Prasad's Conjecture about dualizing involutions

TL;DR

This paper proves a criterion for when the duality involution on characters of finite groups of Lie type sends a character to its dual, linking it to Frobenius eigenvalues of associated unipotent characters, thus confirming a conjecture of D. Prasad.

Contribution

It establishes a precise condition involving Frobenius eigenvalues under which the duality involution acts as the duality on irreducible characters, resolving a finite group analogue of Prasad's conjecture.

Findings

Duality involution acts as the dual on characters if associated unipotent characters have eigenvalues ±1.

The result applies to groups with no exceptional factors and trivial H^1 condition.

Confirms a finite group version of Prasad's conjecture.

Abstract

Let $G$ be a connected reductive group defined over a finite field $\mathbb{F}_q$ with corresponding Frobenius $F$. Let $\iota_G$ denote the duality involution defined by D. Prasad under the hypothesis $2\mathrm{H}^1(F,Z(G))=0$, where $Z(G)$ denotes the center of $G$. We show that for each irreducible character $\rho$ of $G^F$, the involution $\iota_G$ takes $\rho$ to its dual $\rho^{\vee}$ if and only if for a suitable Jordan decomposition of characters, an associated unipotent character $u_\rho$ has Frobenius eigenvalues $\pm$ 1. As a corollary, we obtain that if $G$ has no exceptional factors and satisfies $2\mathrm{H}^1(F,Z(G))=0$, then the duality involution $\iota_G$ takes $\rho$ to its dual $\rho^{\vee}$ for each irreducible character $\rho$ of $G^F$. Our results resolve a finite group counterpart of a conjecture of D.~Prasad.