
On Orbit Closures of Symmetric Subgroups in Flag Varieties
We study $K$orbits in $G/P$ where $G$ is a complex connected
reductive group, $P \subseteq G$ is a parabolic subgroup, and $K
\subseteq G$ is the fixed point subgroup of an involutive
automorphism $\theta$. Generalizing work of Springer, we
parametrize the (finite) orbit set $K \setminus G \slash P$ 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 $B \subseteq P$ is a Borel subgroup.
Finally, for certain $K$orbit closures $X \subseteq G/B$, and for
any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero
global sections, we show that the restriction map $\res_X \colon
H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and
that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we
describe the $K$module $H^0 (X, \mathcal{L})$. This gives
information on the restriction to $K$ of the simple $G$module $H^0
(G/B, \mathcal{L})$. Our construction is a geometric analogue of
Vogan and Sepanski's approach to extremal $K$types.
Keywords:flag variety, symmetric subgroup Categories:14M15, 20G05 