A Cauchy Harish-Chandra integral for a dual pair over a \(p\)-adic field, the definition and a conjecture (Q6581814)

For an irreducible dual pair \((G,G')\) inside a symplectic group over a local field, the theta correspondence sets up a bijection \(\Pi\leftrightarrow\Pi'\) between certain sets of irreducible admissible representations \(\Pi\) and \(\Pi'\) of double covers of \(G\) and \(G'\). In the Archimedean case, the second author [Invent. Math. 141, No. 2, 299--363 (2000; Zbl 0953.22014)] introduced an integral kernel operator called \emph{Cauchy Harish-Chandra integral} which maps the distribution character \(\Theta_\Pi\) of \(\Pi\) to the distribution character \(\Theta_{\Pi'}\) of \(\Pi'\) under a certain stability assumption.\N\NThe purpose of this article is to transfer the construction of [loc. cit.] to the \(p\)-adic case. The authors provide the precise definition of the integral kernel operator and conjecture that, under certain technical assumptions, this operator relates the distribution characters of representations corresponding to each other by the theta correspondence.