Chapter 8 - Spherical Functions on Ordered Spaces

Abstract

This chapter describes the theory of spherical functions, and spherical Laplace transform on noncompactly causal symmetric spaces. The theory is motivated by the classical theory. The second motivation is the Harish-Chandra–Helgason theory of spherical functions on Riemannian symmetric spaces. The material is standard, but usually “Γ” is defined as the set. In that case the product for invariant kernels becomes F#G = G*F. Spherical functions on symmetric spaces of the form Gc/Gwere introduced by J. Faraut are used to diagonalize certain integral equations with symmetry and causality conditions. The proof of the theorem used the relation to the principal series, and the formula of the H-spherical character due to P. Delorme. A more explicit formula for the spherical function and the cΩ(λ) function for Cayley-type spaces was obtained by J. Faraut. The inversion formula for Laplace transform was proved by using the explicit formula for the spherical functions. An inversion formula was proved by using the Abel transform. A Laplace transform associated with the Legendre functions of the second kind was introduced. This transform was related to harmonic analysis of the unit disc. The Poisson transformation is embedded in generalized principal series representations into spaces of eigen functions on M. A further application is the construction of the spherical distributions θλ.