Structure of induced module of $\mathrm{SL}_2(\mathbb{C})$

$\DeclareMathOperator\SL{SL}$Let $G=\SL_2(\mathbb{C})$, $T$ be the diagonal matrices, $B$ be the upper triangular matrices, and $U$ be the strictly upper triangular matrices. Let $\theta$ be a rational character of $T$, then it can be viewed as a $B$ character naturally. Let $\mathbb{C}_\theta$ be the corresponding $B$-module. I'm interested in the induced module $M(\theta)=\mathbb{C}G\otimes_{\mathbb{C}B}\mathbb{C}_\theta$.

When $\theta$ is a dominant character, Weyl module $V(\theta)$ is a quotient module of $M(\theta)$, and I want to figure out the structure of its kernel.

Here are my attempts:

Let $u(a)=(\begin{smallmatrix} 1&a\\0&1 \end{smallmatrix})$, $h(t)=(\begin{smallmatrix} t&0\\0&t^{-1} \end{smallmatrix})$ and $s=(\begin{smallmatrix} 0&1\\-1&0 \end{smallmatrix})$. Let $1_\theta$ be a nonzero element in $\mathbb{C}_\theta$. Then elements of $M(\theta)$ are of the form $\xi=a_01_\theta+\sum_{i\geq1} a_iu(b_i)s1_\theta$. The homomorphism from $M(\theta)$ to $V(\theta)$ is just sending $1_\theta$ to the highest weight vector of $V(\theta)$.

When $\theta(h(t))=t$, the kernel is $$N(\theta)=\{\xi=a_01_\theta+\sum_{i\geq1} a_iu(b_i)s1_\theta | \sum_{i\geq1}a_i=0, a_0=\sum_{i\geq1}a_ib_i\}$$ and has $\mathbb{C}$-basis $f(c)=c1_\theta+s1_\theta-u(c)s1_\theta$ with $c\in \mathbb{C}^*$, $f(0)=0$.

We have \begin{aligned}h(t)f(c)&=t^{-1}f(ct^2),\quad t\in \mathbb{C}^*\\u(b)f(c)&=f(b+c)-f(b)\quad ,b\in\mathbb{C}\\sf(c)&=-cf(-c^{-1}). \end{aligned}

But I cannot calculate it out, even though it can be written down clearly. I guess it is not irreducible but I cannot prove it.

Here are some questions:

1. I cannot find any helpful method to prove it. What I do is try to show that any nonzero element in $N(\theta)$ can generate $N(\theta)$. Could you please point out how to calculate it effectively?

2. Is there some way to associate this representation with its Lie algebra? For any rational $G$ module $V$(even infinite dimensional), it can be viewed as an $L(G)$ module since $V$ has a natural $K[G]^*$-module structure and $L(G)\subset K[G]^*$.

3. What do we know about the infinite-dimensional abstract irreducible representation of $\SL_2(\mathbb{C})$? Is there some paper helpful?

Thanks for your help.