Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces

We study the absolute continuity of the measures and of on the Riemannian symmetric spaces X of noncompact type for nonzero elements Xj, X∈a. For m,l⩾r+1, where r is the rank of X, the considered convolutions have a density. We conjecture that the condition m,l⩾r+1 is necessary. The conjecture is proved for the symmetric spaces of type An−1. Moreover, the minimal value of l is determined, in function of the irregularity of X.