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Unipotent $\ell$-blocks for simply connected $p$-adic groups

Thomas Lanard

Vol. 17 (2023), No. 9, 1533–1572
DOI: 10.2140/ant.2023.17.1533
Abstract

Let F be a nonarchimedean local field and G the F-points of a connected simply connected reductive group over F. We study the unipotent -blocks of G, for p. To that end, we introduce the notion of (d,1)-series for finite reductive groups. These series form a partition of the irreducible representations and are defined using Harish-Chandra theory and d-Harish-Chandra theory. The -blocks are then constructed using these (d,1)-series, with d the order of q modulo , and consistent systems of idempotents on the Bruhat–Tits building of G. We also describe the stable -block decomposition of the depth zero category of an unramified classical group.

Keywords
l-blocks, modular representations of p-adic groups, unipotent representations, Deligne–Lusztig, d-Harish-Chandra theory, stable l-blocks
Mathematical Subject Classification
Primary: 22E50
Secondary: 20C20, 20C33, 20G05, 20G25, 20G40
Milestones
Received: 26 July 2021
Revised: 27 June 2022
Accepted: 4 October 2022
Published: 9 September 2023
Authors
Thomas Lanard
Faculty of Mathematics
University of Vienna
Vienna
Austria

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