Simple and standard bases of K0(O)

Given: The setting above and the hypotheses in the statement.

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. For the abelian category thm-category-o-is-abelian-and-extension-closed, define $K_0(\mathcal{O})$ as the free abelian group on isomorphism classes $[M]$, modulo $[B]=[A]+[C]$ for every short exact sequence $0\to A\to B\to C\to 0$. A set of representatives suffices: every finitely generated $U(\mathfrak{g})$-module is a quotient of some $U(\mathfrak{g}{)}^n$, and those quotients form a set up to isomorphism. Define the formal character by $\mathrm{ch}M={\sum }_{\mu }(\dim M_{\mu })e^{\mu }$. By prop-equivalent-support-description-of-category-o, its integer coefficients are finite and supported in finitely many downward cones. Let $\mathcal{R}$ be the group of all such integer coefficient families, with pointwise addition. It is a ring with $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$: at a fixed resulting weight, in any pair of cones the equation $\beta +\gamma =\eta$ with $\beta ,\gamma \in Q^{+}$ has finitely many solutions, since every simple-root coefficient is bounded. Taking weight spaces is exact, so character gives a well-defined homomorphism $K_0(\mathcal{O})\to \mathcal{R}$. The zero object's class and character are zero. (The Grothendieck group and character of O)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. Every object of $\mathcal{O}$ has a finite composition series and is both Noetherian and Artinian. The length of zero is zero. (Every object of O has finite length)

If an object $A$ in an abelian category has two composition series, then the two series have the same length and the same composition factors up to permutation and isomorphism. (Jordan-Holder theorem in an abelian category)

Let ${\chi }_{\lambda }$ and ${\chi }_{\mu }$ be the central characters obtained from highest weights $\lambda$ and $\mu$. Then ${\chi }_{\lambda }={\chi }_{\mu }\text{if and only if}\mu \in W\cdot \lambda ,$ where $W\cdot \lambda :=\{w(\lambda +\rho )-\rho :w\in W\}$. (Central characters are dot-Weyl orbits)

The proper submodule $J(\lambda )$ which is the sum of all proper submodules is the unique maximal submodule of $M(\lambda )$. The quotient $L(\lambda ):=M(\lambda )\mathrm{/}J(\lambda )$ is simple and is its unique simple quotient. (A Verma module has a unique simple quotient)

The weights of $M(\lambda )$ are exactly $\lambda -\beta$ for $\beta \in Q^{+}$; every weight space is finite dimensional, and $M(\lambda {)}_{\lambda }=\mathbb{C}{v}_{\lambda }$. (Weights of a Verma module lie below lambda)

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Write $Q^{+}={\sum }_i{\mathbb{Z}}_{\ge 0}{\alpha }_i$, $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$, and $w\cdot \lambda =w(\lambda +\rho )-\rho$. The simple objects of $\mathcal{O}$ are exactly the modules $L(\lambda )$, $\lambda \in {\mathfrak{h}}^{\ast }$, and $L(\lambda )\cong L(\mu )$ if and only if $\lambda =\mu$. Simplicity here excludes zero. (The simple objects of O)

Every central element acts on a cyclic highest-weight module by a scalar. In particular, each cyclic highest-weight module has a well-defined central character in the sense of def-central-character-of-a-lie-algebra-module. (Central elements act by scalars on cyclic highest-weight modules)