# Quiver varieties and the affine Grassmannian

Recently I was watching a talk: http://media.cit.utexas.edu/math-grasp/Ben_Webster.html and at the end the lecturer gave a correspondence (I was having trouble with subscripts so changed the notation a bit):

$$\mathfrak{Q} (\lambda, \mu) \Leftrightarrow \mathfrak{M} (\lambda, \mu)$$

Between Quiver varieties and slices of affine Grassmannian. I would like to learn how this exactly works (like precise definition of both objects is a start!). If you have the time and the patience you can post a lengthy response here or you could just point me towards the right place to start reading on my own. Thank you!

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## 1 Answer

Sadly, as the speaker, I would also like to know how that correspondence works. Long story short, some collaborators and I came up with a conjecture that certain pairs of symplectic varieties (algebraic varieties with an algebraic symplectic form) were in some kind of duality with each other. There is still not a good definition of this duality at the moment, but it seems to coincide with certain kinds of S-duality in physics. Unfortunately, you're going to have to wait a few months to read our paper on this stuff, and even then it will be a lot more conjecture than theorem, especially on this point.

So, how do we justify this conjecture? By certain remarkable coincidences. In the case of the quiver varieties and affine Grassmannians, these coincidences are manifestations of fact that both varieties are geometric avatars of the representations of semi-simple Lie algebras (read Nakajima for quiver varieties and Mirkovic-Vilonen for affine Grassmannians).

The most important bit of the conjecture is that these varieties are actually geometric avatars for the representations of categorifications of Lie algebras in the sense of Rouquier or Khovanov-Lauda, by which I mean that the simple module categories over these categorifications should occur as categories of modules over quantizations of either the quiver variety, or the slice. Actually many other module categories, like those corresponding to tensor products should show up as well. This is not written down anywhere in the literature, but the type A special case can be cobbled together from some different sources, and will be in a forthcoming paper of myself, Tom Braden, Tony Licata and Nick Proudfoot. Which I should now get back to writing.

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Thanks for the answer and the pointers. Now here is what I am actually interested in (also why I asked the affine Grassmannian vs affine flag variety question earlier). What happens if one replaces the semi-simple Lie algebra with some arbitrary Kac-Moody algebra? Do nakajima's Quiver varieties still work in that context? Even if they you don't have the affine Grassmannian, because there is no preferred subset of simple roots like the one you have when you are dealing with an affine Kac-Moody algebra ... –  Najdorf Jan 24 '11 at 17:08
@Ben Have you or anyone else written up some notes on "categories of modules over quantizations of either the quiver variety, or the slice." This is a lot more elementary than the corresponding you are talking about, but I would like to understand this categorification. –  B. Bischof Jan 24 '11 at 20:03