Geodesic
The strong Macdonald conjecture and Hodge theory on the loop Grassmannian
Susanna Fishel 1 ; Ian Grojnowski2 ; Constantin Teleman3
1 School of Mathematical and Statistical Science<br/>Arizona State University<br/>Tempe, AZ 85287<br/>United States
2 Department of Pure Mathematics and Mathematical Statistics<br/>University of Cambridge<br/>Cambridge CB3 0WB<br/>United Kingdom
3 Department of Mathematics<br/>University of California<br/>Berkeley, CA 94720<br/>United States
Annals of mathematics, Tome 168 (2008) no. 1, pp. 175-220.

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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].
DOI : 10.4007/annals.2008.168.175

Susanna Fishel  1 ; Ian Grojnowski 2 ; Constantin Teleman 3

1 School of Mathematical and Statistical Science<br/>Arizona State University<br/>Tempe, AZ 85287<br/>United States
2 Department of Pure Mathematics and Mathematical Statistics<br/>University of Cambridge<br/>Cambridge CB3 0WB<br/>United Kingdom
3 Department of Mathematics<br/>University of California<br/>Berkeley, CA 94720<br/>United States 
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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. http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.175/

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