Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Abstract

Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak{g}$, let $G_0\subset G$ be the symmetric subgroup defined by an algebraic involution $\sigma$ and let ${\mathfrak{g}}_1\subset \mathfrak{g}$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak{a}$ of $\mathfrak{g}$ contained in ${\mathfrak{g}}_1$ and stable under the action of some Borel subgroup $B_0\subset G_0$, we classify the $B_0$-orbits in $\mathfrak{a}$ and we characterize the sphericity of $G_0\mathfrak{a}$. Our main tool is the combinatorics of $\sigma$-minuscule elements in the affine Weyl group of $\mathfrak{g}$ and that of strongly orthogonal roots in Hermitian symmetric spaces.

Latex file, 29 pages, minor revision, to appear in Journal of the London Mathematical Society