I am interested in the following analogy with noncommutative Fourier analysis. Take $G=SL(2,\mathbb R)$, $\Gamma = SL(2, \mathbb Z)$, $N$ the upper triangular unipotent matrices (i.e. translations) and $\Gamma_\infty = \Gamma \cap N$. Let $\hat \phi$ be the average over $\Gamma$ (for decaying enough functions) $$\hat \phi (g) = \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \phi(\gamma g)$$ and $f_0$ the zeroth Fourier coefficient $$f_0 (g) = \int_{\Gamma_\infty \backslash N} f(ng) dn.$$
If $\phi$ is $N$-invariant and $f$ is $\Gamma$-invariant, then these two operators are dual in the corresponding $L^2$ spaces, i.e. $$\langle \hat \phi, f \rangle = \langle \phi, f_0 \rangle$$
Is there an analogous situation in the case of classical commutative Fourier analysis? (should we take $N=\mathbb R$-invariant functions, i.e. constant ones?)
In other terms: is there a dual operator to the zeroth Fourier coefficient?