Homological duality for covering groups of reductive $p$-adic groups

TL;DR

This paper extends homological duality properties of Hecke algebras from reductive p-adic groups to their finite central extensions, revealing key duality behaviors and conditions for their equivalence.

Contribution

It generalizes the homological duality functor properties to central extensions of reductive p-adic groups and clarifies when various dualities coincide.

Findings

RHom_H(-,H) is concentrated in a single degree for irreducible representations

Schneider--Stuhler duality for Ext groups is established

On admissible modules, the duality functor reduces to contragredient duality

Abstract

In this largely expository paper we extend properties of the homological duality functor $RHom_{\mathcal H}(-,{\mathcal H})$ where ${\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\mathcal H}(-,{\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre duality with respect to the Bernstein center and provide a proof of the folklore result that on admissible modules this functor is nothing but the contragredient duality. We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block. In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche, on all blocks with trivial stabilizer in the relative Weyl group.