On the restriction of representations of GL₂(F) to a Borel subgroup

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 $E$.