Reduction by stages for finite W-algebras

Let \(\mathfrak {g}\) $\mathfrak{g}$ be a simple Lie algebra: its dual space \(\mathfrak {g}^*\) ${\mathfrak{g}}^{\ast }$ is a Poisson variety. It is well known that for each nilpotent element f in \(\mathfrak {g}\) $\mathfrak{g}$ , it is possible to construct a new Poisson structure by Hamiltonian reduction which is isomorphic to some subvariety of \(\mathfrak {g}^*\) ${\mathfrak{g}}^{\ast }$ , the Slodowy slice \(S_f\) $S_f$ . Given two nilpotent elements \(f_1\) $f_1$ and \(f_2\) $f_2$ with some compatibility assumptions, we prove Hamiltonian reduction by stages: the slice \(S_{f_2}\) $S_{f_2}$ is the Hamiltonian reduction of the slice \(S_{f_1}\) $S_{f_1}$ . 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 Ph.D. 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.