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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)].

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