# Is there a Chevalley map for spherical varieties?

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,

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 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:

1. 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.

2. 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.

• Jason Starr had a few comments on that nCat blog post that you link, so I guessed which one you meant and edited accordingly. I hope that it was correct. May 5, 2022 at 14:12
• @LSpice Exactly, thank you very much :-) May 5, 2022 at 14:16
• In your Question 1. and Question 3., could you please clarify how $H$ is acting on $G/H$? Is it just the left regular action? If I understand correctly, the subgroup $H$ contains a unipotent radical of a Borel subgroup. In that case, the action of $H\times H$ on the open Bruhat cell is "almost transitive" in the sense that the multiplication map from $H\times T \times H$ to the Bruhat cell is surjective. So the quotient variety for the left $H$-action on the open orbit in $G/H$ is already a quotient torus of $T$. May 5, 2022 at 14:31
• I see now that $H$ need not contain a unipotent radical, e.g., a maximal torus in $\textbf{SL}_2$ has a dense orbit on $\textbf{SL}_2/B \cong \mathbb{P}^1$. May 5, 2022 at 14:39
• @JasonStarr Yes! It's the natural left action on cosets! Thanks for your comment, I'll think about it. May 5, 2022 at 14:47

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 cotangent 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.

• Isn't your first paragraph (the case where $H$ is a symmetric variety) the special case already mentioned in the question? May 6, 2022 at 14:48
• Yes, don‘t know how I missed that. Thanks for pointing that out. I‘ll fix that later. May 6, 2022 at 16:26