Good filtrations and F -purity of invariant subrings

Let k be an algebraically closed field of positive characteristic, G a reductive group over k, and V a finite dimensional G-module. Let B be a Borel subgroup of G, and U its unipotent radical. We prove that if S = SymV has a good filtration, then SU is F -pure. Throughout this paper, p denotes a prime number. Let k be an algebraically closed field of characteristic p, and G a reductive group over k. Let B be a Borel subgroup of G, and U its unipotent radical. We fix a maximal torus T contained in B, and fix a base of the root system Σ of G so that B is negative. For any weight λ ∈ X(T ), we denote the induced module indGB(λ) by ∇G(λ). We denote the set of dominant weights by X. For λ ∈ X, we call ∇G(λ) the dual Weyl module of highest weight λ. We say that a G-module W has a good filtration [1] if H(G,W ⊗ ∇G(λ)) = 0 for any λ ∈ X. Let V be a finite dimensional G-module, and S = SymV . The objective of this paper is to prove the following. Theorem 1. If S has a good filtration, then S is F -pure. 2010 Mathematics Subject Classification. Primary 13A50; Secondary 13A35.