  We present a Langlands dual realization of the putative category of affine
character sheaves. Namely, we calculate the categorical center and trace (also
known as the Drinfeld center and trace, or categorical Hochschild cohomology
and homology) of the affine Hecke category starting from its spectral
presentation. The resulting categories comprise coherent sheaves on the
commuting stack of local systems on the two-torus satisfying prescribed support
conditions, in particular singular support conditions as appear in recent
advances in the Geometric Langlands program. The key technical tools in our
arguments are: a new descent theory for coherent sheaves or D-modules with
prescribed singular support; and the theory of integral transforms for coherent
sheaves developed in the companion paper [BNP].
