Modular representations of finite groups and Lie theory (Q6568825)

Conjectures of Alperin and Broué predict that the modular representation theory of general finite groups shares many features with that of finite groups of Lie type. \textit{J. Alperin}'s conjecture [Proc. Symp. Pure Math. 47, 369--379 (1987; Zbl 0657.20013)] is inspired by finite groups of Lie type in defining characteristic. On the other hand, \textit{M. Broué}'s conjecture [Astérisque 181--182, 61--92 (1990; Zbl 0704.20010)] predicts a behavior similar to that of finite groups of Lie type in non-defining characteristic.\N\NThe paper under review discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broué's abelian defect group conjecture. The author discusses both the defining characteristic case, the inspiration for Alperin's weight conjecture, and the non-defining case, the inspiration for Broué's conjecture. The author introduces 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 [\textit{D. A. Craven} et al., J. Eur. Math. Soc. (JEMS) 22, No. 9, 2821--2877 (2020; Zbl 1508.20010)].