%0 Generic %1 gorsky2022trace %A Gorsky, Eugene %A Neguţ, Andrei %D 2022 %K Hall Hecke affine elliptic knots %T The Trace of the affine Hecke category %U http://arxiv.org/abs/2201.07144 %X We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an äffine" version of the construction of 14. Explicitly, we show that the aforementioned trace is generated by the objects $E_d = Tr(Y_1^d_1 Y_n^d_n T_1 \dots T_n-1)$ as $d = (d_1,\dots,d_n) Z^n$, where $Y_i$ denote the Wakimoto objects of 9 and $T_i$ denote Rouquier complexes. We compute certain categorical commutators between the $E_d$'s and show that they match the categorical commutators between the sheaves $E_d$ on the flag commuting stack, that were considered in 27. At the level of $K$-theory, these commutators yield a certain integral form $\mathcalA$ of the elliptic Hall algebra, which we can thus map to the $K$-theory of the trace of the affine Hecke category.

@misc{gorsky2022trace, abstract = {We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an "affine" version of the construction of [14]. Explicitly, we show that the aforementioned trace is generated by the objects $E_{\textbf{d}} = \text{Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 \dots T_{n-1})$ as $\textbf{d} = (d_1,\dots,d_n) \in \mathbb{Z}^n$, where $Y_i$ denote the Wakimoto objects of [9] and $T_i$ denote Rouquier complexes. We compute certain categorical commutators between the $E_{\textbf{d}}$'s and show that they match the categorical commutators between the sheaves $\mathcal{E}_{\textbf{d}}$ on the flag commuting stack, that were considered in [27]. At the level of $K$-theory, these commutators yield a certain integral form $\widetilde{\mathcal{A}}$ of the elliptic Hall algebra, which we can thus map to the $K$-theory of the trace of the affine Hecke category.}, added-at = {2022-04-27T07:44:15.000+0200}, author = {Gorsky, Eugene and Neguţ, Andrei}, biburl = {https://www.bibsonomy.org/bibtex/2f217b33cf0d7cc3da5a7de419ed58505/dragosf}, description = {The Trace of the affine Hecke category}, interhash = {575e8f84c02222c2490450adb2260f2b}, intrahash = {f217b33cf0d7cc3da5a7de419ed58505}, keywords = {Hall Hecke affine elliptic knots}, note = {cite arxiv:2201.07144}, timestamp = {2022-04-27T07:44:15.000+0200}, title = {The Trace of the affine Hecke category}, url = {http://arxiv.org/abs/2201.07144}, year = 2022 }