Lifting representations of finite reductive groups: a character relation

TL;DR

This paper establishes a lifting of representation packets from a fixed-point subgroup to the original reductive group over finite fields, demonstrating a character relation similar to Shintani's for Deligne-Lusztig representations.

Contribution

It introduces a new lifting mechanism for packets of representations between groups related by automorphisms, extending the understanding of character relations in finite reductive groups.

Findings

Existence of a lifting from packets of G(k) to (k)

Lifting satisfies a character relation analogous to Shintani

Applicable to Deligne-Lusztig representations

Abstract

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.