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

TL;DR

This paper establishes explicit geometric correspondences between quiver varieties, loop Grassmannians, and nilpotent cones in type A, leading to new compactifications and duality insights.

Contribution

It provides explicit formulas linking these geometries, embeds quiver varieties into Grassmannians, and introduces a geometric perspective on classical dualities.

Findings

Embedded quiver varieties into Beilinson-Drinfeld Grassmannians

Provided a compactification of Nakajima varieties

Decomposed affine Grassmannians into Nakajima varieties

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.