Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

Abstract

We study the derived representation scheme DRep _n(A) parametrizing the n -dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep _n(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra \mathfrak {gl}_n^*(\bar{C}) defined over a Koszul dual coalgebra of A . This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras \mathfrak {gl}_n(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra \mathfrak g , we define the derived affine scheme DRep _{\mathfrak g}(\mathfrak a) parametrizing the representations (in \mathfrak g ) of a Lie algebra \mathfrak{a} ; we show that the homology of DRep_ {\mathfrak g}(\mathfrak a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra \mathfrak g^*(\bar{C}) , where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra \mathfrak a . We construct a canonical DG algebra map \Phi_{\mathfrak g}(\mathfrak a): \mathrm {DRep}_{\mathfrak g}(\mathfrak a)^G \to \mathrm {DRep}_{\mathfrak h}(\mathfrak a)^W , relating the G -invariant part of representation homology of a Lie algebra \mathfrak a in \mathfrak g to the W -invariant part of representation homology of \mathfrak a in a Cartan subalgebra of \mathfrak g . We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map. We conjecture that, for a two-dimensional abelian Lie algebra \mathfrak{a} , the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide some evidence for this conjecture, including proofs for \mathfrak {gl}_2 and \mathfrak {sl}_2 as well as for \mathfrak {gl}_n, \mathfrak {sl}_n, \mathfrak{so}_n and \mathfrak{sp}_{2n} in the inductive limit as n \to \infty . For any complex reductive Lie algebra \mathfrak g , we compute the Euler characteristic of DRep _{\mathfrak g}(\mathfrak a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep _{\mathfrak h}(\mathfrak a)^W . This yields an interesting combinatorial identity, which we prove for \mathfrak {gl}_n and \mathfrak{sl}_n (for all n ). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proposed in [Ha1, F] and proved in [FGT]. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.