We define a quantum version of Hamiltonian reduction by stages, producing a construction in type A for a quantum Hamiltonian reduction from the W-algebra $U(\mathfrak{ g},e_1)$ to an algebra conjecturally isomorphic to $U(\mathfrak{ g},e_2)$, whenever $e_2 \ge e_1$ in the dominance ordering. This isomorphism is shown to hold whenever $e_1$ is subregular, and in $\mathfrak{ sl}_n$ for all $n \le 4$. We next define embeddings of various categories $\mathcal{ O}$ for the W-algebras associated to $e_1$ and $e_2$, amongst them the embeddings $\mathcal{ O}(e_2,\mathfrak{ p}) \hookrightarrow \mathcal{ O}(e_1,\mathfrak{ p})$, where $\mathfrak{ p}$ is a parabolic subalgebra containing both $e_1$ and $e_2$ in its Levi subalgebra.
Categories: math.RT math.QA