Suppose I have a finite-dimensional highest weight representation $V$ of $\mathfrak{gl}(N)$ with highest weight $\lambda=(\lambda_1,\dots,\lambda_N)$, ie there exists a vector $v\in V$ such that $$E_{ij}v=0, \quad 1\leq i<j\leq N$$$$E_{ii}v=\lambda_i v,\quad i=1,\dots,N$$
What are the necessary and sufficient conditions for the V to also be a finite-dimensional $GL(N)$ representation?