The triangular restriction on projective Verma flags

If $(P(\lambda ):\mathrm{\Delta }(\mu ))\mathrm{\ne }0$, then by [F1] the simple module $L(\lambda )$ is a composition factor of $M(\mu )$, hence its highest weight $\lambda$ is a weight of $M(\mu )$; by [F2] every weight of $M(\mu )$ lies in $\mu -Q^{+}$, so $\lambda \le \mu$, that is, $\mu -\lambda \in Q^{+}$.

$(P(\lambda ):\mathrm{\Delta }(\lambda ))=[M(\lambda ):L(\lambda )]=1$: the kernel $J(\lambda )$ of the quotient map is the sum of all proper submodules, so $J(\lambda )$ contains no highest-weight vector of weight $\lambda$ and $J(\lambda {)}_{\lambda }=0$; since $M(\lambda )={\mathfrak{n}}^{-}M(\lambda )\oplus \mathbb{C}{v}_{\lambda }$ by [F2], this gives $J(\lambda )\subseteq {\mathfrak{n}}^{-}M(\lambda )$. The highest weight of any composition factor of $J(\lambda )$ is a weight of $J(\lambda )$ and is therefore different from $\lambda$, so no factor is isomorphic to $L(\lambda )$; from $0\to J(\lambda )\to M(\lambda )\to L(\lambda )\to 0$ the multiplicity $[M(\lambda ):L(\lambda )]$ is exactly one.

Thus a nonzero multiplicity $(P(\lambda ):\mathrm{\Delta }(\mu ))$ forces $\mu \ge \lambda$, and the label $\lambda$ occurs exactly once in every Verma flag of $P(\lambda )$.