Selecta Mathematica

, Volume 22, Issue 1, pp 341–387 | Cite as

Geometric realizations of the basic representation of \(\widehat{\mathfrak {gl}}_r\)

  • Joel LemayEmail author


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.


Affine Lie algebra Basic representation Equivariant cohomology Fock space Geometric invariant quotient Geometric operator Quiver variety 

Mathematics Subject Classification

17B65 14F43 05E10 



I would like to thank Alistair Savage for his invaluable help throughout the writing of this paper. I would also like to thank Yuly Billig for his helpful advice.


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.University of OttawaOttawaCanada

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