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 $P_0$ and a one-parameter group $\mathbb R^*_+$ of automorphisms of $P_0$, a new model of representations of the current groups $P^X$ is constructed. The construction is applied to the maximal parabolic subgroups of all simple groups of rank 1. In the case of the groups $G=\mathrm{SO}(n,1)$ and $G=\mathrm{SU}(n,1)$, an extension is constructed of representations of the current groups of their maximal parabolic subgroups to representations of the current groups $G^X$. The key role in the construction is played by a certain $\sigma$-finite measure (the infinite-dimensional Lebesgue measure) in the space of distributions.
Bibliography: 32 titles.