Jordan-Holder factors of Verma modules dominate the head (BGG 8.12)

Let $L(\mu )$ be a composition factor of $M(w\circ \lambda )$. By [F1] $\mu \uparrow (w\circ \lambda )$ and ${\chi }_{\mu }={\chi }_{w\circ \lambda }$; by the orbit description of central characters, $\mu =u\circ (w\circ \lambda )$ for some $u\in W$. Since $u\circ (w\circ \lambda )=u(w(\lambda +\rho ))-\rho =(uw)\circ \lambda$, we may write $\mu =u^{\mathrm{\prime }}\circ \lambda$ with $u^{\mathrm{\prime }}=uw\in W$.

For the multiplicity of the head, write $J$ for the sum of all proper submodules of $M(w\circ \lambda )$, the unique maximal submodule, so $M\mathrm{/}J\cong L(w\circ \lambda )$ is simple by [F5]. The highest weight vector $v$ of $M(w\circ \lambda )$ spans the one-dimensional weight space $M(w\circ \lambda {)}_{w\circ \lambda }$ and generates the module, so $v\mathrm{\notin }J$ and hence $J_{w\circ \lambda }=0$. Refine the filtration $0\subset J\subset M(w\circ \lambda )$ to a composition series; its top factor is $M\mathrm{/}J\cong L(w\circ \lambda )$, and any further factor isomorphic to $L(w\circ \lambda )$ would be a subquotient $X\mathrm{/}Y$ of $J$ with $(X\mathrm{/}Y{)}_{w\circ \lambda }\mathrm{\ne }0$, hence would force $X_{w\circ \lambda }\mathrm{\ne }0$ and so $J_{w\circ \lambda }\mathrm{\ne }0$, a contradiction. Therefore $[M(w\circ \lambda ):L(w\circ \lambda )]=1$.

Use the witnessing linkage chain of [F2]: $w\circ \lambda ={\eta }_0\succ {\eta }_1\succ \cdots \succ {\eta }_r=\mu$ with ${\eta }_j=s_{{\alpha }_j}\circ {\eta }_{j-1}$ and positive integral pairings. By step 1.1 and induction each ${\eta }_j$ lies in $W\circ \lambda$; write ${\eta }_j=v_j\circ \lambda$. Then $v_j\circ \lambda =s_{{\alpha }_j}\circ (v_{j-1}\circ \lambda )=(s_{{\alpha }_j}{v}_{j-1})\circ \lambda$, so $v_j=s_{{\alpha }_j}{v}_{j-1}$, with $v_0=w$ and $v_r=u^{\mathrm{\prime }}$. Moreover ${\eta }_{j-1}+\rho =v_{j-1}(\lambda +\rho )$, so the pairing condition reads $⟨\lambda +\rho ,v_{j-1}^{-1}{\alpha }_j^{\vee }⟩=⟨v_{j-1}(\lambda +\rho ),{\alpha }_j^{\vee }⟩\in {\mathbb{Z}}_{>0}$. If $v_{j-1}^{-1}{\alpha }_j$ were a negative root $-\beta$ with $\beta >0$, then $⟨\lambda +\rho ,(-\beta {)}^{\vee }⟩=-⟨\lambda +\rho ,{\beta }^{\vee }⟩<0$ by [F4], contradiction; hence $v_{j-1}^{-1}{\alpha }_j>0$ and the length criterion of Finite Weyl strong exchange and deletion gives $\mathrm{\ell }(s_{{\alpha }_j}{v}_{j-1})>\mathrm{\ell }(v_{j-1})$.

Thus $w=v_0,\dots ,v_r=u^{\mathrm{\prime }}$ is an increasing reflection chain, so $w\le u^{\mathrm{\prime }}$ in Bruhat order by [F3] and $\mathrm{\ell }(u^{\mathrm{\prime }})\ge \mathrm{\ell }(w)$. This proves the domination and length assertions.

Combining steps: every composition factor of $M(w\circ \lambda )$ is $L(u\circ \lambda )$ with $u\ge w$ in Bruhat order, $\mathrm{\ell }(u)\ge \mathrm{\ell }(w)$, and the factor $L(w\circ \lambda )$ occurs exactly once.