Convolution of orbital measures on symmetric spaces of type $C_p$ and $D_p$

Abstract: We study the absolute continuity of the convolution $\delta_{e^X}^\natural \star\delta_{e^Y}^\natural$ of two orbital measures on the symmetric spaces ${\bf SO}_0(p,p)/{\bf SO}(p)\times{\bf SO}(p)$, $\SU(p,p)/{\bf S}({\bf U}(p)\times{\bf U}(p))$ and $\Sp(p,p)/{\bf Sp }(p)\times\Sp(p)$. We prove sharp conditions on $X$, $Y\in\a$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.