The duality correspondence of infinitesimal characters

Tomasz Przebinda

Colloquium Mathematicae (1996)

  • Volume: 70, Issue: 1, page 93-102
  • ISSN: 0010-1354

Abstract

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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.

How to cite

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Przebinda, Tomasz. "The duality correspondence of infinitesimal characters." Colloquium Mathematicae 70.1 (1996): 93-102. <http://eudml.org/doc/210401>.

@article{Przebinda1996,
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.},
author = {Przebinda, Tomasz},
journal = {Colloquium Mathematicae},
keywords = {infinitesimal characters; representations; Howe's duality theorem; dual pairs},
language = {eng},
number = {1},
pages = {93-102},
title = {The duality correspondence of infinitesimal characters},
url = {http://eudml.org/doc/210401},
volume = {70},
year = {1996},
}

TY - JOUR
AU - Przebinda, Tomasz
TI - The duality correspondence of infinitesimal characters
JO - Colloquium Mathematicae
PY - 1996
VL - 70
IS - 1
SP - 93
EP - 102
AB - 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.
LA - eng
KW - infinitesimal characters; representations; Howe's duality theorem; dual pairs
UR - http://eudml.org/doc/210401
ER -

References

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  1. [1] J. D. Adams, Discrete spectrum of the reductive dual pair (O(p,q), Sp(2m)), Invent. Math. 74 (1983), 449-475. Zbl0561.22011
  2. [2] J. D. Adams, Unitary highest weight modules, preprint. Zbl0623.22014
  3. [3] N. Bourbaki, Groupes et Algèbres de Lie, Hermann, Paris, 1968. 
  4. [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. Zbl0535.22012
  5. [5] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, 1962. 
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  10. [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. Zbl0558.22018
  11. [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. [12] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, Berlin, 1972. Zbl0254.17004
  13. [13] N. Jacobson, Basic Algebra I, W. H. Freeman, 1974. 
  14. [14] N. Jacobson, Basic Algebra II, W. H. Freeman, 1980. 
  15. [15] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representation and harmonic polynomials, Invent. Math. 44 (1978), 1-97. Zbl0375.22009
  16. [16] A. Knapp, Representation Theory of Semisimple Groups - an Overview Based on Examples, Princeton University Press, Princeton, N.J., 1986. Zbl0604.22001
  17. [17] A. Knapp and D. Vogan, Jr., Duality theorems in the relative Lie algebra cohomology, preprint. 
  18. [18] R. Parthasarathy, Criteria for the unitarizability of some highest weight modules, Proc. Indian Acad. Sci. 89 (1980), 1-24. Zbl0434.22011
  19. [19] D. Vogan, Jr., Representation Theory of Real Reductive Lie Groups, Birkhäuser, Boston, 1981. 
  20. [20] D. Vogan, Classifying representations by lowest K-types, in: Lectures in Appl. Math. 21, Amer. Math. Soc., 1985, 179-207. 
  21. [21] H. Weyl, The Classical Groups, Princeton University Press, Princeton, N.J., 1946. Zbl1024.20502

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