On the product formula on noncompact Grassmannians

Piotr Graczyk ¹   ; Patrice Sawyer ²

¹ Laboratoire de Mathématiques LAREMA Université d'Angers 49045 Angers, France
² Department of Mathematics and Computer Science Laurentian University P3E 2C6 Sudbury, Ontario, Canada

Colloquium Mathematicum, Tome 133 (2013) no. 2, pp. 145-167

Résumé

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 $.

Détail

DOI : 10.4064/cm133-2-1

Keywords: study absolute continuity convolution delta natural star delta natural orbital measures symmetric space times prove sharp conditions mathfrak existence density convolution measure measure intervenes product formula spherical functions sharp criterion developed times serves spaces times times moreover apply results study absolute continuity convolution powers orbital measure delta natural