Antoine Touzé: Cohomology of algebraic groups with twisted coefficients

Abstract:
Let G be a connected reductive algebraic group over an algebraically closed field k (e.g GL_n(k)). If k has characteristic zero, its algebraic representations are semi-simple. This is not the case if k has positive characteristic, and there is a cohomology theory (Ext-groups) which measures the default of semi-simplicity.

In the first part of the talk, I will recall some basic facts on the cohomology of algebraic groups over a field of positive characteristic, in particular the action of the Frobenius on the cohomology. I will recall some classical theorems which illustrate why this action plays an important role in many applications and connections with other fields of mathematics (invariant theory, cohomology of finite groups...). I will sketch the classical approach to the difficult problem of understanding this action.

In the second part of the talk I will present a recent partial solution to the problem of understanding the action of the Frobenius on the cohomology. This solution is valid for classical matrix groups, and relies on the use of "strict polynomial functors" - a very natural generalization of Schur functors.