$L^{p}$-estimates for matrix coefficients of irreducible representations of compact groups

Abstract:

The following result is proved. Theorem. Let G be a compact connected semisimple Lie group. For any $p > 0$ there exist two positive numbers ${A_p}$ and ${B_p}$ such that (up to equivalence) for any continuous irreducible unitary representation $\pi$ of G there exists a matrix coefficient ${a_\pi }$ of $\pi$ such that \[ {A_p} < {d_\pi }\int _G {|{a_\pi }{|^p} < {B_p}} \] where ${d_\pi }$ is the degree of $\pi$. As an application we show the nonexistence of infinite local ${\Lambda _q}$-sets.