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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 $\operatorname{mult} \alpha$ and $\operatorname{mult} \beta$.

My question is: The number $k = \operatorname{mult} \alpha \times \operatorname{mult} \beta$ has 'any' interpretation in Lie theory like $k$ is also the dimension of some weight spaces of some other Lie algebras some way connected to $\mathfrak{g}$ or in some tensor products of $\mathfrak{g}$ like that.

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

Thanks for your valuable time.

Silvia Ghinassi
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GA316
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  • My knowledge of Lie theory is a bit rusty, so apologies if my question is trivial or worse, stupid. Before going to the infinite dimensional case, what do you know about your $k$ in the case of finite dimensional (semisimple) Lie algebras? – Silvia Ghinassi Feb 01 '16 at 16:04
  • @Silvia Ghinassi even I dont know any interpretation of $k$ for the finite dimensional lie algebras. if you can think of some think it will be very much helpful. – GA316 Feb 01 '16 at 16:07
  • I'm not sure it would be helpful but maybe you can look up something about multiplication of formal characters. – Silvia Ghinassi Feb 01 '16 at 16:14
  • This is a copy of http://mathoverflow.net/questions/229258/ – YCor Feb 01 '16 at 22:25
  • @SilviaGhinassi certainly it is a good point. I will see it in the sense of charatcers in Kac Moody algebra theory. in finite dimensional simple Lie algebra cases multiplicities are one. So there is not much to under stand about $k$ in f.d. Lie algebras. thanks a lot. – GA316 Feb 02 '16 at 00:03

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