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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?

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  • $\begingroup$ Nice question. Naive answer - is clear - take Casimirs and write Casimirs = 0. But as far as I understand you mean that since everything is infinite-dimensional - such things does not make big sense... People always speak about analogies between nilpotent cone in finite-dim g and "global nilpotent cone" for Hitchin system. Which in certain sense can be understood as det(l-L(z)) = 0, but now L(z) is NOT "full current" but contains only positive powers of z... But I guess you know this better than me. $\endgroup$ – Alexander Chervov Dec 27 '11 at 4:24
  • $\begingroup$ So probably the part of yours question can be reformulated "in what sense nilpotent cone in fin-dim g is analogous to the "global nilpotent cone ? " $\endgroup$ – Alexander Chervov Dec 27 '11 at 5:02
  • $\begingroup$ Actually nilpotent cone is the "biggest" nilpotent orbit. So may be you interested in those of smaller dimension ? So question is what would be "global" analogs of the smaller dimensional nilpotent orbits ? Assuming we already believe than biggest- global nilpotent cone... $\endgroup$ – Alexander Chervov Dec 27 '11 at 5:57
  • $\begingroup$ Ah, thank you for pointing out the words "global nilpotent cone". I'm just learning Hitchin systems etc. and trying to formulate my physics questions in terms of something sensible to mathematicians, to see if what I want to know is already known... $\endgroup$ – Yuji Tachikawa Dec 27 '11 at 9:09
  • $\begingroup$ What orbit are You interested in ? Full dimensional or smaller ? About Hitchin - it can be defined in many ways, one which is quite simple - take GL(z), lie algebra acts as vector left (or right) invariant fields, universal enveloping as differentical operators on (GL(z)). The center of U(gl(z)) is "universal Hitchin" hamiltonians. Meaning that Bun = G_{in}\GL(z)/G_{out}. And bi-invariant dif.opers can be pushed down on Bun - they provide quantum Hitchin hamiltonians. $\endgroup$ – Alexander Chervov Dec 27 '11 at 10:13
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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.

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