Representation Homology, Lie Algebra Cohomology and Derived HarishChandra Homomorphism
Abstract
We study the derived representation scheme DRep_n(A) parametrizing the ndimensional 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 ChevalleyEilenberg homology of the current Lie coalgebra gl_n^*(C) defined over a Koszul dual coalgebra of A. We extend this isomorphism to representation schemes of Lie algebras: for a finitedimensional reductive Lie algebra g, we define the derived affine scheme DRep_g(a) parametrizing the representations (in g) of a Lie algebra a; we show that the homology of DRep_g(a) is isomorphic to the ChevalleyEilenberg homology of the Lie coalgebra g^*(C), where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra a. We construct a canonical DG algebra map \Phi_g(a) : DRep_g(a)^G > DRep_h(a)^W, which is a homological extension of the classical restriction homomorphism. We call \Phi_g(a) a derived HarishChandra homomorphism. We conjecture that, for a twodimensional abelian Lie algebra a, the derived HarishChandra homomorphism is a quasiisomorphism, and provide some evidence for this conjecture. For any complex Lie algebra g, we compute the Euler characteristic of DRep_g(a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep_h(a)^W.This yields an interesting combinatorial identity, which we prove for gl_n and sl_n (for all n). Our identity is analogous to the classical Macdonald identity, and our quasiisomorphism conjecture is analogous to the strong Macdonald conjecture proved by S.Fishel, I.Grojnowski and C.Teleman. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our HarishChandra quasiisomorphism conjecture.
 Publication:

arXiv eprints
 Pub Date:
 September 2014
 DOI:
 10.48550/arXiv.1410.0043
 arXiv:
 arXiv:1410.0043
 Bibcode:
 2014arXiv1410.0043B
 Keywords:

 Mathematics  Representation Theory
 EPrint:
 61 pages