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 $H=\mathfrak{p}⋊K$. In this paper, 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.

arXiv comment21 pages. arXiv admin note: substantial text overlap with arXiv:2409.07944