Stabilization of periods of Eisenstein series and Bessel distributions on \(GL(3)\) relative to \(U(3)\). (English) Zbl 0983.11022

Lapid, Erez; Rogawski, Jonathan

Doc. Math. 5, 317-350 (2000).

Let \(E/F\) be a quadratic extension of number fields, let \(G = \text{Res}_{E/F} \text{GL}(3)\) and let \(H\) be the quasi-split \(\text{U}(3)\) relative to \(E/F\). In the relative trace formula for the pair \((G,H)\), distributions involving (regularized) periods of automorphic forms on \(G\) appear at the spectral side. These “relative Bessel distributions” are matched with “Bessel distributions” on \(G' = \text{GL}(3)\) in case they correspond to cuspidal representations. In the present paper, a stabilization of the relative Bessel distributions corresponding to Eisenstein series is defined, and it is proved that the stabilized distributions match with Bessel distributions on \(G'\).
The stabilization is effectuated writing the regularized period of an Eisenstein series as a sum of four factorizable distributions. The local factors at the unramified places are expressed in terms of \(L\)-functions, a computation for which the authors used Mathematica.

Reviewer: J.G.M.Mars (Utrecht)

MSC:

11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F72	Spectral theory; trace formulas (e.g., that of Selberg)
11F30	Fourier coefficients of automorphic forms

Keywords:

Eisenstein series; relative Bessel distributions; stabilization; periods of automorphic forms; stability distributions