The Trace of the affine Hecke category

TL;DR

This paper establishes a connection between the trace of the affine Hecke category and the elliptic Hall algebra, providing explicit generators and computing categorical commutators that relate to sheaves on the flag commuting stack.

Contribution

It explicitly describes the generators of the trace of the affine Hecke category and relates their categorical commutators to those of sheaves on the flag commuting stack, linking to the elliptic Hall algebra.

Findings

Generated the trace by objects E_d involving Y_i and T_i.

Computed categorical commutators matching sheaves on the flag commuting stack.

Mapped these commutators to an integral form of the elliptic Hall algebra.

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.