Non-trivial weight spaces of finite-dimensional irreducible $\frak{g}$-modules

Question

Let $\lambda \in \mathcal{P}^+$ be a dominant weight for $\frak{sl}(n,\mathbb{C})$. When does it hold that the zero weight space, of the associated finite-dimensional $L(\lambda)$, is non-trivial?

What about the same question for the other seires?

Answer 1

I'm not quite sure this question rises to the level of MathOverflow, which is why I initially posted only a comment, but at the request of the question-asker I am converting my comment to an answer.

For any semisimple Lie algebra $\mathfrak{g}$ and dominant weights $\mu,\lambda \in P^+$, the condition for the $\mu$-weight space of the irreducible finite dimensional $\mathfrak{g}$-representation $L(\lambda)$ to be nonzero is that $\lambda - \mu$ is a nonnegative sum of positive roots (equivalently, simple roots). This is normally just denoted $\mu \leq \lambda$ (or $\mu \preceq \lambda$). This is the most common partial order on (dominant) weights. See, e.g., Humphreys' "Introduction to Lie Algebras and Representation Theory," Chapter VI.

There is an equivalent, but more geometric, criterion too: we have $\mu \leq \lambda$ if and only if $\mu$ and $\lambda$ lie in the same coset of $P/Q$ (the weight lattice modulo the root lattice) and the convex hull of $W\mu$ is contained in the convex hull of $W\lambda$. (These convex hulls are sometimes called $W$-permutohedra.) This condition is discussed in Stembridge's "The Partial Order of Dominant Weights" (https://doi.org/10.1006/aima.1998.1736), and as he mentions there they probably also appear in Bourbaki's book on Lie groups and Lie algebras.