Harish-Chandra modules revisited

Abstract The main subject of study of this paper is general properties of Harish-Chandra algebras and modules, with special focus on the transfer of properties to a “spherical subalgebra”. We discuss in some detail the ring theoretical property of being a quasicommutative algebra, showing it to be Morita invariant, among other things. Then we obtain the main result of our paper: if an algebra $U$ and its spherical subalgebra $eUe$, where $e$ is a suitable idempotent, are Morita equivalent, and their Harish-Chandra subalgebras are compatible in a suitable way, we obtain an equivalence for their categories of Harish-Chandra modules. Along the way we realize an important category of modules for rational Cherednik algebras, the category $\widehat{\mathcal O}_{\mathfrak{c}}$, as a Harish-Chandra category, and as an application we obtain the equivalence of the categories of Harish-Chandra modules for two important algebras in Coulomb branch theory. Then we prove some general theorems about Galois rings and orders and later specialize our discussion to invariants of rings of differential operators on the torus and the Weyl algebra. In the last section we show that category $\mathcal{O}$ for fixed rings of the Weyl algebras in the case of complex reflection groups belongs to Harish-Chandra module categories with respect to two very different Harish-Chandra subalgebras. To Pan for all the smiles that she gives me.