Macdonald polynomials and counting parabolic bundles

Abstract: It is well known that Hall-Littlewood polynomials naturally arise from the problem of counting partial flags preserved by a nilpotent matrix over a finite field. I give an explicit interpretation of the modified Macdonald polynomials in a similar spirit, via counting parabolic bundles with nilpotent endomorphism over a curve over finite field. The result can also be interpreted as a formula for a certain truncated weighted counting of points in the affine Springer fiber over a constant nilpotent matrix. This leads to a confirmation of a conjecture of Hausel, Letellier and Rodriguez-Villegas about Poincare polynomials of character varieties. On the other hand, it naturally leads to interesting expansions of Macdonald polynomials and related generating functions that appear in the shuffle conjecture and its generalizations.

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.