Paley–Wiener Theorems for a p–Adic Spherical Variety

Let S pXq be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X, and let C pXq 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 pXq and C pXq. When X" a reductive group, our theorem for C pXq specializes to the well-known theorem of Harish-Chandra, and our theorem for S pXq corresponds to a first step - enough to recover the structure of the Bernstein center - towards the well-known theorems of Bernstein er and Heiermann [Hei01].