Lifting representations of finite reductive groups: a character relation

Abstract

Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\epsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\epsilon$-fixed points. Two of the authors have previously shown that there exists a natural lifting from series of representations of $G\left(k\right)$ to series for $\tilde{G}\left(k\right)$. In the case of Deligne–Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.