  We prove a sharp general inequality estimating the distance of two
probability measures on a compact Lie group in the Wasserstein metric in terms
of their Fourier transforms. We use a generalized form of the Wasserstein
metric, related by Kantorovich duality to the family of functions with an
arbitrarily prescribed modulus of continuity. The proof is based on smoothing
with a suitable kernel, and a Fourier decay estimate for continuous functions.
As a corollary, we show that the rate of convergence of random walks on
semisimple groups in the Wasserstein metric is necessarily almost exponential,
even without assuming a spectral gap. Applications to equidistribution and
empirical measures are also given.
