Induced representations, reproducing kernels and the conformal group

Abstract

The properties of “induced” (or multiplier) representations of groups which act in Hilbert spaces with a reproducing kernel are investigated. A resume of earlier work is followed by a discussion of criteria for the irreducibility of such representations. The notions of reproducing kernel and positive definite spherical function are found to overlap. As a result, functional equations (analogous to those of Godement for spherical functions) are found for the reproducing kernel. The abstract theory is illustrated by certain discrete series representations of the conformal group and by their “limit points”. In particular the so-called ladder representations (which give rise to the conformal symmetry of zero mass particles) are analysed from this viewpoint.