The Cauchy Harish-Chandra character correspondence conjecture

Let $(\widetilde{\mathrm{G}},\widetilde{\mathrm{G}'})$ be a reductive dual pair over a non-archimedean local field, with $\Pi'$ an irreducible admissible representation of $\widetilde{\mathrm{G}'}$ occurring in Howe's correspondence and $\Pi_1$ its corresponding maximal Howe quotient representation of $\widetilde{\mathrm{G}}$. Write $\Theta_{\Pi'}$ for the distribution character of $\Pi'$, and let $\mathop{\text{Chc}}\nolimits_{\tilde h'}$ denote the Cauchy Harish-Chandra integral kernel. For a test function $\Psi\in C_c(\widetilde{\mathrm{G}})$, assume the integral below is finite:

$$\int_{{\mathrm{H}'}^{\mathrm{reg}}}\left|\Theta_{\Pi'}(\tilde h')D(h')\mathop{\text{Chc}}\nolimits_{\tilde h'}(\Psi)\right|\,d h'<\infty.$$

Define $\check\Theta_{\Pi'}(\tilde h')=\Theta_{\Pi'}(\tilde h'^{-1})$ and

$$C_{\Pi'}=\bigl(\text{the central character of $\Pi'$ evaluated at $\widetilde{-1}$}\bigr)^{-1}\cdot\Theta(\widetilde{-1}).$$

Cauchy Harish-Chandra character correspondence conjecture. Assuming the character conjecture referred to in the source, define

$$\Theta_{\Pi'}'(\Psi)=C_{\Pi'}\sum_{\mathrm{H}'}\frac{1}{|W(\mathrm{H}')|}\int_{{\mathrm{H}'}^{\mathrm{reg}}}\check\Theta_{\Pi'}(\tilde h')|D(h')|\frac{1}{\mu(\mathrm{H}'/\mathrm{A}') }\mathop{\text{Chc}}\nolimits_{\tilde h'}(\Psi)\,d\tilde h'$$

Then, as distributions, $\Theta_{\Pi'}'=\Theta_{\Pi_1}$.

This conjecture proposes that the Cauchy Harish-Chandra integral transforms the character of a representation on the smaller member of the dual pair into the character of its Howe correspondent. The preceding prose merely motivates the conjectural construction and does not state a separate mathematical claim.