Representations of the group $SL(2,\mathbf R)$, where $\mathbf R$ is a ring of functions

We obtain a construction of the irreducible unitary representations of the group of continuous transformations $X\to G$, where $X$ is a compact space with a measure $m$ and $G=PSL(2,\mathbf R)$, that commute with transformations in $X$ preserving $m$. This construction is the starting point for a non-commutative theory of generalized functions (distributions). On the other hand, this approach makes it possible to treat the representations of the group of currents investigated by Streater, Araki, Parthasarathy, and Schmidt from a single point of view.