Functions on the commuting stack via Langlands duality

For a complex reductive group, we construct a semi-orthogonal decomposition of the cocenter of the universal variant of its affine Hecke category. We use this to calculate the endomorphisms of a Whittaker object in the cocenter via a diagram organizing parabolic induction of character sheaves. Assuming a universal variant of Bezrukavnikov's spectral description of the affine Hecke category, we deduce a formula for the dg algebra of global functions on commuting stacks of complex reductive groups. In particular, the formula shows that the ring of invariant functions on the commuting scheme is reduced.