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

DOI10.4310/PAMQ.2022.V18.N5.A2arXiv2106.00437MaRDI QIDQ6369123

Dipendra Prasad, Dragos Fratila

Publication date: 1 June 2021

Abstract: In this largely expository paper we extend properties of the homological duality functor $RHo{m}_{mathcalH}(-,mathcalH)$ where $mathcalH$ 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 $RHo{m}_{mathcalH}(-,mathcalH)$ 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.

Mathematics Subject Classification ID

Representation-theoretic methods; automorphic representations over local and global fields (11F70) Representations of Lie and linear algebraic groups over global fields and adèle rings (22E55)

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