Homological duality for covering groups of reductive p-adic groups

In this largely expository paper, we extend properties of the homological duality functor RHomH(-, H) where 1-1 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 are that RHomH(-, 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). Our simple proof is self-contained and bypasses the localization techniques of [SS97, Bez04] improving slightly on [NP20]. 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 else but the contragredient duality. We single out a nec-essary 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 [Roc02], on all blocks with trivial stabilizer in the relative Weyl group.