# product of root multiplicities in Kac Moody Algebras

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.

• This question looks very open-ended, so it would be better motivated if you could point to some specific example where there is an interesting interpretation of the product. (By the way, I added one tag, since the usual finite dimensional semisimple Lie algebras are examples of Kac-Moody algebras. Of course, in those cases $k$ is always 1.) – Jim Humphreys Feb 1 '16 at 18:47
• Trivial comment: the Lie bracket goes from $\mathfrak g \otimes \mathfrak g \to \mathfrak g$, so your appeal to "tensor products of $\mathfrak g$" is not completely unreasonable. – Allen Knutson Feb 1 '16 at 20:54