Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras

Abstract

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