Character formulas for reductive algebraic groups over fields of positive characteristic

In the (algebraic) representation theory of reductive algebraic groups, the classification of simple representations is a classical and well understood question, but any "concrete" information about these representations (e.g. the description of their characters) turns out to be extremely difficult to obtain. In particular, the computation of these characters is the subject of a classical conjecture of G. Lusztig, which was proved in the 1990's under the assumption that the characteristic of the base field is large enough, but for which counterexamples were recently found by G. Williamson in smaller characteristics.

In this talk I will explain recent progress on this and related questions, obtained in joint works with G. Williamson and R. Bezrukavnikov.