  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.
