First cohomology groups of Chevalley groups in cross characteristic. (Q640794)

Let \(G\) be a finite simple group of Lie type constructed over a field \(\mathbb F_q\) of characteristic \(p\). Let \(k\) be an algebraically closed field of characteristic \(r\) distinct from \(p\). (The cross characteristic case.) It is shown that very few irreducible \(kG\)-modules \(V\) have nonzero \(H^1(G,V)\). The authors also give an explicit upper bound for \(\dim H^1(G,V)\) that does not depend on \(q\) or \(V\), but only on the rank of the group. They obtain extremely strong bounds in the case that a Borel subgroup \(B\) has no fixed points in \(V\). They also bound the sum over all irreducible modules \(V\) of the product \(\dim V^B\cdot\dim H^1(G,V)\). Recall that Cline, Parshall and Scott showed that some bound on \(\dim H^1(G,V)\) exists in the equal characteristic case when \(r\mid q\).