Department of Mathematics

Indian Institute of Science

Bangalore 560 012






Ajay Ramadoss, Indiana University
Affiliation : Indiana University

Subject Area






Department of Mathematics, Lecture Hall I




10:00 am.




July 28, 2016 (Thursday)



"Representation homology of Lie algebras and the Macdonald conjectures"


For a finite-dimensional reductive Lie algebra g, we will introduce and study a derived representation scheme DRep(A,g) parametrizing the representations of a given Lie algebra A in g. We relate the homology of DRep(A,g) to the classical (Chevalley-Eilenberg) cohomology of current Lie algebras. This allows us to construct a canonical map F from DRep(A,g)^G to DRep(A,h)^W, relating the G-invariant part of representation homology of A in g to the W-invariant part of representation homology of A in a Cartan subalgebra of 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 if A is a two dimensional abelian Lie algebra, then F is a quasi-isomorphism for any finite-dimensional reductive Lie algebra g. We provide some evidence for this conjecture and explain the relation to the strong Macdonald conjectures proposed by P.Hanlon and B.Feigin in the late 80s and recently proved (in full generality) by S. Fishel, I. Grojnowski and C. Teleman.?