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On orbit closures of spherical subgroups in flag varieties. (English) Zbl 1043.14012
Let \(G\) be a connected reductive linear algebraic group (over \(\mathbb{C}\)), \(B\) a Borel subgroup and \(H\) a spherical subgroup. Then for the natural action of \(H\) on the flag variety \(G/B\) there are only finitely many orbits which have their corresponding counterparts in orbits of \(B\) in \(G/H\). In this article the author studies the closures of these orbits. The set of \(B\)-orbits in \(G/H\) is partially-ordered by \(Y\leq Z\) if there is a sequence \(P_1,\dots, P_n\) of parabolic subgroups containing \(B\) such that \(Z= P_1\cdots P_n Y\). Investigation of the structure of this partial order constitutes the core of this article. The main result establishes rationality of the singularities of any multiplicity-free \(B\)-orbit closure \(Y\) in an equivariant embedding \(X\) of \(G/H\) and furthermore if \(X\) is regular (for the action of \(G\)), then the scheme theoretic intersection of \(Y\) with a \(G\)-orbit closure is proved to be reduced. This is a generalization of the corresponding results for the Schubert subvarieties of \(G/B\).

14M15 Grassmannians, Schubert varieties, flag manifolds
14L30 Group actions on varieties or schemes (quotients)
14B05 Singularities in algebraic geometry
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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