Let $\mathfrak{g}$ be a compact simple Lie algebra of dimension $d$. It is a classical result that the convolution of any $d$ non-trivial, $G$-invariant, orbital measures is absolutely continuous with respect to Lebesgue measure on $\mathfrak{g}$, and the sum of any $d$ non-trivial orbits has non-empty interior. The number $d$ was later reduced to the rank of the Lie algebra (or rank +1 in the case of type ${{A}_{n}}$). More recently, the minimal integer $k\,=\,k\left( X \right)$ such that the $k$-fold convolution of the orbital measure supported on the orbit generated by $X$ is an absolutely continuous measure was calculated for each $X\,\in \,\mathfrak{g}$.

In this paper $\mathfrak{g}$ is any of the classical, compact, simple Lie algebras. We characterize the tuples $\left( {{X}_{1}},\,.\,.\,.\,,\,{{X}_{L}} \right)$, with ${{X}_{i}}\,\in \,\mathfrak{g}$, which have the property that the convolution of the $L$-orbital measures supported on the orbits generated by the ${{X}_{i}}$ is absolutely continuous, and, equivalently, the sum of their orbits has non-empty interior. The characterization depends on the Lie type of $\mathfrak{g}$ and the structure of the annihilating roots of the ${{X}_{i}}$. Such a characterization was previously known only for type ${{A}_{n}}$.

We prove that in any compact symmetric space, G/K, there is a dense set of a1,a2∈G such that if μj=mK*δaj*mk is the K-bi-invariant measure supported on KajK, then μ1*μ2 is absolutely continuous with respect to Haar measure on G. Moreover, the product of double cosets, Ka1Ka2K, has nonempty interior in G.