These questions might be elementary for I just started to learn affine Kac-Moody algebra.
It is well known that if we consider finite dimensional Lie algebra, we have the folloing projection:
$R(\lambda)\otimes R(\nu)\rightarrow R(\lambda+\nu)$
where $\lambda$ and $\nu$ are dominant highest weights, $R(\lambda)$ and $R(\nu)$ are fnite dimensional irreducible representations.
First question:
I wonder whether this projection(or some analogue)still survives in affine Kac-Moody case? To be precisely, if we have $R(\lambda)$ and $R(\nu)$ as irreducible integral highest weight representations. Do we have the projection in the same form? (in classical case, we can prove this by using Weyl Character formula or equivalent multiplicity formula, does the affine case follows from the Weyl-Kac formula. Of course I can try to write the argument, but if somebody can point out the reference, I will be happy.
Second question:
It is related to the question I asked several days ago on extreme vector here:tensor product of extreme vector, Now I think I know the proof based on the comment of Peter Tingley and Jim.
Argument based on Peter
$e_{w\lambda}\otimes e_{w\nu}$ has the weight $w(\lambda+\nu)$ which belongs to the same weight space as $e_{w(\lambda+\nu)}$. So if one need to show $e_{w\lambda}\otimes e_{w\nu}=ke_{w(\lambda+\nu)}$ for some nonzero invertible elements $k$, it is sufficient to show the multiplicity of weight space is 1($dimV[w(\lambda+\nu)]$=1). But this just follows from multiplicity of weight space of highest weight is equal to 1 and the fact that multiplicity of weight space is invariant under action of Weyl group.
My second question is:
Can this argument extend to affine Kac-Moody case? Jim made a comment here, he pointed out the extreme vector can be extend to the integral representations of affine Kac-Moody algebra. So if we pick up two extreme vector of intergal highest weight, then take the tesnor product, can we do the similar argument as above(for finite dimensional Lie algebra)to obtain the same result?
Comments: Jim pointed out the reference by S.Kumar on solutions of PRV conjectures. But I think the situation there is slightly different.(maybe much more difficult), because the situation there, one need to consider multiplicity of $V(\lambda+w\nu)$, where just one weight moved by $w$, but in the situation here, it is much simpler.
All comments are welcome.
Thanks!
Best Answer
First question: Peter's (and Emerton's) argument works not only for affine algebras, but for arbitrary Kac-Moody algebras. Decompose your integrable representations into weights, and take the tensor product as the sum of tensor products of weight spaces. It is straightforward to check that the result is still an integrable representation. The tensor products live in Category O, where irreducible quotients are unique for each highest weight. See Kac, Infinite dimensional Lie algebras, sections 9.1-9.3.
Second question: Yes. See previous paragraph.