  The goal of this paper is to present some results and (more importantly)
state a number of conjectures suggesting that the representation theory of
symplectic reflection algebras for wreath products categorifies certain
structures in the representation theory of affine Lie algebras (namely,
decompositions of the restriction of the basic representation to finite
dimensional and affine subalgebras). These conjectures arose from the insight
due to R. Bezrukavnikov and A. Okounkov on the link between quantum connections
for Hilbert schemes of resolutions of Kleinian singularities and
representations of symplectic reflection algebras.
