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

In this largely expository paper, we extend properties of the homological duality functor $\mathrm{RHom}_\mathcal{H} (-,\mathcal{H})$ 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 are that $\mathrm{RHom}_\mathcal{H} (-,\mathcal{H})$ 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). 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 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 [Roc02], on all blocks with trivial stabilizer in the relative Weyl group.

2010 Mathematics Subject Classification

Primary 11F70. Secondary 22E55.

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D.F. would like to thank IIT Mumbai, where this work has started, for their hospitality. The second author thanks SERB, India for its support through the JC Bose Fellowship, JBR/2020/000006. His work was also supported by a grant of the Government of the Russian Federation for the state support of scientific research carried out under the agreement 14.W03.31.0030 dated 15.02.2018.

Received 1 June 2021

Received revised 2 June 2022

Accepted 18 July 2022

Published 12 January 2023