Central actions on nilradical cohomology factor through the Harish–Chandra projection

Multiplication by a central element is a cochain map: if $\omega \in C^q({\mathfrak{n}}^{+},V)$ and $(z\cdot \omega )(x_1,\dots ,x_q)=z\cdot \omega (x_1,\dots ,x_q)$, then $z\cdot (d\omega )=d(z\cdot \omega )$, because in the two sums of the differential the operator $z$ commutes with every $x_i$ and with every bracket on $V$; hence $z$ acts on each $H^q({\mathfrak{n}}^{+},V)$.

An injective $U(\mathfrak{g})$-module is injective over $U({\mathfrak{n}}^{+})$. Let $u:A\rightarrowtail B$ be an embedding of $U({\mathfrak{n}}^{+})$-modules and $f:A\to I$ a $U({\mathfrak{n}}^{+})$-linear map. By [F3], $U(\mathfrak{g}){\otimes }_{U({\mathfrak{n}}^{+})}-$ preserves monomorphisms, so $U(\mathfrak{g})\otimes u$ is injective, and the map $F:U(\mathfrak{g}){\otimes }_{U({\mathfrak{n}}^{+})}A\to I$, $s\otimes a\mapsto s\cdot f(a)$, is a well-defined $U(\mathfrak{g})$-linear map by the extension-of-scalars adjunction ${\mathrm{Hom}}_{U({\mathfrak{n}}^{+})}(A,I)\cong {\mathrm{Hom}}_{U(\mathfrak{g})}(U(\mathfrak{g}){\otimes }_{U({\mathfrak{n}}^{+})}A,I)$, which is the universal property of the tensor product over $U({\mathfrak{n}}^{+})$. Since $I$ is injective over $U(\mathfrak{g})$, $F$ extends along $U(\mathfrak{g})\otimes u$ to a $U(\mathfrak{g})$-linear $\tilde{F}$; the formula $g(b)=\tilde{F}(1\otimes b)$ defines a $U({\mathfrak{n}}^{+})$-linear map $B\to I$ extending $f$, because $1\otimes u(a)$ is the image of $1\otimes a$ and $\tilde{F}$ extends $F$. Hence $I$ has the extension property over $U({\mathfrak{n}}^{+})$.

Let $0\to V\to I\to I\mathrm{/}V\to 0$ be the short exact sequence obtained from an injective embedding $V\hookrightarrow I$ of [F4]. The connecting maps of its long exact sequence commute with multiplication by $z$ and with the $\mathfrak{h}$-action of [F7]: multiplication by $z$ is a $U({\mathfrak{n}}^{+})$-linear endomorphism of each of $V$, $I$, $I\mathrm{/}V$ commuting with the inclusion and the quotient, so naturality of the long exact sequence of [F6] gives $z\cdot \delta =\delta \cdot z$. For the Cartan action, represent a class by a cocycle $c$ in $C^{p-1}({\mathfrak{n}}^{+},I\mathrm{/}V)$ and lift it to $b$ in $C^{p-1}({\mathfrak{n}}^{+},I)$; then $\delta [c]=[db]$, with $db$ valued in $V$. The coefficient inclusion and quotient intertwine the cochain operators ${\theta }_h$, so ${\theta }_{h}b$ lifts ${\theta }_{h}c$; since ${\theta }_h$ commutes with $d$ by [F7], $\delta [{\theta }_{h}c]=[d{\theta }_{h}b]=[{\theta }_{h}db]={\theta }_h\delta [c]$. This proves Cartan equivariance directly, and therefore equivariance for $\mathrm{pr}(z)\in U(\mathfrak{h})$.

The identity holds in degree zero. Let $v\in H^0({\mathfrak{n}}^{+},V)=V^{{\mathfrak{n}}^{+}}$. By [F1] and [F2], $z-\mathrm{pr}(z)\in U(\mathfrak{g}){\mathfrak{n}}^{+}$, so $(z-\mathrm{pr}(z))v\in U(\mathfrak{g})({\mathfrak{n}}^{+}v)=0$; hence $z\cdot v=\mathrm{pr}(z)\cdot v$, where $\mathrm{pr}(z)\in U(\mathfrak{h})$ acts through the $\mathfrak{h}$-action of [F7] and the identification $U(\mathfrak{h})=S(\mathfrak{h})$.

Positive cohomology with injective coefficients vanishes: $H^q({\mathfrak{n}}^{+},I)=0$ for every $q>0$ and every injective $U(\mathfrak{g})$-module $I$. By step 1.2, $I$ is injective over $U({\mathfrak{n}}^{+})$, so by [F5] $H^q({\mathfrak{n}}^{+},I)\cong {\mathrm{Ext}}_{U({\mathfrak{n}}^{+})}^q(k,I)=0$ for $q>0$.

For every $p\ge 1$ the connecting map $\delta :H^{p-1}({\mathfrak{n}}^{+},I\mathrm{/}V)\to H^p({\mathfrak{n}}^{+},V)$ is surjective: in the long exact sequence the map $H^p({\mathfrak{n}}^{+},V)\to H^p({\mathfrak{n}}^{+},I)$ has image contained in $H^p({\mathfrak{n}}^{+},I)=0$ by step 2.2, and exactness identifies $\mathrm{im}\delta$ with its kernel, which is all of $H^p({\mathfrak{n}}^{+},V)$.

Induction on $p$ proves the identity for every coefficient module $V$, hence the lemma. The case $p=0$ is step 2.1, which uses only that $z$ is central and therefore applies to every module, in particular to $I\mathrm{/}V$. Suppose the identity holds in degree $p-1$ for all modules. Given $\omega \in H^p({\mathfrak{n}}^{+},V)$, write $\omega =\delta \eta$ with $\eta \in H^{p-1}({\mathfrak{n}}^{+},I\mathrm{/}V)$ by step 3.1; then $z\cdot \omega =z\cdot \delta \eta =\delta (z\cdot \eta )=\delta (\mathrm{pr}(z)\cdot \eta )=\mathrm{pr}(z)\cdot \delta \eta =\mathrm{pr}(z)\cdot \omega$ by step 1.3 and the inductive hypothesis. Since every class in degree $p$ arises this way, the identity holds in degree $p$; the cases $V=0$ and ${\mathfrak{n}}^{+}=0$ are immediate (all groups involved are zero or the action is the given one). The proof inherits the Axiom of Choice through the injective embedding of [F4] and the free-resolution interface of [F5].