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; 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 1 ψ1 sum. For simply laced root systems at level 1, we also ﬁnd a ‘strong form’ of Bailey’s 4 ψ4 sum. ...
Characteristic cohomology classes, deﬁned in modulo 2 coeﬃcients by Stiefel  and Whitney  and with integral coeﬃcients by Pontrjagin , make up the primary source of ﬁrst-order invariants of smooth manifolds. When their utility was ﬁrst recognized, it became an obvious goal to study the ways in which they admitted extensions to other categories, such as the categories of topological or PL manifolds; perhaps a clean description of characteristic classes for simplicial complexes could even give useful computational techniques.
We study the Radon transform Rf of functions on Stiefel and Grassmann manifolds. We establish a connection between Rf and G˚ arding-Gindikin fractional integrals associated to the cone of positive deﬁnite matrices. By using this connection, we obtain Abel-type representations and explicit inversion formulae for Rf and the corresponding dual Radon transform. We work with the space of continuous functions and also with Lp spaces.
We describe a Schubert induction theorem, a tool for analyzing intersections on a Grassmannian over an arbitrary base ring. The key ingredient in the proof is the Geometric Littlewood-Richardson rule of [V2]. As applications, we show that all Schubert problems for all Grassmannians are enumerative over the real numbers, and suﬃciently large ﬁnite ﬁelds. We prove a generic smoothness theorem as a substitute for the Kleiman-Bertini theorem in positive characteristic.