Mathematische Annalen

, Volume 327, Issue 4, pp 671-721

First online:

Reflection functors for quiver varieties and Weyl group actions

  • Hiraku NakajimaAffiliated withDepartment of Mathematics, Kyoto University Email author 

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We introduce reflectionfunctors on quiver varieties. They are hyper-Kähler isometries between quiver varieties with different parameters, related by elements in the Weyl group. The definition is motivated by the origial reflection functor given by Bernstein-Gelfand-Ponomarev [1], but they behave much nicely. They are isomorphisms and satisfy the Weyl group relations. As an application, we define Weyl group representations of homology groups of quiver varieties. They are analogues of Slodowy’s construction of Springer representations of the Weyl group.