I think the formula you are looking for is $V(\mu)\cong V(\mu)\otimes g\cong kV(\mu) \oplus \oplus_\alpha V(\mu+\alpha)$ where the sum is over roots $\alpha$ such that $\mu+\alpha$ is a dominant weight. Here $V(\lambda)$ is irreducible with highest weight $\lambda$ for $\lambda$ an integral dominant weight.
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I think the formula you are looking for is $V(\mu)\cong kV(\mu) \oplus \oplus_\alpha V(\mu+\alpha)$ where the sum is over roots $\alpha$ such that $\mu+\alpha$ is a dominant weight. Here $V(\lambda)$ is irreducible with highest weight $\lambda$ for $\lambda$ an integral dominant weight. |
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