Deformations of symplectic singularities and Orbit method for semisimple Lie algebras

TL;DR

This paper classifies filtered quantizations of symplectic singularities, demonstrating their relation to symplectic reflection algebras, and establishes an Orbit method for semisimple Lie algebras linking orbits to primitive ideals.

Contribution

It provides a classification of filtered quantizations of symplectic singularities and introduces an Orbit method connecting adjoint orbits to primitive ideals in Lie algebras.

Findings

Filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras.

A natural injective map from adjoint orbits to primitive ideals for classical Lie algebras.

Classification results for filtered Poisson deformations by Namikawa.

Abstract

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of adjoint orbits in a semisimple Lie algebra to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras.