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 $w\cdot \lambda =w(\lambda +\rho )-\rho$.

For the abelian category Category O is abelian and extension closed among weight modules, 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 The support description of category O with finite generation, 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.