We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of coherent sheaves, the Fourier-Deligne transform in the setting of $\ell$-adic sheaves, and so on.
In the Fourier-Deligne transform, they are in general defined by taking a vector bundle $E$ over a suitable base scheme $S$. Complexes on $E$ are then pulled back to complexes on $E\times_SE^{\vee}$, tensored with $\mu^*\cal{L}_{\psi}$ ($\cal{L}_{\psi}$ the Artin-schreier sheaf associated to an additive character $\psi$, and $\mu:E\times_SE^{\vee}\rightarrow\mathbb{A}^1_S$ is the natural pairing). I have only seen affine lines or trivial bundles come up in applications that are closely related to concrete objects (like Fourier transforms of functions with respect to $\psi$ in analytic number theory).
The first question is the following: Do nontrivial bundles show up in applications pertaining to the Fourier-Deligne transform?
The second question is: are there applications in geometry/analysis where we do the above kind of Fourier transform with respect to the tangent bundle of a smooth manifold? So we take a function on the tangent bundle $TM$ of a smooth manifold, pull it back to to $TM\times_M T^*M$, multiply with $e^{-2\pi i\bullet}$ composed with the natural pairing $TM\times_MT^*M\rightarrow\mathbb{R}$, and then integrate over $TM$ to obtain a function on the cotangent bundle of $M$. Are there other Fourier transforms in geometry where nontrivial vector bundles show up?
