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, 1956b) classified them for all semisimple Lie groups.

Martens (1975) and Hecht (1976) described the characters of holomorphic discrete series representations.

• Quaternionic discrete series representation

• 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, JSTOR 1969129, MR 0021942
• 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, JSTOR 89268, MR 0064780, PMC 1063968, PMID 16578441
• Harish-Chandra (1955a), "Integrable and square-integrable representations of a semisimple Lie group" (PDF), 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, JSTOR 89123, MR 0070957, PMC 528085, PMID 16589671
• Harish-Chandra (1955c), "Representations of semisimple Lie groups. IV", American Journal of Mathematics, 77 (4): 743–777, doi:10.2307/2372596, ISSN 0002-9327, JSTOR 2372596, MR 0072427
• Harish-Chandra (1956a), "Representations of semisimple Lie groups. V", American Journal of Mathematics, 78 (11): 1–41, doi:10.2307/2372481, ISSN 0002-9327, JSTOR 2372481, MR 0082055, PMC 1063967, 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, JSTOR 2372674, MR 0082056
• Hecht, Henryk (1976), "The characters of some representations of Harish-Chandra", Mathematische Annalen, 219 (3): 213–226, doi:10.1007/BF01354284, ISSN 0025-5831, MR 0427542, S2CID 120850258
• 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, JSTOR 65377, MR 0419687, PMC 432971, PMID 16592271

• Garrett, Paul (2004), Some facts about discrete series (holomorphic, quaternionic) (PDF)