View source for Quaternionic discrete series representation

In mathematics, a '''quaternionic discrete series representation''' is a [[discrete series representation]] of a [[semisimple Lie group]] ''G'' associated with a quaternionic structure on the [[symmetric space]] of ''G''. They were introduced by {{harvs|txt|last=Gross|last2=Wallach|year1=1994|year2=1996}}. 

Quaternionic discrete series representations exist when the [[maximal compact subgroup]] of the group ''G'' has a [[normal subgroup]] isomorphic to [[SU(2)]]. Every complex simple [[Lie group]] has a real form with quaternionic discrete series representations. In particular the classical groups SU(2,''n''), SO(4,''n''), and Sp(1,''n'') have quaternionic discrete series representations. 

Quaternionic representations are analogous to [[holomorphic discrete series representation]]s, which exist when the symmetric space of the group has a complex structure. The groups SU(2,''n'') have both holomorphic and quaternionic discrete series representations.

==See also==
*[[Quaternionic symmetric space]]

==References==
*{{Citation | last1=Gross | first1=Benedict H. | last2=Wallach | first2=Nolan R | editor1-last=Brylinski | editor1-first=Jean-Luc | editor2-last=Brylinski | editor2-first=Ranee | editor2-link = Ranee Brylinski | editor3-last=Guillemin | editor3-first=Victor | editor4-last=Kac | editor4-first=Victor  | title=Lie theory and geometry | chapter-url=https://books.google.com/books?ei=MIB-Tp3wO4XkiAKwy6G6Aw | publisher=Birkhäuser Boston | location=Boston, MA | series=Progr. Math. | isbn=978-0-8176-3761-3  |mr=1327538 | year=1994 | volume=123 | chapter=A distinguished family of unitary representations for the exceptional groups of real rank =4 | pages=289–304}}
*{{Citation | last1=Gross | first1=Benedict H. | last2=Wallach | first2=Nolan R | title=On quaternionic discrete series representations, and their continuations | doi=10.1515/crll.1996.481.73 |mr=1421947 | year=1996 | journal=[[Journal für die reine und angewandte Mathematik]] | issn=0075-4102 | volume=1996 | issue=481 | pages=73–123| s2cid=116031362 }}

==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}}

[[Category:Representation theory]]