${\rm SL}_2({\mathbb R})$

Abstract

The group ${\rm SL}_2({\mathbb R})$ is the simplest case of a so called reductive Lie group. Harmonic analysis on these groups turns out to be more complex then the previous cases of abelian, compact, or nilpotent groups. On the other hand, the applications are more rewarding. For example, via the theory of automorphic forms, in particular the Langlands program, harmonic analysis on reductive groups has become vital for number theory. In this chapter we prove an explicit Plancherel Theorem for functions in the Hecke algebra of the group $G={\rm SL}_2({\mathbb R})$. We apply the trace formula to a uniform lattice and as an application derive the analytic continuation of the Selberg zeta function.