Highest weight categories and Macdonald polynomials

The goal of the talk is to explain an approach to the problem of categorification of Macdonald polynomials based on derived categories of modules over Lie algebra of currents. First, I recall the definition of Macdonald polynomials as the orthogonalisation of the linear monomial basis in the ring of symmetric functions with respect to the certain given pairing depending on two parameters. I will give the relationship of the latter pairing with the Grothendieck ring of the category of modules over the Lie algebra of currents. Second, I will explain the orthogonalisation procedure in derived categories and give a hint on categorification problem. Third, I discuss when it is possible to avoid the derived setting and get different applications for the category of modules.
In particular, we will prove the BGG reciprocity for the category of modules over the Lie algebra $g\otimes \mathbb{C}[x]$ with $g$-semisimple.