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Paley–Wiener theorems for a $p$-adic spherical variety
About this Title
Patrick Delorme, Pascale Harinck and Yiannis Sakellaridis
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 269, Number 1312
ISBNs: 978-1-4704-4402-0 (print); 978-1-4704-6462-2 (online)
DOI: https://doi.org/10.1090/memo/1312
Published electronically: March 5, 2021
Keywords: Harmonic analysis,
Paley–Wiener,
Schwartz space,
symmetric spaces,
spherical varieties,
relative Langlands program
Table of Contents
1. Structure, notation and preliminaries
2. Discrete and cuspidal summands
3. Eisenstein integrals
4. Scattering
5. Paley–Wiener theorems
- 1. Characterization of strongly factorizable spherical varieties
Abstract
Let $\mathcal S(X)$ be the Schwartz space of compactly supported smooth functions on the $p$-adic points of a spherical variety $X$, and let $\mathscr C(X)$ 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 $\mathcal S(X)$ and $\mathscr C(X)$. When $X=$ a reductive group, our theorem for $\mathscr C(X)$ specializes to the well-known theorem of Harish-Chandra, and our theorem for $\mathcal S(X)$ corresponds to a first step — enough to recover the structure of the Bernstein center — towards the well-known theorems of Bernstein \cite{BePadic} and Heiermann \cite{Heiermann}.- Avraham Aizenbud, Nir Avni, and Dmitry Gourevitch, Spherical pairs over close local fields, Comment. Math. Helv. 87 (2012), no. 4, 929–962. MR 2984577, DOI 10.4171/CMH/274
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