Unipotent ℓ-blocks for simply connected p-adic groups

Lanard, Thomas

doi:10.2140/ant.2023.17.1533

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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 $ℓ$ -blocks of $G$ , for $ℓ \neq p$ . 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 $ℓ$ -blocks are then constructed using these $(d, 1)$ -series, with $d$ the order of $q$ modulo $ℓ$ , and consistent systems of idempotents on the Bruhat–Tits building of $G$ . We also describe the stable $ℓ$ -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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Vol. 17, No. 9, 2023