Let $M$ be an orientable, compact smooth manifold with a metric $g$ and $H^{-1}(M)$ be the dual space of $$H^{1}(M)=\{f:\int |f|^2+(\nabla f)^2 d\mu<\infty\}$$
I know it is well known that (see Julien Dubedat, page 18, https://arxiv.org/pdf/0712.3018.pdf, for example) we can put a Gaussian measure on $H^{-1}(M)$ induced by the Gaussian free field. Further the (induced) Gaussian measure is a Radon measure. This is essentially because $H^{-1}(M)$ is the dual space of a nuclear space. However, $H^{-1}(M)$ is infinite dimensional. Unless I am mistaken, it is not locally compact.
My naive question is whether it is possible to construct a Poisson summation type formula on a discrete subgroup $H$ of $G=H^{-1}(M)$. Since Fourier transform (in general) switches measure with functions, I should get some Gaussian function on the space $\sim \hat{G/H}\sim H^{1}(M)/\hat{H}$. Is this something well known? I am willing to "scale down" to smaller spaces that contain $C^{\infty}(M)$, but the same issue of non-local compactness still exists.
Some Googling showed results in model theory, which I am not familiar. So I apologize in advance in case this has been resolved using mathematical logic. Alternatively it is possible that we can use certain version of adeles to "recover locally compactness", but I do not know if anyone did this either.
