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 $H^q(X;\Omega^p)$ of the loop Grassmannian $X$ is freely generated by de Rham’s forms on the disk coupled to the indecomposables of $H^\bullet (BG)$. Equating the two Euler characteristics gives an identity, independently known to Macdonald [M], which generalises Ramanujan’s ${}_1\psi_1$ sum. For simply laced root systems at level $1$, we also find a ‘strong form’ of Bailey’s ${}_4\psi_4$ sum. Failure of Hodge decomposition implies the singularity of $X$, and of the algebraic loop groups. Some of our results were announced in [T2].