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ArticleOriginal scientific text

Title

The duality correspondence of infinitesimal characters

Authors 1

Affiliations

  1. Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019, U.S.A.

Abstract

We determine the correspondence of infinitesimal characters of representations which occur in Howe's Duality Theorem. In the appendix we identify the lowest K-types, in the sense of Vogan, of the unitary highest weight representations of real reductive dual pairs with at least one member compact.

Bibliography

  1. J. D. Adams, Discrete spectrum of the reductive dual pair (O(p,q), Sp(2m)), Invent. Math. 74 (1983), 449-475.
  2. J. D. Adams, Unitary highest weight modules, preprint.
  3. N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1968.
  4. T. Y. Enright, R. Howe and N. R. Wallach, A classification of unitary highest weight modules, in: Representation Theory of Reductive Groups, P. C. Trombi (ed.), Birkhäuser, Boston, 1983, 97-143.
  5. S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962.
  6. R. Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), 539-570.
  7. R. Howe, Transcending the classical invariant theory, J. Amer. Math. Soc. 74 (1989), 449-475.
  8. R. Howe, θ-series and invariant theory, in: Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, R.I., 1979, 275-285.
  9. R. Howe, manuscript in preparation on dual pairs.
  10. R. Howe, Dual pairs in physics: harmonic oscillators, photons, electrons, and singletons, in: Lectures in Appl. Math. 21, Amer. Math. Soc., Providence, R.I., 1985, 179-207.
  11. R. Howe, On a notion of rank for unitary representations of the classical groups, in: Harmonic Analysis and Group Representations, Liguori, Napoli, 1982, 223-331.
  12. J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, Berlin, 1972.
  13. N. Jacobson, Basic Algebra I, W. H. Freeman, 1974.
  14. N. Jacobson, Basic Algebra II, W. H. Freeman, 1980.
  15. M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representation and harmonic polynomials, Invent. Math. 44 (1978), 1-97.
  16. A. Knapp, Representation Theory of Semisimple Groups - an Overview Based on Examples, Princeton University Press, Princeton, N.J., 1986.
  17. A. Knapp and D. Vogan, Jr., Duality theorems in the relative Lie algebra cohomology, preprint.
  18. R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. 89 (1980), 1-24.
  19. D. Vogan, Jr., Representation Theory of Real Reductive Lie Groups, Birkhäuser, Boston, 1981.
  20. D. Vogan, Classifying representations by lowest K-types, in: Lectures in Appl. Math. 21, Amer. Math. Soc., 1985, 179-207.
  21. H. Weyl, The Classical Groups, Princeton University Press, Princeton, N.J., 1946.
Colloquium Mathematicum
1996/70/1
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Pages:
93-102
Main language of publication
English
Received
1995-05-17
Published
1996
Exact and natural sciences