Let $\mathfrak{gl}(n,\mathbb{C})$ be the general linear Lie algebra. Let $\{E_{s,t}\}_{1\leq s,t,\leq n}$ be the standard basis for it. And set its Cartan subalgebra $\mathfrak{h}$ to be $\bigoplus_{i=1}^nE_{ii}$. As usual, we have the highest weight theory for representations of $\mathfrak{gl(n,\mathbb{C})}$.
$\bf My$ $\bf question:$ Let $\mathfrak{h}^* = \bigoplus_{i=1}^n\mathbb{C}\epsilon_i$ as usual. For a weight $\lambda:=\sum_{i=1}^na_i\epsilon_i$ with $a_1 \geq a_2 \geq \cdots \geq a_n>0$, how can we prove the weight spaces of weights $-\epsilon_i$'s of the irreducible representation $L(\lambda)$ are zero without using the Weyl character formula?