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

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.