Surjectivity modulo n-minus for free weight-generated modules (BGG 10.5)

If $\phi$ is surjective, then $\bar{\phi }$ is surjective, because $\phi ({\mathfrak{n}}^{-}M)={\mathfrak{n}}^{-}\phi (M)={\mathfrak{n}}^{-}N$ by $U({\mathfrak{n}}^{-})$-linearity, so $\phi$ induces a surjection of the quotients.

Conversely assume $\bar{\phi }$ surjective; we prove that every weight vector of $N$ lies in $\mathrm{im}\phi$ by descending induction on the weight. Since the weight set of $N$ is contained in finitely many cones ${\lambda }_i-Q^{+}$, the set of weights $\nu$ of $N$ with $\nu -\mu \in Q^{+}\setminus \{0\}$ is finite for every weight $\mu$ (only the finitely many cones with ${\lambda }_i-\mu \in Q^{+}$ contribute, and there the coefficients of $\nu -\mu$ are bounded by those of ${\lambda }_i-\mu$).

Inductive step. Fix a weight $\mu$ and $u\in N_{\mu }$, and assume all weight vectors of $N$ of weight ${>}_{\mu }$ lie in $\mathrm{im}\phi$. Since $\bar{\phi }$ is surjective, $\bar{u}$ is a linear combination of the classes $\overline{\phi (v_i)}$, and each nonzero $\overline{\phi (v_i)}$ is a weight vector because $\phi (v_i)$ is a weight vector by hypothesis and the quotient map is $\mathfrak{h}$-equivariant. As $N\mathrm{/}{\mathfrak{n}}^{-}N$ is $\mathfrak{h}$-semisimple, taking the weight-$\mu$ component of the relation lets us discard every generator whose class has weight different from $\mu$; hence $\bar{u}={\sum }_i{c}_i\overline{\phi (v_i)}$ with $c_i=0$ whenever $\mathrm{wt}\phi (v_i)\mathrm{\ne }\mu$ (for the surviving indices $c_i$ is the original coefficient and the corresponding vectors have weight $\mu$). Therefore $u-{\sum }_i{c}_i\phi (v_i)\in N_{\mu }\cap {\mathfrak{n}}^{-}N$.

By [F3] the element $u-{\sum }_i{c}_i\phi (v_i)$ has weight $\mu$ and lies in ${\mathfrak{n}}^{-}N={\sum }_{\alpha }{f}_{\alpha }N$, so its weight-$\mu$ component is a sum ${\sum }_{\alpha }{f}_{\alpha }{w}_{\alpha }$ with $w_{\alpha }\in N_{\mu +\alpha }$: indeed $f_{\alpha }$ lowers weights by $\alpha$. Each $w_{\alpha }$ has weight $\mu +\alpha {>}_{\mu }$, so $w_{\alpha }\in \mathrm{im}\phi$ by the induction hypothesis, and then $f_{\alpha }{w}_{\alpha }\in \mathrm{im}\phi$ because $\mathrm{im}\phi$ is a $U({\mathfrak{n}}^{-})$-submodule. Hence $u\in \mathrm{im}\phi$.

The base of the induction is the case of a maximal weight, where the sum in step 3.1 is empty and $u={\sum }_i{c}_i\phi (v_i)\in \mathrm{im}\phi$; the induction is well founded by step 1.2. Since $N$ is spanned by its weight vectors, $\mathrm{im}\phi =N$, so $\phi$ is surjective.