A few years ago Tina Kanstrup and me proposed another realization of the affine Hecke category by means of equivariant matrix factorizations. We followed a certain intuition relating the derived stack of loops with values in a quotient stack to derived Hamiltonian reduction of the corresponding cotangent bundle. In the present talk we discuss the tools to make this relation precise. Given a scheme X acted by an algebraic group G, we prove that the category of LG-equivariant quasicoherent sheaves on LX is equivalent to the category of quasicoherent sheaves on the derived Hamiltonian reduction. I will use topological intuition to demonstrate the steps in the proof.