 Highest weight of a dual representation

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.