Fourier Inversion for Unipotent Invariant Integrals

Abstract

Consider G a semisimple Lie group and $\Gamma \subseteq G$ a discrete subgroup such that ${\text {vol(}}G/\Gamma ) < \infty$. An important problem for number theory and representation theory is to find the decomposition of ${L^2}(G/\Gamma )$ into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of ${L^2}(G/\Gamma )$ in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank $G = 1$. In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.