Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$

Abstract: Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel'd center of the (graded) finite Hecke category $\mathsf{H}_W$ in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi-Nadler and Bezrukavninov-Finkelberg-Ostrik. In type $A$, we relate the categorical trace to the category of 2-periodic coherent sheaves on the Hilbert schemes of points on $\mathbb{C}^2$ (equivariant with respect to the natural $\mathbb{C}^* \times \mathbb{C}^*$ action), yielding a proof of a conjecture of Gorsky-Negut-Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on Hilbert schemes. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of $\mathsf{H}_W$. This is a joint work with Quoc P. Ho.

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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