Geometric Realizations of the Basic Representation of $\widehat\mathfrak{gl}_r$

TL;DR

This paper provides a geometric interpretation of the basic representation of the affine Lie algebra ffgl_r, connecting algebraic realizations with geometric operators on Nakajima quiver varieties' cohomology.

Contribution

It introduces a novel geometric perspective on the basic representation of ffgl_r, linking algebraic vertex operator constructions to geometric operators on quiver varieties.

Findings

Geometric operators correspond to algebraic realizations of the basic representation.

Explicit description of the geometric interpretation in terms of Nakajima quiver varieties.

Bridges algebraic and geometric frameworks for affine Lie algebra representations.

Abstract

The realizations of the basic representation of $\widehat\mathfrak{gl}_r$ are known to be parametrized by partitions of r and have an explicit description in terms of vertex operators on the bosonic/fermionic Fock space. In this paper, we give a geometric interpretation of these realizations in terms of geometric operators acting on the equivariant cohomology of certain Nakajima quiver varieties.