The trace of the affine Hecke category

We compare the (horizontal) trace of affine Hecke category with elliptic Hall algebra, thus obtaining an “affine” version construction Gorsky et al. (Int. Math. Res. Not. IMRN 2022 (2022) 11304–11400). Explicitly, we show that aforementioned is generated by objects E d = Tr ( Y 1 ⋯ n T − ) $E_{\mathbf {d}} {\rm Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 T_{n-1})$ as , ∈ Z $\mathbf {d}= (d_1,\dots ,d_n) \in \mathbb {Z}^n$ where i $Y_i$ denote Wakimoto Elias and $T_i$ Rouquier complexes. compute certain categorical commutators between {d}}$ 's they match sheaves $\mathcal {E}_{\mathbf on flag commuting stack were considered in Neguț (Publ. Inst. Hautes Études Sci. 135 337–418). At level K $K$ -theory, these yield a integral form A ∼ $\widetilde{\mathcal {A}}$ which can map to -theory category.