Modular representation theory of reductive algebraic groups and local Geometric Langlands duality

In the recent years the PI has been involved in several breakthrough results in the representation theory of reductive algebraic groups (in particular related to the computation of character formulas for simple and indecomposable tilting modules), obtained using various techniques (in particular geometry and categorification).

The present proposal aims at:1. exploring the new perspectives offered by these results, which go beyond the computation of characters, and by the techniques we have already developed;2. developing new geometric tools to support these advances.Our main geometric input will be the development of a modular Local Geometric Langlands duality, in the spirit of work of Bezrukavnikov for characteristic-0 coefficients, and of a modular ""ramified"" geometric Satake equivalence.

We expect in particular applications in the study of tilting modules (e.g. their behaviour under restriction to reductive subgroups, and their multiplicative properties), and to the description of the center of the distribution algebra (with a view towards understanding the ""higher linkage"" phenomena).