Asymptotic $K$-support and restrictions of representations

The restriction, from a compact Lie group $K$ to a closed subgroup, of a polynomially bounded representation remains polynomially bounded provided a geometric assumption on the asymptotic $K$-support of the representation is satisfied. This is a theorem of T. Kobayashi. We give a proof of this theorem using microlocal analysis in the setting of distribution rather than hyperfunction theory. The proof is based on a characterization, up to the natural $K\times K$ action, of the wavefront set of a distribution on $K$ in terms of the asymptotic behavior of its Fourier coefficients.