Endoscopy for representations of disconnected reductive groups over finite fields

TL;DR

This paper develops a canonical Jordan decomposition for irreducible representations of disconnected reductive groups over finite fields, extending Lusztig and Yun's work to more general group schemes.

Contribution

It generalizes the Jordan decomposition to disconnected reductive groups and establishes canonical equivalences between representation categories and unipotent representations of endoscopic groups.

Findings

Canonical Jordan decomposition for disconnected groups

Equivalence between G-representations and endoscopic unipotent representations

Extension of results to smooth group schemes with reductive neutral component

Abstract

Let G be the group of rational points of a connected reductive group over a finite field. Based on work of Lusztig and Yun, we make the Jordan decomposition for irreducible G-representations canonical. It comes in the form of an equivalence between the category of G-representations with a fixed semisimple parameter s and the category of unipotent representations of an endoscopic group of G, enriched with an equivariant structure with respect to the component group of the centralizer of s. Next we generalize these results, replacing the connected reductive group by a smooth group scheme with reductive neutral component. Again we establish canonical equivalences for both the rational and the geometric series of G-representations, in terms of unipotent representations of endoscopic groups.