In the early 1980s, I. Macdonald discovered a number of highly non-trivial combinatorial identities related to a semisimple complex Lie algebra $\mathfrak{g}$. These identities were under intensive study for a decade until they were proved by I.Cherednik using his theory of double affine Hecke algebras. One of the key identities in Macdonald's 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 $\mathfrak{g}$ is a free exterior algebra with generators of prescribed degree (depending on $\mathfrak{g}$). This last fact (called the strong Macdonal conjecture) was proposed by P. Hanlon and B. Feigin in the 80s and proved only recently by S. Fishel, I. Grojnowski and C. Teleman.

I will present analogues (in fact, generalizations) of strong Macdonald conjectures arising from homology of derived representation schemes. In the first talk, I will explain what classical and derived representation schemes are and how they can be constructed. In the second talk, I will explain the relation to Lie algebra cohomology and focus on conjectures and examples.