If the fixed rings U(g)^W and U(g')^W' of enveloping algebras are derived equivalent, then g is isomorphic to g' and W is isomorphic to W'.
Montgomery, Fixed rings of finite automorphism groups of associative rings , (1980) Lecture Notes in Math
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Derived invariants of the fixed ring of enveloping algebras of semisimple Lie algebras
If the fixed rings U(g)^W and U(g')^W' of enveloping algebras are derived equivalent, then g is isomorphic to g' and W is isomorphic to W'.