Integral models of representations of the current groups of simple Lie groups

For the class of locally compact groups P that can be written as the semidirect product of a locally compact subgroup P0 and a oneparameter group R+ of automorphisms of P0, a new model of representations of the current groups P is constructed. The construction is applied to the maximal parabolic subgroups of all simple groups of rank 1. In the case of the groups G = SO(n, 1) and G = SU(n, 1), an extension is constructed of representations of the current groups of their maximal parabolic subgroups to representations of the current groups G . The key role in the construction is played by a certain σ-finite measure (the infinitedimensional Lebesgue measure) in the space of distributions.