What is the analogue of the Jacobi theta function in the Weyl representation?

Question

It is known (see for example the associated Wikipedia entry) that the Jacobi theta function $$\vartheta(z; \tau) = \sum_{n\in\mathbb{Z}} \exp(\pi in^2\tau + 2\pi inz)$$ arises from a certain representation of the Heisenberg group $H_3(\mathbb{R})$. Namely, we take the theta representation of $H_3(\mathbb{R})$, and look at the action of the discrete subgroup $H_3(\mathbb{Z})$. Then $\vartheta$ is invariant under this action.

By Stone–von Neumann, the theta representation is isomorphic to the Weyl representation of the Heisenberg group on $L^2(\mathbb{R})$. This seems to imply that there should be a function in $L^2(\mathbb{R})$ analogous to the Jacobi theta function. Using the Segal–Bargmann transform, one can try calculating the corresponding function explicitly, but the integrals involved seem to get very messy.

Question. Is there any short, explicit description of the function $f\in L^2(\mathbb{R})$ corresponding to $z\mapsto \vartheta(z;\tau)$?

Answer 1

As noted in the comments, the element you're seeking is not in $L^2(\mathbb{R})$. Rather, it is a distribution (aka generalized function) that acts on Schwarz functions. Since you know it has to be invariant by $H_3(\mathbb{Z})$, and unique up to scale, it is immediate to verify that the answer is the Dirac comb: a $\delta$-function at each integer.