Poisson and Fourier Transforms for Tensor Products

For the group $G=\operatorname{SL}(2,\mathbb{R})$, we write out explicitly differential operators intertwining irreducible finite-dimensional representations $T_k$ of $G$ with tensor products $T_{l}\otimes T_{m}$ (we call them Poisson and Fourier transforms); we also describe an analogue of harmonic analysis and write explicit expressions for compositions of these transforms with Lie operators of the overgroup $G\times G$. The constructions are based on a differential-difference relation for the Poisson kernel.