Quiver varieties and Beilinson-Drinfeld Grassmannians of type A

TL;DR

This paper constructs a geometric framework linking Nakajima's quiver varieties of type A with Beilinson-Drinfeld Grassmannians, offering new compactifications and decompositions that facilitate duality applications.

Contribution

It introduces a novel geometric construction of quiver varieties within affine Grassmannians, enabling compactification and decomposition, and applies this to duality theories.

Findings

Embedded quiver varieties into affine Grassmannians.

Provided a compactification of Nakajima's quiver varieties.

Achieved a geometric interpretation of duality between groups.

Abstract

We construct Nakajima's quiver varieties of type A in terms of conjugacy classes of matrices and (non-Slodowy's) transverse slices naturally arising from affine Grassmannians. In full generality quiver varieties are embedded into Beilinson-Drinfeld Grassmannians of type A. Our construction provides a compactification of Nakajima's quiver varieties and a decomposition of an affine Grassmannian into a disjoint union of quiver varieties. As an application we provide a geometric version of skew and symmetric $(GL(m), GL(n))$ duality.