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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 $$G/B$$.

##### MSC:
 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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