Quantum Hamiltonian reductions for W-algebras

In this paper, we establish a general criterion for good pairs, namely pairs consisting of a nilpotent orbit and an even good grading in a simple Lie algebra, which guarantees the existence of a quantum Hamiltonian reduction between associated affine W-algebras. In particular, we show that for type A, any two affine W-algebras associated with two adjacent nilpotent orbits are related by quantum Hamiltonian reductions in full generality.