On a class of multiplicity-free nilpotent $K_{\mathbb{C}}$-orbits

Abstract

Let $G$ be a real, connected, noncompact, semisimple Lie group, let $K_{\mathbb{C}}$ be the complexification of a maximal compact subgroup $K$ of $G$, and let $\mathfrak{g} = \mathfrak{k} + \mathfrak{p}$ be the corresponding Cartan decomposition of the complexified Lie algebra of $G$. Sequences of strongly orthogonal noncompact weights are constructed and classified for each real noncompact simple Lie group of classical type. We show that for each partial subsequence {$\gamma _{1},\ldots ,\gamma _{i}$} there is a corresponding family of nilpotent $K_{\mathbb{C}}$-orbits in $\mathfrak{p}$, ordered by inclusion and such that the representation of $K$ on the ring of regular functions on each orbit is multiplicity-free. The $K$-types of regular functions on the orbits and the regular functions on their closures are both explicitly identified and demonstrated to coincide, with one exception in the Hermitian symmetric case. The classification presented also includes the specification of a base point for each orbit and exhibits a corresponding system of restricted roots with multiplicities. A formula for the leading term of the Hilbert polynomials corresponding to these orbits is given. This formula, together with the restricted root data, allows the determination of the dimensions of these orbits and the algebraic-geometric degree of their closures. In an appendix, the location of these orbits within D. King’s classification of spherical nilpotent orbits in complex symmetric spaces is depicted via signed partitions and Hasse diagrams.