# Pseudoreflection groups in affine varieties

Suppose $$\mathsf{k}$$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result:

(C-S-T): Let $$G$$ be a finite subgroup of $$GL(V)$$, where $$V$$ is a finite dimensional vector space. Then the quotient variety $$V/G$$ is smooth if and only if $$G$$ is a group generated by pseudoreflections.

My first question is the following:

(Question 1) Let $$G$$ be a finite group of automorphisms of an irreducible affine smooth variety $$X$$. When is the quotient variety $$X/G$$ smooth?

Of course this holds if the action of $$G$$ is free, but I'm looking for a condition that is similar in nature to that of C-S-T Theorem. I've heard something among these lines: $$G$$ is generated by elements $$g$$ such that $$X^g = \{ x \in X| g.x=x \}$$ is a divisor, etc; but I could not find an exact statement or reference.

I would greatly appreciate an answer, either in form of an adequate reference, or a proof of an exact statement.

My second question is related to the first one as follows. We have a result by Kantor (1977) and Levasseur (1981) that is:

(K-L): Let $$V$$ be a finite dimensional vector space, $$G$$ a finite subgroup of $$GL(V)$$. Then the quotient map $$V \mapsto V/G$$ induces an isomorphism $$\mathcal{D}(V)^G \simeq \mathcal{D}(V/G)$$ if and only if $$G$$ contains no pseudoreflections. (cf. Levasseur, Thm 5, MR633520)

Here of course $$\mathcal{D}(.)$$ is the ring of differential operators, and the action of $$G$$ on this ring is by conjugation.

(Question 2) Let $$X$$ be an irreducible affine smooth variety, and $$G$$ a finite group of automorphisms of it. The quotient map $$X \mapsto X/G$$ induces an injective map $$\mathcal{D}(X)^G \rightarrow \mathcal{D}(X/G)$$. When is this map an isomorphism?

Again there are sufficient conditions for this to hold, such as the action of $$G$$ being free (cf. Cannings, Holland, 3.7, MR1305263, 1994). I wonder if there is a condition for this to hold analogue to the one considered by Kantor and Levasseur, i.e., $$G$$ having no "pseudoreflections", in the sense I suggested in Question (1).

• Should $X$ be affine in Question 1 (judging by title looks like that's what you intended)? – Sam Hopkins Jun 29 at 18:24
• Yes, it is, thank you for pointing out. I will edit it. – jg1896 Jun 29 at 23:14
• I do not know about the general case, but over the complex numbers, $X/G$ is smooth if and only if the action on $T_xX$ of the stabilizer $G_x$ of each $x\in X$ satisfies the CST condition. The key is Cartan's theorem on local holomorphic linearization, see the reference I gave here. – Moishe Kohan Jul 15 at 16:34
• This is extremely helpful! I will look Cartan's work. Thank you very much @MoisheKohan – jg1896 Jul 15 at 16:39