CONVOLUTION OF ORBITAL MEASURES ON SYMMETRIC SPACES OF TYPE AND

Abstract

AbstractWe study the absolute continuity of the convolution ${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$ of two orbital measures on the symmetric spaces $\mathbf{SO}_{0}(p,p)/\mathbf{SO}(p)\times \mathbf{SO}(p)$, $\mathbf{SU}(p,p)/\mathbf{S}(\mathbf{U}(p)\times \mathbf{U}(p))$ and $\mathbf{Sp}(p,p)/\mathbf{Sp}(p)\times \mathbf{Sp}(p)$. We prove sharp conditions on $X$, $Y\in \mathfrak{a}$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.