Let $G$ be a reductive group over $\mathbb{C}$, and $H=H_r\ltimes H_u$ be a subgroup of $G$. Here, $H_u$ is unipotent and $H_r$ is reductive.
Question: Is it true that when $G/H$ is open in its affine closure $\overline{H\backslash G}^{\text{aff}}:=\text{Spec}(\mathbb{C}[G])^H$, we have
$\overline{H\backslash G}^{\text{aff}} \simeq H_r\backslash\overline{(H_u\backslash G)}^{\text{aff}}$?
Or, is it true that the quotient stack $H_r\backslash\overline{(H_u\backslash G)}^{\text{aff}}$ is an affine scheme, under the above condition? (actually, I am interested in the case when $H\backslash G$ is a spherical $G$-variety, but I do not know if this condition helps.)
If that is not true, may I ask if there are some conditions to ensure the quotient $H_r\backslash\overline{(H_u\backslash G)}^{\text{aff}}$ is an affine scheme?
A further question, for an affine scheme $X$ and a reductive group, are there some conditions which can guarantee the quotient stack $G\backslash X$ is a scheme?
