论文标题
Aggine Hecke类别的痕迹
The Trace of the affine Hecke category
论文作者
论文摘要
我们将Aggine Hecke类别的（水平）痕迹与椭圆形代数进行比较，从而获得了[14]的构造的“仿射”版本。明确地表明，上述跟踪是由对象生成的$ e _ {\ textbf {d}} = \ text {tr}（y_1^{d_1} \ dots y_n^{d_n^{d_n} t_1 （d_1，\ dots，d_n）\ in \ mathbb {z}^n $，其中$ y_i $表示[9]的wakimoto对象和$ t_i $ neote rouquier complexes。我们计算$ e _ {\ textbf {d}} $之间的某些分类换向器，并证明它们与吊带排线$ \ natercal {e} _ {\ textbf {d textbf {d}} $之间的分类换向器匹配，这些s = spack inf the flag ponsconting stack in [27]中都考虑了。在$ k $ - 理论的级别上，这些换向器产生了椭圆形代数的一定积分形式$ \ widetilde {\ mathcal {a}} $，因此我们可以将其映射到offine hecke类别的$ k $ - $ k $的理论。
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.
