Are quotient varieties local complete intersections? Let $G$ be a reductive group acting on the smooth affine variety $X$ such that the stabilizers are finite. Is it true that the quotient $X/G$ is a local complete intersection (LCI)? In particular, is the quotient of a smooth affine variety to the algebraic action of a finite group LCI? If no, is there any condition on the action that guarantees such a property? 
 A: As discussed in the comments, $X/G$ need not in general even be Gorenstein, let alone a local complete intersection. 
Actually, if the ground field has positive characteristic, $X/G$ may not even be Cohen-Macaulay (let alone Gorenstein, let alone LCI). For example, if $X = \mathbb{A}_k^4$, where $k = \overline{\mathbb{F}}_2$, and $G$ is $\mathbb{Z}/4$, with the action given by cyclically permuting the coordinate axes, then $X/G$ is not Cohen-Macaulay.
If the stabilizers all have order prime to the characteristic, then $X/G$ is Cohen-Macaulay, though, by the Hochster-Eagon theorem (Proposition 13 here). Without this condition, even Cohen-Macaulayness is a very delicate question, so from this point on I will assume this condition (orders of all stabilizers are coprime to the ground field characteristic) is met. For example, perhaps you are working over a characteristic zero field.
In this situation, an overly strong sufficient condition is that the stabilizers are generated by elements acting as pseudoreflections (i.e. elements $g$ in a stabilizer $G_x$, $x\in X$ whose fixed-point sets have codimension $1$ in the neighborhood of $x$). By the Chevalley-Shephard-Todd theorem, this forces $X/G$ to be smooth.
I am not aware of a more precise sufficient condition. However, it seems to me a necessary condition is that the stabilizers be generated by bireflections, i.e. elements $g$ in a stabilizer $G_x$, $x\in X$ whose fixed point sets have codimension $2$ in a neighborhood of $x$. See this paper by Kac and Watanabe, which proves this in the situation of a linear action by a finite group on affine space, but it seems to me the argument works in general.
(Stabilizers-generated-by-bireflections is not sufficient. For examples of finite linear groups generated by bireflections whose action on affine 3-space over $\mathbb{C}$ has a quotient that is not a complete intersection, see here.)
