Poisson and Fourier transforms for tensor products (Q280795)

The author studies the decomposition of tensor products of finite dimensional simple \(\mathrm{SL}(2,\mathbb{R})\)-modules. The main result of the paper gives explicit differential operators (which the author calls \textit{Poisson} and \textit{Fourier transforms}) that intertwine a fixed simple \(\mathrm{SL}(2,\mathbb{R})\)-module with a tensor product of two simple modules. The author describes an analogue of harmonic analysis and obtains explicit expressions for the composition of these transforms with certain Lie operators of the overgroup. The constructions are based on differential-difference relations for the Poisson kernel.