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 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.