A note on sign of a self-dual representation

TL;DR

This paper explores the sign of self-dual representations in broader contexts than previously studied, building on D. Prasad's work on detecting signs via central elements in finite and p-adic groups.

Contribution

It extends Prasad's results to more general settings and discusses implications of these extended findings.

Findings

Extended the understanding of signs in self-dual representations.

Identified new conditions under which signs can be detected.

Discussed consequences of these generalizations.

Abstract

D. Prasad showed that the sign of a self-dual representation of a finite or $p$-adic reductive group is often detected by a central element. We study the extension of his results to some more general situations and make some observations about the consequences of his results.