Extensions for finite Chevalley groups II

Let G be a semisimple simply connected algebraic group defined and split over the field F-p with p elements, let G (F-q) be the finite Chevalley group consisting of the F-q-rational points of G where q=p(r), and let G(r) be the rth Frobenius kernel. The purpose of this paper is to relate extensions between modules in Mod(G(F-q)) and Mod(G(r)) with extensions between modules in Mod(G). Among the results obtained are the following: for r>2 and pgreater than or equal to3(h-1), the G(F-q)-extensions between two simple G(F-q)-modules are isomorphic to the G-extensions between two simple p(r)-restricted G-modules with suitably "twisted" highest weights. For pgreater than or equal to3(h-1), we provide a complete characterization of H-1(G(F-q), H-0 (lambda)) where H-0 (lambda)=ind(B)(G) lambda and lambda is p(r)-restricted. Furthermore, for pgreater than or equal to3(h-1), necessary and sufficient bounds on the size of the highest weight of a G-module V are given to insure that the restriction map H-1(G, V)-->H-1(G(F-q),V) is an isomorphism. Finally, it is shown that the extensions between two simple p(r)-restricted G-modules coincide in all three categories provided the highest weights are "close" together.