Nonzero highest-weight images survive modulo n-minus (BGG 10.6b)

Choose a weight $\mu$ of $M$ which is maximal in the weight poset, and a nonzero $u\in M_{\mu }$. Then ${\mathfrak{n}}^{+}u=0$: otherwise some $u_{\alpha }:=e_{\alpha }u\mathrm{\ne }0$ of weight $\mu +\alpha$ would be a weight of $M$ above $\mu$, contradicting maximality. Hence $N:=U(\mathfrak{g})u\subseteq M$ is a highest weight module with head $L(\mu )$ by [F2], so $L(\mu )\in \mathrm{JH}(N)\subseteq \mathrm{JH}(M)$ and in particular the hypothesis of the statement forces $\mu =v\circ \lambda$ with $\mathrm{\ell }(v)\ge \mathrm{\ell }(w_0)$.

Case 1: $\phi (v)\in N$. Then $U(\mathfrak{g})\phi (v)\subseteq N$ is a highest weight module with highest weight $w_0\circ \lambda$, so its head is $L(w_0\circ \lambda )$ and $L(w_0\circ \lambda )\in \mathrm{JH}(N)\subseteq \mathrm{JH}(M(\mu ))$ because $N$ is a quotient of $M(\mu )$. By [F3] the factors of $M(v\circ \lambda )=M(\mu )$ are $L(u^{\mathrm{\prime }}\circ \lambda )$ with $u^{\mathrm{\prime }}\ge v$. Hence $w_0\ge v$ in Bruhat order. Since this gives $\mathrm{\ell }(v)\le \mathrm{\ell }(w_0)$ and $\mathrm{\ell }(v)\ge \mathrm{\ell }(w_0)$, we get $\mathrm{\ell }(v)=\mathrm{\ell }(w_0)$ and therefore $v=w_0$; so $\mu =w_0\circ \lambda$.

Case 2: $\phi (v)\mathrm{\notin }N$. Let $\pi \text{\hspace{0.17em}⁣}:M\to M\mathrm{/}N$. Then $\pi \phi (v)\mathrm{\ne }0$, and $M\mathrm{/}N$ again has all composition factors of the form $L(u\circ \lambda )$ with $\mathrm{\ell }(u)\ge \mathrm{\ell }(w_0)$, because $\mathrm{JH}(M\mathrm{/}N)\subseteq \mathrm{JH}(M)$ by additivity [F1]; moreover $\mathrm{JH}(M)=\mathrm{JH}(N)\bigsqcup \mathrm{JH}(M\mathrm{/}N)$ with $\mathrm{JH}(N)\mathrm{\ne }\mathrm{\varnothing }$ since $N\mathrm{\ne }0$ has finite length, so $M\mathrm{/}N$ has strictly fewer composition factors. By induction on the number of composition factors (the base case being Case 1, which needs no induction hypothesis) we may assume $\pi \phi (v)\mathrm{\notin }{\mathfrak{n}}^{-}(M\mathrm{/}N)$. Since $\pi ({\mathfrak{n}}^{-}M)={\mathfrak{n}}^{-}(M\mathrm{/}N)$, this implies $\phi (v)\mathrm{\notin }{\mathfrak{n}}^{-}M$.

In Case 1, $\phi (v)$ lies in the $\mu$-weight space $M_{\mu }$ with $\mu =w_0\circ \lambda$ maximal among the weights of $M$; hence $M_{\mu +\alpha }=0$ for every $\alpha \in {\mathrm{\Phi }}^{+}$, and by [F4] the weight-$\mu$ part of ${\mathfrak{n}}^{-}M$ is ${\sum }_{\alpha }{f}_{\alpha }{M}_{\mu +\alpha }=0$. As $\phi (v)$ has weight $\mu$ by $\mathfrak{h}$-equivariance, $\phi (v)\mathrm{\notin }{\mathfrak{n}}^{-}M$.

Every $M$ of finite length falls into Case 1 or Case 2, and in Case 2 the reduction terminates; hence in all cases $\phi (v)\mathrm{\notin }{\mathfrak{n}}^{-}M$, as claimed.