Large annihilator category O for sl_{\infty}, o_{\infty}, sp_{\infty}

Abstract

We construct a new analogue of the BGG category $\mathcal{O}$ for the infinite-dimensional Lie algebras $\fg=\mathfrak{sl}(\infty),\mathfrak{o}(\infty), \mathfrak{sp}(\infty)$. A main difference with the categories studied in \cite{Nam} and \cite{CP} is that all objects of our category satisfy the large annihilator condition introduced in \cite{DPS}. Despite the fact that the splitting Borel subalgebras $\fb$ of $\fg$ are not conjugate, one can eliminate the dependency on the choice of $\fb$ and introduce a universal highest weight category $\mathcal{O}\mathcal{L}\mathcal{A}$ of $\fg$-modules, the letters $\mathcal{L}\mathcal{A}$ coming from "large annihilator". The subcategory of integrable objects in $\mathcal{O}\mathcal{L}\mathcal{A}$ is precisely the category $\mathbb T_{\fg}$ studied in \cite{DPS}. We investigate the structure of $\mathcal{O}\mathcal{L}\mathcal{A}$, and in particular compute the multiplicities of simple objects in standard objects and the multiplicities of standard objects in indecomposable injectives.