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 $2\textrm{H}<^>{1}(F,Z(G))=0$, 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 $2\textrm{H}<^>{1}(F,Z(G))=0$, 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.