Colloquium
3:00 p.m., Friday (Oct. 12)
Math 100
Mark Haiman
University of California, Berkeley
The geometric significance of Macdonald positivity
By classical results of Springer, Hotta, Lusztig and others, the
symmetric functions known as HallLittlewood 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 ``qKostka coefficients'' K_{\lambda \mu }(q)
relating HallLittlewood polynomials to Schur functions are
polynomials with positive integer coefficients.
In 1988, I.G. Macdonald introduced generalized HallLittlewood
polynomials, now known as Macdonald polynomials, which depend on two
parameters q and t. He showed that the resulting ``q,tKostka
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
representationtheoretic 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.
