Weight vector in irreducible representation

Question

Let $V$ be a representation of a lie algebra $\mathfrak g$ that decomposes into a direct sum of countably many distinct finite-dimensional irreducible representations. Is it true that if $v \in V$ is a weight vector whose weight $\lambda$ is a dominant integral element that $v \in V_{\lambda}$?

Answer 1

No. This would say that vectors in a decomposition $V=\bigoplus_\lambda V_\lambda$ with a particular dominant integral weight are unique, and appear only as the highest weight vectors of the irreducible components. This is not the case.

For a quick example, let $\mathfrak{g}=\mathfrak{sl}_2$, and consider the representation $V=V_5 \otimes V_3$. By the Clebsch-Gordan rule, we have $$ V \cong V_8 \oplus V_6 \oplus V_4 \oplus V_2 $$

In particular, there is a weight vector of weight $2$ in each of these irreducible components.

This is taking "countably many" to include finitely many. But the idea generalizes; if you have countably many irreducible $\mathfrak{sl}_2$ representations in a decomposition, any with highest weights of the same parity will have weight vectors of the same weight.