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Tensor Products of Holomorphic Discrete Series Representations

Published online by Cambridge University Press:  20 November 2018

Joe Repka
Joe Repka*
Affiliation:
University of Toronto, Toronto, Ontario
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We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.

For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.

The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 4 , 01 August 1979 , pp. 836 - 844
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Bruhat, F., Travaux de Harish-Chandra , Séminaire Bourbaki, t. 9, 1956/57.Google Scholar
2. Garland, H. and Lepowsky, J., Lie algebra homology and the Macdonald-Kac formulas , to appear.Google Scholar
3. Harish-Chandra, , Representations of a semisimple Lie group on a Banach space I, Trans. Amer. Math. Soc. 75 (1953), 185–243.Google Scholar
4. Harish-Chandra, , Representations of semisimple Lie groups II, Trans. Amer. Math. Soc. 76 (1954), 26–65.Google Scholar
5. Harish-Chandra, , Representations of semisimple Lie groups IV, Amer. J. Math. 77 (1955), 743–777.Google Scholar
6. Harish-Chandra, , Representations of semisimple Lie groups V, Trans. Amer. Math. Soc. 78 (1956), 1–41.Google Scholar
7. Harish-Chandra, , Representations of semisimple Lie groups VI, Amer. J. Math. 78 (1950), 564–028.Google Scholar
8. Lang, S., SL2(R), (Addison-Wesley, Reading, Mass., 1975).Google Scholar
9. Repka, J., Tensor products of unitary representations of SL2(R), Amer. J. Math. 100 (1978), 747-774; summary in Bull. Amer. Math. Soc. 82 (1976), 930–932.Google Scholar
10. Wallach, N., Analytic continuation of holomorphie discrete series , Trans. Amer. Math. Soc, to appear.Google Scholar