Real Adjoint representations of complex type

Question

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.

Answer 1

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).

Answer 2

Let $\pi$ be a complex representation of the compact connected Lie group $G$ (no need for semisimplicity here) on a (finite-dimensional) vector space $V$. We say that $\pi$ is of real type if it comes from a representation of $G$ on a real vector space by extension of scalars, and we say that $\pi$ is of quaternionic type if it comes from a representation of G on a quaternionic vector space by restriction of scalars. If $\pi$ is neither of real type nor of quaternionic type, we say that $\pi$ is of complex type.

Let $\rho$ be a real irreducible representation of $G$ on a real vector space $W$. By Schur’s lemma, the centralizer of $\rho(G)$ in $\mathrm{End}(W)$ is an associative real division algebra, thus, by Frobenius’ theorem, isomorphic to one of (a) $\mathbb R$, (b) $\mathbb H$ or (c) $\mathbb C$.

The relation between real and complex representations is, respectively, that: (a) the complexification $\rho^c$ is irreducible (and we say that $\rho$ is absolutely irreducible) and $\rho^c = \pi$ is a representation of real type; (b) the complexification $\rho^c$ is reducible and $\rho^c=\pi\oplus\pi$ where $\pi$ is an irreducible representation of quaternionic type; (c) the complexification $\rho^c$ is reducible and $\rho^c=\pi\oplus\pi^*$ where $\pi$ is an irreducible representation of complex type and $\pi^*$ is not equivalent to $\pi$ (where $\pi^*$ denotes the induced representation on $V^*=\bar V$).

Also $\rho $ is a real form of $\pi$ in the first case ($\rho^c=\pi$), but $\rho$ is $\pi$ viewed as a real representation in the other two cases ($\rho=\pi^r$).

Regarding the adjoint representation of a compact simple Lie group, it is always absolutely irreducible. Simplicity of the group is equivalent to irreducibility of the representation. Even in the semisimple case, admiting and invariant complex structure would mean that the Lie group is a complex Lie group viewed as real, not possible due to the compactness.

Cartan's theory of real representations of semisimple Lie algebras is masterfully presented in modern form in

ESI Lectures in Mathematics and Physics

Arkady L. Onishchik (Yaroslavl State University, Russia)

Lectures on Real Semisimple Lie Algebras and Their Representations ISBN print 978-3-03719-002-9, ISBN online 978-3-03719-502-4 DOI 10.4171/002 February 2004, 95 pages, softcover, 17 x 24 cm. 24.00 Euro