Let $F$ be a local field, and let $G = \textrm{GL}_2(F)$. Let $B \subseteq G$ be the subgroup of upper triangular matrices, $T$ the diagonal matrices, and $N$ the upper triangular matrices with $1$s on the diagonal.
I'm trying to understand the basics of representations of $\textrm{GL}_2$. The following lemma is from Local Langlands Conjecture for $\textrm{GL}_2$ by Henniart and Bushnell.
Let $V$ be the representation space of $\pi$. First, I don't understand why if $V$ contains the trivial character of $N$ (that is, if $V$ has an $N$-fixed vector), then $V$ contains an irreducible representation of $B$ containing an $N$-fixed vector.
If $v$ is an $N$-fixed vector, and $w = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, then since $V$ is irreducible, it should be generated as a $T$-space by $v$ and $\pi(w)v$, by the Bruhat decomposition.
I also don't understand how Frobenius reciprocity is coming in. If $\chi$ is a character of $T$, extended trivially to $B$, then there should be some $B$-map $\pi|_B \rightarrow \chi$ induces an inclusion of $G$-spaces $\pi \rightarrow \textrm{Ind}_B^G(\chi)$.
I would appreciate a hint, not a full solution.