Local methods for blocks of finite simple groups

TL;DR

This survey reviews local methods in the modular representation theory of finite simple groups, highlighting classical and recent results, techniques, and conjectures related to blocks, subgroups, and Morita equivalences.

Contribution

It provides a comprehensive overview of local methods in the modular representation theory of finite simple groups, including recent advances and open problems.

Findings

Connections between p-local subgroups and parabolic subgroups in defining characteristic

Results on simple modules and blocks, including the Alperin weight conjecture

Morita equivalences between blocks preserving defect groups and local structures

Abstract

This survey is about old and new results about the modular representation theory of finite reductive groups with a strong emphasis on local methods. This includes subpairs, Brauer's Main Theorems, fusion, Rickard equivalences. In the defining characteristic we describe the relation between $p$-local subgroups and parabolic subgroups, then give classical consequences on simple modules and blocks, including the Alperin weight conjecture in that case. In the non-defining characteristics, we sketch a picture of the local methods pioneered by Fong-Srinivasan in the determination of blocks and their ordinary characters. This includes the relationship with Lusztig's twisted induction and the determination of defect groups. We conclude with a survey of the results and methods by Bonnaf\'e-Dat-Rouquier giving Morita equivalences between blocks that preserve defect groups and the local structures. The text grew out of the course and talks given by the author in July and September 2016 during the program "Local representation theory and simple groups" at CIB Lausanne. Written Oct 2017, to appear in a proceedings volume published by EMS.