Cartan motion groups: regularity of K-finite matrix coefficients

TL;DR

This paper investigates the regularity of K-finite matrix coefficients of unitary representations of Cartan motion groups, establishing a connection with the regularity of the original semisimple Lie group, using stationary phase techniques.

Contribution

It demonstrates that the optimal Hölder continuity exponent for K-finite coefficients of Cartan motion groups matches that of the original group, and introduces a framework to relate regularity properties.

Findings

Optimal regularity exponent matches between Cartan motion groups and original groups.

Stationary phase techniques effectively analyze regularity of matrix coefficients.

Framework reduces regularity questions from K-finite to K-bi-invariant 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 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 $\kappa(G)$-H\"older 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.