Regularity of K-finite matrix coefficients of semisimple Lie groups

We consider a semisimple Lie group G with finite center and K a maximal compact subgroup of G. We study the regularity of K-finite matrix coefficients of unitary representations of G. More precisely, we find the optimal value \(\kappa (G)\) $\kappa (G)$ such that all such coefficients are \(\kappa (G)\) $\kappa (G)$ -Hölder continuous. The proof relies on analysis of spherical functions of the symmetric Gelfand pair (G, K),  using stationary phase estimates from Duistermaat, Kolk and Varadarajan. If U is a compact form of G,  then (U, K) is a compact symmetric pair. Developing similar tools in this context, we study the regularity of K-finite coefficients of unitary representations of U,  improving on previous results obtained by the author.