The classical BGG category O

Fix a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{h}$, and a positive Borel $\mathfrak{b}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}$. Let $\{{\alpha }_i{\}}_{i\in I}$ be the simple roots of the chosen positive system and write $Q^{+}={\sum }_{i\in I}{\mathbb{Z}}_{\ge 0}{\alpha }_i$ and $\mu \le \lambda$ when $\lambda -\mu \in Q^{+}$. Let $W$ be the Weyl group of Root reflections and the Weyl group action and let $\rho$ be the Weyl vector of The Weyl vector rho for a chosen positive system; for $w\in W$ set $w\cdot \lambda =w(\lambda +\rho )-\rho$.

The BGG category $\mathcal{O}$ is the full subcategory of left $U(\mathfrak{g})$-modules $M$ satisfying all three conditions: $M$ is finitely generated; $M={⨁}_{\mu \in {\mathfrak{h}}^{\ast }}{M}_{\mu }$ where $M_{\mu }=\{v:hv=\mu (h)v\text{ for all }h\in \mathfrak{h}\}$; and $U({\mathfrak{n}}^{+})v$ is finite dimensional for each $v\in M$. Its morphisms are all $\mathfrak{g}$-linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of PBW gives an ordered monomial basis for the enveloping algebra and Triangular decomposition from a chosen positive root system.