Weight conjectures for $\ell$-compact groups and spetses

TL;DR

This paper explores the connection between modular representation theory of finite groups, fusion systems, and $ ext{ell}$-compact groups, providing evidence for weight conjectures and proposing new formulations for spetses.

Contribution

It establishes a link between fusion systems of $ ext{ell}$-compact groups and weight conjectures, and introduces a new conjecture for spetses based on fusion systems.

Findings

Fusion systems satisfy equations related to Alperin's Weight Conjecture.

Proved Robinson's conjecture analogue for certain spetses cases.

Provided evidence supporting the weight conjectures' validity.

Abstract

Fundamental conjectures in modular representation theory of finite groups, more precisely, Alperin's Weight Conjecture and Robinson's Ordinary Weight Conjecture, can be expressed in terms of fusion systems. We use fusion systems to connect the modular representation theory of finite groups of Lie type to the theory of $\ell$-compact groups. Under some mild conditions we prove that the fusion systems associated to homotopy fixed points of $\ell$-compact groups satisfy an equation which for finite groups of Lie type is equivalent to Alperin's Weight Conjecture. For finite reductive groups, Robinson's Ordinary Weight Conjecture is closely related to Lusztig's Jordan decomposition of characters and the corresponding results for Brauer $\ell$-blocks. Motivated by this, we define the principal block of a spets attached to a spetsial ${\mathbb Z}_\ell$-reflection group, using the fusion system related to it via $\ell$-compact groups, and formulate an analogue of Robinson's conjecture for this block. We prove this formulation for an infinite family of cases as well as for some groups of exceptional type. Our results not only provide further strong evidence for the validity of the weight conjectures, but also point toward some yet unknown structural explanation for them purely in the framework of fusion systems.