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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.

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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.

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