Proceedings of the Japan Academy, Series A, Mathematical Sciences

Proper actions of $SL(2,\mathbf{R})$ on semisimple symmetric spaces

Takayuki Okuda

Full-text: Open access


We classify semisimple symmetric spaces $G/H$ for which there exist proper $SL(2,\mathbf{R})$-actions via $G$. This leads us to the classification of semisimple symmetric spaces that admit surface groups as discontinuous groups.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 87, Number 3 (2011), 35-39.

First available in Project Euclid: 3 March 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22F30: Homogeneous spaces {For general actions on manifolds or preserving geometrical structures, see 57M60, 57Sxx; for discrete subgroups of Lie groups, see especially 22E40}
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 53C35: Symmetric spaces [See also 32M15, 57T15] 57S30: Discontinuous groups of transformations

Proper action symmetric space surface group nilpotent orbit weighted Dynkin diagram Satake diagram


Okuda, Takayuki. Proper actions of $SL(2,\mathbf{R})$ on semisimple symmetric spaces. Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 3, 35--39. doi:10.3792/pjaa.87.35.

Export citation


  • P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups. I, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 3, 401–425.
  • P. Bala and R. W. Carter, Classes of unipotent elements in simple algebraic groups. II, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 1, 1–17.
  • Y. Benoist, Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2) 144 (1996), no. 2, 315–347.
  • M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.
  • E. Calabi and L. Markus, Relativistic space forms, Ann. of Math. (2) 75 (1962), 63–76.
  • D. Ž. Djoković, Classification of $\mathbf{Z}$-graded real semisimple Lie algebras, J. Algebra 76 (1982), no. 2, 367–382.
  • E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates).
  • F. Kassel, Deformation of proper actions on reductive homogeneous spaces, arXiv:0911.4247.
  • T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989), no. 2, 249–263.
  • T. Kobayashi, Discontinuous groups acting on homogeneous spaces of reductive type, in Representation theory of Lie groups and Lie algebras (Fuji-Kawaguchiko, 1990), World Sci. Publ., River Edge, NJ, 1992, 59–75.
  • T. Kobayashi, A necessary condition for the existence of compact Clifford-Klein forms of homogeneous spaces of reductive type, Duke Math. J. 67 (1992), no. 3, 653–664.
  • T. Kobayashi, On discontinuous groups acting on homogeneous spaces with noncompact isotropy subgroups, J. Geom. Phys. 12 (1993), no. 2, 133–144.
  • T. Kobayashi, Discontinuous groups and Clifford-Klein forms of pseudo-Riemannian homogeneous manifolds, in Algebraic and analytic methods in representation theory (Sønderborg, 1994), Perspect. Math., 17 Academic Press, San Diego, CA, 1996, 99–165.
  • T. Kobayashi, Discontinuous groups for non-Riemannian homogeneous spaces, in Mathematics unlimited–-2001 and beyond, Springer, Berlin, 2001, 723–747.
  • T. Kobayashi, Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds, Acta Appl. Math. 73 (2002), no. 1–2, 115–131.
  • T. Kobayashi, On discontinuous groups acting on non-Riemannian homogeneous spaces (Translation of Sūgaku), 57 (2005), no. 3, 267–281.v1.
  • T. Kobayashi and S. Nasrin, Multiplicity one theorem in the orbit method, in Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, 210, Amer. Math. Soc., Providence, RI, 2003, 161–169.
  • T. Kobayashi and T. Yoshino, Compact Clifford-Klein forms of symmetric spaces–-revisited, Pure Appl. Math. Q. 1 (2005), no. 3, part 2, 591–663.
  • B. Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.
  • I. Kra, On lifting Kleinian groups to $\mathrm{SL}(2,\mathbf{C})$, in Differential geometry and complex analysis, Springer, Berlin, 1985, 181–193.
  • F. Labourie and R. J. Zimmer, On the non-existence of cocompact lattices for $\mathrm{SL}(n)/\mathrm{SL}(m)$, Math. Res. Lett. 2 (1995), no. 1, 75–77.
  • A. I. Malcev, On semi-simple subgroups of Lie groups, Amer. Math. Soc. Translation 1950 (1950), no. 33, 43 pp.
  • G. Margulis, Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients, Bull. Soc. Math. France 125 (1997), no. 3, 447–456.
  • G. Margulis, Problems and conjectures in rigidity theory, in Mathematics: frontiers and perspectives, Amer. Math. Soc., Providence, RI, 161–174.
  • H. Oh and D. Witte, Compact Clifford-Klein forms of homogeneous spaces of $\mathrm{SO}(2,n)$, Geom. Dedicata 89 (2002), 25–57.
  • T. Ōshima and J. Sekiguchi, The restricted root system of a semisimple symmetric pair, in Group representations and systems of differential equations (Tokyo, 1982), Adv. Stud. Pure Math., 4 North-Holland, Amsterdam, 1984, 433–497.
  • J. Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), no. 1, 127–138.
  • K. Teduka, Proper actions of $\mathrm{SL}(2,\mathbf{C})$ on irreducible complex symmetric spaces, Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 7, 107–111.
  • R. J. Zimmer, Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc. 7 (1994), no. 1, 159–168.