A trace Paley-Wiener theorem for GLn(F)backslash GLn(E), with E/F a quadratic extension of non-Archimedean local fields
Abstract
We extend the classical trace Paley-Wiener theorem for p-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when E/F is a quadratic extension of p-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on GLn(F)backslash GLn(E). These distributions are relative character of the form π↦ Iπ(f), f∈ Cc∞(GLn(E)), as π ranges over GLn(F)-distinguished irreducible tempered representations, and are constructed from a GLn(F)-invariant functional and a Whittaker functional. Using the local Langlands correspondence, and the base-change from a unitary group, we show that the relative characters can be described as elements of the "generic" Bernstein center of the unitary group U(n).
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BibTeX
@misc{coutens2026trace,
title = {{A trace Paley-Wiener theorem for $\mathrmGL_n(F)\backslash \mathrmGL_n(E)$, with $E/F$ a quadratic extension of non-Archimedean local fields}},
author = {Juliette Coutens},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15432},
eprint = {2610.15432},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15432},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A trace Paley-Wiener theorem for GL_n(F)backslash GL_n(E), with E/F a quadratic extension of non-Archimedean local fields
AU - Juliette Coutens
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15432
UR - https://arxiv.org/abs/2610.15432
AB - We extend the classical trace Paley-Wiener theorem for p-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when E/F is a quadratic extension of p-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on GL_n(F)backslash GL_n(E). These distributions are relative character of the form π↦ I_π(f), f∈ C_c^∞(GL_n(E)), as π ranges over GL_n(F)-distinguished irreducible tempered representations, and are constructed from a GL_n(F)-invariant functional and a Whittaker functional. Using the local Langlands correspondence, and the base-change from a unitary group, we show that the relative characters can be described as elements of the "generic" Bernstein center of the unitary group U(n).
ER -