Cohomology of SL2 and related structures

Let SL2 be an algebraic group defined over an algebraically closed field k of characteristic p > 0. In this paper, we provide a closed formula for dimHn(SL2,V(m)) for Weyl SL2-modules V(m) when n ≤ 2p − 3. For n > 2p − 3, an exponential bound, only depending on n, is obtained for dimHn(SL2,V(m)). Analogous results are also established for the extension spaces ExtSL2n(V(m2),V(m1)) between Weyl modules V(m1) and V(m2). As a by-product, our results and techniques give explicit upper bounds for the dimensions of cohomology of the Specht modules of symmetric groups, and the cohomology of simple modules of SL2 and the finite group of Lie type SL2(ps).