Geodesic
A Sharp Criterion for the Existence of the Density in the Product Formula on Symmetric Spaces of Type An
Journal of Lie theory, Tome 20 (2010) no. 4, pp. 751-766.

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\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 ${\bf E}_6/{\bf F}_4$.
Classification : 43A90, 53C35, 15A18
Mots-clés : Product formula, convolution of measures, semisimple Lie groups 
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     author = {P. Graczyk and P. Sawyer },
     title = {A {Sharp} {Criterion} for the {Existence} of the {Density} in the {Product} {Formula} on {Symmetric} {Spaces} of {Type} {A\protect\textsubscript{n}}},
     journal = {Journal of Lie theory},
     pages = {751--766},
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     volume = {20},
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P. Graczyk; P. Sawyer . A Sharp Criterion for the Existence of the Density in the Product Formula on Symmetric Spaces of Type An. Journal of Lie theory, Tome 20 (2010) no. 4, pp. 751-766. http://geodesic.mathdoc.fr/item/JLT_2010_20_4_JLT_2010_20_4_a7/