The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian

By S. Fishel, I. Grojnowski, C. Teleman. 2004

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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;Op) 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].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d5881b26fc2607570562125e3d3593bc75a26a4e