On the Completeness of the System of Projections for the Tensor Product Decomposition of Continuous Series Representations of the Group SL(2, ℝ)

Abstract

As is well known, in the case of the group SL(2, ℝ), the tensor product of two continuous series representations can be decomposed into a direct sum of representations corresponding to the discrete and continuous spectra. The general theory implies the completeness of the system of projections that realize this decomposition. The main purpose of this paper is to check the corresponding relation in the sense of generalized functions. Performing the calculations, we develop a technique for working with projections, in particular, construct operators that realize the unitary equivalence between representations. Our results can be useful in various applications, for example, in calculating 6j-symbols.