- Source: 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 Gross and Wallach (1994, 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 representations, 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
Gross, Benedict H.; Wallach, Nolan R (1994), "A distinguished family of unitary representations for the exceptional groups of real rank =4", in Brylinski, Jean-Luc; Brylinski, Ranee; Guillemin, Victor; Kac, Victor (eds.), Lie theory and geometry, Progr. Math., vol. 123, Boston, MA: Birkhäuser Boston, pp. 289–304, ISBN 978-0-8176-3761-3, MR 1327538
Gross, Benedict H.; Wallach, Nolan R (1996), "On quaternionic discrete series representations, and their continuations", Journal für die reine und angewandte Mathematik, 1996 (481): 73–123, doi:10.1515/crll.1996.481.73, ISSN 0075-4102, MR 1421947, S2CID 116031362
External links
Garrett, Paul (2004), Some facts about discrete series (holomorphic, quaternionic) (PDF)
Kata Kunci Pencarian:
- Quaternionic discrete series representation
- Discrete series representation
- Holomorphic discrete series representation
- List of representation theory topics
- Glossary of representation theory
- Kazhdan's property (T)
- Quaternion-Kähler symmetric space
- Spinor
- Classical group
- Glossary of areas of mathematics
