Let $G$ be a semi-simple compact Lie group. Let $V$ be a real vector space and let:


$\rho : G \to Aut\_{\mathbb{R}}(V)$


be an irreducible real representation of $G$ on $V$. We say that $\rho$ is a real representation of complex type if and only if there exist a $J\in Aut\_{\mathbb{R}}(V)$ satisfying:


$J^{2} = - Id\, , \qquad \rho\circ J = J \circ \rho$.


My question is the following: Is there any sort of classification of real irreducible representations of complex type for $G$ compact and semi-simple? I am particularly interested in the case in which $\rho$ is the Adjoint representation of $G$ on its real Lie algebra $\mathfrak{g}$.


Thanks.



Real Adjoint representations of complex type
Irreducible *real* representations of *complex* type of a compact group correspond to irreducible complex representations that do not admit an invariant bilinear form. Irreducible real representations of *quaternionic* type correspond to irreducible complex representations that admit an alternating invariant bilinear form. See Serre, Linear representations of finite groups, Section 13.2, Prop. 38.


Concerning invariant bilinear forms on irreducible complex representations of simple groups, see Onishchik and Vinberg, Lie groups and algebraic groups, Table 3 (page 297).


