Spherical characters: the supercuspidal case

We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $\pi$ of a connected reductive $p$-adic group, subject to the assumption that $\pi$ is obtained via induction from a representation of an open compact mod centre subgroup. We derive an integral formula for each spherical character belonging to the basis. This formula involves integration of a particular kind of matrix coefficient of $\pi$. We also obtain a similar formula for the function realizing the spherical character. In addition, we determine, subject to some conditions, which of these spherical characters vanish identically on an open neighbourhood of the identity. We verify that the requisite conditions are always satisfied for distinguished tame supercuspidal representations of groups that split over tamely ramified extensions.