Let $G$ be a compact Lie group and Let $G\_\mathbb{C}$ be its complexification. Let $T$ be a maximal torus of $G$ and let $X$ be the quotient $G/T$.


Consider $H$ to be a Lie subgroup of $G$ and denote by $H\_\mathbb{C}$ its complexification. Let $x \in X$, denote by $H\_x$ and ${(H\_{\mathbb{C}})}\_x$ the stabilizers of $x$ in $H$ and in $H\_\mathbb{C}$ respectively.


Question : is $H/H\_x$ isomorphic to $H\_\mathbb{C}/{(H\_{\mathbb{C}})}\_x$?



Generalization of $G/T \simeq G_\mathbb{C}/B$
No.


Take $G = SU(2)$, $G\_{\mathbb C} = SL\_2(\mathbb C)$, $G/B$ the complex projective line alias the sphere, $H$ the diagonal $U(1)$, $x$ any point other than the two fixed points of $H$, so that the orbit $H/H\_x$ is a circular slice of the sphere, and $H\_{\mathbb C}$ the diagonal $GL\_1(\mathbb C)$, so that the orbit $H\_{\mathbb C}/ (H\_{\mathbb C})\_x$ is the whole sphere minus two points.


The circle is not isomorphic to the whole sphere minus two points.


