Complexification of homogeneous spaces

Question

Let $X=G/H$ be a homogeneous space, where $G$ is a Lie group and $H$ is a closed subgroup of $G$. Assume that $G/H$ has a complexification $\hat X$. Is $\hat X$ also a homogeneous space i.e $\hat G/\hat H$? What is the relation between $G$ and $\hat G$ and $H$ and $\hat H$?

Answer 1

There is a discussion about this in the paper On Complexifications of Differentiable Manifolds by R.S. Kulkarni. When $G$ is compact, the complexification of $G/H$ is $G_\mathbb{C}/H_\mathbb{C}$, and they are homotopic to each other.