On the cohomology of SL2 with coefficients in a simple module

摘要

Let G be the simple algebraic group SL2 defined over an algebraically closed field k of characteristic p > 0. Using results of A. Parker, we develop a method which gives, for any q is an element of N, a closed form description of all simple modules M such that H-q(G,M) not equal 0, together with the associated dimensions dim H-q(G,M). We apply this method for arbitrary primes p and for q q, the dimension of the cohomology H-q(G,M) is at most 1, for any simple module M. Based on this evidence we pose a question for general semisimple algebraic groups.