Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
it's not isomorphic to its dual (in which case we call it 'complex')
it has a nondegenerate symmetric bilinear form (in which case we call it 'real')
it has a nondegenerate antisymmetric bilinear form (in which case we call it 'quaternionic')
It's 'real' in this sense iff it's the complexification of a representation on a real vector space, and it's 'quaternionic' in this sense iff it's the underlying complex representation of a representation on a quaternionic vector space.
Offhand, I know just four compact Lie groups where all continuous irreducible representations are either real or quaternionicfour compact Lie groups whose continuous irreducible representations on complex vector spaces are all either real or quaternionic in the above sense:
the group Z/2
the trivial group
the group SU(2)
the group SO(3)
Note that I'm so desperate for examples that I'm including 0-dimensional compact Lie groups, i.e. finite groups!
Note that 1) is the group of unit-norm real numbers, 2) is a group covered by that, 3) is the group of unit-norm quaternions, and 4) is a group covered by that. That probably explains why these are all the examples I know. For 2) and 4), all the continuous irreducible representations are in fact real.
- is the group of unit-norm real numbers, 2) is a group covered by that, 3) is the group of unit-norm quaternions, and 4) is a group covered by that. This probably explains why these are all the examples I know. For 1), 2) and 4), all the continuous irreducible representations are in fact real.
What are all the examples?