### 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) |
---|---|

Pages (from-to) | 175-220 |

Number of pages | 46 |

Journal | Annals of Mathematics |

Volume | 168 |

Issue number | 1 |

State | Published - 2008 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Annals of Mathematics*,

*168*(1), 175-220.

**The strong Macdonald conjecture and Hodge theory on the loop Grassmannian.** / Fishel, Susanna; Grojnowski, Ian; Teleman, Constantin.

Research output: Contribution to journal › Article

*Annals of Mathematics*, vol. 168, no. 1, pp. 175-220.

}

TY - JOUR

T1 - The strong Macdonald conjecture and Hodge theory on the loop Grassmannian

AU - Fishel, Susanna

AU - Grojnowski, Ian

AU - Teleman, Constantin

PY - 2008

Y1 - 2008

N2 - 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; Ωp) 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 1Ψ1 sum. For simply laced root systems at level 1, we also find a 'strong form' of Bailey's 4Ψ4 sum. Failure of Hodge decomposition implies the singularity of X, and of the algebraic loop groups. Some of our results were announced in [T2].

AB - 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; Ωp) 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 1Ψ1 sum. For simply laced root systems at level 1, we also find a 'strong form' of Bailey's 4Ψ4 sum. Failure of Hodge decomposition implies the singularity of X, and of the algebraic loop groups. Some of our results were announced in [T2].

UR - http://www.scopus.com/inward/record.url?scp=49749107524&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49749107524&partnerID=8YFLogxK

M3 - Article

VL - 168

SP - 175

EP - 220

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -