Multiplicity-free actions and the geometry of nilpotent orbits

Abstract

Let G be a reductive Lie group. Take a maximal compact subgroup K of G and denote their Lie algebras by $ {\mathfrak g}_0 $ and ${\mathfrak k}_0 $ respectively. We get a Cartan decomposition $ {\mathfrak g}_0 ={\mathfrak k}_0 \oplus{\mathfrak s}_0 $ . Let $ {\mathfrak g} $ be the complexification of ${\mathfrak g}_0 $ , and $ {\mathfrak g} ={\mathfrak k} \oplus{\mathfrak s} $ the complexified decomposition. The adjoint action restricted to K preserves the space $ {\mathfrak s}_0 $ , hence $ K_{\mathbb C} $ acts on ${\mathfrak s}$ , where $ K_{\mathbb }C $ denotes the complexification of K. In this paper, we consider a series of small nilpotent $ K_{\mathbb C}$ -orbits in ${\mathfrak s} $ which are obtained from the dual pair $(G, G') = (O(p, q), Sp(2 n, \R)) $ ([R. Howe, Transcending classical invariant theory. J. Amer. Math. Soc. 2 (1989), no. 3, 535–552]). We explain astonishing simple structures of these nilpotent orbits using generalized null cones. For example, these orbits have a linear ordering with respect to the closure relation, and $ K_{\mathbb C} $ acts on them in multiplicity-free manner. We clarify the $ K_{\mathbb C} $ -module structure of the regular function ring of the closure of these nilpotent orbits in detail, and prove the normality. All these results naturally comes from the analysis on the null cone $ {\mathfrak N} $ in a matrix spaceW , and the double fibration of nilpotent orbits in $ {\mathfrak s} $ and ${\mathfrak s}' $ . The classical invariant theory assures that the regular functions on our nilpotent orbits are coming from harmonic polynomials on W with repspect to $ K_{\mathfrak C} $ or $ K'_{\mathfrak C} $ . We also provide many interesting examples of multiplicity-free actions on conic algebraic varieties.