Abstract
We compare the (horizontal) trace of the affine Hecke category with the
elliptic Hall algebra, thus obtaining an äffine" version of the construction
of 14. Explicitly, we show that the aforementioned trace is generated by the
objects $E_d = Tr(Y_1^d_1 Y_n^d_n T_1 \dots
T_n-1)$ as $d = (d_1,\dots,d_n) Z^n$, where $Y_i$
denote the Wakimoto objects of 9 and $T_i$ denote Rouquier complexes. We
compute certain categorical commutators between the $E_d$'s and show
that they match the categorical commutators between the sheaves
$E_d$ on the flag commuting stack, that were considered in
27. At the level of $K$-theory, these commutators yield a certain integral
form $\mathcalA$ 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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