On the stable rationality of $X/G$
Let $G$ be a connected, reductive algeraic group whose Dynkin diagram contains no components of type $G_2,$ $F_4,$ $E_6,$ $E_7$ or $E_8.$ That is, all the components are of classical type. Suppose $X$ is an affine variety, and suppose $G$ acts freely on $X.$ Then $X$ and $X/G$ are stably birationally equivalent.
alg-geom↗