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Minh-Tam Trinh (U Chicago)
Artin braids, annular invariants, and point-counting
Let beta be a positive braid on n strands. Using microlocal techniques, Shende-Treumann-Zaslow showed the following are proportional up to a q-factor: (1) the coefficient of the highest a-power in the HOMFLY series of beta; (2) the coefficient of 1 when we write beta in the standard basis of the Iwahori-Hecke algebra of S_n. We give a new proof that moreover generalizes this result to any Weyl group W. To do this, we introduce a new class function on the Artin braid group of W that refines the Markov trace of Jones-Ocneanu-Gomi. Its output is a sum of q-graded total Springer representations, weighted by point-counts on certain stacks over F_q. If time permits, we will show that special values of this class function are characters of virtual A_W-modules, where A_W is the rational Cherednik algebra of W.
