On momentum images of representations and secant varieties

Let $K$ be a connected compact semisimple group and $V_{\lambda }$ be an irreducible unitary representation with highest weight $\lambda$. We study the momentum map $\mu :\mathbb{P}(V_{\lambda })\to {\mathfrak{k}}^{\ast }$. The intersection $\mu (\mathbb{P}(V_{\lambda }){)}^{+}=\mu (\mathbb{P}(V_{\lambda }))\cap {\mathfrak{t}}^{+}$ of the momentum image with a fixed Weyl chamber is a convex polytope called the momentum polytope of $V_{\lambda }$. We construct an affine rational polyhedral convex cone ${\mathrm{{\rm Y}}}_{\lambda }$ with vertex $\lambda$, such that $\mu(\mathbb P(V_\lambda))^+\subset\Upsilon_\lambda \cap {\mathfrak t}^+$. 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 $\lambda$. We also present some results on the critical points of $\mathrm{\mid }\mathrm{\mid }\mu \mathrm{\mid }{\mathrm{\mid }}^2$. 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 $\mathbb{X}\subset \mathbb{P}(V_{\lambda })$, and prove a relation between the momentum images of the secant varieties and the degrees of $K$-invariant polynomials on $V_{\lambda }$.