On Frobenius splitting of orbit closures of spherical subgroups in flag varieties.

*(English)*Zbl 1264.14066Let \(G\) 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.

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.

Reviewer: Jacopo Gandini (Göttingen)

##### MSC:

14M15 | Grassmannians, Schubert varieties, flag manifolds |

14L30 | Group actions on varieties or schemes (quotients) |

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\textit{X. He} and \textit{J. F. Thomsen}, Transform. Groups 17, No. 3, 691--715 (2012; Zbl 1264.14066)

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##### References:

[1] | D. Barbasch, S. Evens, K-orbits on Grassmannians and a PRV conjecture for real groups, J. Algebra 167 (1994), no. 2, 258–283. · Zbl 0823.14037 |

[2] | M. Brion, On orbit closures of spherical subgroups in flag varieties, Comment. Math. Helv. 76 (2001), 263–299. · Zbl 1043.14012 |

[3] | M. Brion, Multiplicity-free subvarieties of flag varieties, in: Commutative Algebra (Grenoble/Lyon, 2001), Contemp. Math., Vol. 331, American Mathematical Society, Providence, RI, 2003, pp. 13–23. · Zbl 1052.14055 |

[4] | M. Brion, S. Kumar, Frobenius Splittings Methods in Geometry and Representation Theory, Progress in Mathematics, Vol. 231, Birkhäuser Boston, Boston, MA, 2005. · Zbl 1072.14066 |

[5] | J. Brundan, Dense orbits and double cosets, in: Algebraic Groups and Their Representations (Cambridge, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 259–274. · Zbl 0933.20038 |

[6] | S. Donkin, Rational Representations of Algebraic Groups, Lecture Notes in Mathematics, Vol. 1140, Springer-Verlag, Berlin, 1985. · Zbl 0586.20017 |

[7] | X. He, J. F. Thomsen, Frobenius splitting and geometry of G-Schubert varieties, Adv. Math. 219 (2008), 1469–1512. · Zbl 1160.14035 |

[8] | M. Hochster, C. Huneke, Tight closure and strong F-regularity, in: Colloque en L’Honneur de Pierre Samuel (Orsay, 1987). Mém. Soc. Math. France (N.S.), No. 38, 1989, pp. 119–133. · Zbl 0699.13003 |

[9] | A. Knutson, Frobenius splitting, point-counting, and degeneration, arXiv:0911.4941. |

[10] | N. Lauritzen, U. Raben-Pedersen, J. F. Thomsen, F-regularity of Schubert varieties with applications to $ \(\backslash\)mathcal{D} $ -modules J. Amer. Math. Soc. 19 (2006), no. 2, 345–355. · Zbl 1098.14038 |

[11] | V. B. Mehta, A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40. · Zbl 0601.14043 |

[12] | O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole Norm. Sup. 23 (1990), 625–644. · Zbl 0748.20026 |

[13] | S. Ramanan, A. Ramanathan, Projective normality of flag varieties and Schubert varieties, Invent. Math. 79 (1985), 217–224. · Zbl 0553.14023 |

[14] | N. Ressayre, Spherical homogeneous spaces of minimal rank, Adv. Math. 224 (2010), 1784–1800. · Zbl 1203.14054 |

[15] | R. W. Richardson, T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), nos. 1–3, 389–436. · Zbl 0704.20039 |

[16] | R. W. Richardson, T. A. Springer, Complements to: ”The Bruhat order on symmetric varieties”, Geom. Dedicata 49 (1994), no. 2, 231–238. · Zbl 0826.20045 |

[17] | K. E. Smith, Globally F-regular varieties: applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553–572. · Zbl 0994.14012 |

[18] | T. A. Springer, Remarks on groups with involution, preprint. |

[19] | J. F. Thomsen, Frobenius splitting of equivariant closures of regular conjugacy classes, Proc. London Math. Soc. 93 (2006), 570–592. · Zbl 1111.14051 |

[20] | W. van der Kallen, Steinberg modules and Donkin pairs, Transform. Groups 6 (2001), no. 1, 87–98. · Zbl 0985.20031 |

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