Featured Speaker :  Minh-Tam Trinh (MIT)

Title :  From the Hecke Category to the Unipotent Locus

Abstract :  When W is the Weyl group of a reductive group G, we can categorify its Hecke algebra by means of equivariant sheaves on the double flag variety of G. We will define a functor from the resulting category to a certain category of modules over a polynomial extension of C[W]. We will prove that, on objects called Rouquier complexes, our functor yields the equivariant Borel-Moore homology of a generalized Steinberg variety attached to a positive element in the braid group of W. Some reasons this may be interesting: (1) In type A, the triply-graded Khovanov-Rozansky homology of the link closure of the braid is a summand of the weight-graded equivariant homology of this variety. This extends previously-known results for the top and bottom "a-degrees" of KR homology. (2) The "Serre duality" of KR homology under insertion of full twists leads us to conjecture a mysterious homeomorphism between pieces of different Steinbergs. (3) We find evidence for a rational-DAHA action on the (modified) homology of the Steinbergs of periodic braids. It seems related to conjectures of Broué-Michel and Oblomkov-Yun in rather different settings.

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