Principal series for $\operatorname{GL}_2$, question about an exact sequence

Question

I have a question about Proposition 7.2 of these notes by Gordon Savin.

Here $G = \operatorname{GL}_2(F)$ for $F$ a $p$-adic field, $w = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, and $\delta$ is the modulus character $\delta \begin{pmatrix} a_1 \\ & a_2 \end{pmatrix} = \frac{|a_1|}{|a_2|}$.

I follow all the steps in the proof, except for the very last sentence. Maybe I would understand things if I knew what the map

$$0 \rightarrow \delta^{\frac{1}{2}} \chi^w \rightarrow Ind_B^G(\chi)_N$$

was in the first place.

Answer 1

$\DeclareMathOperator{\Ind}{Ind}$Oh wait, I think I get it. The kernel of $\alpha_w: \Ind_B^G(\chi)_w \rightarrow \mathbb C$ is exactly $\Ind_B^G(\chi)_w(N)$, so we get an isomorphism of the Jacquet module $\Ind_B^G(\chi)_{w,N}$ with $\mathbb C$. This is how we get an embedding

$$\mathbb C \rightarrow \Ind_B^G(\chi)_{w,N} \rightarrow \Ind_B^G(\chi)_N$$