TITLE: On the definition of highest weight
QUESTION [5 upvotes]: Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $M$ a finite-dimensional representation of $\mathfrak{g}$ (or more generally an element of the category $\mathcal{O}$).
Let $\mathfrak{h}$ be a Cartan subalgebra, $\Phi^+$ a system of positive root and consider the partial ordering on $\mathfrak{h}^*$ induced by $\Phi^+$. Let $\Psi(M)\subseteq\mathfrak{h}^*$ be the set of weights of $M$. Here are two standard definitions:

$(1)$ A highest weight is a maximal element of $\Psi(M)$ with respect to this partial ordering. That is, $\lambda$ is a highest weight if for all $\mu\in\Psi(M)$, either $\mu$ doesn't compare to $\lambda$, or $\mu\leq\lambda$.
$(2)$ $\lambda\in\Psi(M)$ is a highest weight if there exists a nonzero $v\in M_\lambda$ such that $e_\alpha\cdot v=0$ for all $\alpha\in\Phi^+$ where $e_\alpha$ is a root vector of $\alpha$.

I can easily see that $(1)\Rightarrow(2)$. Is the converse also true?

REPLY [3 votes]: Choose $\lambda$ such that both $\lambda$ and $\lambda-\alpha$ are dominant. Then $L(\lambda)\oplus L(\lambda-\alpha)$ is a counter example.