  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 $\mathbb{C}^* \times \mathbb{C}^*$
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 $\mathsf{Hilb}_n(\mathbb{C}^2)$. As an important
computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich
on the formality of the Hochschild homology of $\mathsf{H}_W^\mathsf{gr}$.

Representation Theory;Algebraic Geometry;Geometric Topology;Quantum Algebra