  Let $G$ be a split reductive group over a local field $\bK$, and let $G((t))$
be the corresponding loop group. In \cite{GK} we have introduced the notion of
a representation of (the group of $\bK$-points) of $G((t))$ on a pro-vector
space. In addition, we have defined an induction procedure, which produced
$G((t))$-representations from usual smooth representations of $G$. We have
conjectured that the induction of a cuspidal irreducible representation of $G$
is irreducible. In this paper we prove this conjecture for $G=SL_2$.
