On the Category of Harish-Chandra Block Modules

TL;DR

This paper generalizes the structure theory of Harish-Chandra modules by removing the quasicommutativity assumption on the subalgebra, introducing an equivalence relation on maximal ideals, and describing modules via block decompositions and category equivalences.

Contribution

It extends previous results by defining Harish-Chandra block modules with respect to an equivalence relation, providing a decomposition of module categories, and establishing category equivalences under broader conditions.

Findings

Decomposition of the category of Harish-Chandra block modules.

Equivalence between Harish-Chandra block modules and profinite modules over a category.

Conditions for finiteness of simple Harish-Chandra block modules with a given support.

Abstract

If $\Gamma$ is a subalgebra of $A$, then an $A$-module is called a Harish-Chandra module if it is the direct sum of its generalized weight spaces with respect to $\Gamma$. In 1994, Drozd, Futorny, and Ovsienko defined a generalization of a central subalgebra called a Harish-Chandra subalgebra and showed that when $\Gamma$ is a Harish-Chandra subalgebra of $A$ the structure of Harish-Chandra $A$-modules can be described using information about the relationship between $A$ and the cofinite maximal ideals of $\Gamma$. We extend these results by dropping the assumption that $\Gamma$ is quasicommutative. We facilitate this by introducing an equivalence relation $\sim$ on the set $\mathrm{cfs}(\Gamma)$ of cofinite maximal ideals of $\Gamma$. We define Harish-Chandra block modules with respect to $\sim$ to be $A$-modules that are the direct sum of so called block spaces corresponding to the equivalence classes $\mathrm{cfs}(\Gamma)/{\sim}$. If $\Gamma$ is a Harish-Chandra block subalgebra of $A$ with respect to $\sim$, then the structure of Harish-Chandra block modules can be described based on the relationship between $A$ and $\mathrm{cfs}(\Gamma)/{\sim}$. In particular, we give a decomposition of the category of Harish-Chandra block modules and the collection of isomorphism classes of irreducible Harish-Chandra block modules. Furthermore, we define a category $\mathcal{A}$ on $\mathrm{cfs}(\Gamma)/{\sim}$, and show the category of profinite $\mathcal{A}$-modules is equivalent to the category of Harish-Chandra block modules. Taking $\Gamma$ to be noetherian and quasicommutative, and $\sim$ to be the equality relation, we recover (in fact, a slight refinement of) results from Drozd, Futorny, and Ovsienko. Lastly, we provide a sufficient condition for when there are a finite number of isoclasses of simple Harish-Chandra block modules with a given support.