Pure and Applied Mathematics Quarterly

Volume 18 (2022)

Number 5

Special Issue in honor of Professor Benedict Gross’s 70th birthday

Guest Editors: Zhiwei Yun, Shouwu Zhang, and Wei Zhang

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

Pages: 1867 – 1950

DOI: https://dx.doi.org/10.4310/PAMQ.2022.v18.n5.a2


Dragoş Frăţilă (Université de Strasbourg, IRMA, Strasbourg, France)

Dipendra Prasad (Indian Institute of Technology Bombay, Mumbai, India)


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.

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