In Bump's Automorphic Forms and Representations, p. 200, he gives the definition of a $(\mathfrak{g},K)$-module for $\mathfrak{g}=\mathfrak{gl}_n\mathbb{R}$ and $K=O(n)$ being the maximal compact subgroup of $\text{GL}_n(\mathbb{R})$.
He then says this definition can be generalized for all reductive Lie groups. Does he mean all real reductive Lie groups?
So let's say I have the reductive Lie group $G = \text{GL}_n(\mathbb{C})$ with $\mathfrak{g} =\mathfrak{gl}_n(\mathbb{C})$ and $K= U(n)$, and I want to define a $(\mathfrak{g},K)$-module with these parameters, do I have to first view $\mathfrak{g}$ as a real Lie algebra via $\mathbb{R}\hookrightarrow\mathbb{C}$, and $K$ as a real Lie group?
I've been told that $U(n)$ is not the complex points of any affine algebraic group, so does that mean when I say $K=U(n)$ is the maximal compact subgroup of $G=\text{GL}_n(\mathbb{C})$, I am already treating $G$ as a real Lie group?
My knowledge of Lie groups is limited, so I apologize in advance if this question is ill-defined.