On Kazhdan-Yom Din asymptotic Schur orthogonality for K-finite matrix coefficients

In a recent article, D. Kazhdan and A. Yom Din conjectured the validity of an asymptotic form of Schur’s orthogonality for tempered irreducible unitary representations of semisimple groups defined over local fields. In the non-Archimedean case, they established such an orthogonality for $K$-finite matrix coefficients. Building on their work, and exploiting the admissibility of irreducible unitary representations, we prove the analogous result in the Archimedean case.