Signs, involutions and Jacquet modules

TL;DR

This paper investigates how involutions affect the signs of bilinear forms on representations of p-adic groups, expressing these signs via Jacquet modules and establishing that for GL(n), the signs are always positive.

Contribution

It introduces a method to compute signs of bilinear forms on representations using Jacquet modules and reduces the problem to tempered representations, with a specific result for GL(n).

Findings

Signs are expressed via Jacquet modules.

Reduction to tempered representations.

Signs for GL(n) are always positive.

Abstract

Let $G$ be a connected reductive $p$-adic group and let $\theta$ be an automorphism of $G$ of order at most two. Suppose $\pi$ is an irreducible smooth representation of $G$ that is taken to its dual by $\theta$. The space $V$ of $\pi$ then carries a non-zero bilinear form $(\mspace{7mu},\mspace{6mu})$, unique up to scaling, with the invariance property $(\pi(g)v, \pi({}^{\theta}g)w) = (v,w)$, for $g \in G$ and $v, w \in V$. The form is easily seen to be symmetric or skew-symmetric and we set $\varepsilon_\theta(\pi) = \pm1$ accordingly. We use Cassleman's pairing (in commonly observed circumstances) to express $\varepsilon_\theta(\pi)$ in terms of certain Jacquet modules of $\pi$ and thus, via the Langlands classification, reduce the problem of determining the sign to the case of tempered representations. For the transpose-inverse involution of the general linear group, we show that the associated signs are always one.