Components of affine Springer fibers

Let $\mathbf{G}$ be a connected split reductive group over a field of characteristic zero or sufficiently large characteristic, ${\gamma }_0\in (\mathrm{Lie}\mathbf{G})((t))$ be any topologically nilpotent regular semisimple element, and $\gamma =t{\gamma }_0$. Using methods from $p$-adic orbital integrals, we show that the number of components of the Iwahori affine Springer fiber over $\gamma$ modulo $Z_{\mathbf{G}((t))}(\gamma )$ is equal to the order of the Weyl group.