On the representations of disconnected reductive groups over F_q

TL;DR

This paper proves a conjecture about how unipotent representations of a connected reductive group over a finite field extend to a larger group formed by a semidirect product with a cyclic automorphism, advancing understanding of representation theory over finite fields.

Contribution

It establishes the extension properties of unipotent representations for disconnected reductive groups over finite fields, confirming G. Malle's conjecture.

Findings

Proves G. Malle's conjecture on representation extensions.

Characterizes extensions of unipotent representations in the context of disconnected groups.

Provides new insights into the structure of representations of semidirect products over finite fields.

Abstract

Let \hat G be the semidirect product of a connected reductive group G over F_q with a finite cyclic group generated by a quasisemisimple automorphism of G defined over F_q. In this paper we prove a conjecture of G. Malle concerning the extensions of unipotent representations of G(F_q) to \hat G(F_q).