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Equivalence of categories between coefficient systems and systems of idempotents


Author: Thomas Lanard
Journal: Represent. Theory 25 (2021), 422-439
MSC (2020): Primary 22E50; Secondary 20E42
DOI: https://doi.org/10.1090/ert/572
Published electronically: June 2, 2021
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Abstract: The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $\operatorname {Rep}_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline {\mathbb {Z}}_{\ell }$, $\ell \neq p$, by Dat for $GL_{n}$ and the author for a more general group. Wang proved in the case of $GL_{n}$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_{n}$ and a unipotent block of another group. In this paper, we generalize Wang’s equivalence of category to a connected reductive group on a non-archimedean local field.


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Additional Information

Thomas Lanard
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
MR Author ID: 1288084
Email: thomas.lanard@univie.ac.at

Received by editor(s): August 4, 2020
Received by editor(s) in revised form: January 14, 2021, and February 8, 2021
Published electronically: June 2, 2021
Article copyright: © Copyright 2021 American Mathematical Society