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

Let $k$ be the number field and $G$ a $k$-split reductive group, with complex dual group $G^\vee(\mathbb{C})$. Let $\rho\colon G^\vee(\mathbb{C})\to\operatorname{GL}_n(\mathbb{C})$ be a finite-dimensional representation, let $\sigma\in\mathcal{A}_{\mathrm{cusp}}(G)$, and let $\widetilde{\sigma}$ be its contragredient. Write $\mathcal{S}_{\sigma,\rho}(\mathbb{A}^{\times})$ and $\mathcal{S}_{\widetilde{\sigma},\rho}(\mathbb{A}^{\times})$ for the corresponding Schwartz spaces, and let $\mathcal{F}_{\sigma,\rho,\psi}$ be the $(\sigma,\rho)$-Fourier operator from the former to the latter. A $k^\times$-invariant linear functional is a linear functional invariant under the multiplicative action of $k^\times$. The $(\sigma,\rho)$-Poisson summation formula. For every such $\rho$ and every $\sigma\in\mathcal{A}_{\mathrm{cusp}}(G)$, there exist nontrivial $k^\times$-invariant linear functionals $\mathcal{E}_{\sigma,\rho}$ and $\mathcal{E}_{\widetilde{\sigma},\rho}$ on $\mathcal{S}_{\sigma,\rho}(\mathbb{A}^{\times})$ and $\mathcal{S}_{\widetilde{\sigma},\rho}(\mathbb{A}^{\times})$, respectively, such that

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

for every $\phi\in\mathcal{S}_{\sigma,\rho}(\mathbb{A}^{\times})$. This would generalize the classical Poisson summation formula and is intended to provide a harmonic-analytic framework for automorphic $L$-functions and the Langlands program. The source presents it as a conjectural formula; no resolution is supplied here.