Inductive local-global conditions and generalized Harish-Chandra theory

TL;DR

This paper develops a generalized Harish-Chandra theory compatible with Clifford theory and automorphisms, aiding the verification of local-global conjectures in the representation theory of finite groups, especially groups of Lie type.

Contribution

It introduces a version of generalized Harish-Chandra theory that incorporates Clifford theory and automorphisms, facilitating the verification of inductive conditions for local-global conjectures.

Findings

Provides a framework for verifying inductive conditions in representation theory.

Extends the parametrization of Harish-Chandra series to non-unipotent cases.

Shows impact on Dade's Conjecture verification.

Abstract

We work towards a version of generalized Harish-Chandra theory compatible with Clifford theory and with the action of automorphisms on irreducible characters. This provides a fundamental tool to verify the inductive conditions for the so-called local-global conjectures in representation theory of finite groups in the crucial case of groups of Lie type in non-defining characteristic. In particular, as shown by the author in an earier paper, this as a strong impact on the verification of the inductive condition for Dade's Conjecture. As a by-product, we also show how to extend the parametrization of generalized Harish-Chandra series given by Brou\'e, Malle and Michel to the non-unipotent case by assuming maximal extendibility.