The strong Macdonald conjecture and Hodge theory on the Loop Grassmannian

2004/11/16 by Susanna Fishel, Fishel, Susanna, I. Grojnowski +3 · 1 citation
Mathematics · #14C30 #17B67 #33D52 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.math/0411355

openalex publication_date 2004/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 H^q(X;Ω^p) of the loop Grassmannian X is freely generated by de Rham's forms on the disk coupled to algebra generators of H*(BG). Equating Euler characteristics of the two gives an identity, independently known to Macdonald [M], which generalises Ramanujan's₁ψ₁ sum. Simply laced root systems at level 1 are related to a `strong'₄ψ₄ sum. Failure of Hodge decomposition implies the singularity of X, and of the algebraic loop groups.

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• On a theorem of Narasimhan and Ramanan on deformations

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