Modular representations of finite groups and Lie theory

TL;DR

This paper explores the modular representation theory of finite groups of Lie type, proposing new methods and conjectures to understand their behavior in different characteristics, aiming to advance decomposition matrix calculations.

Contribution

It introduces a degeneration method and leverages perverse equivalences to study decomposition matrices in non-defining characteristic.

Findings

Proposes a new degeneration method for modular representations.

Links the behavior of finite groups of Lie type to Broue's conjecture.

Provides a framework for two-variable decomposition matrices in large characteristic.

Abstract

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broue's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broue's conjecture. The modular representation theory of general finite groups is conjectured to behave both like that of finite groups of Lie type in defining characteristic, and in non-defining characteristic, to a large extent. The expected behaviour of modular representation theory of finite groups of Lie type in defining characteristic is particularly difficult to grasp along the lines of Broue's conjecture and we raise a new question related to the change of central character. We introduce a degeneration method in the modular representation theory of finite groups of Lie type in non-defining characteristic. Combined with the rigidity property of perverse equivalences, this provides a setting for two variable decomposition matrices, for large characteristic. This should help make progress towards finding decomposition matrices, an outstanding problem with few general results beyond the case of general linear groups. This last part is based on joint work with David Craven and Olivier Dudas.