Proceedings of the Japan Academy, Series A, Mathematical Sciences

Semisimple symmetric spaces without compact manifolds locally modelled thereon

Yosuke Morita

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Let $G$ be a real reductive Lie group and $H$ a closed subgroup of $G$ which is reductive in $G$. In our earlier work it was shown that, if the homomorphism $i : H^{\bullet}(\mathfrak{g}_{\mathbf{C}}, \mathfrak{h}_{\mathbf{C}}; \mathbf{C}) \to H^{\bullet}(\mathfrak{g}_{\mathbf{C}},(\mathfrak{k}_{H})_{\mathbf{C}}; \mathbf{C})$ is not injective, there does not exist a compact manifold locally modelled on $G/H$. In this paper, we give a classification of the semisimple symmetric spaces $G/H$ for which $i$ is not injective. We also study the case when $G$ cannot be realised as a linear group.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 2 (2015), 29-33.

First available in Project Euclid: 2 February 2015

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Primary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 57S30: Discontinuous groups of transformations
Secondary: 17B56: Cohomology of Lie (super)algebras 22F30: Homogeneous spaces {For general actions on manifolds or preserving geometrical structures, see 57M60, 57Sxx; for discrete subgroups of Lie groups, see especially 22E40} 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

Local model $(G,X)$-structure Clifford–Klein form symmetric space relative Lie algebra cohomology invariant polynomial


Morita, Yosuke. Semisimple symmetric spaces without compact manifolds locally modelled thereon. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 2, 29--33. doi:10.3792/pjaa.91.29.

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  • M. Atiyah and W. Schmid, A geometric construction of the discrete series for semisimple Lie groups, Invent. Math. 42 (1977), 1–62; Erratum: Invent. Math. 54 (1979), no. 2, 189–192.
  • Y. Benoist, Actions propres sur les espaces homogènes réductifs, Ann. of Math. (2) 144 (1996), no. 2, 315–347.
  • Y. Benoist and F. Labourie, Sur les espaces homogènes modèles de variétés compactes, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 99–109.
  • M. Berger, Les espaces symétriques noncompacts, Ann. Sci. École Norm. Sup. (3) 74 (1957), 85–177.
  • A. Borel, Compact Clifford–Klein forms of symmetric spaces, Topology 2 (1963), 111–122.
  • H. Cartan, La transgression dans un groupe de Lie et dans un espace fibré principal, in Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège, Paris, 1951, pp. 57–71.
  • D. Constantine, Compact Clifford–Klein forms–geometry, topology and dynamics, arXiv:1307.2183. (to appear in Proceedings of the conference Geometry, Topology and Dynamics in Negative Curvature (Bangalore, 2010), London Mathematical Society Lecture Notes Series).
  • B. Klingler, Complétude des variétés lorentziennes à courbure constante, Math. Ann. 306 (1996), no. 2, 353–370.
  • 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, 1992.
  • 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, 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, On discontinuous group actions on non-Riemannian homogeneous spaces [translation of Sūgaku 57 (2005), no. 3, 267–281 (in Japanese)], Sugaku Expositions 22 (2009), no. 1, 1–19.
  • T. Kobayashi and K. Ono, Note on Hirzebruch's proportionality principle, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 37 (1990), no. 1, 71–87.
  • 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.
  • F. Labourie, Quelques résultats récents sur les espaces localement homogènes compacts, in Manifolds and geometry (Pisa, 1993), 267–283, Sympos. Math., XXXVI, Cambridge Univ. Press, Cambridge, 1996.
  • 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.
  • 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.
  • Y. Morita, A topological necessary condition for the existence of compact Clifford–Klein forms, arXiv:1310.0796. (to appear in J. Differential Geom.).
  • T. Okuda, Classification of semisimple symmetric spaces with proper $SL(2,\mathbf{R})$-actions, J. Differential Geom. 94 (2013), no. 2, 301–342.
  • R. J. Zimmer, Discrete groups and non-Riemannian homogeneous spaces, J. Amer. Math. Soc. 7 (1994), no. 1, 159–168.