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 $H^\bullet (BG)$. 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 $1$, we also find a ‘strong form’ of Bailey’s ${}_4\psi_4$ sum. Failure of Hodge decomposition implies the singularity of $X$, and of the algebraic loop groups. Some of our results were announced in [T2].
Susanna Fishel 1 ; Ian Grojnowski 2 ; Constantin Teleman 3
@article{10_4007_annals_2008_168_175,
author = {Susanna Fishel
and Ian Grojnowski and Constantin Teleman},
title = {The strong {Macdonald} conjecture and {Hodge} theory on the loop {Grassmannian}},
journal = {Annals of mathematics},
pages = {175--220},
year = {2008},
volume = {168},
number = {1},
doi = {10.4007/annals.2008.168.175},
mrnumber = {2415401},
zbl = {1186.17010},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.175/}
}
TY - JOUR AU - Susanna Fishel AU - Ian Grojnowski AU - Constantin Teleman TI - The strong Macdonald conjecture and Hodge theory on the loop Grassmannian JO - Annals of mathematics PY - 2008 SP - 175 EP - 220 VL - 168 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.175/ DO - 10.4007/annals.2008.168.175 LA - en ID - 10_4007_annals_2008_168_175 ER -
%0 Journal Article %A Susanna Fishel %A Ian Grojnowski %A Constantin Teleman %T The strong Macdonald conjecture and Hodge theory on the loop Grassmannian %J Annals of mathematics %D 2008 %P 175-220 %V 168 %N 1 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.175/ %R 10.4007/annals.2008.168.175 %G en %F 10_4007_annals_2008_168_175
Susanna Fishel ; Ian Grojnowski; Constantin Teleman. The strong Macdonald conjecture and Hodge theory on the loop Grassmannian. Annals of mathematics, Tome 168 (2008) no. 1, pp. 175-220. doi: 10.4007/annals.2008.168.175
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