On momentum images of representations and secant varieties

Abstract

Let $K$ be a connected compact semisimple group and $V_λ$ be an irreducible unitary representation with highest weight $λ$. We study the momentum map $μ:\mathbb P(V_λ)\to\mathfrak k^*$. The intersection $μ(\mathbb P(V_λ))^+=μ(\mathbb P(V_λ))\cap{\mathfrak t}^+$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_λ$. We construct an affine rational polyhedral convex cone $Υ_λ$ with vertex $λ$, such that $μ(\mathbb P(V_λ))^+\subsetΥ_λ\cap {\mathfrak t}^+$. We show that equality holds for a class of representations, including those with regular highest weight. For those cases, we obtain a complete combinatorial description of the momentum polytope, in terms of $λ$. We also present some results on the critical points of $||μ||^2$. Namely, we consider the existence problem for critical points in the preimages of Kirwan's candidates for critical values. Also, we consider the secant varieties to the unique complex orbit $\mathbb X\subset\mathbb P(V_λ)$, and prove a relation between the momentum images of the secant varieties and the degrees of $K$-invariant polynomials on $V_λ$.