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# Some Convexity Results for the Cartan Decomposition

Published:2003-10-01
Printed: Oct 2003
• P. Graczyk
• P. Sawyer
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## Abstract

In this paper, we consider the set $\mathcal{S} = a(e^X K e^Y)$ 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 $\SL(3,\mathbf{F})$ where $\mathbf{F} = \mathbf{R}$, $\mathbf{C}$ or $\mathbf{H}$. In particular, we show that $\mathcal{S}$ 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.
 Keywords: convexity theorems, Cartan decomposition, spherical functions, product formula, semisimple Lie groups, singular values
 MSC Classifications: 43A90 - Spherical functions [See also 22E45, 22E46, 33C55] 53C35 - Symmetric spaces [See also 32M15, 57T15] 15A18 - Eigenvalues, singular values, and eigenvectors

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