  The Deligne-Langlands correspondence parametrizes irreducible representations
of the affine Hecke algebra $\mathcal{H}^{\text{aff}}$ by certain perverse
sheaves. We show that this can be lifted to an equivalence of triangulated
categories. More precisely, we construct for each central character $\chi$ of
$\mathcal{H}^{\text{aff}}$ an equivalence of triangulated categories between a
perfect derived category of dg-modules
$D_{\text{perf}}(\mathcal{H}^{\text{aff}}/(\text{ker}(\chi)) - \text{dgMod})$
and the triangulated category generated by the corresponding perverse sheaves.
The main step in this construction is a formality result that we prove for a
wide range of `Springer sheaves'.
