A Laplace transform on the Lorentz groups

Abstract

As a first step in the generalisation of the Laplace transform to a non abelian group, we examine the representations of the groupsSO(n, 1) by means of transformations of (not necessarily integrable) functions defined over the hyperboloidsO(n, 1)/O(n). We define a regularised version of the Gel'fand-Graev transformation from then-dimensional hyperboloid to its associated cone, which is valid (under certain restrictions) for polynomially bounded functions. Upon the cone we then carry out a pair of classical Laplace transforms parallel to a generator. We give inversion formulas for both these procedures, and express the Laplace transform/inversion pair directly in terms of the function on the hyperboloid.

For integrable functions our results reduce to those already known; in the nonintegrable case they are new. New features include the divergence of the transform for certain discrete asymptotic behaviours; the existence of a finite dimensional kernel subspace which is annihilated; good asymptotic behaviour of both Laplace projection and inversion formulas; and the existence of discrete terms contributing to the inversion formula for even dimension. Our results are valid for all dimensions and are completely independent of the usual “Laplace transforms” involving projection by means of “second-kind representation functions”; in a final section of the paper we examine briefly the significance of that approach in the light of our own.