Let X be a Poisson variety with a Hamiltonian G-action and H be a normal subgroup of G. Then X//G is obtained by a (Hamiltonian) reduction of X//H by the induced G/H-action under suitable assumptions, called reduction by stages. We apply for the Slodowy slices and show that the Slodowy slice associated to (𝔤,𝒪) is obtained by a reduction of the Slodowy slice associated to (𝔤,𝒪’) for a simple Lie algebra 𝔤 and nilpotent orbits 𝒪, 𝒪’ such that 𝒪 > 𝒪’ with some conditions. The quantum cases imply that the finite/affine W-algebras associated to (𝔤,𝒪) are obtained by W-algebras associated to (𝔤,𝒪’), which proves a conjecture of Morgan in finite cases and gives a conjectural generalization of results of Madsen and Ragoucy in affine cases.
This is a joint work with Thibault Juillard.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop Geometric Representation Theory and W-algebras.
