extension of p-adic fields. We compute the value of is an element of(1/2, pi, r,psi) for a square integrable representation pi of GL(n)(E), which is (Galois) conjugate self-dual, where r denotes the Asai representation. This is the twisted version of a well-known result due to Bushnell and Henniart. The proof makes use of a result on the corresponding global root number, which is proved by a method conceived by Lapid and Rallis

ON THE DEGREE OF CERTAIN LOCAL L-FUNCTIONS

Let pi be an irreducible supercuspidal representation of GL(n)(F), where F is a p-adic field. By a result of Bushnell and Kutzko, the group of unramified self-twists of pi has cardinality n/e, where e is the o(F)-period of the principal o(F)-order in M-n(F) attached to pi. This is the degree of the local Rankin-Selberg L-function L(s, pi x pi(boolean OR)). In this paper, we compute the degree of the Asai, symmetric square, and exterior square L-functions associated to pi. As an application, assuming p is odd, we compute the conductor of the Asai lift of a supercuspidal representation, where we also make use of the conductor formula for pairs of supercuspidal representations due to Bushnell, Henniart, and Kutzko (1998)

Test vectors for local periods

Let E/F be a quadratic extension of non-Archimedean local fields of characteristic zero. An irreducible admissible representation pi of GL(n, E) is said to be distinguished with respect to GL(n, F) if it admits a non-trivial linear form that is invariant under the action of GL(n, F). It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when pi is unitary and generic. In this paper, we prove that the essential vector of [14] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case

Iwahori-Hecke model for supersingular representations of GL(2)(Q(p))

In this paper, we realize a regular supersingular representation pi of GL(2)(Q(p)) as a quotient of a representation induced from the Iwahori subgroup of GL(2)(Q(p)). We also show that this realization provides a uniform way of looking at all the self-extensions of pi which have a four dimensional space of I(1)-invariants. (C) 2014 Elsevier Inc. All rights reserved

On the conductor of certain local L-functions

The conductor formula of Bushnell, Henniart and Kutzko [BHK98] computes the conductor of a pair of supercuspidal representations of general linear groups over a p-adic field. This is the conductor of the tensor product lift. In this paper, we give an explicit formula for the conductors of the symmetric and exterior square lifts, under the assumption that p not equal 2

On distinguished square-integrable representations for Galois pairs and a conjecture of Prasad

International audienceWe prove an integral formula computing multiplicities of square-integrable representations relative to Galois pairs over $p$-adic fields and we apply this formula to verify two consequences of a conjecture of Dipendra Prasad. One concerns the exact computation of the multiplicity of the Steinberg representation and the other the invariance of multiplicities by transfer among inner forms