Skip to main content
SpringerLink
Log in

On Orbits in Double Flag Varieties for Symmetric Pairs

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.

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

References

  1. M. Brion, A. G. Helminck, On orbit closures of symmetric subgroups in flag varieties, Canad. J. Math. 52 (2000), no. 2, 265–292.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. Benson, G. Ratcliff, A classification of multiplicity free actions, J. Algebra 181 (1996), no. 1, 152–186.

    Article  MathSciNet  MATH  Google Scholar 

  3. Michel Brion, Quelques propriétés des espaces homogènes sphériques, Manuscripta Math. 55 (1986), no. 2, 191–198.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, Wiley, New York, 1985.

  5. M. Finkelberg, V. Ginzburg, On mirabolic D-modules, Int. Math. Res. Not. IMRN (2010), no. 15, 2947–2986.

  6. M. Finkelberg, V. Ginzburg, R. Travkin, Mirabolic affine Grassmannian and character sheaves, Selecta Math. (N.S.) 14 (2009), no. 3-4, 607–628.

  7. A. Henderson, P. E. Trapa, The exotic Robinson-Schensted correspondence, J. Algebra 370 (2012), 32–45.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. G. Helminck, S. P. Wang, On rationality properties of involutions of reductive groups, Adv. Math. 99 (1993), no. 1, 26–96.

    Article  MathSciNet  MATH  Google Scholar 

  9. V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Kato, An exotic Deligne–Langlands correspondence for symplectic groups, Duke Math. J. 148 (2009), no. 2, 305–371.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progress in Mathematics, Vol. 140, Birkhäuser Boston, Boston, MA, 2002.

  12. K. Kondo, K. Nishiyama, H. Ochiai, K. Taniguchi, Closed orbits on partial flag varieties and double flag variety of finite type, Kyushu J. Math. 68 (2014), no. 1.

    Google Scholar 

  13. A. S. Leahy, A classification of multiplicity free representations, J. Lie Theory 8 (1998), no. 2, 367–391.

    MathSciNet  MATH  Google Scholar 

  14. P. Littelmann, On spherical double cones, J. Algebra 166 (1994), no. 1, 142–157.

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Lusztig, Character sheaves. I–V, Adv. in Math., 56 (1985), 193–237; 57 (1985), 226–265; 57 (1985), 266–315; 59 (1986), 1–63; 61 (1986), 103–155.

  16. G. Lusztig, D. A. Vogan, Jr., Singularities of closures of K-orbits on flag manifolds, Invent. Math. 71 (1983), no. 2, 365–379.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Magyar, J. Weyman, A. Zelevinsky, Multiple flag varieties of finite type, Adv. Math. 141 (1999), no. 1, 97–118.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Magyar, J. Weyman, A. Zelevinsky, Symplectic multiple flag varieties of finite type, J. Algebra 230 (2000), no. 1, 245–265.

    Article  MathSciNet  MATH  Google Scholar 

  19. K. Nishiyama, H. Ochiai, Double flag varieties for a symmetric pair and finiteness of orbits, J. Lie Theory 21 (2011), no. 1, 79–99.

    MathSciNet  MATH  Google Scholar 

  20. D. I. Panyushev, Complexity and rank of double cones and tensor product decompositions, Comment. Math. Helv. 68 (1993), no. 3, 455–468.

    Article  MathSciNet  MATH  Google Scholar 

  21. D. I. Panyushev, On the conormal bundle of a G-stable subvariety, Manuscripta Math. 99 (1999), no. 2, 185–202.

    Article  MathSciNet  MATH  Google Scholar 

  22. A. Petukhov, A geometric approach to (g, k)-modules of finite type, arXiv: 1105.5020 (2011).

  23. R. W. Richardson, T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1–3, 389–436.

    MathSciNet  MATH  Google Scholar 

  24. R. W. Richardson, T. A. Springer, Combinatorics and geometry of K-orbits on the flag manifold, in: Linear Algebraic Groups and their Representations (Los Angeles, CA, 1992), Contemp. Math., Vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 109–142.

  25. R. W. Richardson, T. A. Springer, Complements to: “The Bruhat order on symmetric varieties” [Geom. Dedicata 35(1990), no. 1–3, 389–436], Geom. Dedicata 49 (1994), no. 2, 231–238.

  26. T. A. Springer, Algebraic groups with involutions, in: Proceedings of the 1984 Vancouver Conference in Algebraic Geometry CMS Conf. Proc., Vol. 6, American Mathematical Society, Providence, RI , 1986, pp. 461–471.

  27. T. A. Springer, Linear Algebraic Groups, 2nd ed., Progress in Mathematics, Vol. 9, Birkhäuser Boston, Boston, MA, 1998.

  28. R. Steinberg, Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, no. 80, American Mathematical Society, Providence, RI, 1968.

  29. J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404–439 (electronic).

  30. Y. Tanaka, Classification of visible actions on flag varieties, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 6, 91–96.

    Article  MathSciNet  MATH  Google Scholar 

  31. R. Travkin, Mirabolic Robinson–Schensted–Knuth correspondence, Selecta Math. (N.S.) 14 (2009), no. 3–4, 727–758.

  32. Э. Б. Винберг, Cлoжнoсмь дeйсmвuй рeдyкmuвных групп, фyнкц. анализ и его прилож 20 (1986), no. 1, 1–13. Engl. transl.: È. B. Vinberg, Complexity of action of reductive groups, Funct. Anal. Appl. 20 (1986), no. 1, 1–11.

  33. Э. Б. Винберг, Б. H. Кимельфельд, Однородные обласmu на флаговых многообразuях u cферческuе nодгруnnы nолуnроmых групп Лu, функц. анализ и его прилоҗ. 12 (1978), no. 3, 12–19. Engl. transl.: È. B. Vinberg, B. N. Kimel’fel’d, Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funct. Anal. Appl. 12 (1979), no. 3, 168–174.

    Google Scholar 

  34. T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France 102 (1974), 317–333.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hong-Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong-Kong

    XUHUA HE

  2. Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    HIROYUKI OCHIAI

  3. Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1, Sagamihara, 229-8558, Japan

    KYO NISHIYAMA

  4. Kavli IPMU (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan

    YOSHIKI OSHIMA

Authors
  1. XUHUA HE
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. HIROYUKI OCHIAI
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. KYO NISHIYAMA
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. YOSHIKI OSHIMA
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to XUHUA HE.

Additional information

(XUHUA HE) Partially supported by HKRGC grant 602011

(KYO NISHIYAMA) Supported by JSPS Grant-in-Aid for Scientific Research (B) #21340006

(HIROYUKI OCHIAI) Supported by JSPS Grant-in-Aid for Scientific Research (A) #19204011, and JST CREST.

(YOSHIKI OSHIMA) Supported by Grant-in-Aid for JSPS Fellows (10J00710).

Dedicated to Jiro Sekiguchi on the occasion of his sixtieth birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

HE, X., OCHIAI, H., NISHIYAMA, K. et al. On Orbits in Double Flag Varieties for Symmetric Pairs. Transformation Groups 18, 1091–1136 (2013). https://doi.org/10.1007/s00031-013-9243-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-013-9243-8

Keywords

Search

Navigation