representations of real forms of simple lie algebra

Question

This might be a basic question but it's been a while since I looked at representation theory. If I have a finite dimensional matrix representation of a simple lie algebra (say su(4) for example) when is it possible to get a corresponding representation (of same dimension) for one of it's real forms (su(2,2) for example)? If it helps you can assume that the su(4) representation consists of real matrices or that it's an irreducible highest weight representation.

Answer 1

This is possible for complex representations and it is a version of Weyl's unitary trick. You can realize both $\mathfrak{su}(4)$ and $\mathfrak{su}(2,2)$ as real forms of the same complex simple Lie algebra (in this case $\mathfrak{sl}(4,\mathbb C)$). Given a representation of $\mathfrak{su}(4)$ on a complex vector space $V$, you can uniquely extend it to a complex representation of $\mathfrak{sl}(4,\mathbb C)$ on $V$ and then restrict the result to $\mathfrak{su}(2,2)$. The general version is that complex representations of a real semisimple Lie algebra are "the same as" complex representations of the compact real form of its complexification.