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, ℒ_χ) ^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.

Key Words:

• Coxeter elements
• Indecomposable dominant characters

2000 Mathematics Subject Classification:

• 14L24

Notes

Communicated by S. Kleiman.