Harish-Chandra modules over invariant subalgebras in a skew-group ring (Q2121818)

The authors construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Tseitlin algebras and rational Galois algebras. The algebras are defined using a geometric realization in terms of sheaves of functions invariant under an action of a finite group. That is, let \(G\) and \(V\) be complex-analytic or linear algebraic Lie groups such that \(G\) acts on \(V\). Letting \(\gimel\) be a \(G\)-invariant subgroup in \(V\) and \(v\in V\), consider the \(G\)-invariant subset \((G \ltimes \gimel) \cdot v\). Let \(\mathfrak{E}^G\) be the subring of \(G\)-invariant sections of the direct sum \(\mathfrak{E}\) of certain fibration of vector spaces over \(V\). Then \(\mathfrak E^G|_{(G \ltimes \gimel) \cdot v/G}\) is defined. Let \(\mathcal{S}:= \mathcal{S}(\gimel)\), the vector space of global meromorphic or rational sections of the trivial vector bundle \(V\times \mathbb{C} \gimel\rightarrow V\). Take an element \(A\) in \(\mathcal S\) preserving the vector space \(\mathfrak O^{G}|_{(G \ltimes \gimel) \cdot v/G}\), where \(\mathfrak O\) is the algebra \(\bigoplus_{x\in V} \mathcal O_x\) of germs of holomorphic functions. Such \(A\) has the form \(A|_{\mathfrak O^{G}|_{(G \ltimes \gimel) \cdot v/G}}= \sum_i \sum\limits_{h\in G} h \cdot (f_i \phi_{\xi_i})\) on \(\mathfrak O^{G}|_{(G \ltimes \gimel) \cdot v/G}\) (here, \(A\) may be meromorphic). Letting \(G\) to be a finite group and letting \(A\) to send \(\mathfrak O^{G}|_{(G \ltimes \gimel) \cdot v/G}\) to itself, the action of \(A\) on \(\mathfrak O^{G}|_{(G \ltimes \gimel) \cdot v/G}\) induces an action of \(A\) on \(\mathfrak E^G|_{(G \ltimes \gimel) \cdot v/G}\) (Theorem 4, page 436). A natural class of modules over these algebras can be constructed using a similar geometric construction (Lemma 5, page 437, and Theorem 10, page 439). In the special case of a local reflection group (Section 4.1, page 441), these modules are shown to have an explicit basis (Theorem 19, page 444), generalizing similar results for orthogonal Gelfand-Zeitlin algebras and for rational Galois algebras (Section 4.2, page 442). The authors also construct a family of canonical simple Harish-Chandra modules and give sufficient conditions for simplicity of some modules (Proposition 28, page 451): assuming that \(H^0(V, \mathcal O^G)\) separates orbits in \((G\ltimes \gimel) \cdot v\) and that the \(H^0(V, \mathcal O^G)\)-module \(\mathbb E^G_{\bar w}\) is generated by \(\tilde 1_{\bar w}\), \(N_{\bar w}\) has a unique maximal submodule.