In the early 1980s, I. Macdonald discovered a number of highly non-trivial combinatorial identities related to a semisimple complex Lie algebra g. These identities were under intensive study for a decade until they were proved by I. Cherednik using representation theory of his double affine Hecke algebras. One of the key identities in the Macdonald list - the so-called constant term identity - has a natural homological interpretation: it formally follows from the fact that the Lie algebra cohomology of a truncated current Lie algebra over g is a free exterior algebra with generators of prescribed degree (depending on g). This last fact (called the strong Macdonald conjecture) was proposed by P. Hanlon and B. Feigin in the 80s and proved only recently by
S. Fishel, I. Grojnowski and C. Teleman (2008).
In this talk, I will discuss analogues (in fact, generalizations) of strong Macdonald conjectures arising from homology of derived representations schemes.