Highest weight categories of $\mathfrak{gl}(\infty)$-modules

TL;DR

This paper introduces a new highest weight category of modules over the infinite-dimensional Lie algebra rak{gl}(\u221e), characterizes its structure, and establishes key properties such as BGG reciprocity and block decomposition.

Contribution

It defines a novel category al_{ ext{LA}}^{rak l} rak{gl}() modules, proves it is a highest weight category, and computes multiplicities and block decompositions.

Findings

Proves al_{ ext{LA}}^{rak l} rak{gl}() is a highest weight category.

Computes simple and standard multiplicities within the category.

Establishes a form of BGG reciprocity and describes the block decomposition.

Abstract

We study a category of modules over $\mathfrak{gl}(\infty)$ analogous to category $\mathcal O$. We fix adequate Cartan, Borel and Levi-type subalgebras $\mathfrak h, \mathfrak b$ and $\mathfrak l$ with $\mathfrak l \cong \mathfrak{gl}(\infty)^n$, and define $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$ to be the category of $\mathfrak h$-semisimple, $\mathfrak n$-nilpotent modules that satisfy a large annihilator condition as $\mathfrak l$-modules. Our main result is that these are highest weight categories in the sense of Cline, Parshall and Scott. We compute the simple multiplicities of standard objects and the standard multiplicities in injective objects, and show that a form of BGG reciprocity holds in $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$. We also give a decomposition of $\mathcal O_{\mathsf{LA}}^{\mathfrak l}{\mathfrak{gl}(\infty)}$ into irreducible blocks.