Decomposition of the Unitary Representation of $SL_{2}(\mathbb{R})$ on the Upper Half Plane into Irreducible Components

Abstract

The main purpose of the paper is to find the inversion formula for the covariant transform. This formula is equivalent to the decomposition of the unitary representation of $SL_{2}(\mathbb{R})$ on the upper half plane into irreducible components. We consider an eigenvalue $1+s^{2}$ of the Casimir operator: $$d\rho_{k}(C)=-4v^{2}\left(\partial_{u}^{2}+\partial_{v}^{2}\right),\quad \text{where}\, k=0.$$ To find the inversion formula, first we study the representation of  $SL_{2}(\mathbb{R})$, $\rho_{k}$ and $\rho_{\tau}$, induced from the complex characters of $K$ and $N$ respectively. Then, we find the induced covariant transform $\mathcal{W}_{\varphi_{0}}^{\rho_{k}}$ with $N$-eigenvector to obtain a transform in the space $\FSpace{L}{2}(SL_{2}(\mathbb{R}/N)$. Thereafter, we compute the contravariant transform with $K$-eigenvector  $$\mathcal{M}_{\phi_{0}}^{\rho_{\tau}}:\FSpace{L}{2}(SL_{2}(\mathbb{R})/N)\rightarrow \FSpace{L}{2}(SL_{2}(\mathbb{R})/K).$$