Highest weight module of direct sum of two Lie algebras Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two complex reductive Lie algebras. For each $\mathfrak{g}_i$ ($i=1,2$), fix a Cartan subalgebra and a positive system. Suppose that $V$ is a simple (not necessarily finite dimensional) module of $\mathfrak{g}_1\oplus\mathfrak{g}_2$, it is known that $V=V_1\otimes V_2$ with $V_i$ a simple module of $\mathfrak{g}_i$ for $i=1,2$.
$\mathbf{QUESTION}$ If $V$ is a simple $\mathbf{highest weight}$ module of $\mathfrak{g}_1\oplus\mathfrak{g}_2$, is $V_i$ a simple $\mathbf{highest weight}$ module of $\mathfrak{g}_i$ for $i=1,2$?
 A: Firstly, since $V$ is a weight module of $\mathfrak{g}_1\oplus\mathfrak{g}_2$, it is also a weight module upon restriction to $\mathfrak{g}_i$ for $i=1,2$. Moreover, because $\mathfrak{g}_1$ (respectively, $\mathfrak{g}_2$) acts trivially on $V_2$ (respectively, $V_1$), it follows that $V_i$ is a weight module of $\mathfrak{g}_i$ for $i=1,2$.
Secondly, let $v=\displaystyle{\sum_{j=1}^n}v_j^{(1)}\otimes v_j^{(2)}$ be a highest weight vector in $V_1\otimes V_2$. Since $V_i$ is a weight module of $\mathfrak{g}_i$, one may assume that each $v_j^{(i)}$ is a weight vector of $\mathfrak{g}_i$ for $i=1,2$. Moreover, because $v$ is a weight vector of   $\mathfrak{g}_1\oplus\mathfrak{g}_2$, it follows that all $v_j^{(i)}$ have the same weight of $\mathfrak{g}_i$ for $i=1,2$.
Thirdly, take an arbitrary summand in $v$, say $v_1^{(1)}\otimes v_1^{(2)}$, and it has the same weight with $v$ of $\mathfrak{g}_1\oplus\mathfrak{g}_2$. Assume that the sum of the positive root spaces of $\mathfrak{g}_i$ does not vanish $v_1^{(1)}\otimes v_1^{(2)}$ for $i=1,2$, and then one obtains a higher weight space than the weight space of $v$, which is a contradiction because $V$ is a simple $\mathfrak{g}_1\oplus\mathfrak{g}_2$-module and $v$ is a highest weight vector.
Finally, $v_1^{(1)}\otimes v_1^{(2)}\in V$ is vanished by the sum of the positive root spaces of $\mathfrak{g}_i$ for $i=1,2$, and so is $v_1^{(i)}$. By the simplicity of $V_i$, it is concluded that $V_i$ is a highest weight module of $\mathfrak{g}_i$ with a highest weight vector $v_1^{(i)}$ for $i=1,2$.
