Highest weight of a dual representation
= Highest weight of a dual representation
{title2=$-w_0\lambda$}
For a finite-dimensional <irreducible representation> $L(\lambda)$ of a complex <semisimple Lie algebra>, its <dual Lie algebra representation> has <highest weight> $-w_0\lambda$, where $w_0$ is the <longest Weyl-group element>. Indeed duality negates all <weights>, and $w_0\lambda$ is the lowest weight of the original <module>. In particular $w_0=-I$ makes every such <irreducible module> self-dual.