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](https://eudml.org/doc/142882), 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,


1. 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](https://golem.ph.utexas.edu/category/2008/12/the_toric_variety_associated_t.html#c021093) in the blog post [The Toric Variety Associated to the Weyl Chambers](https://golem.ph.utexas.edu/category/2008/12/the_toric_variety_associated_t.html), 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:


2. 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.
3. 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.



Is there a Chevalley map for spherical varieties?
**Edit:** The answer to question 1 is yes if $G/H$ is a symmetric variety as the OP pointed out.


For arbitrary spherical varieties the answer is no in general. If my memory serves me right, the spherical variety $Sp(4,\mathbb C)/(\mathbb C^\*\times SL(2,\mathbb C))$ is a counterexample. As far as I know, the $H$-orbit structure of $G/H$ is still unknown in full generality.


**Added:** The actual theorem of Chevalley is not a statement about conjugacy classes on the group $G$ but rather on its Lie algebra $\mathfrak g$: there is an isomorphism $\mathbb C[\mathfrak g]^G\overset\sim\to\mathbb C[\mathfrak t]^W$ where $\mathfrak t\subseteq\mathfrak g$ is a Cartan subalgebra. This theorem has been extended by Kostant-Rallis to symmetric spaces in the form $\mathbb C[\mathfrak p]^H\overset\sim\to\mathbb C[\mathfrak a]^{W\_X}$ where $X=G/H$ is a symmetric variety. Here $\mathfrak p$ is the tankent space of $X$ in $eH$.


The point is that Kostant-Rallis does generalize to arbitrary spherical varieties $X=G/H$ if one replaces the tangent space by the **co**tangent space $\mathfrak h^\perp=(\mathfrak g/\mathfrak h)^\*\subseteq\mathfrak g^\*$ of $X$ in $eH$. Of course this makes only a difference if $H$ is not reductive but then it is essential.


**Theorem:** There is a subspace $\mathfrak a^\*\subseteq\mathfrak h^\perp$ and an action of the little Weyl group $W\_X$ of $X$ on $\mathfrak a^\*$ such that $\mathbb C[\mathfrak h^\perp]^H\overset\sim\to\mathbb C[\mathfrak a]^{W\_X}$.


The proof of this theorem is much more involved than Kostant-Rallis. First of all, the little Weyl group $W\_X$ is not easy to define since it is in general not a subquotient of $H$. It was discovered by Brion while studying compactifications of $X$. Second, the subspace $\mathfrak a^\*$ is not at all canonical (even up to conjugation by $H$). Finally, the action of $W\_X$ on $\mathfrak a^\*$ is not induced by elements of $H$. It is rather a monodromy action.


References: My Inventiones papers in vols. 99 and 116. In the first paper the theorem is proved where $W\_X$ is some monodromy group. The second paper shows that $W\_X$ coincides with the little Weyl group defined by Brion in J. Algebra vol. 134.


Coming back to the original question: The change from the tangent space to the cotangent space prevented so far all attempts to get a global Chevalley theorem. This change makes only a difference if $H$ is not reductive but that affects even the reductive case since most methods involve non-reductive subgroups in an essential manner.


