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

We prove that if m_a = m_K*d_a*m_K{mu _{a},{=},m_{K}*delta _{a}*m_{K}} is the K-bi-invariant measure supported on the double coset KaK í SU(n){KaKsubseteq SU(n)} , for K = SO(n), then m_a^k{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 m_a^n-1{mu _{a}^{n-1}} is singular to the Haar measure.