Quantized Symplectic Actions and W -algebras

With a nilpotent element in a semisimple Lie algebra g one associates a finitely generated associative algebra W called a W -algebra of finite type. This algebra is obtained from the universal enveloping algebra U(g) by a certain Hamiltonian reduction. We observe that W is the invariant algebra for an action of a reductive group G with Lie algebra g on a quantized symplectic affine variety and use this observation to study W . Our results include an alternative definition of W , a relation between the sets of prime ideals of W and of the corresponding universal enveloping algebra, the existence of a one-dimensional representation of W in the case of classical g and the separation of elements of W by finite dimensional representations.