Regularity of K-finite matrix coefficients of semisimple Lie groups

TL;DR

This paper determines the optimal Hölder continuity exponent for K-finite matrix coefficients of unitary representations of semisimple Lie groups, using spherical functions and stationary phase analysis.

Contribution

It establishes the precise regularity exponent for these coefficients and extends the analysis to compact forms, improving previous results.

Findings

Identified the optimal Hölder exponent κ(G) for matrix coefficients.

Demonstrated regularity results for coefficients of compact forms.

Enhanced understanding of the smoothness properties of matrix coefficients.

Abstract

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 $\kappa(G)$ such that all such coefficients are $\kappa(G)$-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.