Skip to main content
SpringerLink
Log in

Orbits of strongly solvable spherical subgroups on the flag variety

  • Published:
Journal of Algebraic Combinatorics Aims and scope Submit manuscript

Abstract

Let G be a connected reductive complex algebraic group and B a Borel subgroup of G. We consider a subgroup \(H \subset B\) which acts with finitely many orbits on the flag variety G / B, and we classify the H-orbits in G / B in terms of suitable root systems. As well, we study the Weyl group action defined by Knop on the set of H-orbits in G / B, and we give a combinatorial model for this action in terms of weight polytopes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Notice that when \({{\mathrm{supp}}}(\beta )\) is of type \(\mathsf F_4\), our enumeration of the set \({{\mathrm{supp}}}(\beta )\) in Table 1 differs from the one in [2]: following [4] for us \(\alpha _1\) and \(\alpha _2\) are the long simple roots.

References

  1. Arzhantsev, I.V., Zaidenberg, M.G., Kuyumzhiyan, K.: Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity. Mat. Sb. 203(7):3–30 (2012); translation in: Sb. Math. 203 no. 7–8, (2012), 923–949

  2. Avdeev, R.S.: On solvable spherical subgroups of semisimple algebraic groups. Trans. Moscow Math. Soc. 72, 1–44 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avdeev, R.: Strongly solvable spherical subgroups and their combinatorial invariants. Selecta Math. 21(3), 931–993 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie, Chapitres IV, V, VI, Actualités Scientifiques et Industrielles 1337, Hermann, Paris (1968)

  5. Bravi, P., Luna, D.: An introduction to wonderful varieties with many examples of type \(\sf F_4\). J. Algebra 329(1), 4–51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bravi, P., Pezzini, G.: Primitive wonderful varieties. Math. Z. 282(3–4), 1067–1096 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brion, M.: Quelques propriétés des espaces homogènes sphériques. Manuscripta Math. 55(2), 191–198 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brion, M.: Classification des espaces homogènes sphériques. Compositio Math. 63(2), 189–208 (1987)

    MathSciNet  MATH  Google Scholar 

  9. Brion, M.: Rational smoothness and fixed points of torus actions. Transform. Groups 4(2–3), 127–156 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brion, M.: On orbit closures of spherical subgroups in flag varieties. Comment. Math. Helv. 76(2), 263–299 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carrell, J.B.: Torus actions and cohomology, In: Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action, 83–158, Encyclopaedia Math. Sci. 131, Springer, Berlin (2002)

  12. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties, Graduate Studies in Mathematics 124. American Mathematical Society, Providence, RI (2011)

    Google Scholar 

  13. Cupit-Foutou, S.: Wonderful Varieties: A Geometrical Realization. arXiv:0907.2852

  14. Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. École Norm. Sup. 3, 507–588 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fulton, W., MacPherson, R., Sottile, F., Sturmfels, B.: Intersection theory on spherical varieties. J. Algebr. Geom. 4(1), 181–193 (1995)

    MathSciNet  MATH  Google Scholar 

  16. Gandini, J.: Spherical orbit closures in simple projective spaces and their normalizations. Transform. Groups 16(1), 109–136 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hashimoto, T.: \(B_{n-1}\)-orbits on the flag variety \({{\rm GL}}_n/B_n\). Geom. Dedicata 105, 13–27 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Humphreys, J.E.: Linear Algebraic Groups, Graduate Texts in Mathematics 21. Springer, Berlin (1975)

    Book  Google Scholar 

  19. Knop, F.: The Luna-Vust Theory of spherical embeddings. In: Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), pp. 225–249, Manoj Prakashan, Madras (1991)

  20. Knop, F.: On the set of orbits for a Borel subgroup. Comment. Math. Helv. 70(2), 285–309 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Knop, F.: Automorphisms, root systems, and compactifications of homogeneous varieties. J. Amer. Math. Soc. 9, 153–174 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Krämer, M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compositio Math. 38(2), 129–153 (1979)

    MathSciNet  MATH  Google Scholar 

  23. Llendo, A.: Affine \(T\)-varieties of complexity one and locally nilpotent derivations. Transform. Groups 15(2), 389–425 (2010)

    Article  MathSciNet  Google Scholar 

  24. Losev, I.V.: Uniqueness property for spherical homogeneous spaces. Duke Math. J. 147(2), 315–343 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Luna, D.: Sous-groupes sphériques résolubles. Prépubl. Inst. Fourier 241, 20 (1993)

    Google Scholar 

  26. Luna, D.: Variétés sphériques de type \({ A}\). Publ. Math. Inst. Hautes Études Sci. 94, 161–226 (2001)

    Article  MATH  Google Scholar 

  27. Mikityuk, I.V.: Integrability of invariant Hamiltonian systems with homogeneous configuration spaces. Mat. Sb. (N.S.) 129(4), 514–534 (1986) English transl.: Math. USSR–Sb. 57 (1987), 527–546

  28. Montagard, P.L.: Une nouvelle proprieté de stabilité du pléthysme. Comment. Math. Helv. 71, 475–505 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Oda, T.: Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 15, Springer, Berlin (1988)

  30. Ressayre, N.: About Knop’s action of the Weyl group on the set of orbits of a spherical subgroup in the flag manifold. Transform. Groups 10(2), 255–265 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedicata 35, 389–436 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  32. Richardson, R.W., Springer, T.A.: Combinatorics and geometry of K-orbits on the flag manifold. In: Elman, R., Schacher, M., Varadarajan, V. (eds.) Linear Algebraic Groups and their Representations, Contemp. Math. vol. 153, Providence, Amer. Math. Soc. pp. 109–142 (1993)

  33. Springer, T.A.: Schubert varieties and generalizations. In: Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol 514, Kluwer Acad. Publ., Dordrecht, pp. 413–440 (1998)

  34. Timashev, D.A.: A generalization of the Bruhat decomposition. Izv. Ross. Akad. Nauk Ser. Mat. 58(5), 110–123 (1995) Translation in: Russian Acad. Sci. Izv. Math. 45(2), 339–352 (1994)

  35. Vinberg, E.B.: Complexity of actions of reductive groups. Funct. Anal. Appl. 20(1), 1–11 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Vinberg, E.B.: On certain commutative subalgebras of a universal enveloping algebra. Math. USSR Izv. 36, 1–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wolf, J.: Admissible representations and geometry of flag manifolds, In: Eastwood, M., Wolf J., Zierau R. (eds.), The Penrose transform and analytic cohomology in representation theory, Contemp. Math. 154, Providence, Amer. Math. Soc., pp. 21–45 (1993)

Download references

Acknowledgements

We thank P. Bravi, M. Brion, F. Knop and A. Maffei for useful conversations on the subject, and especially R.S. Avdeev for numerous remarks and suggestions on previous versions of the paper which led to significant improvements. This work originated during a stay of the first named author in Friedrich-Alexander-Universität Erlangen-Nürnberg during the fall of 2012 partially supported by a DAAD fellowship, and he is grateful to F. Knop and to the Emmy Noether Zentrum for hospitality. Both the authors were partially supported by the DFG Schwerpunktprogramm 1388—Darstellungstheorie.

Author information

Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126, Pisa, Italy

    Jacopo Gandini

  2. Dipartimento di Matematica Sapienza, Università di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy

    Guido Pezzini

Authors
  1. Jacopo Gandini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guido Pezzini
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guido Pezzini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gandini, J., Pezzini, G. Orbits of strongly solvable spherical subgroups on the flag variety. J Algebr Comb 47, 357–401 (2018). https://doi.org/10.1007/s10801-017-0779-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10801-017-0779-x

Keywords

Mathematics Subject Classification

Search

Navigation