Frobenius splitting of certain rings of invariants

Abstract

Let k be an algebraically closed field of characteristic p > 0, and V an n-dimensional k-vector space together with a non-degenerate symmetric bilinear form. Let G denote one of the groups G = SL(V ) or SO(V ) where we assume that p > 2 if G=SO(V ). Let R denote the coordinate ring of V m,q := V ⊕m . V ⊕*q (resp. V m := V ⊕m ) if G = SL(V ) (resp. if G = SO(V )), V* being the dual of V . The defining representation of G on V induces the diagonal action of G on V m,q (resp. V m ). Let S=R G . In this paper, we show that S is Frobenius split.