### 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; Ω^{p}) 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}Ψ_{1} sum. For simply laced root systems at level 1, we also find a 'strong form' of Bailey's _{4}Ψ_{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].

Original language | English (US) |
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Pages (from-to) | 175-220 |

Number of pages | 46 |

Journal | Annals of Mathematics |

Volume | 168 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2008 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Mathematics*,

*168*(1), 175-220. https://doi.org/10.4007/annals.2008.168.175