On the completeness of projectors for tensor product decomposition of continuous series representations groups $SL (2,\mathbb{R})$

As is well known, the tensor product of two representations of continuous series in the case of the $SL(2,\mathbb{R})$ group can be decomposed into a direct sum of representations corresponding to the discrete and continuous spectrum. The completeness of the projectors performing decomposition also follows from the general theory. The main aim of the work is to check this equality in a particular case in the sense of generalized functions. It is worth noting that in the course of calculations, a technique for working with projectors is developed, in particular, operators for unitary equivalence are built. This work can be useful in various applications, for example, for calculating of $6j$-symbols.