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 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.