On the product formula on noncompact Grassmannians

Volume 133 / 2013

Piotr Graczyk, Patrice Sawyer Colloquium Mathematicum 133 (2013), 145-167 MSC: Primary 43A90; Secondary 53C35. DOI: 10.4064/cm133-2-1


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 $.


  • Piotr GraczykLaboratoire de Mathématiques LAREMA
    Université d'Angers
    49045 Angers, France
  • Patrice SawyerDepartment of Mathematics
    and Computer Science
    Laurentian University
    P3E 2C6 Sudbury, Ontario, Canada

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