Geodesic
On Orbit Closures of Symmetric Subgroups in Flag Varieties
Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 265-292

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2000-012-9
Mots-clés : 14M15, 20G05, flag variety, symmetric subgroup 
Brion, Michel; Helminck, Aloysius G. On Orbit Closures of Symmetric Subgroups in Flag Varieties. Canadian journal of mathematics, Tome 52 (2000) no. 2, pp. 265-292. doi: 10.4153/CJM-2000-012-9
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