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 $H^q(X;\Omega^p)$ 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].


Susanna Fishel

School of Mathematical and Statistical Science
Arizona State University
Tempe, AZ 85287
United States

Ian Grojnowski

Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge CB3 0WB
United Kingdom

Constantin Teleman

Department of Mathematics
University of California
Berkeley, CA 94720
United States