Cartan motion groups: regularity of K -finite matrix coefficients

Abstract If G is a connected semisimple Lie group with finite center and K is a maximal compact subgroup of G , then the Lie algebra of G admits a Cartan decomposition $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ . This allows us to define the Cartan motion group $H=\mathfrak {p}\rtimes K$ . In this article, we study the regularity of K -finite matrix coefficients of unitary representations of H . We prove that the optimal exponent $\kappa (G)$ for which all such coefficients are $\kappa (G)$ -Hölder continuous coincides with the optimal regularity of all K -finite coefficients of the group G itself. Our approach relies on stationary phase techniques that were previously employed by the author to study regularity in the setting of $(G,K)$ . Furthermore, we provide a general framework to reduce the question of regularity from K -finite coefficients to K -bi-invariant coefficients.