"Let SpXq be the Schwartz space of compactly supported smooth functions on the p-adic points of a spherical variety X, and let CpXq 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 SpXq and C pXq. When X " a reductive group, our theorem for CpXq specializes to the well-known theorem of Harish-Chandra, and our theorem for SpXq 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]"--.
ISBN: | 9781470444020 |
Publication date: | 30th September 2021 |
Author: | Patrick Delorme, Pascale Harinck, Yiannis Sakellaridis |
Publisher: | American Mathematical Society |
Format: | Paperback |
Series: | Memoirs of the American Mathematical Society |
Genres: |
Algebraic geometry Algebra |