Regularity of K-finite matrix coefficients of semisimple Lie groups

We consider G a semisimple Lie group 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 κ(G) such that all such coefficients are κ(G)-H\"older 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. Using the same tools, we study the regularity of K-finite coefficients of unitary representations of U, improving on previous results obtained by the author.