Deformations of symplectic singularities and orbit method for semisimple Lie algebras

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 coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.