Lp-estimates for matrix coefficients of irreducible representations of compact Lie groups

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 π of G there exists a matrix coefficient a_π of π such that A_p < d_π ∫|a_π|^p < B_p where d_π is the degree of π. As an application we show the nonexistence of infinite local Λ_q-sets.