The Cauchy–Harish-Chandra character conjecture for p-adic dual pairs

Let $(\mathrm{G},\mathrm{G}')$ be a reductive dual pair over a non-Archimedean local field, let $\Pi'$ be an irreducible admissible representation of $\widetilde{\mathrm{G}'}$ occurring in Howe's correspondence, and let $\Pi_1$ be the corresponding maximal Howe quotient representation of $\widetilde{\mathrm{G}}$. Write $\Theta_{\Pi'}$ for the distribution character of $\Pi'$, and let $Z'$ and $\mathrm{G}'{}^0$ denote the center and the Zariski identity component of $\mathrm{G}'$, respectively. For a test function $\Psi\in C_c(\widetilde{\mathrm{G}})$, assume the defining integral is absolutely convergent and define $\Theta'_{\Pi'}$ by the Weyl–Harish-Chandra integration formula using the Cauchy–Harish-Chandra kernel $\mathop{\text{Chc}}\nolimits_{\tilde h'}$. Cauchy–Harish-Chandra character conjecture. If the character $\Theta_{\Pi'}$ is supported in $Z'\mathrm{G}'{}^0$, then, as distributions,

$$\Theta'_{\Pi'}=\Theta_{\Pi_1}.$$

The conjecture asserts that the p-adic Cauchy–Harish-Chandra integral recovers the character of the representation paired with $\Pi'$ by Howe correspondence; the source formulates it under the preceding convergence condition and assuming the stated character conjecture used in the construction. Its general status is not resolved in the supplied text.