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

Abstract

We study a category of modules over gl(∞) analogous to category O. We fix adequate Cartan, Borel and Levi-type subalgebras h, b and l with l ∼ = gl(∞) n , and define O l LA gl(∞) to be the category of h-semisimple, n-nilpotent modules that satisfy a large annihilator condition as 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 O l LA gl(∞). We also give a decomposition of O l LA gl(∞) into irreducible blocks.