The strong Macdonald conjecture and Hodge theory on the loop Grassmannian

• 成果类型: Article
• 署名作者: Fishel, Susanna; Grojnowski, Ian; Teleman, Constantin
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2008.168.175
• 发表日期: 2008
• 页码: 175-220
• 关键词: g-bundles; HOMOLOGY; MODULI; bott

摘要

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 the indecomposables of H degrees(BG). Equating the two Euler characteristics gives an identity, independently known to Macdonald [M], which generalises Ramanujan's 1 psi 1 sum. For simply laced root systems at level I., we also find a. 'strong form' of Baileys 4 psi 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].