A trace Paley-Wiener theorem for GL(n, F)\GL(n, E)
Date(s) : 26/05/2026 iCal
14h00 - 15h00
This talk is related to the relative Langlands’ program, which aims to extend the classical Langlands’ program to spherical varieties. In the classical case, a well-known trace Paley-Wiener theorem was given by Bernstein, Deligne and Kazhdan in 1986. It gives a characterization of the functions
π ↦ Trace(π(f))
with G a reductive p-adic group, and where π ranges over isomorphism classes of smooth irreducible representations of G and f is a smooth function with compact support on G.
We will explain how to extend this to a specific relative case. That is when E/F is a quadratic extension of p-adic fields, our main result is a scalar Paley-Wiener theorem for relative Bessel distributions on GL(n, F)\GL(n, E). These distributions are relative characters of the form
π ↦ I_π(f),
where f is a smooth function with compact support on GL(n, E), and π ranges over GL(n, F)-distinguished irreducible tempered representations. Those relative characters are constructed from a GL(n, F)-invariant functional and a Whittaker functional. We will explain how by using the local Langlands correspondence, and the base-change from a unitary group, the relative characters can be described as elements of the “generic” Bernstein center of the quasi-split unitary group U(n).
Emplacement
I2M Luminy - TPR2, Salle de Séminaire 304-306 (3ème étage)
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