Decomposition of the Unitary Representation of SL₂(R) on the Upper Half Plane into Irreducible Components

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 SL₂(R) on the upper half plane into irreducible components. We consider an eigenvalue 1+s² of the Casimir operator: dρₖ(C)=-4v²(∂ᵤ²+∂ᵥ²), where\, k=0. To find the inversion formula, first we study the representation of SL₂(R), ρₖ and ρτ, induced from the complex characters of K and N respectively. Then, we find the induced covariant transform W_φ₀^ρₖ with N-eigenvector to obtain a transform in the space L2(SL₂(R/N). Thereafter, we compute the contravariant transform with K-eigenvector M_φ₀^ρτ:L2(SL₂(R)/N)→ L2(SL₂(R)/K).