Hi all I have a question, let $\mathfrak g=\mathfrak{gl}(n)$ be the general linear Lie algebra with the usual Cartan and Borel subalgebra. Then it is well-known that all finite-dimensional (integer weight) $\mathfrak g$-modules are parametrized by partitions. Let us denote they by $L(\lambda)$ as usual ($\lambda$ is the highest weight of $L(\lambda)$, the corresponding partiion).
${\bf My ~Question}$: What is the sufficient and necessary condition for $\lambda$ being such that the zero weight space of $L(\lambda)$ is nonzero? Thanks!
It is not true that the set of partitions parametrize the finite dimensional irreducible integral weight $\mathfrak{g}$-modules. The partitions parametrize the subset consisting of irreducible representations that are polynomial as representations of $\mathrm{GL}_n$ (see e.g. Macdonald's book Symmetric functions and Hall polynomials).
So I assume what you meant to ask was: for a non-increasing sequence of integers $\lambda=(\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n)$, what is a necessary and sufficient condition for the zero weight space of $L(\lambda)$ to be non-zero?
The answer then is that $$\lambda_1+\lambda_2+\cdots+\lambda_n=0$$ is the condition you want.