A family of highest weight categories of 𝔤𝔩 ( ∞ ) -modules

We study a family of categories of 𝔤𝔩 ( ∞ ) -modules depending on the choice of a Levy-type subalgebra. More precisely for a fixed choice of Cartan, Borel and Levi-type subalgebras 𝔥 , 𝔟 and 𝔩 with 𝔩 ≅ 𝔤𝔩 ( ∞ ) n for some n ∈ ℕ , we define 𝒪 LA 𝔩 𝔤𝔩 ( ∞ ) to be the category of 𝔥 -semisimple, 𝔫 -torsion modules that satisfy a certain “large annihilator condition” when seen as 𝔩 -modules. We see these categories as various analogues of the BGG category 𝒪 of 𝔤𝔩 ( n , ℂ ) . Our main result is that the category 𝒪 LA 𝔩 𝔤𝔩 ( ∞ ) 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 𝒪 LA 𝔩 𝔤𝔩 ( ∞ ) .