Special Representations of the Groups $U(\infty,1)$ and $O(\infty,1)$ and the Associated Representations of the Current Groups $U(\infty,1)^X$ and $O(\infty,1)^X$ in Quasi-Poisson Spaces

A method for constructing representations of the current groups $O(n,1)^X$ and $U(n,1)^X$, $n\in\mathbb N$, developed in the previous papers by the authors is generalized to the case of infinite $n$. This leads to an interesting difference in construction (absent for finite $n$) between the cases of the orthogonal and unitary groups, which is due to the different character of special representations of the groups of coefficients.