Speaker:  Minh-Tam Trinh (MIT) 

Title: Φ-Harish-Chandra Series, Level-Rank Duality, and Affine Springer Fibers

Abstract: Broué–Malle–Michel noticed that for any m, the unipotent irreducible characters of a finite reductive group G can be partitioned into “m-twisted” Harish-Chandra (HC) series with “m-twisted” Howlett–Lehrer parametrizations by irreducible characters of relative Weyl groups, recovering the usual notions when m = 1. Ting Xue and I conjecture that for any G and ℓ and m, the intersection of an ℓ-twisted HC series and an m-twisted HC series is simultaneously parametrized by a union of m-blocks for a certain Hecke algebra on the ℓ side and a union of ℓ-blocks for a certain Hecke algebra on the m side, in a way that matches up blocks. We show that when G = GL(n), this is Uglov’s level-rank duality in disguise. More surprisingly, we conjecture that these bijections are (essentially) realized by bimodules that Oblomkov–Yun and Boixeda Alvarez–Losev construct from the cohomology of affine Springer fibers.