The strong Macdonald conjecture and Hodge theory on the loop Grassmannian

Abstract

We prove the strong Macdonald conjecture of Hanlon and Feigin for reductive groups G. In a geometric reformulation, we show that the Dolbeault cohomology Hq(X; Ωp) of the loop Grassmannian X is freely generated by de Rham's forms on the disk coupled to the indecomposables of H.(BG). Equating the two Euler characteristics gives an identity, independently known to Macdonald [M], which generalises Ramanujan's 1Ψ1 sum. For simply laced root systems at level 1, we also find a 'strong form' of Bailey's 4Ψ4 sum. Failure of Hodge decomposition implies the singularity of X, and of the algebraic loop groups. Some of our results were announced in [T2].