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The Trace of the affine Hecke category (2201.07144v1)

Published 18 Jan 2022 in math.GT, math.AG, and math.RT

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.

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