holomorphic discrete series representation
{{Use American English|date = March 2019}}
{{Short description|Representation of semisimple Lie groups}}
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.
{{harvtxt|Bargmann|1947}} found the first examples of holomorphic discrete series representations, and {{harvs|txt|last=Harish-Chandra|year1=1954|year2=1955a|year3=1955c|year4=1956a|year5=1956b}} classified them for all semisimple Lie groups.
{{harvtxt|Martens|1975}} and {{harvtxt|Hecht|1976}} described the characters of holomorphic discrete series representations.
See also
References
- {{Citation | last1=Bargmann | first1=V | author1-link=Valentine Bargmann | title=Irreducible unitary representations of the Lorentz group | jstor=1969129 | mr=0021942 | year=1947 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=48 | issue=3 | pages=568–640 | doi=10.2307/1969129}}
- {{Citation | last1=Harish-Chandra | title=Representations of semisimple Lie groups. VI | jstor=89268 | mr=0064780 | year=1954 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=40 | issue=11 | pages=1078–1080 | doi=10.1073/pnas.40.11.1078| pmc=1063968 | pmid=16578441| doi-access=free }}
- {{Citation | last1=Harish-Chandra | title=Integrable and square-integrable representations of a semisimple Lie group | jstor=89123 | mr=0070957 | year=1955a | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=41 | issue=5 | pages=314–317 | doi=10.1073/pnas.41.5.314| pmid=16589671 | pmc=528085 | url=http://repository.ias.ac.in/30821/1/322.pdf | doi-access=free }}
- {{Citation | last1=Harish-Chandra | title=Representations of semisimple Lie groups. IV | jstor=2372596 | mr=0072427 | year=1955c | journal=American Journal of Mathematics | issn=0002-9327 | volume=77 | issue=4 | pages=743–777 | doi=10.2307/2372596}}
- {{Citation | last1=Harish-Chandra | title=Representations of semisimple Lie groups. V | jstor=2372481 | mr=0082055 | year=1956a | journal=American Journal of Mathematics | issn=0002-9327 | volume=78 | issue=11 | pages=1–41 | doi=10.2307/2372481| pmc=1063967 | pmid=16578440}}
- {{Citation | last1=Harish-Chandra | title=Representations of semisimple Lie groups. VI. Integrable and square-integrable representations | jstor=2372674 | mr=0082056 | year=1956b | journal=American Journal of Mathematics | issn=0002-9327 | volume=78 | issue=3 | pages=564–628 | doi=10.2307/2372674}}
- {{Citation | last1=Hecht | first1=Henryk | title=The characters of some representations of Harish-Chandra | doi=10.1007/BF01354284 | mr=0427542 | year=1976 | journal=Mathematische Annalen | issn=0025-5831 | volume=219 | issue=3 | pages=213–226| s2cid=120850258 }}
- {{Citation | last1=Martens | first1=Susan | title=The characters of the holomorphic discrete series | jstor=65377 | mr=0419687 | year=1975 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=72 | issue=9 | pages=3275–3276 | doi=10.1073/pnas.72.9.3275| pmc=432971 | pmid=16592271| doi-access=free }}
External links
- {{citation |title=Some facts about discrete series (holomorphic, quaternionic) |url=http://www.math.umn.edu/~garrett/m/v/facts_discrete_series.pdf |first= Paul |last=Garrett |year=2004}}