For $k=1, \dots, l$, let $V_k$ be an irreducible $\frak{g}$-module, where $\frak{g}$ is a simple complex Lie algebra. Moreover, for each $k$, let $h_k \in V_k$ be a choice of highest weight vector, and $l_k \in V_k$ a choice of lowest weight vector. As is easily checked, the element $$ h := h_1 \otimes \cdots \otimes h_k \in V_1 \otimes \cdots \otimes V_k, $$ is a highest weight vector, and $$ l:= l_1 \otimes \cdots \otimes l_k \in V_1 \otimes \cdots \otimes V_k. $$ is a lowest weight vector.
Is there an easy way to see if $h$ and $l$ are contained in the same irreducible component of $V_1 \otimes \cdots \otimes V_k$.
Recall that the Weyl group acts on the set of weights, and in a finite dimensional representation Weyl-conjugate weights appear with the same multiplicity.
In particular the irreducible sub-representation containing $h_1 \otimes \dots \otimes h_k$ must also contain $l_1 \otimes \dots \otimes l_k $ as they have Weyl-conjugate weights and are the unique vectors (up to scaling) of those weights.