Dominant integral dot translates embed canonically in the Verma module

We prove by induction on $\mathrm{\ell }(w)$ that $w\circ \lambda \uparrow \lambda$ and that $M(w\circ \lambda )$ embeds in $M(\lambda )$ with image generated by a singular vector of weight $w\circ \lambda$. For $w=1$ the chain is empty and the identity embeds $M(\lambda )$ in itself. If $w\mathrm{\ne }1$, choose a reduced word $w=s_{i}u$ with $\mathrm{\ell }(w)=\mathrm{\ell }(u)+1$; then $w\circ \lambda =s_i\circ (u\circ \lambda )$ and $u\circ \lambda \uparrow \lambda$ by induction. Since $\mathrm{\ell }(s_{i}u)=\mathrm{\ell }(u)+1$, step [F5] gives $u^{-1}{\alpha }_i>0$; hence $⟨u\circ \lambda +\rho ,{\alpha }_i^{\vee }⟩=⟨u(\lambda +\rho ),{\alpha }_i^{\vee }⟩=⟨\lambda +\rho ,u^{-1}{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{>0}$ by [F4]. So [F2] provides the one-step chain $u\circ \lambda \succ s_i\circ (u\circ \lambda )=w\circ \lambda$, which concatenated with the inductive chain gives $w\circ \lambda \uparrow \lambda$, and [F1] gives an embedding $M(w\circ \lambda )\hookrightarrow M(u\circ \lambda )$ that we compose with $M(u\circ \lambda )\hookrightarrow M(\lambda )$.

Now let $w\ge w^{\mathrm{\prime }}$. By [F6] fix a saturated chain $w=v_0,\dots ,v_r=w^{\mathrm{\prime }}$ with $v_j=s_{{\alpha }_j}{v}_{j-1}$ and $\mathrm{\ell }(v_j)=\mathrm{\ell }(v_{j-1})-1$. At each step the pairing $⟨v_j\circ \lambda +\rho ,{\alpha }_j^{\vee }⟩=⟨\lambda +\rho ,v_j^{-1}{\alpha }_j^{\vee }⟩$ is a positive integer: $v_j^{-1}{\alpha }_j=(s_{{\alpha }_j}{v}_{j-1}{)}^{-1}{\alpha }_j=v_{j-1}^{-1}{s}_{{\alpha }_j}{\alpha }_j=-v_{j-1}^{-1}{\alpha }_j$, and $\mathrm{\ell }(s_{{\alpha }_j}{v}_{j-1})=\mathrm{\ell }(v_{j-1})-1$ gives $v_{j-1}^{-1}{\alpha }_j<0$ by [F5], so $v_j^{-1}{\alpha }_j>0$. Hence [F1] gives embeddings $M(v_{j-1}\circ \lambda )\hookrightarrow M(v_j\circ \lambda )$ for all $j$, whose composite embeds $M(w\circ \lambda )$ in $M(w^{\mathrm{\prime }}\circ \lambda )$; the pair $(w,w^{\mathrm{\prime }})$ is nonzero in $\mathrm{Hom}$, which is one dimensional by [F3].

By the embeddings constructed in step 1.1, ${\mathrm{Hom}}_{\mathfrak{g}}(M(w\circ \lambda ),M(\lambda ))\mathrm{\ne }0$, of dimension exactly $1$ by [F3]; its image is the submodule generated by the image of the highest weight vector, which by [F7] is the unique-up-to-scalar singular vector of weight $w\circ \lambda$ in $M(\lambda )$. This proves the first two assertions.

Both the composite $M(w\circ \lambda )\hookrightarrow M(w^{\mathrm{\prime }}\circ \lambda )\hookrightarrow M(\lambda )$ and the embedding of step 2.1 are nonzero elements of the one-dimensional space ${\mathrm{Hom}}_{\mathfrak{g}}(M(w\circ \lambda ),M(\lambda ))$; rescaling the chosen embedding $M(w\circ \lambda )\hookrightarrow M(w^{\mathrm{\prime }}\circ \lambda )$ by the reciprocal scalar makes the composite equal to the canonical embedding, so after this normalisation the image of $M(w\circ \lambda )$ lies inside the image of $M(w^{\mathrm{\prime }}\circ \lambda )$ inside $M(\lambda )$, i.e. $M(w\circ \lambda )\subseteq M(w^{\mathrm{\prime }}\circ \lambda )\subseteq M(\lambda )$ for the canonical singular-vector submodules. The stated uniqueness is exactly the one-dimensionality of [F3].