Let $n$ be an integer $\geq 3$ and let $\mathfrak{g}$ be the Kac-Moody algebra with cartan matrix $C$ given by $C_{ij} = 2 \delta_{i,j} - \delta_{i,j+1} - \delta_{i,j-1} - \delta_{|i-j|,n-2}$. For example for $n = 4$ the Cartan matrix is :
$$ \begin{pmatrix} 2 & -1 & 0 & -1 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ -1 & 0 & -1 & 2 \end{pmatrix} $$
This cartan matrix is clearly of rank $n-1$ so the Cartan algebra of $\mathfrak{h}$ of $\mathfrak{g}$ is of dimension $n+1$. Let $\alpha_1,\cdots, \alpha_n \in \mathfrak{h}^*$ and $\alpha_i^\vee,\cdots,\alpha_n^\vee \in \mathfrak{h}$ be the roots and coroots of $\mathfrak{g}$. Let $\omega$ be an element of $\mathfrak{h}^*$ such that $\omega(\alpha_i^\vee)= \delta_{i,1} - \delta_{i,2}$ for all $i \in \{1,...,n\}$ (this is not unique but we choose one).
For every integer $i \in \mathbf{Z}$ define $[i]$ to be the unique integer in $\{1,\cdots,n\}$ such that $i -[i] \in n\mathbf{Z}$ (just a representing element of $i \mod n$).
I am asked (past exam) to show that there exists a unique representation $V$ of $\mathfrak{g}$ such that
(1) $V$ is not zero and sum of its weight spaces
(2) the weight spaces are one dimensional
(3) If $\lambda \in \mathfrak{h}^*$ is such that $V_\lambda \neq 0$ then there exists an integer $i \geq 1$ such that
$$\lambda = \omega + \alpha_{[2]} + \cdots + \alpha_{[i]} \text{ or } \lambda = \omega - (\alpha_{[1]} + \alpha_{[0]} + \cdots + \alpha_{[2-i]}) $$
My idea was to set $V = \bigoplus_{i \in \mathbf{Z} - 0} \mathbf{C}v_i$ with $Hv_i = (\omega + \alpha_{[2]} + \cdots + \alpha_{[i]})(H)v_i$ if $i\geq 1$ and $Hv_i = (\omega - (\alpha_{[1]} + \alpha_{[0]} + \cdots + \alpha_{[2-i]}))(H)v_i$ if $i \leq 1$.
But that's just a guess and i'm not sure how to actually define the actions of $X_i$ and $Y_i$ and to how show that it is a representation (if it is I think it's clear that it verifies (1),(2),(3)).
Since we are asked to show that this representation is unique up to isomorphism I guess that we should be able to read the actions of $X_i$ and $Y_i$ from the conditions but I don't see how. In fact I don't see how to make use of the specific form of the Cartan matrix nor the definition of $\omega$ so I guess I really don't understand the problem.
Any ideas on how to do this ?