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Coefficient Systems on the Bruhat-Tits Building and Pro-$p$ Iwahori-Hecke Modules
About this Title
Jan Kohlhaase
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 279, Number 1374
ISBNs: 978-1-4704-5376-3 (print); 978-1-4704-7228-3 (online)
DOI: https://doi.org/10.1090/memo/1374
Published electronically: July 27, 2022
Table of Contents
Chapters
- Introduction
- 1. A reminder on the Bruhat-Tits building
- 2. Coefficient systems
- 3. The equivalence of categories
- 4. Applications to representation theory
Abstract
Let $G$ be the group of rational points of a split connected reductive group over a nonarchimedean local field of residue characteristic $p$. Let $I$ be a pro-$p$ Iwahori subgroup of $G$ and let $R$ be a commutative quasi-Frobenius ring. If $H=R[I\backslash G/I]$ denotes the pro-$p$ Iwahori-Hecke algebra of $G$ over $R$ we clarify the relation between the category of $H$-modules and the category of $G$-equivariant coefficient systems on the semisimple Bruhat-Tits building of $G$. If $R$ is a field of characteristic zero this yields alternative proofs of the exactness of the Schneider-Stuhler resolution and of the Zelevinski conjecture for smooth $G$-representations generated by their $I$-invariants. In general, it gives a description of the derived category of $H$-modules in terms of smooth $G$-representations and yields a functor to generalized $(\varphi ,\Gamma )$-modules extending the constructions of Colmez, Schneider and VignĂ©ras.- Noriyuki Abe, A comparison between pro-$p$ Iwahori-Hecke modules and $\textrm {mod}\,p$ representations, Algebra Number Theory 13 (2019), no. 8, 1959â1981. MR 4017541, DOI 10.2140/ant.2019.13.1959
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