Kazhdan-Lusztig map and Langlands duality

Let $G$ be a connected reductive group over $\mathbb{C}$ with Weyl group $W$. Following a suggestion of Bezrukavnikov, we define a map from two-sided cells to conjugacy classes in $W$ using the geometry of the affine flag variety. This is an affine analog of the classical story of two-sided cells of $W$, special nilpotent orbits and special representations of $W$. The proof involves studying Springer representations appearing in the cohomology of affine Springer fibers. This relies on tools developed by Yun in his papers on Global Springer theory. Our result provides evidence for a conjecture of Lusztig on strata in a connected reductive group.