Representations of the Groups SU(1,1) and SL(2, ℝ) in Mixed Bases. The Hypergeometric Function

In the preceding chapter we have introduced representations Tχ, x = (τ, e), of the group SU(1,1) and have studied their matrix elements in the basis einθ which diagonalizes the operators Tχ(g(t)), g(t) = diag(eit/2, e-it/2). Now we study other realizations of these representations. It will be convenient for us to consider representations Tχ of the group SU(2, ℝ) which is isomorphic to SU(1, 1). Subgroups and decompositions, considered below, have simpler form for S L(2, ℝ).