Consider a continuous irreducible representation of a compact Lie group on a finite-dimensional complex Hilbert space. There are three mutually exclusive options:
1) it's not isomorphic to its dual (in which case we call it 'complex')
2) it has a nondegenerate symmetric bilinear form (in which case we call it 'real')
3) 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 whose continuous irreducible representations on complex vector spaces are all either real or quaternionic in the above sense:
1) the group Z/2
2) the trivial group
3) the group SU(2)
4) 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 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?
