Algebraic groups over a 2-dimensional local field: irreducibility of certain induced representations

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 $G=SL_2$.