Weight conjectures for ℓ-compact groups and spetses
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 ℓ-compact groups. Under some mild conditions we prove that the fusion systems associated to homotopy fixed points of ℓ-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 ℓ-blocks. Motivated by this, we define the principal block of a spets attached to a spetsial Zℓ-reflection group, using the fusion system related to it via ℓ-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.
History
School
- Science
Published in
Annales Scientifiques de l'Ecole Normale SuperieureVolume
57Issue
3Pages
841 - 894Publisher
Société Mathematique de FranceVersion
- AM (Accepted Manuscript)
Rights holder
© The Author(s)Publisher statement
Authors retain the right to distribute their author accepted manuscript (AAM), such as via an institutional and/or subject repository (e.g. arXiv or HAL), under a Creative Commons Attribution 4.0 International (CC BY 4.0) licence for release no later than the date of first online publication. The submitted version can be distributed in the same kind of repositories (or authors' personal webpages) without licensing conditions. This is in accordance with the SMF publishing agreement.Publication date
2024-05-01Copyright date
2024ISSN
0012-9593eISSN
1873-2151Publisher version
Language
- en