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.