The BGG differential induces an injection into kernel coinvariants (BGG 10.6)

Given: The Axiom of Choice, a dominant integral weight $\lambda \in {\mathrm{\Lambda }}^{+}$, an integer $k\ge 0$, the BGG complex $C_{\bullet }(\lambda )$ together with the hypothesis that it is exact in degrees $0,\dots ,k-1$, and the induced map ${\bar{d}}_{k+1}$ on ${\mathfrak{n}}^{-}$-coinvariants.

$C_{k+1}(\lambda )={⨁}_{\mathrm{\ell }(w)=k+1}M(w\circ \lambda )$; each $M(w\circ \lambda )\cong U({\mathfrak{n}}^{-})v_w$ is free over $U({\mathfrak{n}}^{-})$ on its highest weight vector $v_w$, and $M(w\circ \lambda )\mathrm{/}{\mathfrak{n}}^{-}M(w\circ \lambda )=\mathbb{C}{\bar{v}}_w$ with ${\bar{v}}_w$ of weight $w\circ \lambda$; the weights $w\circ \lambda$ for $w\in W_{k+1}$ are pairwise distinct, so $\{{\bar{v}}_w:w\in W_{k+1}\}$ is a basis of $C_{k+1}(\lambda )\mathrm{/}{\mathfrak{n}}^{-}{C}_{k+1}(\lambda )$ consisting of $\mathfrak{h}$-eigenvectors of distinct weights (The PBW model of a Verma module, The Bruhat graph and the BGG Verma sum in degree k, Positive coroot pairings of a dominant integral weight, The classical BGG category O).

$d_{k+1}$ is a $\mathfrak{g}$-homomorphism, hence $\mathfrak{h}$-equivariant, and $d_k\circ d_{k+1}=0$; therefore $d_{k+1}$ maps $C_{k+1}(\lambda )$ into $\ker d_k$, and its restriction ${\phi }_w:=d_{k+1}{\mathrm{\mid }}_{M(w\circ \lambda )}$ to each summand is a $\mathfrak{g}$-homomorphism $M(w\circ \lambda )\to \ker d_k$; the induced map ${\bar{d}}_{k+1}$ on coinvariants is $\mathfrak{h}$-equivariant because ${\mathfrak{n}}^{-}{C}_{k+1}(\lambda )$ and ${\mathfrak{n}}^{-}\ker d_k$ are $\mathfrak{h}$-stable (The BGG differential squares to zero, The BGG differential from signed Verma maps, The classical BGG category O).

For $w\in W_{k+1}$ the vector $d_{k+1}(v_w)$ is nonzero. Its component in the summand $M(w^{\mathrm{\prime }}\circ \lambda )$ of $C_k(\lambda )$ is $\epsilon (w,w^{\mathrm{\prime }}){\iota }_{w\to w^{\mathrm{\prime }}}(v_w)$ for every cover $w⊳w^{\mathrm{\prime }}$; since $\mathrm{\ell }(w)=k+1\ge 1$ there is at least one such cover, because for a suitable simple reflection $s_i$ one has $\mathrm{\ell }(w{s}_i)=\mathrm{\ell }(w)-1$ and then $w{s}_i⊲w$ is a cover; each ${\iota }_{w\to w^{\mathrm{\prime }}}$ is injective, so each displayed component is nonzero, and a tuple of vectors in a direct sum is nonzero as soon as one component is (The BGG differential from signed Verma maps, Bruhat covers give canonical Verma embeddings, and composites are inclusions, Finite Weyl strong exchange and deletion).

BGG 10.6b (Nonzero highest-weight images survive modulo n-minus (BGG 10.6b)): if $M\in \mathcal{O}$ has all composition factors of the form $L(u\circ \lambda )$ with $\mathrm{\ell }(u)\ge \mathrm{\ell }(w_0)$, and $\phi \text{\hspace{0.17em}⁣}:M(w_0\circ \lambda )\to M$ satisfies $\phi (v)\mathrm{\ne }0$ for a highest weight vector $v$, then $\phi (v)\mathrm{\notin }{\mathfrak{n}}^{-}M$.

BGG 10.6a (Composition factors of the BGG kernel lie above the degree (BGG 10.6a)): under the present hypothesis, every composition factor of $\ker d_k$ is of the form $L(u\circ \lambda )$ with $\mathrm{\ell }(u)\ge k+1$; moreover $\ker d_k$ is an object of $\mathcal{O}$, being a subobject of $C_k(\lambda )\in \mathcal{O}$ (The classical BGG category O).

If an $\mathfrak{h}$-equivariant linear map between $\mathfrak{h}$-semisimple modules is nonzero on each vector of a basis consisting of eigenvectors of pairwise distinct weights, then it is injective: the images are nonzero eigenvectors of pairwise distinct weights, hence linearly independent. This is ordinary linear algebra (The classical BGG category O).