# Harish-Chandra modules and Galois orders revisited

## TL;DR

This paper explores properties of Harish-Chandra modules and Galois algebras, focusing on transfer properties, invariants, and the construction of irreducible modules, linking various approaches and introducing infinite rank generalized Weyl algebras.

## Contribution

It provides new insights into transfer properties of Harish-Chandra modules, links different approaches to Galois rings, and introduces the concept of infinite rank generalized Weyl algebras.

## Key findings

- Established transfer properties to spherical subalgebras.
- Analyzed freeness over Harish-Chandra subalgebras.
- Constructed concrete irreducible Harish-Chandra modules.

## 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.