Olivier Schiffmann (ENS Paris)

Homological realization of Nakajima varieties and Weyl group actions   

In this joint work with I. Frenkel and M. Khovanov, we first give a description of Nakajima quiver varieties M0(v,w) associated to a quiver Q (with moment map parameter l = 0) in terms of (derived category of) representations of the zig-zag algebra A(Q) of Q. This (finite-dimensional) algebra is Koszul dual to the preprojective algebra; for instance, if Q is an affine Dynkin diagram correspondiong to the finite subgroup G Ì SL(2,C) then A(Q) is Morita equivalent to L C² ×G. Our result follows from the standard Koszul duality. However, in order to treat the case of arbitrary moment map parameter l it is necessary to introduce a deformation (of the derived category of representations) of A(Q) (which can be seen as a deformation of A(Q) as an A_¥-algebra), and to use a deformed version of the Koszul duality.

  In this realization we show that the action of the Weyl group on the collection of varieties M_l(v,w) is simply obtained as a tensor product with certain ``tilting modules''. These titling modules bear a strong ressemblance to the ones used by Khovanov, Seidel and Thomas to define the action of braid groups on certain derived categories.