  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.
