We study a family of categories of $\mathfrak{gl}(\infty )$-modules depending on the choice of a Levy-type subalgebra. More precisely for a fixed choice of Cartan, Borel and Levi-type subalgebras $\mathfrak{h}, \mathfrak{b}$ and $\mathfrak{l}$ with $\mathfrak{l} \cong \mathfrak{gl}(\infty )^n$ for some $n \in \mathbb{N}$, we define $\mathcal{O}_{\mathsf {LA}}^{\mathfrak{l}}{\mathfrak{gl}(\infty )}$ to be the category of $\mathfrak{h}$-semisimple, $\mathfrak{n}$-torsion modules that satisfy a certain “large annihilator condition” when seen as $\mathfrak{l}$-modules.
We see these categories as various analogues of the BGG category $\mathcal{O}$ of $\mathfrak{gl}(n,\mathbb{C})$. Our main result is that the category $\mathcal{O}_{\mathsf {LA}}^{\mathfrak{l}}{\mathfrak{gl}(\infty )}$ is a highest weight category in the sense of Cline, Parshall and Scott. We compute the simple multiplicities of standard objects and the standard multiplicities in injective objects explicitly, prove a version of BGG reciprocity, and present the irreducible blocks of $\mathcal{O}_{\mathsf {LA}}^{\mathfrak{l}}{\mathfrak{gl}(\infty )}$.
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Keywords: Infinite dimensional Lie algebras, representations, $\mathfrak{gl}(\infty )$, large annihilator condition
Zadunaisky, Pablo 1
CC-BY 4.0
Zadunaisky, Pablo. A family of highest weight categories of $\mathfrak{gl}(\infty )$-modules. Annals of Representation Theory, Volume 3 (2026) no. 2, pp. 299-334. doi: 10.5802/art.42
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