Journal of the Mathematical Society of Japan

On representations of real Nash groups

Francesco GUARALDO

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Some basic results on compact affine Nash groups related to their Nash representations are given. So, first a Nash version of the Peter-Weil theorem is proved and then several more results are given: for example, it is proved that an analytic representation of such a group is of class Nash and that the category of the classes of isomorphic embedded compact Nash groups is isomorphic with that of the classes of isomorphic embedded algebraic groups. Moreover, given a compact affine Nash group G, a closed subgroup H and a homogeneous Nash G-manifold X, it is proved that the twisted product G ×H X is a Nash G-manifold which is Nash G-diffeomorphic to an algebraic G-variety; besides, this algebraic structure is unique.

Article information

J. Math. Soc. Japan, Volume 64, Number 3 (2012), 927-939.

First available in Project Euclid: 24 July 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20G05: Representation theory
Secondary: 57S15: Compact Lie groups of differentiable transformations 14P20: Nash functions and manifolds [See also 32C07, 58A07]

representation theory equivariant Nash conjecture


GUARALDO, Francesco. On representations of real Nash groups. J. Math. Soc. Japan 64 (2012), no. 3, 927--939. doi:10.2969/jmsj/06430927.

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  • J. Bochnak, M. Coste and M.-F. Roy, Géométrie Algébrique Réelle, Ergeb. Math. Grenzgeb. (3), 12, Springer-Verlag, Berlin, Heidelberg, New York, 1987.
  • G. E. Bredon, Introduction to Compact Transformation Groups, Pure and Applied Mathematics, 46, Academic Press, New York, London, 1972.
  • T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Grad. Texts in Math., 98, Springer-Verlag, Berlin, Heidelberg, New York, 1985.
  • K. H. Dovermann and M. Masuda, Algebraic realization of manifolds with group actions, Adv. Math., 113 (1995), 304–338.
  • K. H. Dovermann and M. Masuda, Uniqueness questions in real algebraic transformation groups, Topology Appl., 119 (2002), 147–166.
  • K. H. Dovermann, M. Masuda and D. Y. Suh, Algebraic realization of equivariant vector bundles, J. Reine Angew. Math., 448 (1994), 31–64.
  • P. Flondor and F. Guaraldo, Some results on real Nash groups, Rév. Roumaine Math. Pures Appl., 43 (1998), 137–145.
  • P. Heinzner, A. T. Huckleberry and F. Kutzschebauch, A Real Analytic Version of Abel's Theorem and Complexifications of Proper Lie Group Actions, Complex Analysis and Geometry, Lectures Notes in Pure and Appl. Math., 173, Marcel Dekker, New York, 1996.
  • T. Kawakami, Nash $G$ manifold structures of compact or compactifiable $C^\infty$ $G$ manifolds, J. Math. Soc. Japan, 48 (1996), 321–331.
  • T. Kawakami, Definable $C^r$ groups and proper definable actions, Bull. Fac. Ed. Wakayama Univ. Natur. Sci., 58 (2008), 9–18.
  • J. J. Madden and C. M. Stanton, One-dimensional Nash groups, Pacific J. Math., 154 (1992), 331–344.
  • A. L. Onishchik (Ed.), Lie Groups and Lie Algebras. I, Encyclopaedia Math. Sci., 20, Springer-Verlag, Berlin, Heidelberg, New York, 1993.
  • A. L. Onishchik and E. B. Vinberg, Lie Groups and Algebraic Groups, Springer Ser. Soviet Math., Springer-Verlag, Berlin, Heidelberg, New York, 1990.
  • R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2), 73 (1961), 295–323.
  • M. Shiota, Nash Manifolds, Lecture Notes in Math., 1269, Springer-Verlag, 1987.
  • M. Shiota, Nash functions and manifolds, In: Lectures in Real Geometry (Madrid, 1994), (ed. F. Broglia), de Gruyter Exp. Math., 23, Walter de Gruyter, Berlin, New York, 1996, pp.,69–112.