Some Convexity Results for the Cartan Decomposition

[摘要] 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 theproduct formula for the spherical functions on symmetric spaces ofnoncompact type. We give a simple description of that support inthe 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 ofsingular values of a product of two arbitrary matrices withprescribed singular values.

[学科分类] 数学（综合）