All modules here are finite dimensional. The field is over complex number. Let $U$ be an irreducible $\mathfrak{sl}_n$-module, and $V$ is a highest weight $\mathfrak{sl}_n$-module. Suppose $U\otimes V$ is not a highest weight representation. Then $U\otimes V$ has an irreducible quotient $W$. My question is: what is the maximal vector of $W$? or if $u^{-}\otimes v^{+}$ can be a maximal vector of $W$? It is known that $u^{-}\otimes v^{+}$ generates $U\otimes V$.

Thank you in advance.