# In category O: weight spaces of tensor products

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 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 $\bigoplus_{\mu + \nu = \lambda} V_\mu \otimes W_\nu$.