The trace of the affine Hecke category

We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN 2022 (2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects Ed=Tr(Y1d1⋯YndnT1⋯Tn−1)$E_{\mathbf {d}} = {\rm Tr}(Y_1^{d_1} \dots Y_n^{d_n} T_1 \dots T_{n-1})$ as d=(d1,⋯,dn)∈Zn$\mathbf {d}= (d_1,\dots ,d_n) \in \mathbb {Z}^n$ , where Yi$Y_i$ denote the Wakimoto objects of Elias and Ti$T_i$ denote Rouquier complexes. We compute certain categorical commutators between the Ed$E_{\mathbf {d}}$ 's and show that they match the categorical commutators between the sheaves Ed$\mathcal {E}_{\mathbf {d}}$ on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of K$K$ ‐theory, these commutators yield a certain integral form A∼$\widetilde{\mathcal {A}}$ of the elliptic Hall algebra, which we can thus map to the K$K$ ‐theory of the trace of the affine Hecke category.