tailieunhanh - Đề tài " The strong Macdonald conjecture and Hodge theory on the loop Grassmannian "

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. . | Annals of Mathematics The strong Macdonald conjecture and Hodge theory on the loop Grassmannian By Susanna Fishel Ian Grojnowski and Constantin Teleman Annals of Mathematics 168 2008 175 220 The strong Macdonald conjecture and Hodge theory on the loop Grassmannian By Susanna Fishel Ian Grojnowski and Constantin Teleman 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 Hq X Qp 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 01 sum. For simply laced root systems at level 1 we also find a strong form of Bailey s -lU -i sum. Failure of Hodge decomposition implies the singularity of X and of the algebraic loop groups. Some of our results were announced in T2 . Introduction This article address some basic questions concerning the cohomology of affine Lie algebras and their flag varieties. Its chapters are closely related but have somewhat different flavours and the methods used in each may well appeal to different readers. Chapter I proves the strong Macdonald constant term conjectures of Hanlon H1 and Feigin F1 describing the cohomologies of the Lie algebras g z zn of truncated polynomials with values in a reductive Lie algebra g and of the graded Lie algebra g z s of g-valued skew polynomials in an even variable z and an odd one s Theorems A and B . The proof uses little more than linear algebra and while Nakano s identity effects a substantial simplification we have included a brutal computational by-pass in Appendix A to avoid reliance on external sources. Chapter II discusses the Dolbeault cohomology Hq Qp of flag varieties of loop groups. In addition to the Macdonald cohomology the methods and proofs draw heavily on T3 . For the loop Grassmannian X G z G z we