Transfer of characters for discrete series representations of the unitary groups in the equal rank case via the Cauchy-Harish-Chandra integral

As conjectured by T. Przebinda, the transfer of characters in the Howe’s correspondence should be obtained via the Cauchy-Harish-Chandra integral. In this paper, we prove that the conjecture holds for the dual pair $(G=U(p,q),G^{\mathrm{\prime }}=U(r,s))$, $p+q=r+s$, starting with a discrete series representation $\mathrm{\Pi }$ of $\tilde{U}(p,q)$.