Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 20 (2010), No. 4, 751--766Copyright Heldermann Verlag 2010 A Sharp Criterion for the Existence of the Density in the Product Formula on Symmetric Spaces of Type An Piotr Graczyk Dép. de Mathématiques, Université d'Angers, 2 Boulevard Lavoisier, 49045 Angers, France piotr.graczyk@univ-angers.fr Patrice Sawyer Dept. of Mathematics and Computer Science, Laurentian University, Sudbury, Ontario, Canada P3E 5C6 psawyer@laurentian.ca [Abstract-pdf] \def\a{{\frak a}} \def\C{{\Bbb C}} \def\F{{\Bbb F}} \def\H{{\Bbb H}} \def\R{{\Bbb R}} We find sharp conditions on $X,Y\in\a$ for the existence of the density of the measure $\delta_{e^X}^\natural \star \delta_{e^Y}^\natural$ intervening in the product formula for the spherical functions on the symmetric spaces of noncompact type ${\bf X}={\bf SL}(n,\F)/{\bf SU}(n,\F)$ where $\F=\R$, $\C$ or $\H$. Our results also apply to the symmetric space ${\bf E}_6/{\bf F}_4$. Keywords: Product formula, convolution of measures, semisimple Lie groups. MSC: 43A90, 53C35, 15A18 [ Fulltext-pdf  (222  KB)] for subscribers only.