Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if ${\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}}$ is the K-bi-invariant measure supported on the double coset ${KaK\subseteq SU(n)}$ , for K = SO(n), then ${\mu _{a}^{k}}$ is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that ${\mu _{a}^{n-1}}$ is singular to the Haar measure.