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

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It is well-known that Catalan numbers C n = 1 n+1 2n n count the number of dominant regions in the Shi arrangement of type A, and that they also count partitions which are both n-cores as well as (n + 1)-cores. These concepts have natural extensions, which we call here the m-Catalan numbers and m-Shi arrangement. In this paper, we construct a bijection between dominant regions of the m-Shi arrangement and partitions which are both n-cores as well as (mn + 1)-cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric group.

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In this paper we present a bijection ωn between two well known families of Catalan objects: the set of facets of the m-generalized cluster complex ∆ m (An) and the set of dominant regions in the m-Catalan arrangement Cat m (An), where m ∈ N >0 . In particular, ωn bijects the facets containing the negative simple root −α to dominant regions having the hyperplane {v ∈ V | v, α = m} as separating wall. As a result, ωn restricts to a bijection between the set of facets of the positive part of ∆ m (An) and the set of bounded dominant regions in Cat m (An). The map ωn is a composition of two bijections in which integer partitions in an m-staircase shape come into play.