A note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup

Abstract

Let G be a connected, semisimple, simply connected, algebraic group over an algebraically closed field K of positive characteristic. The action of G on itself by conjugation induces on the coordinate algebra K[G], and hence on the coordinate algebra K[Gr] of the infinitesimal subgroup Gr, the structure of a rational G-module. In the recent paper, [3], by Bendel, Nakano, Pillen and Sobaje, the authors ask whether the tensor product of K[Gr] with the rth Steinberg module Str has a good filtration. Here we show that this is the case provided that the characteristic of K is good for G. We do so by relating K[Gr] to a certain quotient of the coordinate algebra K[g] of the Lie algebra g of G and using a Koszul resolution. We note that the argument may also be applied to certain polynomial modules occurring in the context of Doty's Conjecture, [17, 4.2.11].