Homological duality for covering groups of reductive p-adic groups

2021/06/01 by Fratila, Dragos, Prasad, Dipendra · 1 citation
#22E45 #22E50 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2106.00437

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.

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