Cartan motion groups: regularity of K-finite matrix coefficients

TL;DR

This work determines the precise Hölder regularity for all $K$-finite matrix coefficients of the Cartan motion group $H=\mathfrak{p}\rtimes K$ attached to a connected semisimple group $G$ with finite center. The authors show that the optimal class is $C^{(r,\delta)}(H_r)$ with $r=\lfloor \kappa(G)\rfloor$ and $\delta=\kappa(G)-r$, where $\kappa(G)=\frac{1}{2}\min_{\lambda\neq0} n(\lambda)$ and $n(\lambda)=\sum_{\alpha\in \Sigma^+:\langle\alpha,\lambda\rangle\neq0} m(\alpha)$; every $K$-finite coefficient belongs to this class and the result is optimal. The approach combines a reduction from $K$-finite to $K$-bi-invariant coefficients with a stationary-phase analysis of spherical functions on the associated Gelfand pair $(H,K)$, and a general framework that leverages a well-behaved $KAK$ decomposition on an open dense subset $H_r$. The findings align the regularity profile of $H$ with that of the ambient group $G$ and its compact form, and they provide a practical method to transfer regularity results from $K$-bi-invariant data to the full $K$-finite setting. This sharp regularity characterization sharpens our understanding of harmonic analysis on Cartan motion groups and has potential implications for rigidity and representation-theoretic questions tied to regularity phenomena.

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 $κ(G)$ for which all such coefficients are $κ(G)$-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.

Theorems & Definitions (34)

• Theorem 1
• Definition 2.1
• Remark 2.2
• Lemma 2.3
• Definition 2.4
• Proposition 2.5
• Lemma 2.6
• Lemma 2.7
• Definition 2.8
• Lemma 2.9