Title: Local methods for blocks of finite simple groups,
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 Bonnafé-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.