Formality in the Deligne-Langlands correspondence

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 $D_{\text{perf}}({\mathcal{H}}^{\text{aff}}\mathrm{/}(\text{ker}(\chi ))-\text{dgMod})$ 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’.