Harmonic analysis on reductive symmetric spaces

Abstract

We give a relatively non-technical survey of some recent advances in the Fourier theory for semisimple symmetric spaces. There are three major results: An inversion formula for the Fourier transform, a Palley-Wiener theorem, which describes the Fourier image of the space of completely supported smooth functions, and the Planchered theorem, which describes the decomposition into irreducibles of the regular representation.