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

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.