I am reading a paper and do not really understand the following.
The setting is $G$ is a compact connected Lie group and $\mathbb{R}^n$ a representation of $G$. Later on in the paper the authors mention the "complexified action of $G$ on $\mathbb{C}^n$". Could someone please explain to me what this means?
If we write the representation as $\pi:G\to GL(n;\mathbb{R})$, is the complexification referring to the Lie group (see https://en.wikipedia.org/wiki/Complexification_(Lie_group)), or is it referring to the ambient space $\mathbb{R}^n$ (so the image of the representation becomes $GL(n;\mathbb{C})$), or is it both?
What would be the proper definition of this complexification?
Comment: I do not think $G$ gets complexified because they use the Peter-Weyl theorem later on in the paper, which is a result for compact Lie groups, and the complexification of $G$ is not compact (see https://en.wikipedia.org/wiki/Complex_Lie_group, the final example).
Any help will be very appreciated! Thanks!
A summary of the discussion in the comments:
It's simply the composition of your $\pi:G\rightarrow \operatorname{GL}(n, \Bbb R)$ with the canonical embedding $\operatorname{GL}(n, \Bbb R)\hookrightarrow\operatorname{GL}(n, \Bbb C)$.
If you prefer group action language, then it's the formula $\rho(g)(v_1 + iv_2) = \pi(g)(v_1) + i\pi_g(v_2)$, but with $v_1, v_2\in \Bbb R^n$.
Note that the canonical $\Bbb R$-basis of $\Bbb R^n$ is also a canonical $\Bbb C$-basis of $\Bbb C^n$, thus under that basis, the matrix of $\rho(g)$ is exactly the same as the matrix of $\pi(g)$, just viewed as complex matrix. Hence it's tautologically in $\operatorname{GL}(n, \Bbb C)$.
In other words, it's simply multiplying a real matrix with complex vectors.