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.](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0350.0359.ocr.pdf)


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? 



Primitive ideals of the universal enveloping algebras of affine Lie algebras
I'm not aware of any reasonable analogue of the nilpotent variety (or related theory of associated varieties) in this infinite dimensional setting. But you may get some inspiration from the work of Joseph, including his book (which has many citations listed on MathSciNet):


Anthony Joseph, *Quantum groups and their primitive ideals*. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 29. Springer-Verlag, Berlin, 1995.


Apart from asking Joseph himself to comment, you might take a look at one of his reviews:


MR2145731 Martino, Maurizio (4-GLAS) The associated variety of a Poisson prime ideal. J. London Math. Soc. (2) 72 (2005), no. 1, 110–120.


