On orbit closures of spherical subgroups in flag varieties

Abstract

Let $\mathcal{F}$ be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on $\mathcal{F}$ with finitely many orbits, and let V be an H-orbit closure in $\mathcal{F}$ . Expanding the cohomology class of V in the basis of Schubert classes defines a union V0 of Schubert varieties in $\mathcal{F}$ 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 $\mathcal{F}$ . Thus, for any effective line bundle L on $\mathcal{F}$ , the restriction map $H^0(\mathcal{F},\mathrm{L})\to H^0(\mathrm{V},\mathrm{L}$ is surjective, and $H^n(\mathrm{V},\mathrm{L})=0$ for all $n\ge 1$ .