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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Intertwining differential operators for $ {\rm Mp}(n,\,{\bf R})$ and $ {\rm SU}(n,\,n)$


Author: Hans Plesner Jakobsen
Journal: Trans. Amer. Math. Soc. 246 (1978), 311-337
MSC: Primary 22E45; Secondary 35L99, 47A15
MathSciNet review: 515541
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Abstract: For each of the two series of groups, three series of representations $ {U_n}$, $ {D_n}$, and $ {H_n}(n \in Z)$ are considered. For each series of representations there is a differential operator with the property, that raised to the nth power $ (n > 0)$, it intertwines the representations indexed by $ - n$ and n. The operators are generalizations of the d'Alembertian, the Diracoperator and a combination of the two. Unitarity of subquotients of representations indexed by negative integers is derived from the intertwining relations.


References [Enhancements On Off] (What's this?)

  • [1] R. Godement, Fonctions automorphes, Séminaire Cartan, Université de Paris, 1957-1958.
  • [2] Kenneth I. Gross and Ray A. Kunze, Bessel functions and representation theory. II. Holomorphic discrete series and metaplectic representations, J. Functional Analysis 25 (1977), no. 1, 1–49. MR 0453928 (56 #12181)
  • [3] Lars Gårding, The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. of Math. (2) 48 (1947), 785–826. MR 0022648 (9,240a)
  • [4] Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York, 1962. MR 0145455 (26 #2986)
  • [5] Hans Plesner Jakobsen and Michele Vergne, Wave and Dirac operators, and representations of the conformal group, J. Functional Analysis 24 (1977), no. 1, 52–106. MR 0439995 (55 #12876)
  • [6] H. P. Jakobsen, Conformal harmonic analysis and intertwining differential operators, Thesis, Massachusetts Institute of Technology, 1976.
  • [7] Bertram Kostant, Verma modules and the existence of quasi-invariant differential operators, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Springer, Berlin, 1975, pp. 101–128. Lecture Notes in Math., Vol. 466. MR 0396853 (53 #713)
  • [8] Hugo Rossi and Michèle Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Functional Analysis 13 (1973), 324–389. MR 0407206 (53 #10989)
  • [9] M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Math. 136 (1976), no. 1-2, 1–59. MR 0480883 (58 #1032)
  • [10] G. Mack and Abdus Salam, Finite-component field representations of the conformal group, Ann. Physics 53 (1969), 174–202. MR 0245267 (39 #6578)
  • [11] Irving Ezra Segal, Mathematical cosmology and extragalactic astronomy, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976. Pure and Applied Mathematics, Vol. 68. MR 0496337 (58 #14894)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0515541-9
PII: S 0002-9947(1978)0515541-9
Keywords: Representation, holomorphic function, upper half-plane, reproducing kernel, intertwining differential operator, subquotient, unitarity
Article copyright: © Copyright 1978 American Mathematical Society