On the failure of spectral synthesis for compact semisimple Lie groups

Let G be a compact connected semisimple Lie group with Lie algebra $\text{g}$. We show that the conjugacy class of a regular element of G is not a set of synthesis for the Fourier algebra of G. Similarly, the Ad(G)-orbit of a regular element of $\text{g}$ is not a set of synthesis for the algebra of Fourier transforms on $\text{g}$. In proving this latter result we demonstrate a regularity property of Ad-invariant Fourier transforms, analogous to the differentiability of radial Fourier transforms.