Pointwise Estimates of the Size of Characters of Compact Lie Groups

Pointwise bounds for characters of representations of the classical, compact, connected, simple Lie groups are obtained which allow us to study the singularity of central measures. For example, we find the minimal integer k such that any continuous orbital measure convolved with itself k times belongs to L. We also prove that if k = rankG then μ ∈ L for all central, continuous measures μ. This improves upon the known classical result which required the exponent to be the dimension of the group G.