Regularity of matrix coefficients of a compact symmetric pair of Lie groups

We consider Gelfand pairs (G, K) (G, K) where G G is a compact Lie group and K K a subgroup of fixed points of an involutive automorphism. We study the regularity of K K -bi-invariant matrix coefficients of unitary representations of G G. The results rely on the analysis of the spherical functions of the Gelfand pair (G, K) (G, K). When the symmetric space G / K G/K is of rank 1 1 or isomorphic to a Lie group, we find the optimal regularity of K K -bi-invariant matrix coefficients of unitary representations. Furthermore, in rank 1 1 we also find the optimal regularity of K K -bi-invariant Herz-Schur multipliers of S p (L 2 (G) ) Sₚ (L² (G) ). We also give a lower bound for the optimal regularity in some families of higher rank symmetric spaces. From these results, we make a conjecture in the general case involving the root system of the symmetric space. Finally, we prove that if all K K -bi-invariant matrix coefficients of unitary representations of G G are α -Hölder continuous for some α > 0 >0, then all K K -finite matrix coefficients of unitary representations are also α -Hölder continuous.