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Harish-Chandra modules and Galois orders revisited

João Schwarz

TL;DR

The paper develops a unifying framework for Harish-Chandra algebras and modules associated with a pair $(U,\Gamma)$, introducing transfer to spherical subalgebras and linking these ideas to Galois rings/orders, generalized Weyl algebras, and invariant theory. It proves Morita-type equivalences between Harish-Chandra categories for $U$ and its spherical counterpart $eUi$, and applies the theory to multiplicative invariants of differential operator rings on tori and fixed rings of GWAs under permutation and complex reflection groups, establishing principal Galois order structures and GK-type results. The authors introduce infinite rank GWAs, construct concrete Harish-Chandra modules, and demonstrate holonomicity and GK-dimension bounds in several invariant settings, thereby advancing the representation theory of noncommutative algebras arising in invariant theory and mathematical physics. The work connects classical Harish-Chandra theory with modern Galois order techniques, offering tools to study category $ ext{O}$ analogues, fixed rings, and spherical subalgebras in a broad noncommutative context. Its results have potential implications for understanding Noether-type problems, rational Cherednik algebras, and the structure of modules over invariant differential operator algebras.

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.

Harish-Chandra modules and Galois orders revisited

TL;DR

The paper develops a unifying framework for Harish-Chandra algebras and modules associated with a pair , introducing transfer to spherical subalgebras and linking these ideas to Galois rings/orders, generalized Weyl algebras, and invariant theory. It proves Morita-type equivalences between Harish-Chandra categories for and its spherical counterpart , and applies the theory to multiplicative invariants of differential operator rings on tori and fixed rings of GWAs under permutation and complex reflection groups, establishing principal Galois order structures and GK-type results. The authors introduce infinite rank GWAs, construct concrete Harish-Chandra modules, and demonstrate holonomicity and GK-dimension bounds in several invariant settings, thereby advancing the representation theory of noncommutative algebras arising in invariant theory and mathematical physics. The work connects classical Harish-Chandra theory with modern Galois order techniques, offering tools to study category analogues, fixed rings, and spherical subalgebras in a broad noncommutative context. Its results have potential implications for understanding Noether-type problems, rational Cherednik algebras, and the structure of modules over invariant differential operator algebras.

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.
Paper Structure (21 sections, 88 theorems, 31 equations)

This paper contains 21 sections, 88 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Harish-Chandra subalgebras and Harish-Chandra modules I
  4. A primer of invariant theory and Morita theory
  5. Galois rings and orders
  6. Generalized Weyl algebras
  7. Harish-Chandra subalgebras and Harish-Chandra modules II
  8. TGWAs and infinite rank generalized Weyl algebras
  9. Harish-Chandra subalgebras and modules in the literature
  10. Quasicommutative algebras and "spherical subalgebras"
  11. Galois rings and orders II
  12. General results
  13. Multiplicative invariant theory of the algebra of differential operators on the torus
  14. Fixed rings of GWA
  15. Permutation groups
  16. ...and 6 more sections

Key Result

Proposition 2.4

$\Gamma$ is quasicommutative if and only if every finite dimensional $\Gamma$-module $M$ is a generalized weight module. In case $\Gamma$ is affine Noetherian, this is the same as to require that $\mathsf m \mathsf n = \mathsf n \mathsf m = \mathsf n \cap \mathsf m$ for $\mathsf n, \mathsf m \in \op

Theorems & Definitions (188)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • proof
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • ...and 178 more