On the spectral Theorem of Langlands

TL;DR

This paper advances the proof of Langlands' spectral theorem by demonstrating the completeness of wave packets of Eisenstein series in the space of automorphic forms, utilizing truncation methods inspired by Arthur.

Contribution

It provides a new proof of Langlands' spectral theorem using wave packets of Eisenstein series and truncation techniques, building on Bernstein and Lapid's work.

Findings

Hilbert subspace generated by Eisenstein series wave packets equals the whole space

Achieves a proof of Langlands' spectral theorem based on Bernstein and Lapid's work

Employs truncation on compact sets similar to Arthur's approach

Abstract

We show that the Hilbert subspace of $L^2(G(F)\backslash G(\A))$ generated by wave packets of Eisenstein series built from discrete series is the whole space. Together with the work of Lapid \cite{L1}, it achieves a proof of the spectral theorem of Langlands based on the work of Bernstein and Lapid \cite{BL} on the meromorphic continuation of Eisenstein series. I have to say that I was unable to complete the proof of an earlier version. Instead, I use truncation on compact sets, as Arthur did to prove the Local Trace Formula in \cite{Alt}.