Let $\frak{g}$ be a simple Lie algebra (over $\mathbb{R}$ or $\mathbb{C}$) and $V\_{\lambda\_i}$ a fundamental representation. What happens if I take the sum, in the dual of the/a Cartan subalgebra $\frak{h} \subseteq \frak{g}$, of the weights of all the non-trivial weight spaces of $V\_{\lambda\_i}$? For $\frak{sl}\_2$ this calculation gives $0$. It also seems to give zero for the standard fundamental representation of $\frak{sl}\_n$, for any $n$. Moreover, for all self-dual representations of $\frak{g}$, it also seems to give zero. Is it always zero?



The sum of the weights of an irreducible simple Lie algebra module
The set weights occurring in a representation of $\mathfrak{g}$ is invariant under the action of the Weyl group. 


Thus the sum you are asking about is fixed by the action of the Weyl group, which means that it must be zero.


