Paley-Wiener Theorems for a p-Adic Spherical Variety

Let S(X) be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X, and let C(X) be the space of Harish-Chandra Schwartz functions. Under assumptions on the spherical variety, which are satisfied when it is symmetric, we prove Paley-Wiener theorems for the two spaces, characterizing them in terms of their spectral transforms. As a corollary, we get relative analogs of the smooth and tempered Bernstein centers rings of multipliers for S(X) and C(X). When X = a reductive group, our theorem for C(X) specializes to the well-known theorem of Harish-Chandra, and our theorem for S(X) corresponds to a first step - enough to recover the structure of the Bernstein center - towards the well-known theorems of Bernstein [Ber] and Heiermann [Hei01].