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.

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.