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 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.