Let $V_{\lambda} = \bigoplus_{i \in \mathbb{Z}_{\geq 0}} \mathbb{C}v_i$, where $\lambda \in \mathbb{C}$. This is a representation of $\mathfrak{sl}(2)$ according to the following action:
$$hv_{i} = (\lambda - 2i)v_{i}, \qquad fv_{i}= v_{i+1}, \qquad ev_{i} = i(\lambda - i +1)v_{i -1}.$$
I want to find all values of $\lambda$ for which $V_{\lambda}$ is irreducible and all the submodules of $V_{\lambda}$, for the values of $\lambda$ for which $V_{\lambda}$ is not irreducible.
So far, I have only noticed that a single subspace $\mathbb{C}v_{i}$ cannot be a submodule of $V_{\lambda}$, since it is not stable under the action. So, if $W$ is a submodule of $V_{\lambda}$, then $W = \bigoplus_{i \in \mathbb{Z}_{\geq 0}} W \cap \mathbb{C}v_i $. However, I can’t seem to understand how the choice of $\lambda$ can determine irreducibility of $V_{\lambda}$. Any hints?
Hint: If $\lambda\in\Bbb N$, then $ev_{\lambda+1}=0$ and $\bigoplus_{i=\lambda+1}^{\infty}\Bbb Cv_{i}$ is a subrepresentation.