Harish-Chandra modules and Galois orders revisited

Abstract: The main subject of study of this paper are general properties of HarishChandra algebras and modules with respect wito a pair of algebra and subalgebra, with special focus on the transfer properties to a "spherical subalgebra". We also discuss general properties of Galois rings and algebras, where the former discussion is specialized, and we obtain an important link between different approaches to it in the literature. Then we focus our study into finite multiplicative invariants on the ring of differential operators on the torus and fixed rings under the action of a finite group of algebra automorphisms of generalized Weyl algebras. We study freeness over the Harish-Chandra subalgebra and the Gelfand-Kirillov Conjecture for them. Our last section construction some concrete irreducible Harish-Chandra modules. This paper also introduces the notion of an infinite rank generalized Weyl algebra.