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On Frobenius splitting of orbit closures of spherical subgroups in flag varieties. (English) Zbl 1264.14066
Let $$G$$ be a semisimple algebraic group over an algebraically closed field $$k$$ and let $$H$$ be a spherical subgroup of $$G$$, namely a subgroup acting with finitely many orbits on the flag variety $$G/B$$ associated with $$G$$. In the present paper, under the assumption that $$H$$ is reductive and connected, the authors establish a criterion for $$H$$-orbit closures in $$G/B$$ to have nice geometric and cohomological properties. The main tools which are used are Frobenius splitting, introduced by V. B. Mehta and A. Ramanathan [Ann. Math. (2) 122, 27–40 (1985; Zbl 0601.14043)], and global F-regularity, introduced by K. E. Smith [Mich. Math. J. 48, Spec. Vol., 553–572 (2000; Zbl 0994.14012)]. In particular, any Frobenius split variety is weakly normal and, if projective, the higher cohomology groups of ample line bundles are zero, whereas any globally F-regular projective variety is normal, Cohen-Macaulay and the higher cohomology groups of nef line bundles are zero.
More precisely, assume that the characteristic of $$k$$ is sufficiently large and let $$B$$ be a Borel subgroup of $$G$$ intersecting $$H$$ in a Borel subgroup of $$H$$. Let $$W$$ be the Weyl group of $$G$$ and, if $$w \in W$$, denote by $$X(w) = \overline{BwB/B}$$ the corresponding Schubert variety in $$G/B$$. Given a parabolic subgroup $$P$$ of $$G$$ containing $$B$$, denote $$W_P$$ the corresponding parabolic subgroup of $$W$$. The authors deal with the orbit closures of the shape $$\overline{HP/B}$$: they give conditions for these varieties to admit a Frobenius splitting along a divisor whose restriction to $$P/B$$ is ample that is compatible with all the subvarieties $$\overline{HX(w)}$$ with $$w \in W_P$$, and they give conditions for these subvarieties to be globally F-regular. Although the subgroup $$H$$ occurring in the criterion is not assumed to be spherical, in many cases the relevant orbit closures $$\overline{HX(w)}$$ coincide with closures of orbits under spherical subgroups.
As a special case, if $$H$$ is trivial then the theorem reduces to the well known results that the flag variety admits a Frobenius splitting along an ample divisor which is compatible with all Schubert varieties, and that any Schubert variety is globally F-regular. Another special case is when $$(G,H)$$ is in the list of minimal rank pairs of N. Ressayre [Adv. Math. 224, No. 5, 1784–1800 (2010; Zbl 1203.14054)]. In this case, every $$H$$-orbit closure in the flag variety is of the shape $$\overline{HX(w)}$$ for some $$w \in W$$, and the theorem implies that the flag variety admits a Frobenius splitting along an ample divisor that is compatible with all the $$H$$-orbit closures.
##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 14L30 Group actions on varieties or schemes (quotients)
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##### References:
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