Given a finite-dimensional semisimple Lie algebra $\frak g$, take an irreducible representation $V$, and let $ann(V)$ the annihilator of $V$ in $U(\mathfrak g)$. Such ideals are called primitive ideals. Then the variety defined by the associated graded ideal $gr(ann(V))$ of $gr U(\mathfrak{g})=S\mathfrak g$ is known to define the closure of a nilpotent orbit of $\frak g$. See the review by Borho in ICM 1986.

Could you suggest a reference where primitive ideals of the universal enveloping algebras of affine Lie algebras and their associated varieties are studied? What replaces the concept of the nilpotent orbit in that case?