Number of left cells in the second lowest two-sided cell

We prove Shi's conjecture that the number of left cells in the second lowest two-sided cell of the affine Weyl group attached to a simply connected simple complex group $G$ is exactly $|W_0|/2$, where $W_0$ is the corresponding finite Weyl group. The same formula holds for the corresponding extended affine Weyl group. The key step is to show that the reductive centralizer in $G$ of a minimal nilpotent element has identity component with simply connected derived subgroup and admits no nontrivial algebraic central extensions by $\mathbb G_m$. This gives an isomorphism of based rings between the asymptotic ring of the extended cell and a matrix ring over the representation ring of the centralizer, with matrix size equal to the Euler characteristic of the Springer fiber, namely $|W_0|/2$.