Complexification of maximal compact subgroup of $GL(2,\mathbb{R})$.

Question

Given the Lie group $G=GL(2,\mathbb{R})$, we have that $K=O(2)$ is a maximal compact subgroup of $G$. I am trying to describe the complexification $K_\mathbb{C}$ of $K$. The Lie algebra $k_0$ of $K$ is $\mathfrak{o}(2) \cong \mathfrak{so}(2)$, and the complexification $k$ of $k_0$ is $$ \mathfrak{so}(2) \bigoplus i \mathfrak{so}(2) \cong \mathfrak{so}(2,\mathbb{C}). $$ Am I correct in thinking that this means $K_\mathbb{C}$ is then $SO(2,\mathbb{C})$? Thanks in advance!

Answer 1

Well, the complexification of a matrix group is just the group defined by the same equations, but taken over the complex numbers. So in this case it is $O(2,\mathbb{C})$. It does have the same Lie algebra as $SO(2,\mathbb{C})$, though.