Let $\mathfrak{g}$ be a complex finite-dimensional semisimple Lie algebra with a fixed Cartan subalgebra $\mathfrak{h}$. Assume that $\lambda \in \mathfrak{h}^*$ such that $\lambda$ is a $\mathbb{Z}$-linear combination of roots of $\mathfrak{g}$.
Question: Is it true that $\lambda$ is a weight of a finite-dimensional module over $\mathfrak{g}$? Thanks!
Yes. If $\alpha_1,\alpha_2,\ldots,\alpha_n$ are the simple roots and $\lambda=\sum_{i=1}^n m_i\alpha_i$, then $\lambda$ is a weight of the $m$-fold symmetric power $S^mL$ of the adjoint representation $L$, where $m=\sum_{i=1}^n|m_i|$.
This is because $\pm\alpha_i$ are weights of $L$, and sums of any $m$ weights of a module $V$ are weights in the symmetric power $S^mV$. The $m$-fold tensor product $V^{\otimes m}$ would do equally well.