Holomorphic discrete series representation
In mathematics, a holomorphic discrete series representation is a discrete series representation of a semisimple Lie group that can be represented in a natural way as a Hilbert space of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian. Holomorphic discrete series representations are the easiest discrete series representations to study because they have highest or lowest weights, which makes their behavior similar to that of finite-dimensional representations of compact Lie groups.
(Bargmann 1947) found the first examples of holomorphic discrete series representations, and Harish-Chandra (1954, 1955a, 1955c, 1956a) classified them for all semisimple Lie groups.
(Martens 1975) and (Hecht 1976) described the characters of holomorphic discrete series representations.
See also
References
- Bargmann, V (1947), "Irreducible unitary representations of the Lorentz group", Annals of Mathematics, Second Series 48 (3): 568–640, doi:10.2307/1969129, ISSN 0003-486X
- Harish-Chandra (1954), "Representations of semisimple Lie groups. VI", Proceedings of the National Academy of Sciences of the United States of America 40 (11): 1078–1080, doi:10.1073/pnas.40.11.1078, ISSN 0027-8424, PMID 16578441
- Harish-Chandra (1955a), "Integrable and square-integrable representations of a semisimple Lie group", Proceedings of the National Academy of Sciences of the United States of America 41 (5): 314–317, doi:10.1073/pnas.41.5.314, ISSN 0027-8424, PMID 16589671, PMC 528085, http://repository.ias.ac.in/30821/1/322.pdf
- Harish-Chandra (1955c), "Representations of semisimple Lie groups. IV", American Journal of Mathematics 77 (4): 743–777, doi:10.2307/2372596, ISSN 0002-9327
- Harish-Chandra (1956a), "Representations of semisimple Lie groups. V", American Journal of Mathematics 78 (11): 1–41, doi:10.2307/2372481, ISSN 0002-9327, PMID 16578440
- Harish-Chandra (1956b), "Representations of semisimple Lie groups. VI. Integrable and square-integrable representations", American Journal of Mathematics 78 (3): 564–628, doi:10.2307/2372674, ISSN 0002-9327
- Hecht, Henryk (1976), "The characters of some representations of Harish-Chandra", Mathematische Annalen 219 (3): 213–226, doi:10.1007/BF01354284, ISSN 0025-5831
- Martens, Susan (1975), "The characters of the holomorphic discrete series", Proceedings of the National Academy of Sciences of the United States of America 72 (9): 3275–3276, doi:10.1073/pnas.72.9.3275, ISSN 0027-8424, PMID 16592271
External links
- Garrett, Paul (2004), Some facts about discrete series (holomorphic, quaternionic), http://www.math.umn.edu/~garrett/m/v/facts_discrete_series.pdf
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