Paley-Wiener theorems for a p-adic spherical variety

Patrick Delorme, Pascale Harinck, Yiannis Sakellaridis

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1 Scopus citations


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 languageEnglish (US)
Pages (from-to)1-114
Number of pages114
JournalMemoirs of the American Mathematical Society
Issue number1312
StatePublished - Jan 2021

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics


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


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