Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let G be a simply connected semisimple algebraic group with Lie algebra g, let G0 ⊂ G be the symmetric subgroup defined by an algebraic involution σ and let g1 ⊂ g be the isotropy representation of G0. Given an abelian subalgebra a of g contained in g1 and stable under the action of some Borel subgroup B0 ⊂ G0, we classify the B0-orbits in a and characterize the sphericity of G0a. Our main tool is the combinatorics of σ-minuscule elements in the affine Weyl group of g and that of strongly orthogonal roots in Hermitian symmetric spaces