On actions of a maximal torus on orbits of highest weight vectors
Let \(G\) be a semi-simple, simply connected algebraic group defined over an algebraically closed field of characteristic 0. Let \(T\) be a maximal torus and \(B\) a Borel subgroup containing \(T\). For a dominant weight \(\lambda\), let \(V(\lambda)\) be the irreducible \(G\)-module with highest weight \(\lambda\). Let \(C(\lambda)\) denote the closure of the \(G\)-orbit of the highest weight vectors in \(V(\lambda)\). The author characterizes the \(\lambda\)'s for which the quotient map \(C(\lambda)\to C(\lambda)//T\) is equidimensional; the author also characterizes the \(\lambda\)'s for which the quotient map \(C(\lambda)//T\) is an affine space, i.e., \(k[C(\lambda)]^T\) is a polynomial algebra. The author also describes the canonical module of \(k[C(\lambda)]^T\).
semi-simple simply connected algebraic groups
scientific article; zbMATH DE number 1271827