Harish-Chandra Induction and Jordan Decomposition of Characters

TL;DR

This paper demonstrates that for finite connected reductive groups, Jordan decomposition can be aligned with Harish-Chandra induction, extending known results to broader classes of groups.

Contribution

It establishes the compatibility of Jordan decomposition with Harish-Chandra induction for all finite connected reductive groups, generalizing previous results.

Findings

Jordan decomposition can be chosen to commute with Harish-Chandra induction

Endomorphism algebra of Harish-Chandra induction of a cuspidal representation is isomorphic to a unipotent counterpart

Results extend known cases from groups with connected center to all finite connected reductive groups

Abstract

We show that for any finite connected reductive group, a Jordan decomposition can always be chosen such that it commutes with Harish-Chandra induction. En route, we show that the endomorphism algebra of the Harish-Chandra induction of a cuspidal representation of a Levi subgroup is isomorphic to a unipotent counterpart. These results generalize the well known results for groups with connected center.