Prasad's Conjecture about dualizing involutions

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.