Comparison of quiver varieties, loop Grassmannians and nilpotent cones in type A

Abstract

In type A we find equivalences of geometries arising in three settings: Nakajima's (``framed'') quiver varieties, conjugacy classes of matrices and loop Grassmannians. These are now all given by explicit formulas. In particular, we embedd the framed quiver varieties into Beilinson-Drinfeld Grassmannians. This provides a compactification of Nakajima varieties and a decomposition of affine Grassmannians into Nakajima varieties. As an application we provide a geometric version of symmetric and skew $(GL(m),GL(n))$ dualities.

Some of the results of this paper have appeared in arXiv:0712.4160. The new results are the explicit isomorphism of quiver varieties and loop Grassmannians and a much better understanding of the relation of loop Grassmannians and nilpotent cones. The paper has also been extensively rewritten. Final published version