I’m studying Verma modules of $\mathfrak{sl}_{2} = \mathfrak{sl}_{2} (\mathbb{C})$. Let’s introduce standard notation. Elements $h,e,f\in \mathfrak{sl}_{2} $ form the basis of $\mathfrak{sl}_{2}$, with the usual bracket relations \begin{gather} [e, f] = h\\ [h, e] = 2e\\ [h, f] = -2f \end{gather}
Let $M(\lambda)$ be a Verma $\mathfrak{sl}_{2} $-module. It has a highest vector $v$, such that $hv = \lambda v$ and $ev = 0$. The following construction is to induce $M(\lambda)$ from a representation $\mathbb{C}v$ of the Borel subalgebra. Its basis is $\left\{v, fv, f^{2}v, f^{3}v,\dots\right\}$. With the relations \begin{gather} f(f^{k}v) = f^{k+1}v\\ h(f^{k}v) = (\lambda - 2k)f^{k}v\\ e(f^{k}v) = k(\lambda + 1 - k)f^{k-1}v \end{gather}
I’m interested in tensor products of $M(\lambda)$ with the finite dimensional representations of $\mathfrak{sl}_{2}$. For example let $V = \langle v_{1} = \binom{1}{0}, v_{2} =\binom{0}{1}\rangle$ be the natural two dimensional representation of $\mathfrak{sl}_{2}$. Then \begin{gather} ev_{1} = 0; & ev_{2} = v_{1};\\ hv_{1} = v_{1}; & hv_{2} = -v_{2};\\ fv_{1} = v_{2}; & fv_{2} = 0 \end{gather}
While considering the module $N = V\otimes M(\lambda)$ I easily found it submodule $D\subseteq N$ starting with vector $v_{1}\otimes v$. This module $D$ is isomorphic to $M(\lambda + 1)$ and has a basis \begin{equation} \left\{kv_{2}\otimes f^{k-1}v +v_{1}\otimes f^{k} v\mid k\in\{0,1,\dots\}\right\} \end{equation} therefore an arbitrary element of $D$ has form \begin{equation} \sum_{k=0}^{\infty} \lambda_{k}\left(kv_{2}\otimes f^{k-1}v +v_{1}\otimes f^{k} v\right) = \sum_{k=0}^{\infty}\left( \lambda_{k+1}(k+1)v_{2} + \lambda_{k}v_{1}\right)\otimes f^k v \end{equation} Here $\{\lambda_{k}\}$ is a set of some complex numbers, from which only a finite number is non-zero.
The general element of $N$ has the form: \begin{equation} \sum_{k=0}^{\infty}\left(\alpha_k v_1 \otimes f^k v+\beta_k v_2 \otimes f^k v\right)=\sum_{k=0}^{\infty}\left(\alpha_k v_1+\beta_k v_2\right)\otimes f^k v \end{equation} with the same condition on $\alpha_k$ and $\beta_k$.
My question is that I don’t understand how to write down the basis and $\mathfrak{sl}_2$ action for the quotient module $N/D$. As I understand it has to be isomorphic to $M(\lambda - 1)$.
Also two consider the products of higher dimensional representations $V$ with $M(\lambda)$ one should use subquotient modules. Maybe someone could recommend a book or lecture notes where these are described.