%0 Generic %1 ho2023graded %A Ho, Quoc P. %A Li, Penghui %D 2023 %K Character homflypt sheaves %T Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $C^2$ %U http://arxiv.org/abs/2305.01306 %X 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 $H_W^gr = Ch^b(SBim_W)$ 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 $Hilb_n(C^2)$ of points on $C^2$ (equivariant with respect to the natural $C^* C^*$ 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 $Hilb_n(C^2)$. As an important computational input, we also establish a conjecture of Gorsky-Hogancamp-Wedrich on the formality of the Hochschild homology of $H_W^gr$.

@misc{ho2023graded, 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 $\mathbb{C}^* \times \mathbb{C}^*$ 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 $\mathsf{H}_W^\mathsf{gr}$.}, added-at = {2023-05-03T13:02:46.000+0200}, author = {Ho, Quoc P. and Li, Penghui}, biburl = {https://www.bibsonomy.org/bibtex/2d30867748f19fccb7b4b47715a4e7679/dragosf}, description = {Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$}, interhash = {125db38ae10877d6f82ae1b3de6ada50}, intrahash = {d30867748f19fccb7b4b47715a4e7679}, keywords = {Character homflypt sheaves}, note = {cite arxiv:2305.01306}, timestamp = {2023-05-03T13:02:46.000+0200}, title = {Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$}, url = {http://arxiv.org/abs/2305.01306}, year = 2023 }