TY - JOUR
T1 - The strong Macdonald conjecture and Hodge theory on the loop Grassmannian
AU - Fishel, Susanna
AU - Grojnowski, Ian
AU - Teleman, Constantin
PY - 2008/7
Y1 - 2008/7
N2 - 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].
AB - 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].
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U2 - 10.4007/annals.2008.168.175
DO - 10.4007/annals.2008.168.175
M3 - Article
AN - SCOPUS:49749107524
SN - 0003-486X
VL - 168
SP - 175
EP - 220
JO - Annals of Mathematics
JF - Annals of Mathematics
IS - 1
ER -