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The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian

Susanna Fishel·Ian Grojnowski·Constantin Teleman
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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 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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Susanna Fishel, Ian Grojnowski, Constantin Teleman. 2004-11-16. The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian. https://arxiv.org/abs/math/0411355

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