  Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and let $W$ be a
finite subgroup of $\mathbb{C}$-algebra automorphisms of the enveloping algebra
$U(\mathfrak{g})$. We show that the derived category of
$U(\mathfrak{g})^W$-modules determines isomorphism classes of both
$\mathfrak{g}$ and $W.$ Our proofs are based on the geometry of the Zassenhaus
variety of the reduction modulo $p\gg 0$ of $\mathfrak{g}.$ Specifically, we
use non-existence of certain \'etale coverings of its smooth locus
