On the product formula on noncompact Grassmannians
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 $SO_0(p,q)/SO(p)\timesSO(q)$, $q>p$. 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. We show that the sharp criterion developed for $\SO_0(p,q)/\SO(p)\times\SO(q)$ will also serve for the spaces $SU(p,q)/S(U(p)\timesU(q))$ and $Sp(p,q)/Sp(p)\timesSp(q)$, $q>p$. We also apply our results to the study of absolute continuity of convolution powers of an orbital measure $\delta_{e^X}^\natural$.
 Publication:

arXiv eprints
 Pub Date:
 November 2012
 arXiv:
 arXiv:1212.0002
 Bibcode:
 2012arXiv1212.0002G
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Probability