A representation of an arbitrary sheaf from derived category of sheaves on a stack G/G (action by conjugacy) in terms of "dual" basis. Abelian case — Fourier-Mukai transform for abelian varieties which relates sheaves on A with sheaves on Av using the correspondence A\times Av.
(there's some important property of Fourier missing above — I'll return to this tomorrow). It's something about relationship between G/G and G^\wedge/G^\wedge.
After passing to functions, you may think about the resulting version as rewriting an arbitrary conjugate-invariant function as a sum by characters. That specializes to abelian case — where all functions are conjugate-invariant.
As an example, for the real line you have to present f(x) as an integral of e^{2\lambda ix} (well, L^2, i and the appearance of integral are some important technical details because of non-compactness). The correspondence after all becomes integration with the kernel K(x, \lambda) = e^{2\lambda ix}.
