Non-zero Ext for Chevalley Groups (via Algebraic Groups)

Take G to be a split, simply-connected semisimple algebraic group (for example, SL(w, K) or Sp(2m, K)) over an algebraic closure K of the field of p elements, where p is prime. Denote by G(n) the finite subgroup consisting of elements of G defined over the field of p elements. The simple (rational) G-modules L(X) are parametrized by the set X of dominant weights, a subset of the character group X(T) of some fixed maximal torus TofG. In turn, the simple #G(w)-modules are precisely the restrictions of those L(X) for which XeXn, the subset of X + consisting of Z-linear combinations of the fundamental dominant weights with coefficients less than p. We consider ExtG (L(M), L(X)) and Extfc(B) (L(//),L(A)) for suitable X,fi. (1.1) In the case when X,neXn, the restriction to G(n) of a non-split extension