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We characterize the absolute continuity of convolution products of orbitalmeasures on the classical, irreducible Riemannian symmetric spaces $G/K$ ofCartan type $III$, where $G$ is a non-compact, connected Lie group and $K$ is acompact, connected subgroup. By the orbital measures, we mean the uniformmeasures supported on the double cosets, $KzK,$ in $G$. The characterizationcan be expressed in terms of dimensions of eigenspaces or combinatorialproperties of the annihilating roots of the elements $z$.  A consequence of our work is to show that the convolution product of anyrank% $G/K,$ continuous, $K$-bi-invariant measures is absolutely continuous inany of these symmetric spaces, other than those whose restricted root system istype $A_{n}$ or $D_{3}$, when rank$G/K$ $+1$ is needed.

The absolute continuity of convolutions of orbital measures in symmetric spaces