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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(R) on the upper half plane into irreducible components. We consider an eigenvalue 1+s^2 of the Casimir operator: dρ_k(C)=-4v^2(_u^2+_v^2), where k=0. To find the inversion formula, first we study the representation of SL_2(R), ρ_k and ρ_τ, induced from the complex characters of K and N respectively. Then, we find the induced covariant transform W_φ_0 ^ρ_k with N-eigenvector to obtain a transform in the space L2(SL_2(R/N). Thereafter, we compute the contravariant transform with K-eigenvector M_φ_0 ^ρ_τ :L2(SL_2(R)/N)→ L2(SL_2(R)/K). , number=3, journal=European Journal of Pure and Applied Mathematics, year=2024, month=Jul., pages=2092–2105