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Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism. (English) Zbl 1406.16014

Summary: We study the derived representation scheme \(\mathrm{DRep}_n(A)\) parametrizing the \(n\)-dimensional representations of an associative algebra \(A\) over a field of characteristic zero. We show that the homology of \(\mathrm{DRep}_n(A) \) 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 the first author et al. [Adv. Math. 245, 625–689 (2013; Zbl 1291.14006)] and the first author and the fourth author [J. Reine Angew. Math. 715, 143–187 (2016; Zbl 1428.16006)], 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 \(\mathrm{DRep}_{\mathfrak{g}}(\mathfrak{a})\) parametrizing the representations (in \(\mathfrak{g}\)) of a Lie algebra \(\mathfrak{a}\); we show that the homology of \(\mathrm{DRep}_{\mathfrak{g}}(\mathfrak{a})\) 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 \(\mathrm{DRep}_{\mathfrak{g}}(\mathfrak{a})^G\) in terms of matrix integrals over \(G\) and compare it to the Euler characteristic of \(\mathrm{DRep}_{\mathfrak{h}}(\mathfrak{a})^W\). 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 [P. Hanlon, Invent. Math. 86, 131–159 (1986; Zbl 0604.17007); B. L. Feigin, Sel. Math. Sov. 7, No. 1, 49–62 (1988; Zbl 0657.17009)] and proved in [S. Fishel et al., Ann. Math. (2) 168, No. 1, 175–220 (2008; Zbl 1186.17010)]. 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.

MSC:

16G99 Representation theory of associative rings and algebras
17B56 Cohomology of Lie (super)algebras
17B63 Poisson algebras
18G55 Nonabelian homotopical algebra (MSC2010)
16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.)
16E45 Differential graded algebras and applications (associative algebraic aspects)
53D30 Symplectic structures of moduli spaces
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
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