TITLE: Properties of singularities that are preserved by categorical quotients
QUESTION [7 upvotes]: Let $G$ be a reductive group acting on an affine singular
variety $X$, and let $X/G$ be the categorical quotient. I know that if
$X$ has rational singularities, then so does $X/G$ (http://link.springer.com/article/10.1007%2FBF01405091). I am curious about
other properties of singularities that pass to the quotient. For
example, if $X$ is Gorenstein, must $X/G$ be Gorenstein? What if $X$
has canonical singularities? More generally, I'd be happy for a
reference where these questions are discussed.

REPLY [10 votes]: Neither Gorenstein nor canonical are preserved.  Already Gorenstein is destroyed for the action of $\mathbb{Z}/3\mathbb{Z}$ on $\mathbb{A}^2$ acting by $(x,y)\mapsto (\zeta\cdot x,\zeta\cdot y)$, where $\zeta$ is a primitive cube root of $1$.  Also, the whole point of the Reid -- Shepherd-Barron -- Tai criterion is to determine when a quotient singularity is canonical.  Typically it is not canonical.