A nonabelian Fourier transform for tempered unipotent representations, II

We extend the construction of the nonabelian Fourier transform proposed in our previous paper from the space of unipotent elliptic characters to the space of unipotent compact characters of a reductive $p$-adic group, and we investigate its relation with the branching to maximal compact open subgroups. We prove that this Fourier transform is compatible with parahoric restriction for certain groups, including ${\mathrm{S}\mathrm{L}}_n,{\mathrm{G}\mathrm{L}}_n$, and $G_2$. We also relate the compact setting to the elliptic setting using parabolic induction.