Meaning of $\overline{V}$ if $V$ is a complex representation of $G$?

Question

In Real and Complex Representations, the goal is to compute the endomorphism ring $End_{\mathbb{R}}(W)$ of a real representation $W$ of a compact Lie group $G$. To start, the author defines the realification of a vector space as follows:

Given a complex representation $V$ of $G$, we may regard $V$ as a real vector space (of twice the dimension) and treat it as a real representation of $G$, the realification $rV$ of $V$. Trivially, $r\overline{V}=rV$.

I have looked through the paper, and can't seem to find the definition of $\overline{V}$. Is this a standard construction, and if so, what is it? If it is not standard, does anyone have any good guesses on what it is?

Thanks

Answer 1

$\overline{V}$ is the complex conjugate of $V$: the action of $G$ is the same, but the action of $\mathbb{C}$ is the complex conjugate of the usual action. If $V$ is finite-dimensional, this representation happens to be isomorphic to the dual representation $V^{\ast}$, but not canonically. Its character is the complex conjugate of $V$'s.