Equal central character does not give every Verma embedding direction

Put $m=⟨\lambda +\rho ,{\alpha }_1^{\vee }⟩\in {\mathbb{Z}}_{>0}$ and $\mu =s_1\cdot \lambda =\lambda -m{\alpha }_1$. The singular-vector lemma The simple-root singular vector in a Verma module gives a nonzero highest vector $f_1^m{v}_{\lambda }$ of weight $\mu$ in $M(\lambda )$. The Verma universal property The universal property of Verma modules therefore gives a nonzero map $M(\mu )\to M(\lambda )$.

Every central element acts by a scalar on each cyclic highest-weight module by The Harish-Chandra projection computes the highest-weight scalar. The nonzero map in step 1.1 intertwines these scalar actions, so ${\chi }_{\mu }={\chi }_{\lambda }$.

The PBW weight description The PBW model of a Verma module puts every weight of $M(\mu )$ in $\mu -Q^{+}$. Since $\lambda =\mu +m{\alpha }_1$ lies strictly above $\mu$, $M(\mu )$ has no vector of weight $\lambda$. A map $M(\lambda )\to M(\mu )$ must therefore kill its highest vector and is zero. Thus equal central characters do not give both embedding directions.