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 dim Hⁿ(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 Hⁿ(SL₂, V(m))). Analogous results are also established for the extension spaces Extⁿ_SL2 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(p^s)..