Cusp Forms and cuspidal and non-cuspidal discrete series on hyperbolic spaces
Nils Byrial Andersen, The Mads Clausen Institute, University of Southern Denmark, Denmark
A function f on a group G is called a cusp form if f belongs to the kernel of a certain family of Radon transforms on G. Cusp forms were defined on semi-simple Lie groups by Harish-Chandra, who also showed that all discrete series for semi-simple Lie groups were cuspidal, in the sense that a Schwartz function f belongs to the discrete series if, and only if, f is a cusp form. The notion of cusp forms has now also become a very important research area for p-adic groups.
Flensted-Jensen has recently proposed a new family of Radon transforms and associated Cusp forms on Reductive Symmetric Spaces. For Reductive Lie groups, the group case, the new definition reduces to the Harish-Chandra definition. In this talk, we will discuss the case of the Hyperbolic Spaces. Here new phenomena occur, for example the existence of non-cuspidal discrete series .
This is joint work with Mogens Flensted-Jensen and Henrik Schlichtkrull.