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Orbits of a spherical subgroup in a flag manifold. (Sur les orbites d’un sous-groupe sphérique dans la variété des drapeaux.) (French) Zbl 1076.14073
Let $$G$$ be a connected reductive complex algebraic group and $$H$$ its spherical subgroup, i.e., the Borel subgroup $$B\subset G$$ acts on $$G/H$$ with finitely many orbits. The author considers $$(B\times H)$$-orbits in $$G$$ and studies the geometry of their closures in $$(G\times G)$$-equivariant embeddings $$X\supset G$$. This work is inspired by a paper of M. Brion [Comment. Math. Helv. 76, No.2, 263–299 (2001; Zbl 1043.14012)], where the $$B$$-orbit closures in $$G$$-equivariant embeddings $$Y\supset G/H$$ are studied. A direct link between these two papers is provided by an observation that there is a natural bijection between the sets of $$B$$-orbits in $$G/H$$ and of $$(B\times H)$$-orbits in $$G$$ and furthermore, certain $$Y$$ (and $$B$$-orbit closures therein) are obtained as geometric quotients by $$H$$ of ($$(B\times H)$$-orbit closures in) open subsets in certain $$X$$ [N. Ressayre, J. Algebra 265, No.1, 1–44 (2003; Zbl 1052.14061)]. First, the author proves that there are finitely many $$(B\times H)$$-orbits in $$X$$. It suffices to prove it for toroidal $$X$$, where it is possible to describe the $$(B\times H)$$-orbits explicitly using the structure of toroidal varieties. The other results concern toroidal embeddings. The author describes the local structure of $$(B\times H)$$-orbit closures and their intersections with $$(G\times G)$$-orbits in $$X$$ (they appear to be proper). For smooth $$X$$, the intersection multiplicities are determined (they are powers of 2). For smooth complete $$X$$, the expression of the cohomology classes of $$(B\times H)$$-orbit closures in terms of the classes of $$(B\times B)$$-orbit closures (which span the cohomology of $$X$$) is found using $$B$$-equivariant cohomology. Finally, the author constructs a smooth toroidal embedding $$Y$$ of $$G/H$$ (namely, the wonderful embedding of $$\text{PGL}_4/\text{PO}_4$$) containing a $$G$$-orbit $$\mathcal{O}$$ (namely, the unique closed orbit) and a $$B$$-orbit $$V$$ such that there exists an irreducible component of $$\overline{V}\cap\overline{\mathcal{O}}$$ consisting of singular points of $$\overline{V}$$. This answers negatively a question of M. Brion. The main tools used in the paper are the structure of toroidal embeddings of $$G$$ (orbits, isotropy groups, transversal slices, see [M. Brion, Comment. Math. Helv. 73, No.1, 137–174 (1998; Zbl 0935.14029)] and the oriented graph of $$(B\times H)$$-orbits in $$G$$ [see M. Brion, Comment. Math. Helv. 76, No.2, 263–299 (2001; Zbl 1043.14012)].

MSC:
 14M17 Homogeneous spaces and generalizations 14M15 Grassmannians, Schubert varieties, flag manifolds 55N91 Equivariant homology and cohomology in algebraic topology
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