On the Category of Harish-Chandra Block Modules

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.