Functional transforms and their kernels on symmetric spaces and Lie groups

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A symmetric space refers, in plain words, to the kind of "universe" I am studying. The general context of my work is harmonic analysis of this universe. The basic idea of harmonic analysis, itself a generalization of Fourier analysis, is to represent general functions in terms of more basic ones, i.e. spherical functions. Historically, such studies concerned spaces such as the real line (the one-dimensional space); the plane (the two-dimensional universe); or space (the three-dimensional universe). A natural extension was then the Euclidean n-dimensional space. There are more complicated objects mathematicians want to work with (the sphere being a natural example). Symmetric spaces are sufficiently complex objects to be of interest as a generalization, while sufficiently well-behaved to make their study relevant and interesting (not to mention accessible!). Not only can one consider questions such as the heat diffusion on a symmetric space, but it also makes sense to study probability theory using such objects. As well, the behaviour of the basic functions i.e. the spherical functions becomes important. Natural assumptions as to how a well-known phenomenon in the Euclidean setting will behave in a different situation are either confirmed or contradicted. These questions can be addressed by finding explicit formulae for the spherical functions (when possible) or by using general principles.