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A topological criterion for group decompositions

Published online by Cambridge University Press:  24 October 2008

J. Hebda  and
P. Moylan
J. Hebda
Affiliation:
Department of Mathematics, St Louis University, St Louis, Missouri 63103
P. Moylan
Affiliation:
Department of Science and Mathematics, Parks College of Saint Louis University, Cahokia, Illinois 62206
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Abstract

Given a connected Lie group G and a closed connected subgroup H of G we prove a necessary and sufficient condition that G decomposes into the Cartesian product of H with G/H is that a similar decomposition holds for the maximal compact subgroups of G and H. Our criterion is applied to the three series of groups for which G/H is SO0(p, q)/SO0(p, q − 1), SU(q + 1, q + 1)/S[U(q + 1, q) × U(1)], and SU(q + 1, q + 1)/SL(n, ℂ) ⋊ H(n) (p, q ≥ 1), and we list the values of p and q for which G ≅ H × G/H in each of the three cases. We describe certain decompositions for some of the groups. We show the usefulness of our criterion in obtaining characterization of the space of differentiable vectors for a unitary induced group representation, and, finally, we show by example of SU(2, 2), how the asymptotic properties of certain function spaces for induced group representations are readily obtained using our results. Our results should be of interest to those working in de Sitter and conformal field theories.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 2 , March 1988 , pp. 285 - 298
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Bott, R. and Milnor, J.. On the parallelizability of spheres. Bull. Amer. Math. Soc. 64 (1958), 87–89.CrossRefGoogle Scholar
[2]Dobrev, V. K.. Elementary representations and intertwining operators for SU(2, 2): I. J. Math. Phys. 26 (1985), 235–251.CrossRefGoogle Scholar
[3]Dobrev, V. K. and Moylan, P.. Elementary representations and intertwining operators for SU(2, 2): II. MPI/MAE-PTh 49/85.Google Scholar
[4]Dold, A.. Partitions of unity in the theory of fibrations. Ann. of Math. 78 (1963), 223–255.CrossRefGoogle Scholar
[5]Drechler, W.. Linearly and nonlinearly transforming fields on homogeneous spaces of the (4, 1)-de Sitter group. J. Math. Phys. 26 (1985), 41–54.CrossRefGoogle Scholar
[6]Geroch, R.. Spinor structure of space-times in general relativity: I. J. Math. Phys. 9 (1968), 1739–1744.CrossRefGoogle Scholar
[7]Hannabuss, K. C.. The localizability of particles in de Sitter space. Proc. Cambridge Philos. Soc. 70 (1971), 283–301.CrossRefGoogle Scholar
[8]Helgason, S.. Differential Geometry and Symmetric Spaces (Academic Press, 1962.)Google Scholar
[9]Husemoller, D.. Fibre Bundles (Springer-Verlag, 1975).Google Scholar
[10]Iwasawa, K.. On some types of topological groups. Ann. of Math. 50 (1949), 507–558.CrossRefGoogle Scholar
[11]Ørsted, B.. Conformally invariant differential equations and projective geometry. J. Funct. Anal. 44 (1981), 1–23.CrossRefGoogle Scholar
[12]Paneitz, S. M. and Segal, I. E.. Analysis in space-time bundles: I. J. Funct. Anal. 47 (1982), 78–142.CrossRefGoogle Scholar
[13]Poulsen, N. S.. On C ∞ vectors and intertwining bilinear forms for representations of Lie groups. J. Funct. Anal. 9 (1972), 87–120.CrossRefGoogle Scholar
[14]Rossmann, W.. Analysis on real hyperbolic spaces. J. Funct. Anal. 30 (1978), 448–477.CrossRefGoogle Scholar
[15]Schlichtkrull, H.. Hyperfundions and Harmonic Analysis on Symmetric Spaces (Birkhäuser, 1984).CrossRefGoogle Scholar
[16]Sekiguchi, J.. Eigenspaces of the Laplace—Beltrami operator on a hyperboloid. Nagoya Math. J. 79 (1980), 151–185.CrossRefGoogle Scholar
[17]Steenrod, N.. The topology of fibre bundles (Princeton University Press, 1951).CrossRefGoogle Scholar
[18]Strasburger, A. and Wawrzyńczyk, A.. Automorphic forms for conical representations. Bull. Acad. Polon. Sci. Sér. Sci. Math. 20 (1972), 921–927.Google Scholar
[19]Strichartz, R.. Harmonic analysis on hyperboloids. J. Funct. Anal. 12 (1973), 341–383.CrossRefGoogle Scholar
[20]Takahashi, R.. Sur les représentations unitaires des groups de Lorentz généralisés. Bull. Soc. Math. France 91(1963), 289–433.Google Scholar
[21]Wallach, N.. Harmonic Analysis on Homogeneous Spaces (Marcel Dekker, Inc., 1973).Google Scholar
[22]Warner, F. W.. Foundations of Differential Manifolds and Lie Groups (Scott, Foresman and Co., 1971).Google Scholar
[23]Wells, R. O. and Wolf, J.. Poincaré series and automorphic cohomology on flag domains. Ann. of Math. 105 (1977), 397–448.CrossRefGoogle Scholar
[24]Whitehead, G. W.. Elements of Homotopy Theory (Springer-Verlag, 1978).CrossRefGoogle Scholar
[25]Wolf, J.. Spaces of Constant Curvature, 3rd ed. (Publish or Perish, 1974).Google Scholar
[26]Varadarajan, V.. Geometry of Quantum Theory (Van Nostrand, 1968).Google Scholar
[27]Vogan, D. A. Jr. Representations of Real Reductive Lie Groups (Birkhäuser, 1981).Google Scholar