Torus Invariants of the Homogeneous Coordinate Ring of G/B – Connection with Coxeter Elements

Abstract

In this article, we prove that for any indecomposable dominant character χ of a maximal torus T of a simple adjoint group G over ℂ such that there is a Coxeter element w in the Weyl group W for which , the graded algebra is a polynomial ring if and only if dim(H 0(G/B, ℒχ) T ) ≤rank of G. We also prove that the coordinate ring ℂ[𝔥] of the cartan subalgebra 𝔥 of the Lie algebra 𝔤 of G and are isomorphic if and only if is nonempty for some coxeter element w in W, where α0 denotes the highest long root.