Spherical nilpotent orbits and abelian subalgebras in isotropy representations

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.