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

Representation Theory