Let $V(\lambda)$ be the unique irreducible representation of a Kac—Moody algebra $\mathfrak{g}$ with the highest weight $\lambda$. If $\mathfrak{g}$ is not of finite type, then even for $\lambda$ one of the fundamental weights the weights of $V(\lambda)$ usually have multiplicities $\geqslant2$.

Is there a nice way to pick a basis in each of the weight spaces $V(\lambda)_\mu$?

Here "nice" means that there is an explicit description of the action of the standard generators of $\mathfrak{g}$, so one could actually write (infinite) matrices for the Lie algebra elements. I have checked some sources including *Lie Algebras of Finite and Affine Type* by R. Carter and *Affine Lie Algebras, Weight Multiplicities and Branching Rules* by Kass, Moody, Patera and Slansky, but none contains such details even for the smallest cases like $\tilde{A}_1$. Here is a picture from the second:

As you can see, this diagram gives no hint to how exactly the generators act inside the weight spaces. I guess this can be reconstructed from the desription of $V(\lambda)$ as a quotient of the Verma module, but this seems quite computationaly extensive, so I was wondering whether someone has already done it in some cases (say, for the untwisted affine KM algebras for some particular choice of $\lambda$)?