Quaternionic discrete series representation
Encyclopedia
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 .
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.
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...
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 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
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 ...
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.