## Abstract

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 CpXq. 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].

Original language | English (US) |
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Pages (from-to) | 1-114 |

Number of pages | 114 |

Journal | Memoirs of the American Mathematical Society |

Volume | 269 |

Issue number | 1312 |

DOIs | |

State | Published - Jan 2021 |

## All Science Journal Classification (ASJC) codes

- General Mathematics
- Applied Mathematics

## Keywords

- Harmonic analysis
- Paley-Wiener
- Relative Langlands program
- Schwartz space
- Spherical varieties
- Symmetric spaces