Let $\mathfrak{g}$ be a Kac-Moody Algebra with GCM $A$. Let $\alpha$ and $\beta$ be two roots not necessarily real and $g_\alpha$ and $g_\beta$ be the corresponding weight spaces of dimension $ \text{mult}\, \alpha$ and $\text{mult}\, \beta$.

My question is: Does the number $k = \text{mult}\, \alpha \times \text{mult}\, \beta$ have 'any' interpretation in Lie theory? Like $k$ is also the dimension of some weight spaces of some other Lie algebras which are in some way connected to $\mathfrak{g}$ or in some tensor products of $\mathfrak{g}$ or something like that.

I want to interpret this number $k$ in terms some Lie theory objects. Any suggestion is welcomed.

Thanks for your valuable time.