First cohomology groups of Chevalley groups in cross characteristic

Abstract

Let G be a simple Chevalley group defined over Fq.We show that if r does not divide q and k is an algebraically closed field of characteristic r, then very few irreducible kG-modules have nonzero H 1 (G, V ).We also give an explicit upper bound for dim H 1 (G, V ) for V an irreducible kG-module that does not depend on q, but only on the rank of the group.Cline, Parshall and Scott showed that such a bound exists when r|q.We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on V .

Locations

• Annals of Mathematics - View - PDF