Quantized symplectic actions and $W$-algebras

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