Lifting representations of finite reductive groups: a character relation

Jeffrey D. Adler, Michael Cassel, Joshua M. Lansky, Emma Morgan, Yifei Zhao

Published 2012-05-29, updated 2015-11-02Version 2

Given a connected reductive group $\tilde{G}$ over a finite field $k$, and a semisimple $k$-automorphism $\varepsilon$ of $\tilde{G}$ of finite order, let $G$ denote the connected part of the group of $\varepsilon$-fixed points. Then there exists a lifting from packets of representations of $G(k)$ to packets for $\tilde{G}(k)$. In the case of Deligne-Lusztig representations, we show that this lifting satisfies a character relation analogous to that of Shintani.