Equivariant deformation theory for nilpotent slices in symplectic Lie algebras

TL;DR

This paper studies the deformation theory of nilpotent slices in symplectic Lie algebras, revealing their relation to type D slices and establishing equivariant universal Poisson deformations with non-commutative counterparts.

Contribution

It demonstrates the isomorphism of certain nilpotent Slodowy varieties to type D varieties and establishes the equivariant universal Poisson deformation in type C, including non-commutative aspects.

Findings

Nilpotent Slodowy varieties in type C are isomorphic to those in type D.

The universal Poisson deformation in type C is a slice in type D.

The Slodowy slice in type C is the $bZ_2$-equivariant universal Poisson deformation.

Abstract

The Slodowy slice is a flat Poisson deformation of its nilpotent part, and it was demonstrated by Lehn-Namikawa-Sorger that there is an interesting infinite family of nilpotent orbits in symplectic Lie algebras for which the slice is not the universal Poisson deformation of its nilpotent part. This family corresponds to slices to nilpotent orbits in symplectic Lie algebras whose Jordan normal form has two blocks. We show that the nilpotent Slodowy varieties associated to these orbits are isomorphic as Poisson $\mathbb{C}^\times$-varieties to nilpotent Slodowy varieties in type D. It follows that the universal Poisson deformation in type C is a slice in type D. When both Jordan blocks have odd size the underlying singularity is equipped with a $\mathbb{Z}_2$-symmetry coming from the type D realisation. We prove that the Slodowy slice in type C is the $\mathbb{Z}_2$-equivariant universal Poisson deformation of its nilpotent part. This result also has non-commutative counterpart, identifying the finite W-algebra as the universal equivariant quantization.