Abstract: Spherical functions have always played an important role in harmonic analysis starting with the works of Legendre and Laplace on spherical harmonics on the sphere in the late 18th century. In the modern representation theoretic framework they showed up around 1930 in the work of Cartan and Weyl on compact symmetric spaces and then around 1950 in the works of Gelfand, Harish-Chandra and others for non-compact symmetric spaces.

I will review the classical examples of spherical functions and generalizations thereof, which are important in the study of automorphic forms. In the theory of automorphic forms, spherical functions are one of several ingredients in the local theory. As an application we will discuss a higher rank analogue of a result of Waldspurger that relates a sum of an automorphic form over CM-points to a special value of an L-function at the center of symmetry.

This is joint work with E. Lapid.