The Continuous Part of L²(Г\G)

Abstract

We now look at the orthogonal complement of ⁰ L ²(Г\G) and prove a spectral decomposition theorem following Godemenťs paper [Go 2], using the Poisson summation formula. The method works for arithmetic subgroups Г, and has the advantage of being rapid and easy. It fails for more general discrete subgroups, and the question is reconsidered by other methods in the next and last chapter. The spectral decomposition is achieved by the Eisenstein transform, which maps the orthogonal complement of ⁰ L ²(Г\G) and the constant functions on the L ² space of a positive real line—with our normalization, the upper half of the imaginary line Re s = 1/2.