Harish–Chandra modules over invariant subalgebras in a skew-group ring

We construct a new class of algebras resembling enveloping and generalizing orthogonal Gelfand-Zeitlin rational Galois studied by [EMV,FuZ,RZ,Har]. The are defined via geometric realization in terms sheaves functions invariant under an action finite group. A natural modules over these algebra can be constructed similar realization. In the special case local reflection group, shown to have explicit basis, results for from [EMV] [FuZ]. also family canonical simple Harish-Chandra give sufficient conditions simplicity some modules.