Hello everyone! Please excuse me if this question is too elementary...


Let $M$ and $E$ be modules living in category $\mathcal{O}$, $E$ is finite dimensional, hence $M\otimes E$ also lives in $\mathcal{O}$. 


I'm wondering if the weight spaces of $M\otimes E$ look like this:


$( M\otimes E )\_{\lambda}=\bigoplus\_{\mu+\nu=\lambda} M\_{\mu}\otimes E\_{\nu}$.


The inclusion "from right to left" is obvious, but the other one?


I would think that this has to be well known if it is true - but I could not find it here, on math.stackexchange.com, somewhere else in the web or in the books I have access to.


Thank you very much in advance - any pointers would be helpful to me.



In category O: weight spaces of tensor products
In general, if $V = \bigoplus\_{\mu \in \Lambda} V\_\mu$ and $W = \bigoplus\_{\nu\in \Lambda} W\_\nu$ are $\Lambda$-graded vector spaces (for some abelian group $\Lambda$), then


$V \otimes W = (\bigoplus\_\mu V\_\mu) \otimes (\bigoplus\_\nu W\_\nu) = \bigoplus\_{\mu,\nu} V\_\mu \otimes W\_\nu = \bigoplus\_\lambda \left( \bigoplus\_{\mu + \nu = \lambda} V\_\mu \otimes W\_\nu\right)$


shows that $V \otimes W$ is again a $\Lambda$-graded vector space, and that its $\lambda$-component is $\bigoplus\_{\mu + \nu = \lambda} V\_\mu \otimes W\_\nu$.


