Abstract: The affine Hecke category, defined using affine Soergel
bimodules, categorifies the affine Hecke algebra. I will compare the
derived horizontal trace of the affine Hecke category with the elliptic
Hall algebra, and with the derived category of the commuting stack. In
particular, I will describe certain explicit generators for the trace
category and some categorical commutation relations between these. This
is a joint work with Andrei Negut.