On the Kronecker products of irreducible representations of the 2×2 real unimodular group. I

Abstract

Introduction. The purpose of the present paper is to determine the decomposition of the Kronecker product of two irreducible representations of the real 2X2 unimodular group into a continuous direct sum of irreducible representations. The irreducible unitary representations of this group have been determined first by V. A. Bargmann [l](1), and those of the 2X2 complex unimodular group by I. M. Gel'fand and M. A. Nalmark [3]. In both cases the list of these representations contains two continuous series; first, the principal continuous series, the members of which can be described by a pair (m, p) of two variables, m with a discrete, p with a continuous range; and secondly, the representations of the exceptional interval, characterized by a single parameter, varying over a finite interval. In the real case in addition to these there exists a discrete series of representations characterized by integers. Concerning the representations of the exceptional interval it has been proved that they do not occur in the decomposition of the left regular representations of these groups into a continuous direct sum of irreducible representations. The problem of finding the irreducible parts for the Kronecker product of two of these representations by the Reduction Theory of von Neumann [9] was taken up first by G. W. Mackey, in the complex case, for two factors taken from the principal series [4; 5]. W. F. Stinespring applied the same method to the discussion of the analogous case for the real group(2). Recently, M. A. Nalmark attacked the same problem in the complex case, and gives a complete discussion of all possibilities [10](3). In Parts I, II, and III of the present work we give the decomposition of the product of any two irreducible unitary representations of the real 2X2 unimodular group. To sketch our method, we restrict ourselves, for the sake