My question in the most simple form:


Let $\mathfrak{g}=\mathfrak{g}\_1\oplus \mathfrak{g}\_2$ be a direct sum of simple finite-dimensional Lie algebras over $\mathbb{C}$ and let $M$ be a finite-dimensional simple $\mathfrak{g}$-module (it is known that $M$ is some $V(\lambda)$ highest weight module).


Since $\mathfrak{g}\_1$ is simple, $M$ decomposes into a direct sum of simple $\mathfrak{g}\_1$-modules. What is that decomposition?


More generally, given an irreducible module over $\mathfrak{g}$ how does it decomposes over a subalgebra (simple, semisimple) of $\mathfrak{g}.$ Has it been investigated in some circumstances? I would be grateful for any links. 



An irreducible Lie algebra module decomposition over a subalgebra
A highest weight of $\mathfrak g$ is a pair of highest weights $\lambda=(\lambda\_1,\lambda\_2)$ for the two factors. The highest weight module is just the tensor product $V(\lambda)=V(\lambda\_1)\otimes V(\lambda\_2)$. So when you restrict, that just means you forget that $V(\lambda\_2)$ has an action, and it just becomes the multiplicity space.


