On the Self-dual Representations of a p-adic Group

Abstract

In an earlier paper [P1]; we studied self-dual complex representations of a finite group of Lie type. In this paper; we make an analogous study in thep-adic case. We begin by recalling the main result of that paper. LetG(F) be the group of F rational points of a connected reductive algebraic group G over a finite field F. Fix a Borel subgroup B of G; defined over F; which always exists by Lang’s theorem. LetBDTU be a Levi decomposition of the Borel subgroupB. Suppose that there is an element t02 T(F) that operates byi1 on all the simple roots in U with respect to the maximal split torus inT. It can be seen thatt 2 belongs to the center ofG. We proved in [P1] thatt 2 operates by 1 on a self-dual irreducible complex representation …; which has a Whittaker model; if and only if… carries a symmetricG-invariant bilinear form. If … carries a symmetric G-invariant bilinear form; then we call … an orthogonal representation. If … is an irreducible admissible self-dual representation of a p-adic group G; then there exists a nondegenerateG-invariant bilinear formB: …£…! C. This form is unique up to scalars by a simple application of Schur’s lemma; and it is either symmetric or skew-symmetric. The aim of this paper is to provide for a criterion to decide which of the two possibilities holds in the context of p-adic groups similar to our work in the finite field case in [P1]. This partly answers a question raised by Serre (see [P1]). We are able to say nothing about representations not admitting a Whittaker model. Our method; however; works in some cases even when there is no element inT which operates on each simple root byi1. This is the case; for instance; in some cases of SLn andSpn; where we work instead with the similitude group where such an element exists; then we deduce information on SLn andSpn.