Abstract

I will survey some recent progress in our understanding of the representation theory of reductive algebraic groups (character formulas for simple modules, (derived) equivalences of categories, ...). The situation in characteristic zero is well understood. By contrast the situation in positive characteristic is complicated and many mysteries remain. One of the fascinating aspects of the subject is the richness and diversity of available techniques, as well as the connections to several branches of representation theory (finite groups, Lie algebras, quantum groups). I will survey what is known and not known and then move on to a discussion of application of ideas from categorification as well as connections to topology via perverse sheaves (Lusztig's conjecture and the Finkelberg--Mirkovic conjecture).