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Geometric realizations of the basic representation of \(\widehat{\mathfrak{gl}}_r\). (English) Zbl 1395.17050

Summary: 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.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
05E10 Combinatorial aspects of representation theory
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