A Note on Spectral Analysis for ${\rm GL}_2$: I

TL;DR

This paper establishes Fourier inversion for smooth vectors in L^2(GL_2) over a number field with minimal automorphic theory, highlights gaps in existing proofs, and computes the intertwining operator explicitly.

Contribution

It provides a new minimal approach to Fourier inversion for GL_2 and discusses extending this to broader classes of functions, also clarifying misconceptions in the literature.

Findings

Fourier inversion established for smooth vectors in L^2(GL_2)

Explicit computation of the intertwining operator

Discussion on extending Fourier inversion to larger function classes

Abstract

We establish the Fourier inversion for the smooth vectors in ${\rm L}^2({\rm GL}_2, \omega)$ over a number field $\mathbf{F}$, using minimal knowledge from automorphic representation theory. We point out a possible way to establish Fourier inversion for larger classes of functions. We also point out the incompleteness of some commonly believed "proof" of Fourier inversion in the literature. Moreover, the explicit computation of the intertwining operator has independent interests.