Homological dimension of simple pro-$p$-Iwahori–Hecke modules

Let $G$ be a split connected reductive group defined over a nonarchimedean local field of residual characteristic $p$, and let $\mathcal{H}$ be the pro-$p$-Iwahori–Hecke algebra over $\overline{\mathbb{F}}_p$ associated to a fixed choice of pro-$p$-Iwahori subgroup. We explore projective resolutions of simple right $\mathcal{H}$-modules. In particular, subject to a mild condition on $p$, we give a classification of simple right $\mathcal{H}$-modules of finite projective dimension, and consequently show that “most” simple modules have infinite projective dimension.

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During the preparation of this article, funding was provided by NSF grant DMS-1400779 and an EPDI fellowship.

Received 14 June 2017

Accepted 26 June 2018

Published 25 October 2019