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 étale coverings of its smooth locus.