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Some Convexity Results for the Cartan Decomposition

Published online by Cambridge University Press:  20 November 2018

P. Graczyk  and
P. Sawyer
P. Graczyk
Affiliation:
Département de mathématiques, Université d'Angers, 2, boulevard Lavoisier, 49045 Angers cedex 01, France
P. Sawyer
Affiliation:
Department of Mathematics and Computer Science, Laurentian University, Sudbury, Ontario, P3E 2C6
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Abstract

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In this paper, we consider the set $\text{S}=a\left( {{e}^{X}}K{{e}^{Y}} \right)$ where $a\left( g \right)$ 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 $\text{SL}\left( 3,\,\mathbf{F} \right)\,\text{where}\,\mathbf{F}\,=\,\mathbf{R},\,\mathbf{C}\,\text{or}\,\mathbf{H}$. In particular, we show that $\text{S}$ 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

43A9053C3515A18convexity theoremsCartan decompositionspherical functionsproduct formulasemisimple Lie groupssingular values
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 5 , 01 October 2003 , pp. 1000 - 1018
Copyright
Copyright © Canadian Mathematical Society 2003

References

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