Category O ($\mathcal{O}$)

For a finite-dimensional complex semisimple Lie algebra with a fixed triangular decomposition of a Lie algebra, category $\mathcal{O}$ consists of finitely generated universal enveloping algebra modules that are direct sums of Cartan weight spaces and are locally finite under the positive nilpotent subalgebra. Their weight spaces are finite-dimensional and their weights lie in a finite union of downward positive-root cones. Verma modules, finite-dimensional modules, submodules and quotients belong to this category. Closure under submodules uses the noetherian property of the enveloping algebra, obtained from the Poincaré-Birkhoff-Witt theorem. The category condition does not require the module to be finite-dimensional.

Every object $M$ of Category O has a finite composition series of a module. Choose weight-vector generators and let $T$ be the sum of their finite-dimensional weight spaces. The centre preserves $T$; the kernel of its action on $T$ has finite codimension and, by centrality, annihilates all generators and hence all of $M$. Thus only finitely many central characters can appear in subquotients. The Harish-Chandra isomorphism makes each such character correspond to one finite Weyl orbit of shifted highest weights. Let $S$ be the resulting finite set of possible highest weights.

Every nonzero subquotient has a highest-weight vector from its bounded-above support, and a common central eigenvector can be chosen in the finite-dimensional highest-weight space. Its weight belongs to $S$. The integer $d(N)={\sum }_{\nu \in S}\dim N_{\nu }$ is therefore finite, additive on short exact sequences and positive for every nonzero subquotient $N$. Each strict refinement of a submodule chain consumes at least one unit of $d(M)$, bounding its length and proving the claim.