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On Orbit Decompositions for Symmetric k-Varieties

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Book cover Symmetry and Spaces

Part of the book series: Progress in Mathematics ((PM,volume 278))

Summary

Orbit decompositions play a fundamental role in the study of symmetric k-varieties and their applications to representation theory and many other areas of mathematics, such as geometry, the study of automorphic forms and character sheaves. Symmetric k-varieties generalize symmetric varieties and are defined as the homogeneous spaces G k /H k , where G is a connected reductive algebraic group defined over a field k of characteristic not 2, H the fixed point group of an involution σ and G k (resp., H k ) the set of k-rational points of G (resp., H).

In this contribution we give a survey of results on the various orbit decompositions which are of importance in the study of these symmetric k-varieties and their applications with an emphasis on orbits of parabolic k-subgroups acting on symmetric k-varieties. We will also discuss a number of open problems.

Mathematics Subject Classification (2000): 20G20, 22E15, 22E46, 53C35

Dedicated to Gerry Schwarz on the occasion of his 60th birthday

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References

  1. Abeasis, S.: On a remarkable class of subvarieties of a symmetric variety. Adv. in Math. 71, 113–129 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. van den Ban, E.P.: The principal series for a reductive symmetric space. I. H-fixed distribution vectors. Ann. Sci. École Norm. Sup. (4) 21(3), 359–412 (1988)

    MathSciNet  MATH  Google Scholar 

  3. van den Ban, E.P., Schlichtkrull, H.: Multiplicities in the Plancherel decomposition for a semisimple symmetric space. In: Representation theory of groups and algebras, pp. 163–180. Amer. Math. Soc., Providence, RI (1993)

    MathSciNet  Google Scholar 

  4. van den Ban, E.P., Schlichtkrull, H.: Fourier transform on a semi-simple symmetric space. Invent. Math. 130(3), 517–574 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. van den Ban, E.P., Schlichtkrull, H.: The most continuous part of the Plancherel decomposition for a reductive symmetric space. Ann. of Math. (2) 145(2), 267–364 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barbasch, D., Evens, S.: K-orbits on Grassmannians and a PRV conjecture for real groups. J. Algebra 167, 258–283 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beilinson, A., Bernstein, J.: Localisation de g-modules. C.R. Acad. Sci. Paris 292(I), 15–18 (1981)

    MathSciNet  MATH  Google Scholar 

  8. Beun, S.L., Helminck, A.G.: On the classification of orbits of symmetric varieties acting on flag varieties of SL(2,k) (2007). To appear in Communications in Algebra.

    Google Scholar 

  9. Beun, S.L., Helminck, A.G.: On orbits of symmetric subgroups acting on flag varieties of SL(n,k) (2008). In Preparation

    Google Scholar 

  10. Borel, A., Tits, J.: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27, 55–152 (1965)

    Article  MathSciNet  Google Scholar 

  11. Borel, A., Tits, J.: Compléments a l'article “groupes réductifs”. Inst. Hautes Études Sci. Publ. Math. 41, 253–276 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  12. Brion, M., Helminck, A.G.: On orbit closures of symmetric subgroups in flag varieties. Canad. J. Math. 52(2), 265–292 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. Inst. Hautes Études Sci. Publ. Math. (41), 5–251 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brylinski, J.L., Delorme, P.: Vecteurs distributions H-invariants pour les séries principales généralisées d'espaces symétriques réductifs et prolongement méromorphe d'intégrales d'Eisenstein. Invent. Math. 109(3), 619–664 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Carmona, J., Delorme, P.: Base méromorphe de vecteurs distributions H-invariants pour les séries principales généralisées d'espaces symétriques réductifs: equation fonctionnelle. J. Funct. Anal. 122(1), 152–221 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cartan, E.: Groupes simples clos et ouvert et géométrie Riemannienne. J. Math. Pures Appl. 8, 1–33 (1929)

    MathSciNet  Google Scholar 

  17. Cooley, C., Haas, R., Helminck, A.G., Williams, N.: Combinatorial properties of the Richardson-Springer involution poset (2008). In preparation

    Google Scholar 

  18. De Concini, C., Procesi, C.: Complete symmetric varieties. In: Invariant theory (Montecatini, 1982), Lecture notes in Math., vol. 996, pp. 1–44. Springer Verlag, Berlin (1983)

    Google Scholar 

  19. De Concini, C., Procesi, C.: Complete symmetric varieties. II. Intersection theory. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), pp. 481–513. North-Holland, Amsterdam (1985)

    Google Scholar 

  20. Delorme, P.: Formule de Plancherel pour les espaces symétriques réductifs. Ann. of Math. (2) 147(2), 417–452 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dixmier, J.: Les C *-algèbres et leurs représentations. Gauthier-Villars, Paris (1994)

    Google Scholar 

  22. Faraut, J.: Distributions sphérique sur les espaces hyperboliques. J. Math. pures et appl. 58, 369–444 (1979)

    MathSciNet  MATH  Google Scholar 

  23. Flensted-Jensen, M.: Discrete series for semi-simple symmetric spaces. Annals of Math. 111, 253–311 (1980)

    Article  MathSciNet  Google Scholar 

  24. Grojnowski, I.: Character sheaves on symmetric spaces. Ph.D. thesis, Massachusetts Institute of Technology (1992)

    Google Scholar 

  25. Haas, R., Helminck, A.G.: Algorithms for twisted involutions in Weyl groups (2007). To Appear

    Google Scholar 

  26. Haas, R., Helminck, A.G.: Computing admissible sequences for twisted involutions in Weyl groups. Canadian Math. Bulletin (2007). To Appear

    Google Scholar 

  27. Harish-Chandra: Collected papers. Vol. I-IV. Springer Verlag, New York (1984). 1944–1983, Edited by V. S. Varadarajan

    Google Scholar 

  28. Helminck, A.G.: Algebraic groups with a commuting pair of involutions and semi-simple symmetric spaces. Adv. in Math. 71, 21–91 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Helminck, A.G.: Tori invariant under an involutorial automorphism I. Adv. in Math. 85, 1–38 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. Helminck, A.G.: Tori invariant under an involutorial automorphism II. Adv. in Math. 131(1), 1–92 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Helminck, A.G.: On the classification of k-involutions I. Adv. in Math. 153(1), 1–117 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Helminck, A.G.: Combinatorics related to orbit closures of symmetric subgroups in flag varieties. In: Invariant theory in all characteristics, CRM Proc. Lecture Notes, vol. 35, pp. 71–90. Amer. Math. Soc., Providence, RI (2004)

    Google Scholar 

  33. Helminck, A.G.: Tori invariant under an involutorial automorphism III (2008). In preparation

    Google Scholar 

  34. Helminck, A.G., Helminck, G.F.: Spherical distribution vectors. Acta Appl. Math. 73(1-2), 39–57 (2002). The 2000 Twente Conference on Lie Groups (Enschede)

    Google Scholar 

  35. Helminck, A.G., Helminck, G.F.: Multiplicity one for representations corresponding to spherical distribution vectors of class ρ. Acta Appl. Math. 86(1-2), 21–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Helminck, A.G., Helminck, G.F.: Intertwining operators related to spherical distribution vectors (2007). To appear.

    Google Scholar 

  37. Helminck, A.G., Schwarz, G.W.: Orbits and invariants associated with a pair of commuting involutions. Duke Math. J. 106(2), 237–279 (2001)

    MathSciNet  MATH  Google Scholar 

  38. Helminck, A.G., Schwarz, G.W.: Smoothness of quotients associated with a pair of commuting involutions. Canad. J. Math. 56(5), 945–962 (2004)

    MathSciNet  MATH  Google Scholar 

  39. Helminck, A.G., Wang, S.P.: On rationality properties of involutions of reductive groups. Adv. in Math. 99, 26–96 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  40. Helminck, A.G., Wu, L.: Classification of involutions of SL(2,k). Comm. Algebra 30(1), 193–203 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  41. Helminck, A.G., Wu, L., Dometrius, C.E.: Classification of involutions of SO(n, k, β) In preparation.

    Google Scholar 

  42. Helminck, A.G., Wu, L., Dometrius, C.E.: Involutions of SL(n,k), (n > 2). Acta Appl. Math. 90, 91–119 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. Hirzebruch, F., Slodowy, P.J.: Elliptic genera, involutions, and homogeneous spin manifolds. Geom. Dedicata 35(1-3), 309–343 (1990)

    MathSciNet  MATH  Google Scholar 

  44. Kostant, B., Rallis, S.: Orbits and representations associated with symmetric spaces. Amer. J. Math. 93, 753–809 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  45. Lusztig, G.: Symmetric spaces over a finite field. In: The Grothendieck Festschrift Vol. III, Progr. Math., vol. 88, pp. 57–81. Birkhäuser, Boston, MA (1990)

    MathSciNet  Google Scholar 

  46. Lusztig, G., Vogan, D.A.: Singularities of closures of K-orbits on flag manifolds. Invent. Math. 71, 365–379 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  47. Matsuki, T.: The orbits of affine symmetric spaces under the action of minimal parabolic subgroups. J. Math. Soc. Japan 31, 331–357 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  48. Ōshima, T., Matsuki, T.: A description of discrete series for semi-simple symmetric spaces. In: Group representations and systems of differential equations (Tokyo, 1982), pp. 331–390. North-Holland, Amsterdam (1984)

    Google Scholar 

  49. Ōshima, T., Sekiguchi, J.: Eigenspaces of invariant differential operators in an affine symmetric space. Invent. Math. 57, 1–81 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  50. Ramanan, S., Ramanathan, A.: Projective normality of flag varieties and Schubert varieties. Invent. Math. 79(2), 217–224 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  51. Ramanathan, A.: Equations defining Schubert varieties and Frobenius splitting of diagonals. Inst. Hautes Études Sci. Publ. Math. (65), 61–90 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  52. Richardson, R. W.: Orbits, invariants and representations associated to involutions of reductive groups. Invent. Math. 66, 287–312 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  53. Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedicata 35(1-3), 389–436 (1990)

    MathSciNet  MATH  Google Scholar 

  54. Richardson, R.W., Springer, T.A.: Complements to: “The Bruhat order on symmetric varieties” [Geom. Dedicata 35 (1990), no. 1-3, 389–436; MR 92e:20032]. Geom. Dedicata 49(2), 231–238 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  55. Rossmann, W.: The structure of semi-simple symmetric spaces. Canad. J. Math. 31, 157–180 (1979)

    MathSciNet  MATH  Google Scholar 

  56. Springer, T.A.: Some results on algebraic groups with involutions. In: Algebraic groups and related topics, Adv. Stud. in Pure Math., vol. 6, pp. 525–543. Academic Press, Orlando, FL (1984)

    Google Scholar 

  57. Steinberg, R.: Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc., vol. 80. Amer. Math. Soc., Providence, RI (1968)

    Google Scholar 

  58. Steinberg, R.: Conjugacy classes in Algebraic Groups, Lecture Notes in Mathematics, vol. 366. Springer Verlag, Berlin-Heidelberg-New York (1974)

    Google Scholar 

  59. Tong, Y.L., Wang, S.P.: Geometric realization of discrete series for semi-simple symmetric space. Invent. Math. 96, 425–458 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  60. Vogan, D.A.: Irreducible characters of semi-simple Lie groups IV, Character-multiplicity duality. Duke Math. J. 49, 943–1073 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  61. Vogan, D.A.: Irreducible characters of semi-simple Lie groups III. Proof of the Kazhdan-Lusztig conjectures in the integral case. Invent. Math. 71, 381–417 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  62. Vust, T.: Opération de groupes reductifs dans un type de cônes presque homogènes. Bull. Soc. Math. France 102, 317–334 (1974)

    MathSciNet  MATH  Google Scholar 

  63. Wolf, J.A.: Finiteness of orbit structure for real flag manifolds. Geom. Dedicata 3, 377–384 (1974)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, N.C., 27695, USA

    A. G. Helminck

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  1. A. G. Helminck
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Correspondence to A. G. Helminck .

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Editors and Affiliations

  1. Dept. Mathematics, University of New Brunswick, Fredericton, E3B 5A3, Canada

    H. E. A. Campbell

  2. Dept. Mathematics, North Carolina State University, Harrelson Hall 255, Raleigh, 27695, U.S.A.

    Aloysius G. Helminck

  3. Inst. Mathematik, Universität Basel, Rheinsprung 21, Basel, 4051, Switzerland

    Hanspeter Kraft

  4. Dept. Mathematics & Statistics, Queen's University, Kingston, University Ave. 99, Kingston, K7L 3N6, Canada

    David Wehlau

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Helminck, A.G. (2010). On Orbit Decompositions for Symmetric k-Varieties. In: Campbell, H., Helminck, A., Kraft, H., Wehlau, D. (eds) Symmetry and Spaces. Progress in Mathematics, vol 278. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4875-6_6

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