If $G$ is a reductive group, $T$ a maximal torus and $W$ its Weyl group the Chevalley restriction theorem (in its "multiplicative" version) gives an isomorphism between the GIT quotient of $G$ by the conjugation action on itself and the quotient $T/W$.

This result has several generalisations. In particular, in Orbits, Invariants, and Representations Associated to Involutions of Reductive Groups, Richardson proved a similar theorem: $X//G^{\theta} \cong A/W_{\theta}$ for $X=G/G^{\theta}$ a symmetric variety, $A$ a maximal $\theta$-anisotropic torus and $W_{\theta}$ the so called "little Weyl group".

A well-known generalisation of symmetric varieties are spherical varieties. I was wondering if a similar result exists in this situation, namely, if $G/H$ is a spherical homogeneous space,

- does there exist an isomorphism between the GIT quotient of $X$ by $H$ and the quotient of the torus $A$ associated to the spherical variety and the corresponding Weyl group?

Related to the theory of spherical varieties is the theory of spherical embeddings, and in particular, the wonderful compactification: a projective variety compactifying a spherical variety $X$.

In a comment in the blog post The Toric Variety Associated to the Weyl Chambers, Jason Starr mentions some "extension" of the Chevalley map, from the wonderful compactification of $G$ to the toric $T$-variety defined by the fundamental Weyl chamber. Regarding this I have two questions:

Is there a reference for this last fact? That is, a reference for the fact that there is an isomorphism between the GIT quotient of the wonderful compactification by $G$ and the quotient of that toric variety by the Weyl group.

Can it be generalized to any spherical variety? The result I have in mind is the existence of an isomorphism between the GIT quotient of the wonderful compactification of the spherical homogeneous space $G/H$ by the spherical subgroup $H$ and the toric variety defined by the associated torus and the corresponding dominant cocharacters.

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