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 ${\mathfrak{gl}}_n^{\ast }(\bar{C})$ 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 ${\mathfrak{g}}^{\ast }(\bar{C})$, 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.