  We prove the strong Macdonald conjecture of Hanlon and Feigin for reductive
groups G. In a geometric reformulation, we show that the Dolbeault cohomology
$H^q(X;\Omega^p)$ of the loop Grassmannian X is freely generated by de Rham's
forms on the disk coupled to algebra generators of $H*(BG)$. Equating Euler
characteristics of the two gives an identity, independently known to Macdonald
[M], which generalises Ramanujan's_1\psi_1 sum. Simply laced root systems at
level 1 are related to a `strong'_4\psi_4 sum. Failure of Hodge decomposition
implies the singularity of X, and of the algebraic loop groups.
