JournalscmhVol. 76 , No. 2DOI 10.1007/pl00000379

On orbit closures of spherical subgroups in flag varieties

  • Michel Brion

    Université Grenoble I, Saint-Martin-d'Hères, France
On orbit closures of spherical subgroups in flag varieties cover

Abstract

Let F\cal F be the flag variety of a complex semi-simple group G, let H be an algebraic subgroup of G acting on F\cal F with finitely many orbits, and let V be an H-orbit closure in F\cal F . Expanding the cohomology class of V in the basis of Schubert classes defines a union V0 of Schubert varieties in F\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 F\cal F . Thus, for any effective line bundle L on F\cal F , the restriction map H0(F,L)H0(V,L{H}^0({\cal F}, {\rm L}) \to {H}^0({\rm V, L} is surjective, and Hn(V,L)=0{H}^n({\rm V, L}) = 0 for all n1n \geq 1 .