On Kazhdan–Yom Din asymptotic orthogonalityfor K-finite matrix coefficients of tempered representations

conjectured the validity of an asymptotic form of Schur orthogonality for tempered, irreducible, unitary representations of semisimple groups defined over local fields.In the non-Archimedean case, they established it for K -finite matrix coefficients.In this article we prove the analogous result in the Archimedean case. 1. Introduction 23 2. Recollections on representation theory 31 3. Asymptotic behaviour of representations 42 3.1.Asymptotic expansions of matrix coefficients 42 3.2.The functions ,l 47 3.3.Some properties of the functions ,l 51 3.4.The functions ,l as intertwining operators 54 4. Asymptotic orthogonality 62 Acknowledgements 70 References 71 Recently, D. Kazhdan and A. Yom Din [10] conjectured the validity of an asymptotic version of Schur orthogonality relations.It should hold for matrix coefficients of tempered, irreducible, unitary representations of G, generalising well-known Schur orthogonality relations for discrete series.