How does one define weights for a semisimple Lie group?

representation-theory lie-groups lie-algebras

"Representations" here means "complex representations." A complex representation of a Lie group $G$ gives rise to a complex representation of its Lie algebra $\mathfrak{g}$, and hence to a complex representation of the complexification $\mathfrak{g} \otimes \mathbb{C}$ of its Lie algebra. If $G$ is semisimple then so is this complex Lie algebra. There is no need to bring in either the complexification of $G$ or any of its other real forms.

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