Note on the Classification of the Unitary $\rm{SL}_2(\mathbb{R})$ Representations

Abstract

The aim of this paper is to explain the classification of the unitary SL_2(R) representations done by Gelfand by using the induced representation technique. We induce the SL_2(R) representation from the subgroup N. We get a representation constructed on a space of homogeneous functions in two variables. Then, we move to induce the SL_2(R) representation in stages. Consequently, the representation of SL_2(R) acts on a space of functions of one variable.