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 ${\mathsf{H}}_W^{\mathsf{gr}}={\mathsf{Ch}}^b({\mathsf{SBim}}_W)$ 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 ${\mathsf{Hilb}}_n({\mathbb{C}}^2)$ of points on ${\mathbb{C}}^2$ (equivariant with respect to the natural ${\mathbb{C}}^{\ast }\times {\mathbb{C}}^{\ast }$ action), yielding a proof of (a $2$-periodized version of) a conjecture of Gorsky-Negut-Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on ${\mathsf{Hilb}}_n({\mathbb{C}}^2)$. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of ${\mathsf{H}}_W^{\mathsf{gr}}$.

References in corpus (1)

• Eisenstein series via factorization homology of Hecke categories