The geometric significance of Macdonald positivity

By classical results of Springer, Hotta, Lusztig and others, the symmetric functions known as Hall-Littlewood polynomials provide the solution to three closely related problems of geometry and representation theory: (1) describe the action of S_{n} on the cohomology H^{*}(X^{u}), where X=GL_{n}/B is the flag variety and u\in GL_{n} is a unipotent matrix; (2) describe the action of GL_{n}(q) on the finite flag variety X(q) in terms of irreducible GL_{n}(q) characters evaluated on unipotent matrices u\in GL_{n}(q); (3) describe the local intersection homology of the closure of a unipotent conjugagy class. The geometric interpretations imply that the ``q-Kostka coefficients'' K_{\lambda \mu }(q) relating Hall-Littlewood polynomials to Schur functions are polynomials with positive integer coefficients.

In 1988, I.G. Macdonald introduced generalized Hall-Littlewood polynomials, now known as Macdonald polynomials, which depend on two parameters q and t. He showed that the resulting ``q,t-Kostka coefficients'' K_{\lambda \mu }(q,t) enjoy a pleasant symmetry between the roles of q and t, and he conjectured that they too are polynomials with positive integer coefficients. Recently I proved Macdonald's positivity conjecture. The proof involves a representation-theoretic interpretation which A. Garsia and I conjectured in 1991, together with a geometric interpretation first suggested by C. Procesi, involving the Hilbert scheme of points in the plane.

The geometric interpretation of K_{\lambda \mu }(q,t) is different from the classical Springer/Lusztig interpretation of K_{\lambda \mu }(q). Hints of the link between the two are just emerging, along with clues to a possible extension to other Lie groups and Weyl groups.

Refreshments will be served in Math Annex Room 1115, 2:45 p.m.