Proceedings of the Japan Academy, Series A, Mathematical Sciences

Proper actions of $SL(2,\mathbf{C})$ on irreducible complex symmetric spaces

Katsuki Teduka

Full-text: Open access

Abstract

We classify irreducible complex symmetric spaces that admit proper $SL(2,\mathbf{C})$-actions.$^{*}$

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 84, Number 7 (2008), 107-111.

Dates
First available in Project Euclid: 17 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.pja/1216308251

Digital Object Identifier
doi:10.3792/pjaa.84.107

Mathematical Reviews number (MathSciNet)
MR2450061

Zentralblatt MATH identifier
1156.22018

Subjects
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

Keywords
Proper action symmetric space complex manifold properly discontinuous action Fuchs group weighted Dynkin diagram

Citation

Teduka, Katsuki. Proper actions of $SL(2,\mathbf{C})$ on irreducible complex symmetric spaces. Proc. Japan Acad. Ser. A Math. Sci. 84 (2008), no. 7, 107--111. doi:10.3792/pjaa.84.107. https://projecteuclid.org/euclid.pja/1216308251


Export citation

References

  • Y. Benoist, Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2) 144 (1996), no. 2, 315–347.
  • E. Calabi and L. Markus, Relativistic space forms, Ann. of Math. (2) 75 (1962), 63–76.
  • D. H. Collingwood and W. M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold, New York, 1993.
  • 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), 59–75, World Sci. Publ., River Edge, NJ.
  • 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), 99–165, Perspect. Math., 17, Academic Press, San Diego, CA, 1996.
  • T. Kobayashi, Criterion of proper actions on homogeneous spaces of reductive groups, J. Lie Theory 6 (1996), no. 2, 147–163.
  • T. Kobayashi, Discontinuous groups for non-Riemannian homogeneous spaces, in Mathematics unlimited–-2001 and beyond, 723–747, Springer, Berlin, 2001.
  • 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 and T. Yoshino, Compact Clifford-Klein forms of symmetric spaces–-revisited, Pure Appl. Math. Q. 1 (2005), Special Issue: In Memory of Armand Borel, no. 3, 591–663.
  • T. Kobayashi, On discontinuous groups acting on non-Riemannian homogeneous spaces, Sūgaku 57 (2005), no. 3, 267–281, An English translation to appear Sugaku Exposition (Amer. Math. Soc.), math. DG/0603319.
  • F. Labourie, S. Mozes, and R. J. Zimmer, On manifolds locally modelled on non-Riemannian homogeneous spaces, Geom. Funct. Anal. 5 (1995), no. 6, 955–965.
  • F. Labourie, R. J. Zimmer, On the non-existence of cocompact lattices for SL(n)/SL(m), Math. Res. Lett. 2 (1995), no. 1, 75–77.
  • G. A. Margulis, Free completely discontinuous groups of affine transformations, Dokl. Akad. Nauk SSSR 272 (1983), no. 4, 785–788.
  • G. A. Margulis, Existence of compact quotients of homogeneous spaces, measurably proper actions, and decay of matrix coefficients, Bul. Soc. math. France. 125 (1997), no. 3, 447–456.
  • G. Margulis, Problems and conjectures in rigidity theory, in Mathematics: frontiers and perspectives, 161–174, Amer. Math. Soc., Providence, RI, 2000.
  • J. Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math 25 (1977), no. 2, 178–187.
  • H. Oh and D. Witte, Compact Clifford–Klein forms of homogeneous spaces of $SO(2,n)$, Geom. Dedicata 89 (2002), 25–57.
  • K. Teduka, Proper action of $SL(2,\mathbf{R})$ on $SL(n,\mathbf{R})$-homogeneous spaces, J. Math. Sci. Uni. Tokyo. (to appear).
  • R. J. Zimmer, Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc. 7 (1994), no. 1, 159–168.