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 one-parameter group R*+ of automorphisms of P0, a new model of representations of the current groups PX 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 GX. The key role in the construction is played by a certain σ-finite measure (the infinite-dimensional Lebesgue measure) in the space of distributions. Bibliography: 32 titles.