Spherical functions on ordered symmetric spaces

Abstract

We define on an ordered semi simple symmetric space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ℳ</mml:mi> <mml:mo>=</mml:mo> <mml:mi>G</mml:mi> <mml:mo>/</mml:mo> <mml:mi>H</mml:mi> </mml:mrow> </mml:math> a family of spherical functions by an integral formula similar to the Harish-Chandra integral formula for spherical functions on a Riemannian symmetric space of non compact type. Associated with these spherical functions we define a spherical Laplace transform. This transform carries the composition product of invariant causal kernels onto the ordinary product. We invert this transform when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is a complex group, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> a real form of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , and when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℳ</mml:mi> </mml:math> is a symmetric space of rank one.