On Orbit Closures of Symmetric Subgroups in Flag Varieties

Abstract

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 $K\,\backslash \,G/P$ 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 $B\,\subseteq \,P$ 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 $\text{re}{{\text{s}}_{X}}\,:\,{{H}^{0}}\,\left( G\,/\,B,\,\mathcal{L} \right)\,\to \,{{H}^{0}}\,\left( X,\,\mathcal{L} \right)$ is surjective and that ${{H}^{i}}\,\left( X,\mathcal{L} \right)\,=\,0$ for $i\,\ge \,1$. Moreover, we describe the $K$-module ${{H}^{0}}\left( X,L \right)$. This gives information on the restriction to $K$ of the simple $G$-module ${{H}^{0}}\,\left( G\,/\,B,\mathcal{L} \right)$. Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal $K$-types.