Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-16T22:23:42.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Derived intertwining norms for reducible spherical principal series

Part of: Lie groups Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

J. E. Gilbert ,
R. A. Kunze  and
C. Meaney
J. E. Gilbert
Affiliation:
Dept. Mathematics University of TexasAustin, Texas 78712USA e-mail: [email protected]
R. A. Kunze
Affiliation:
Dept. Mathematics University of GeorgiaAthens, Georgia 30602USA e-mail: [email protected]
C. Meaney
Affiliation:
School of MPCE Macquarie UniversityNSW 2109Australia e-mail [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use the second derivative of intertwining operators to realize a unitary structure for the irreducible subrepresentations in the reducible spherical principal series of U(1, n). These representations can also be realized as the kernels of certain invariant first-order differential operators acting on sections of homogeneous bundles over the hyperboloid (U(1) × U(n))/U(1, n).

MSC classification

Secondary: 22E46: Semisimple Lie groups and their representations 22E30: Analysis on real and complex Lie groups 43A85: Analysis on homogeneous spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 61 , Issue 2 , October 1996 , pp. 171 - 188
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Blank, B. E., ‘Boundary behaviour of limits of discrete series representations of real rank one semisimple groups’, Pacific J. Math. 122 (1986), 299–318.CrossRefGoogle Scholar
[2]Davis, K. M., Gilbert, J. E. and Kunze, R. A., ‘Elliptic differential operators in harmonic analysis, I. Generalized Cauchy-Riemann systems’, Amer. J. Math. 113 (1991), 75–116.CrossRefGoogle Scholar
[3]Fabec, R. C., ‘Homogeneous distributions on the Heisenberg group and representations of SU(2,1)’, Trans. Amer. Math. Soc. 328 (1991), 351–391.Google Scholar
[4]Fabec, R. C., ‘Localizable representations of the de Sitter group’, J. Analyse Math. 35 (1979), 151–208.CrossRefGoogle Scholar
[5]Faraut, J., ‘Noyaux spheriques sur un hyperboloide a une nappe’, Lecture Notes in Math. 497 (Springer, Berlin, 1975), 172–210.Google Scholar
[6]Faraut, J., ‘Distributions sphériques sur les espaces hyperboliques’, J. Math. Pures Appl. 58 (1979), 369–444.Google Scholar
[7]Flensted-Jensen, M., ‘Spherical functions on a simply connected semisimple Lie group’, Amer. J. Math. 99 (1977), 341–361.CrossRefGoogle Scholar
[8]Gaillard, P.-Y., ‘Eigenforms of the Laplacian on real and complex hyperbolic spaces’, J. Funct. Anal. 78 (1988), 99–115.CrossRefGoogle Scholar
[9]Hansen, M. L., ‘Weak amenability of the universal covering group of SU(1, n)’, Math. Ann. 288 (1990), 445–472.CrossRefGoogle Scholar
[10]Johnson, K. D. and Wallach, N. R., ‘Composition series and intertwining operators for the spherical principal series. I’, Trans. Amer. Math. Soc. 229 (1977), 137–173.CrossRefGoogle Scholar
[11]Kaplan, A. and Putz, R., ‘Boundary behavior of harmonic forms on a rank one symmetric space’, Trans. Amer. Math. Soc. 231, 369–384.CrossRefGoogle Scholar
[12]Knapp, A. W., Representation theory of semisimple groups., Princeton Math. Series 36 (Princeton Univ. Press, Princeton, 1986).CrossRefGoogle Scholar
[13]Knapp, A. W. and Stein, E. M., ‘Intertwining operators for semisimple groups’, Ann. of Math. 93 (1971), 489–578.CrossRefGoogle Scholar
[14]Knapp, A. W., ‘Intertwining operators for semisimple groups, II’, Invent. Math. 60 (1980), 9–84.CrossRefGoogle Scholar
[15]Knapp, A. W. and Wallach, N. R., ‘Szegö kernels associated with discrete series’, Invent. Math. 34 (1976), 163–200.CrossRefGoogle Scholar
[16]Koornwinder, T. H., ‘The addition formula for Jacobi polynomials, II. The Laplace type integral representation and the product formula’, Report TW 133, Mathematisch Centrum, Amsterdam, 1972.Google Scholar
[17]Koornwinder, T. H., ‘Jacobi functions and analysis on noncompact semisimple Lie groups’, in: Special functions: group theoretical aspects and applications (Reidel, Dordrecht, 1984), pp. 1–85.Google Scholar
[18]Krein, M. G., ‘Hermitian-positive kernels on homogeneous spaces. I and II’, Ukrain. Mat. Žh. 1 (1949), 64–98. and 2 (1950), 10–59. (Translated in: Amer. Math. Soc. series 2, 34 (1963), 69–164.)Google Scholar
[19]Kunze, R. A., ‘Quotient representations’, Conf. Torino-Milano 1 (1983), 57–80.Google Scholar
[20]Meaney, C., ‘Cauchy-Szegö maps, invariant differential operators and some representations of SU(n+1, 1)’, Trans. Amer. Math. Soc., 313 (1989), 161–186.Google Scholar
[21]Molchanov, V. F., ‘Harmonic analysis on a hyperboloid of one sheet’, Soviet Math. Dokl. 7 (1966), 1553–1556.Google Scholar
[22]Molchanov, V. F., ‘Representations of pseudo-orthogonal groups associated with a cone’, Math. USSR-Sbo. 10 (1970), 337–347.Google Scholar
[23]Rudin, W., Function theory on the unit ball in Cn (Springer, Berlin, 1980).CrossRefGoogle Scholar
[24]Schlichtkrull, H., ‘Eigenspaces of the Laplacian on hyperbolic spaces; composition series and integral transforms’, J. Funct. Anal. 70 (1987), 194–219.CrossRefGoogle Scholar
[25]Shintani, T., ‘On the decomposition of regular representation of the Lorentz group on a hyperboloid of one sheet’, Proc. Japan Acad. 43 (1967), 1–5.Google Scholar
[26]Takahashi, R., ‘Sur les représentations unitaires de groups de Lorentz généralises’, Bull. Soc. Math. France 91 (1963), 289–433.CrossRefGoogle Scholar
[27]van der Ven, H., ‘Vector valued Poisson transforms on Riemannian symmetric spaces of rank one’, J. Funct. Anal. 119 (1994), 358–400.CrossRefGoogle Scholar
[28]Vogan, D. A., ‘Unitarizability of certain series of representations’, Ann. of Math. 120 (1984), 141–187.CrossRefGoogle Scholar