Harish-Chandra modules over Hopf Galois orders

TL;DR

This paper introduces $ ext{H}$-Galois $ ext{Λ}$-orders, a generalization of Galois orders involving Hopf algebras, and studies their simple Harish-Chandra modules, including existence, construction, and finiteness properties.

Contribution

It defines $ ext{H}$-Galois $ ext{Λ}$-orders and spherical Galois orders, extending the theory of Galois orders with new examples and structural results.

Findings

Existence of simple Harish-Chandra modules for each maximal ideal of finite codimension

Construction of canonical simple Harish-Chandra modules from characters of Λ

Finiteness of fibers of simple Harish-Chandra modules under certain conditions

Abstract

The theory of Galois orders was introduced by Futorny and Ovsienko. We introduce the notion of $\mathcal{H}$-Galois $\Lambda$-orders. These are certain noncommutative orders $F$ in a smash product of the fraction field of a noetherian integral domain $\Lambda$ by a Hopf algebra $\mathcal{H}$ (or, more generally, by a coideal subalgebra of a Hopf algebra). They are generalizations of Webster's principal flag orders. Examples include Cherednik algebras, as well as examples from Hopf Galois theory. We also define spherical Galois orders, which are the corresponding generalizations of principal Galois orders introduced by the author. The main results are (1) for every maximal ideal $\mathfrak{m}$ of $\Lambda$ of finite codimension, there exists a simple Harish-Chandra $F$-module in the fiber of $\mathfrak{m}$; (2) for every character of $\Lambda$ we construct a canonical simple Harish-Chandra module as a subquotient of the module of local distributions; (3) if a certain stabilizer coalgebra is finite-dimensional, then the corresponding fiber of simple Harish-Chandra modules is finite; (4) centralizers of symmetrizing idempotents are spherical Galois orders and every spherical Galois order appears that way.