On Orbit Closures of Symmetric Subgroups in Flag Varieties

We study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K\G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle L on G/B having nonzero global sections, we show that the restriction map resX : H0(G/B,L) → H0(X,L) is surjective and that Hi(X,L) = 0 for i ≥ 1. Moreover, we describe the K-module H0(X,L). This gives information on the restriction to K of the simple G-module H0(G/B,L). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.