Correspondence on characters of irreducible unitary representations of \(GL(n,\mathbb{C})\) (Q1923241)

\textit{D. A. Kazhdan} and \textit{S. J. Patterson} [Adv. Math. 60, 161-234 (1986; Zbl 0616.10023)] conjectured the existence of a general correspondence between irreducible representations of general linear groups and irreducible representations of metaplectic coverings, which satisfy a certain character identity. We show that such correspondence in the case of complex general linear groups carries an irreducible unitary representation either to an irreducible unitary representation up to a sign or to 0. Moreover, we obtain an explicit formula for the correspondence in this case.