Reduction by stages for finite W-algebras

Let $\mathfrak{g}$ be a simple Lie algebra: its dual space $\mathfrak{g}^*$ is a Poisson variety. It is well known that for each nilpotent element $f$ in $\mathfrak{g}$, it is possible to construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of $\mathfrak{g}^*$, the Slodowy slice $S_f$. Given two nilpotent elements $f_1$ and $f_2$ with some compatibility assumptions, we prove Hamiltonian reduction by stages: the slice $S_{f_2}$ is the Hamiltonian reduction of the slice $S_{f_1}$. We also state an analogous result in the setting of finite W-algebras, which are quantizations of Slodowy slices. These results were conjectured by Morgan in his PhD thesis. As corollary in type A, we prove that any hook-type W-algebra can be obtained as Hamiltonian reduction from any other hook-type one. As an application, we establish a generalization of the Skryabin equivalence. Finally, we make some conjectures in the context of affine W-algebras.