Scalar field in $\mathrm{AdS}_2$ and representations of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$

We analyse the solutions of the Klein-Gordon equation in the universal covering space of two-dimensional anti-de Sitter spacetime for different ranges of the mass of the scalar field. For certain values of the mass parameter, we impose a suitable set of boundary conditions which make the spatial component of the Klein-Gordon operator self-adjoint. Finally, we use the transformation properties of the mode solutions under the isometry group of the theory, namely, the universal covering group of $\mathrm{SL}(2,\mathbb{R})$, in order to determine which self-adjoint boundary conditions are invariant under the action of this group and which result in positive-frequency solutions forming unitary representations of this group, which are classified in a manner similar to those for $\mathrm{SL}(2,\mathbb{R})$ given by Bargmann.