Schur-Weyl Duality for Heisenberg Cosets

Let $V$ be a simple vertex operator algebra containing a rank $n$ Heisenberg vertex algebra $H$ and let $C=\text{Com}(H,V)$ be the coset of $H$ in $V$. Assuming that the representation categories of interest are vertex tensor categories in the sense of Huang, Lepowsky and Zhang, a Schur-Weyl type duality for both simple and indecomposable but reducible modules is proven. Families of vertex algebra extensions of $C$ are found and every simple $C$-module is shown to be contained in at least one $V$-module. A corollary of this is that if $V$ is rational and $C_2$-cofinite and CFT-type, and $\text{Com}(C,V)$ is a rational lattice vertex operator algebra, then so is $C$. These results are illustrated with many examples and the $C_1$-cofiniteness of certain interesting classes of modules is established.