  Let $F$ be a non-Archimedean local field and let $p$ be the residual
characteristic of $F$. Let $G=GL_2(F)$ and let $P$ be a Borel subgroup of $G$.
In this paper we study the restriction of irreducible representations of $G$ on
$E$-vector spaces to $P$, where $E$ is an algebraically closed field of
characteristic $p$. We show that in a certain sense $P$ controls the
representation theory of $G$. We then extend our results to smooth $\oK[G]$-
modules of finite length and unitary $K$-Banach space representations of $G$,
where $\oK$ is the ring of integers of a complete discretely valued field $K$,
with residue field $E$.
