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

We prove that if μ_a = m_k*δ_a*m_K is the K-bi-invariant measure supported on the double coset K a K {succeeds or equal to} SU (n), for K = SO(n), then μ_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 μ_a^n-1 is singular to the Haar measure.