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 subset of or equal to G is a parabolic subgroup, and K subset of or equal to G is the fixed point subgroup of an involutive automorphism theta. 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 theta-stable (resp. theta-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B subset of or equal to P is a Borel subgroup. Finally for certain K-orbit closures X subset of or equal to G/B, and for any homogeneous line bundle L on G/B having nonzero global sections. we show that the restriction map res(x): H-0(G/B, L) --> H-0(X, L) is surjective and that H-i(X, L) = 0 for i greater than or equal to 1. Moreover, we describe the R-module H-0(X, L). This gives information on the restriction to K of the simple G-module H-0(G/B, L). Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal K-types.