Let $G\_0$ be a connected semisimple algebraic group defined over an *algebraically closed* field $k\_0$. Let $k\supset k\_0$ be a larger algebraically closed field.
We write $G=G\_0\times\_{k\_0} k$ for the base change of $G\_0$ from $k\_0$ to $k$.



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> **Question.** Let $H\subset G$ be a *spherical subgroup* of $G$ (defined over $k$). Is it true that $H$ is always conjugate to a (spherical) subgroup defined over $k\_0$?
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In other words, is it true that there exists an element $g\in G(k)$ and a (spherical) subgroup $H\_0\subset G\_0$ (defined over $k\_0$) such that 
$$ gHg^{-1}=H\_0\times\_{k\_0} k\ ?$$



Is any spherical subgroup conjugate to a subgroup defined over a smaller algebraically closed field?
The morphism $G/H\to\textrm{Spec}\,k$ is defined and flat over over some finitely generated $k\_0$-subalgebra of $k$. Thus, the question amounts to whether a flat family of homomogeneous spherical $k\_0$-varieties is generically trivial in the étale toplogy. In characteristic zero, this is true by Thm. 3.1 of Alexeev, V.; Brion, M.: Moduli of affine schemes with reductive group action. J. Algebraic Geom. 14 (2005), no. 1, 83–117. In positive characteristic, the problem is completely open, as far as I know.


