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

TL;DR

This paper investigates unipotent ll-blocks of simply-connected reductive groups over non-archimedean local fields, introducing new series and describing their structure using Harish-Chandra theory and idempotents.

Contribution

It introduces the notion of ,1-series for finite reductive groups and constructs ll-blocks using these series and idempotents on the Bruhat-Tits building.

Findings

Defined ,1-series partitioning irreducible representations.

Constructed ll-blocks via ,1-series and idempotents.

Described stable ll-block decomposition for unramified classical groups.

Abstract

Let $F$ be a non-archimedean local field and $G$ the $F$-points of a connected simply-connected reductive group over $F$. In this paper, we study the unipotent $\ell$-blocks of $G$, for $\ell \neq p$. 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 $\ell$-blocks are then constructed using these $(d,1)$-series, with $d$ the order of $q$ modulo $\ell$, and consistent systems of idempotents on the Bruhat-Tits building of $G$. We also describe the stable $\ell$-block decomposition of the depth zero category of an unramified classical group.