Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Let G be a simply connected semisimple algebraic group with Lie algebra g, let G(0) subset of G be the symmetric subgroup defined by an algebraic involution sigma and let g(1) subset of g be the isotropy representation of G(0). Given an abelian subalgebra a of g contained in g(1) and stable under the action of some Borel subgroup B-0 subset of G(0), we classify the B-0-orbits in a and characterize the sphericity of G(0)a. Our main tool is the combinatorics of sigma-minuscule elements in the affine Weyl group of g and that of strongly orthogonal roots in Hermitian symmetric spaces.