The support of Kostant's weight multiplicity formula is an order ideal in the weak Bruhat order

For integral weights $\lambda$ and $\mu$ of a classical simple Lie algebra $\mathfrak{g}$, Kostant’s weight multiplicity formula gives the multiplicity of the weight $\mu$ in the irreducible representation with highest weight $\lambda$, which we denote by $m(\lambda ,\mu )$. Kostant’s weight multiplicity formula is an alternating sum over the Weyl group of the Lie algebra whose terms are determined via a vector partition function. The Weyl alternation set $\mathcal{A}(\lambda ,\mu )$ is the set of elements of the Weyl group that contribute nontrivially to the multiplicity $m(\lambda ,\mu )$. In this article, we prove that Weyl alternation sets are order ideals in the weak Bruhat order of the corresponding Weyl group. Specializing to the Lie algebra ${\mathfrak{s}\mathfrak{l}}_{r+1}(\mathbb{C})$, we give a complete characterization of the Weyl alternation sets $\mathcal{A}(\tilde{\alpha },\mu )$, where $\tilde{\alpha }$ is the highest root and $\mu$ is a negative root, answering a question of Harry posed in 2024. We also provide some enumerative results that pave the way for our future work, where we aim to prove Harry’s conjecture that the $q$-analog of Kostant’s weight multiplicity formula is $m_q(\tilde{\alpha },\mu )=q^{r+j-i+1}+q^{r+j-i}-q^{j-i+1}$ when $\mu =-({\alpha }_i+{\alpha }_{i+1}+\cdots +{\alpha }_j)$ is a negative root of ${\mathfrak{s}\mathfrak{l}}_{r+1}(\mathbb{C})$.