Geometric methods in modular representation theory of finite reductive groups – GeRepMod

This project aims at studying the modular representation theory of finite reductive groups (in non-defining characteristic) using the geometric methods that have proven very successful in the ordinary setting (in characteristic zero). Geometric representation theory is a very active area of research in mathematics, with spectacular results obtained in the last three decades in the representation theory of various objects in Lie theory. More recently, more attention has been drawn to the study of representations over a field of positive characteristic, but not so much progress has been made for finite reductive groups, and many questions raised in the 90’s remain open. The aim of this project is to adapt and exploit these geometric methods in the modular case, where algebraic methods have reached their limits.

The proposal is centered on three mains problems, each of which covers different aspects of the representation theory of finite groups: classification, numerical and homological properties of the modular representations. More precisely, it will be focused on the following three tasks:
1 - Study of decomposition matrices and determination of decomposition numbers;
2 - Classification of representations via cuspidal and supercuspidal support;
3 - Construction of perverse derived equivalences predicted by Broué’s abelian defect group conjecture.

The pioneering work of Bonnafé-Rouquier on modular Deligne-Lusztig theory has paved the way for this strategy to be successful. In addition, recent results of members of this project show how to overcome some of the scientific and technical obstacles that will be encountered. Concrete progress has already been made (computation of decomposition numbers, construction of categorical actions for classical groups), and new objects have been introduced (parity sheaves, which behave in many ways like intersection cohomology complexes). These will be at the heart of many questions in this project.

The proposal gathers six young researchers as well as two senior experts with strong background in modular representation theory of finite and p-adic groups, geometric representation theory and representation theory of Hecke algebras. We plan to meet regularly to benefit from each other’s expertise, share any progress, and put together our efforts in order to overcome the most challenging aspects of the project. In addition, leading mathematicians from abroad will be invited to continue or start fruitful collaborations. This shall give a new impetus to modular representation theory of finite reductive groups in France and in Europe, keeping in mind that the overall impact of this proposal will most certainly go beyond the scope of representation theory of finite groups.