On orbit closures of spherical subgroups in flag varieties

Abstract.

Let $\cal F$ be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on $\cal F$ with finitely many orbits, and let V be an H-orbit closure in $\cal F$. Expanding the cohomology class of V in the basis of Schubert classes defines a union V₀ of Schubert varieties in $\cal F$ with positive multiplicities. If G is simply-laced, we show that these multiplicities are equal to the same power of 2. For arbitrary G, we show that V₀ is connected in codimension 1. If moreover all multiplicities are 1, we show that the singularities of V are rational and we construct a flat degeneration of V to V₀ in $\cal F$. Thus, for any effective line bundle L on $\cal F$, the restriction map ${H}^0({\cal F}, {\rm L}) \to {H}^0({\rm V, L}$ is surjective, and ${H}^n({\rm V, L}) = 0$ for all $n \geq 1$.