Cusp Forms and cuspidal and non-cuspidal discrete series on hyperbolic spaces
Nils Byrial Andersen, The Mads Clausen Institute, University of Southern Denmark, Denmark
Abstract
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.