Reduction by stages for finite W-algebras

Naoki Genra, Thibault Juillard

Published 2022-12-12Version 1

Let $\mathfrak{g}$ be a semisimple 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 $S_f$. We prove that, given two nilpotent elements $f_1$ and $f_2$, with some assumptions, it is possible to perform a Hamiltonian reduction by stages: the slice $S_{f_2}$ is a Hamiltonian reduction of the slice $S_{f_1}$. We also state an analogue result in the setting of finite W-algebras, which are quantizations of Slodowy slices. As corollary in type A, we prove the conjecture of Morgan stating 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 about a possible analogue of our results in the context of affine W-algebras.