Mathematics Colloquia and Seminars

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The Macdonald constant term identity for the root system of a semisimple Lie algebra L can be proved by calculating the Lie algebra cohomology of L[z,s], where z is an ordinary variable and s is an odd variable. This calculation was completed by Fishel, Grojnowski, and Teleman using ideas from the Kahler geometry of the loop Grassmannian. I will explain how to extend this calculation to an arbitrary affine flag variety, leading to strong Macdonald theorems for parahoric subalgebras of any affine Kac-Moody algebra. As applications, we will get a proof of the affine constant term identities, and also a proof that the affine Kostka-Foulkes polynomials have positive coefficients.