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

Abstract

In the preceding chapter we have introduced representations T_χ, x = (τ, ε), of the group SU(1,1) and have studied their matrix elements in the basis e^inθ which diagonalizes the operators T_χ(g(t)), g(t) = diag(e^it/2, e^-it/2). Now we study other realizations of these representations. It will be convenient for us to consider representations T_χ 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).