Parabolic induction in characteristic p

TL;DR

This paper investigates how unipotent and pro-p Iwahori invariant functors interact with parabolic induction in characteristic p, revealing their commutation properties and characterizing supercuspidal representations via invariants.

Contribution

It proves the commutation of invariant functors with parabolic induction and its right adjoints, and characterizes supercuspidal representations through invariants in characteristic p.

Findings

Invariant functors commute with parabolic induction and right adjoints.

They do not generally commute with left adjoints unless p is invertible in R.

Supercuspidal representations correspond to supersingular invariants in characteristic p.

Abstract

Let G be the group of rational points of a reductive connected group over a finite field (resp. nonarchimedean local field of characteristic p) and R a commutative ring. The unipotent (resp. pro-p Iwahori) invariant functor takes a smooth representation of G to a module over the unipotent (resp. pro-p Iwahori) Hecke R-algebra H of G. We prove that these functors for G and for a Levi subgroup of G commute with the parabolic induction functors, as well as with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show in the local case that an irreducible admissible R-representation V of G is supercuspidal (or equivalently supersingular) if and only if the H-module V^I of its invariants by the pro-p Iwahori I admits a supersingular subquotient, if and only if V^I is supersingular.