On the cohomology of simple restricted modules for algebraic groups

Abstract

First and second cohomology groups of simple restricted modules for algebraic groups with irreducible root system over an algebraically closed field of positive characteristic are investigated. We show that the cohomology group of simple restricted module for algebraic group is isomorphic to the space of morphisms between the trivial one-dimensional module and the corresponding cohomology group for the Frobenius kernel of this algebraic group. In the second cohomology case assumed that p ≥ 3h − 3, where h is the Coxeter number.