Degenerate affine Grassmannians and loop quivers

Abstract

We study the connection between the affine degenerate Grassmannians in type $A$, quiver Grassmannians for one vertex loop quivers, and affine Schubert varieties. We give an explicit description of the degenerate affine Grassmannian of type ${GL}_n$ and identify it with semi-infinite orbit closure of type $A_{2n-1}$. We show that principal quiver Grassmannians for the one vertex loop quiver provide finite-dimensional appro- ximations of the degenerate affine Grassmannian. Finally, we give an explicit description of the degenerate affine Grassmannian of type $A_1^{\left(1\right)}$, propose a conjectural description in the symplectic case, and discuss the generalization to the case of the affine degenerate flag varieties.