Representations of SL(2,R) in a Hilbert space of analytic functions and a class of associated integral transforms

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Abstract
It is shown that the boson operators of SL(2,R) realized as hyperdifferential operators in Bargmann’s Hilbert space of analytic functions yield, on exponentiation, a parametrized continuum of integral transforms. Each value of the group parameters yields an integral transform pair. For the metaplectic representation the resulting integral transform is essentially the mapping of the Moshinsky–Quesne transform in Bargmann’s Hilbert spaceB(C). The formula for the inversion of this transform is obtained simply by replacing the group element by its inverse. The corresponding Hilbert space for arbitrary representations of the discrete series is B(C 2), where C 2 is the two‐dimensional complex Euclidean space. To carry out the reduction of B(C 2) into the eigenspacesB k (C) (k= 1/2 ,1, (3)/(2) ,...) of irreducible representations of the positive discrete class, the complex polar coordinates (z 1=z cos φ, z 2=z sin φ) in C 2 are introduced. The ‘‘reduced Bargmann space’’ B k (C) has many interesting features. The elements of B k (C) are entire functions of the complex ‘‘radius’’ z analytic in the upper half‐plane. In contrast to the Gaussian measure in B(C 2), the integration measure in the scalar product in B k (C) contains a modified Bessel function of the second kind. The principal vector in B k (C), on the other hand, is a modified Bessel function of the first kind. The resulting integral transform maps B k (C) onto itself and the integral kernel is the product of an exponential and a modified Bessel function of the first kind. The inversion formula for this transform is obtained again by replacing the group element by its inverse.