Mirror Symmetry of the Affine Toda Systems

For a complex reductive group $G$, we prove a homological mirror symmetry between the wrapped Fukaya category of the affine Toda system for $G$ and coherent sheaves on the regular centralizer group scheme for the Langlands dual group $G^{\vee }$. This can be interpreted as a geometric Langlands equivalence for ${\mathbb{P}}^1$ with mildest wild ramification at $0$ and $\mathrm{\infty }$.