The $(\sigma,\rho)$-Poisson summation formula

Let $k$ be a number field, let $G$ be a $k$-split reductive group, and let $\rho\colon G^\vee(\mathbb{C})\to\operatorname{GL}_n(\mathbb{C})$ be a finite-dimensional representation. For $\sigma\in\mathcal{A}_{\mathrm{cusp}}(G)$, let $\mathcal{S}_{\sigma,\rho}(\mathbb{A}^\times)$ and $\mathcal{S}_{\widetilde{\sigma},\rho}(\mathbb{A}^\times)$ be the associated Schwartz spaces, let $\mathcal{F}_{\sigma,\rho,\psi}$ be the associated Fourier operator, and write $\Theta_{\sigma,\rho}(x,\phi)=\sum_{\alpha\in k^\times}\phi(\alpha x)$. Then there exist $k^\times$-invariant linear functionals $\mathcal{E}_{\sigma,\rho}$ and $\mathcal{E}_{\widetilde{\sigma},\rho}$ on the respective Schwartz spaces such that $(\sigma,\rho)$-Poisson summation formula.

$$\mathcal{E}_{\sigma,\rho}(\phi)=\mathcal{E}_{\widetilde{\sigma},\rho}(\mathcal{F}_{\sigma,\rho,\psi}(\phi))$$

for every $\phi\in\mathcal{S}_{\sigma,\rho}(\mathbb{A}^\times)$. Moreover, if $\phi\in\mathcal{S}_{\sigma,\rho}^{\circ\circ}(\mathbb{A}^\times)$, then

$$\mathcal{E}_{\sigma,\rho}(\phi^x)=\Theta_{\sigma,\rho}(x,\phi)=\sum_{\alpha\in k^\times}\phi(\alpha x),$$

where $\phi^x(\cdot)=\phi(\cdot x)$ for $x\in\mathbb{A}^\times$. This is the conjectural global Poisson summation formula underlying the proposed Fourier analysis on $\mathrm{GL}_1$ and the associated automorphic $L$-functions; its status is not resolved in the supplied source.