In mathematics, a quaternionic discrete series representation is a
discrete series representation
In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group ''G'' that is a subrepresentation of the left regular representation of ''G'' on L²(''G''). In the Plancherel mea ...
of a
semisimple Lie group ''G'' associated with a quaternionic structure on the
symmetric space of ''G''. They were introduced by .
Quaternionic discrete series representations exist when the
maximal compact subgroup of the group ''G'' has a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
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
*
*
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
Representation theory