Lien : Université d'Angers LAREMA Lien : CNRS

Prépublication n° 144


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
43A90 Spherical functions [See also 22E45, 22E46, 33C65]
53C35 Symmetric spaces [See also 32M15, 57T15]
15A18 Eigenvalues, singular values, and eigenvectors

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Date d'enregistrement : 31 janvier 2002

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