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Journal of Lie Theory 20 (2010), No. 4, 751--766
Copyright 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

Patrice Sawyer
Dept. of Mathematics and Computer Science, Laurentian University, Sudbury, Ontario, Canada P3E 5C6


\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

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