On Fourier Transforms and Functional Equations on GL(2)

We consider a novel setting for local harmonic analysis on reductive groups motivated by Langlands functoriality conjecture. To this end, we characterize certain non-linear Schwartz spaces on tori and reductive groups in spectral terms, and develop some of their structure in the unramified case, and we derive estimates of their moderate growth at infinity. We also consider non-linear Fourier transforms, and calculate their action on tame supercuspidal representations of $GL_2(F)$ in terms of inducing cuspidal data.