The strong Macdonald conjecture and Hodge theory on the loop Grassmannian

We prove the strong Macdonald conjecture of Hanlon and Feigin for reductive groups G. In a geometric reformulation, we show that the Dolbeault cohomology H q (X; p ) 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 1 1 sum. For simply laced root systems at level 1, we also find a 'strong form' of Bailey's 4 4 sum. Failure of Hodge decomposition implies the singularity of X, and of the algebraic loop groups. Some of our results were announced in T2.