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 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].

Original languageEnglish (US)
Pages (from-to)175-220
Number of pages46
JournalAnnals of Mathematics
Volume168
Issue number1
DOIs
StatePublished - Jul 2008

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ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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