Quantum Hamiltonian reduction of W-algebras and category O

Abstract

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.

arXiv comment22 pages. This is the journal version of my PhD thesis (arXiv:1502.07025 [math.RT]) with extended results