LAREMA Prépublication n° 144

Piotr GRACZYK, Patrice SAWYER

### Some convexity results for the Cartan decomposition

In this paper, we consider the set $\S=a (e^X\,K\,e^Y)$ where $a (g)$ is the abelian part in the Cartan decomposition of $g$. This is exactly the support of the measure intervening in the product formula for the spherical functions on symmetric spaces of noncompact type. We give a simple description of that support in the case of ${\bf SL} (3,\F)$ where $\F=\R$, $\C$ or $\H$. In particular, we show that $\S$ is convex. We also give an application of our result to the description of singular values of a product of two arbitrary matrices with prescribed singular values.

Mots Clés
Convexity theorems ; Cartan decomposition ; spherical functions ; product formula ; semisimple Lie groups ; singular values

Codes MSC