Symplectic Reflection Algebras and Affine Lie Algebras

These are the notes of my talk at the conference “Double affine Hecke algebras and algebraic geometry” (MIT, May 18, 2010). The goal of this talk is to discuss some results and conjectures suggesting that the representation theory of symplectic reflection algebras for wreath products categorifies certain structures in the representation theory for affine Lie algebras. 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, and took a much more definite shape after my conversations with I. Losev. At the moment, these notes are in a very preliminary form. Many proofs are just sketched or have missing details. The notes may (and probably do) contain errors, and should be viewed as no more than a basis for further thinking about these things. The plan is that these notes will be improved as I continue thinking about these questions, and as I receive feedback. So any feedback is very welcome! Acknowledgements. I would never have thought of these things without the encouragement and the vision of R. Bezrukavnikov and A. Okounkov. Also, I am very grateful to R. Bezrukavnikov and I. Losev for numerous discussions, without which I would have gotten nowhere. This work was supported by the NSF grant DMS-0854764.