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

TL;DR

This paper proves a conjecture by Przebinda that the transfer of characters in Howe's correspondence for dual pairs of unitary groups with equal rank can be achieved using the Cauchy-Harish-Chandra integral, specifically for discrete series representations.

Contribution

It establishes the validity of the character transfer conjecture via the Cauchy-Harish-Chandra integral for dual pairs of unitary groups in the equal rank case, confirming a key aspect of Howe's correspondence.

Findings

Confirmed the conjecture for dual pairs (U(p,q), U(r,s)) with p+q = r+s.

Demonstrated the transfer of characters for discrete series representations.

Validated the use of the Cauchy-Harish-Chandra integral in this context.

Abstract

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 $(G = U(p, q), G' = U(r, s))$, $p+q = r+s$, starting with a discrete series representation $\Pi$ of $\widetilde{U}(p, q)$.