Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 102 / 5 / b / Solution

Take $\mathfrak{g}={\mathfrak{sl}}_3(\mathbb{C})$ and its adjoint representation $V=\mathfrak{g}$. This representation is irreducible because $\mathfrak{g}$ is simple. Its zero-weight space is the Cartan subalgebra $\mathfrak{t}$, so

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