The standard induced resolution of the trivial module

Assume the Axiom of Choice (The Axiom of Choice). Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$, the positive system ${\mathrm{\Phi }}^{+}$ and ${\mathfrak{n}}^{-}$ as in Triangular decomposition from a chosen positive root system. The quotient $\mathfrak{g}\mathrm{/}\mathfrak{b}$ is a $\mathfrak{b}$-module for the adjoint action; $\mathfrak{h}$ acts with weights ${\mathrm{\Phi }}^{-}$. Projection along $\mathfrak{g}={\mathfrak{n}}^{-}\oplus \mathfrak{b}$ identifies this quotient with ${\mathfrak{n}}^{-}$ as an $\mathfrak{h}$-module; its $\mathfrak{b}$-action is $b\cdot x={\mathrm{pr}}_{{\mathfrak{n}}^{-}}[b,x]$, which need not vanish for $b\in {\mathfrak{n}}^{+}$. For $k\ge 0$ put

$B_k:=U(\mathfrak{g}){\otimes }_{U(\mathfrak{b})}{\mathrm{\Lambda }}^k(\mathfrak{g}\mathrm{/}\mathfrak{b}),$

the $k$-th exterior power being taken over $\mathbb{C}$ with the induced $\mathfrak{b}$-action. Thus $B_0=U(\mathfrak{g}){\otimes }_{U(\mathfrak{b})}\mathbb{C}\cong M(0)$, the Verma module of highest weight $0$ of Verma modules. By PBW gives an ordered monomial basis for the enveloping algebra one has $U(\mathfrak{g})\cong U({\mathfrak{n}}^{-})\otimes U(\mathfrak{b})$ as vector spaces, so

$B_k\cong U({\mathfrak{n}}^{-}){\otimes }_{\mathbb{C}}{\mathrm{\Lambda }}^k({\mathfrak{n}}^{-})$

as $U({\mathfrak{n}}^{-})$-modules; hence $B_k$ is a free $U({\mathfrak{n}}^{-})$-module with $\left(\genfrac{}{}{0px}{}{\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }}{k}\right)$ generators, and $B_k=0$ for $k>\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$. Each $B_k$ is a finitely generated $\mathfrak{g}$-module in the category $\mathcal{O}$ of The classical BGG category O: it is generated by the image of the finite-dimensional space ${\mathrm{\Lambda }}^k(\mathfrak{g}\mathrm{/}\mathfrak{b})$, it is $\mathfrak{h}$-semisimple: a root-vector exterior basis has the finite multiset of weights ${\mu }_S={\sum }_{\beta \in S}\beta$ for subsets $S\subseteq {\mathrm{\Phi }}^{-}$ with $\mathrm{\#}S=k$, with distinct subsets counted separately even when their sums coincide. For $k=0$ the empty subset has weight $0$. PBW negative-root monomials shift these weights by elements of $-Q^{+}$, so the weights of $B_k$ lie in the finite union ${\bigcup }_{S\subseteq {\mathrm{\Phi }}^{-},\mathrm{\#}S=k}({\mu }_S-Q^{+})$, with finite-dimensional weight spaces by PBW. Raising a fixed weight by positive roots can reach only finitely many weights in that union: their simple-root coefficients are bounded above by the finitely many $\mu$ and below by the starting weight. Hence the $U({\mathfrak{n}}^{+})$-orbit of each weight vector is finite-dimensional, proving local finiteness.

For $k\ge 1$ define the differential $d_k\text{\hspace{0.17em}⁣}:B_k\to B_{k-1}$ on elementary tensors by

$d_k(u\otimes {\xi }_1\wedge \cdots \wedge {\xi }_k)={\sum }_{i=1}^k(-1{)}^{i+1}u{\xi }_i\otimes {\xi }_1\wedge \cdots \widehat{{\xi }_i}\cdots \wedge {\xi }_k+{\sum }_{1\le i<j\le k}(-1{)}^{i+j}u\otimes \overline{[{\xi }_i,{\xi }_j]}\wedge {\xi }_1\wedge \cdots \widehat{{\xi }_i}\cdots \widehat{{\xi }_j}\cdots \wedge {\xi }_k,$

where ${\xi }_i\in \mathfrak{g}$ are representatives of elements of $\mathfrak{g}\mathrm{/}\mathfrak{b}$ and the bar denotes the class in $\mathfrak{g}\mathrm{/}\mathfrak{b}$. The balanced well-definedness, $\mathfrak{g}$-linearity and square-zero identity are proved explicitly in The standard induced complex is a resolution of the trivial module ↗; the formula uses the actual quotient adjoint action above. Finally the counit $\epsilon \text{\hspace{0.17em}⁣}:U(\mathfrak{g})\to \mathbb{C}$ induces a well-defined map $d_0\text{\hspace{0.17em}⁣}:B_0\to \mathbb{C}$, $u\otimes 1\mapsto \epsilon (u)$, because $\epsilon (ub)=0$ for $b\in \mathfrak{b}$ and $\epsilon (1)=1$; it is the augmentation of the complex $(B_{\bullet },d_{\bullet })$, a chain complex in the sense of Chain complex in an abelian category.