The strong Macdonald conjecture and Hodge theory on the loop Grassmannian

Susanna Fishel, Ian Grojnowski, Constantin Teleman

Research output: Contribution to journalArticle

18 Citations (Scopus)

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
StatePublished - 2008

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Hodge Theory
Grassmannian
Hodge Decomposition
Loop Groups
Reductive Group
Euler Characteristic
Root System
Ramanujan
Algebraic Groups
Reformulation
Cohomology
Singularity
Imply
Generalise
Form
Decomposition

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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

In: Annals of Mathematics, Vol. 168, No. 1, 2008, p. 175-220.

Research output: Contribution to journalArticle

Fishel, S, Grojnowski, I & Teleman, C 2008, 'The strong Macdonald conjecture and Hodge theory on the loop Grassmannian', Annals of Mathematics, vol. 168, no. 1, pp. 175-220.
Fishel, Susanna ; Grojnowski, Ian ; Teleman, Constantin. / The strong Macdonald conjecture and Hodge theory on the loop Grassmannian. In: Annals of Mathematics. 2008 ; Vol. 168, No. 1. pp. 175-220.
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