Representation on ⁰L²(Г\G)

Abstract

The algebraic and arithmetic properties of SL ₂ begin to be felt when we consider the representation on Г\G for some discrete subgroup Г. In this chapter, after a general discussion of the nature of the factor space Г\G or Г\G/K = Г\ℌ, which is essentially classical, we prove that on a certain subspace ⁰L²(Г\G) of L²(Г\G) the representation is completely reducible when Г = SL ₂(Z). The method works just as well for any “arithmetic” subgroup, i.e. a subgroup of finite index in SL ₂(Z). It uses the Poisson summation formula, in addition to some estimates. It has the advantage of being very rapid and of using a minimum of analysis.