The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian

Details The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian

2004-11-16

arxiv.org/abs/math/0411355v1

Mathematics

Algebraic Geometry

Massive expansion of our old math.RT/0107072

Scientific paper

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.

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