The speaker will present some results and conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products (in particular, cyclotomic rational Cherednik algebras) 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. Some of these conjectures were recently proved in the works of Shan-Vasserot and Gordon-Losev.