On the product formula on noncompact Grassmannians
Volume 133 / 2013
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 space ${\bf SO}_0(p,q)/{\bf SO}(p)\times {\bf SO}(q)$, $q>p$. 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. We show that the sharp criterion developed for ${\bf SO}_0(p,q)/{\bf SO}(p)\times {\bf SO}(q)$ also serves for the spaces ${\bf SU}(p,q)/{\bf S}({\bf U}(p)\times {\bf U}(q))$ and ${\bf Sp}(p,q)/{\bf Sp}(p)\times {\bf Sp}(q)$, $q>p$. We moreover apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta _{e^X}^\natural $.
