Geometric approach to parabolic induction

TL;DR

This paper introduces a geometric method to understand parabolic induction in reductive groups over local fields, showing stability preservation and independence of parabolic subgroup choice.

Contribution

It constructs a restriction map on cocenters that dualizes to parabolic induction, providing new geometric proofs and extending Lusztig-Spaltenstein's theorem.

Findings

Parabolic induction preserves stability.

The character of normalized induction is independent of the parabolic subgroup.

A new geometric proof for properties of parabolic induction.

Abstract

In this note we construct a "restriction" map from the cocenter of a reductive group G over a local non-archimedean field F to the cocenter of a Levi subgroup. We show that the dual map corresponds to parabolic induction and deduce that parabolic induction preserves stability. We also give a new (purely geometric) proof that the character of normalized parabolic induction does not depend on a parabolic subgroup. In the appendix, we use a similar argument to extend a theorem of Lusztig-Spaltenstein on induced unipotent classes to all infinite fields.