LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Finite Borel-stable generators and weight flags

Statement

Every $M \in O$ has a finite-dimensional, $b$ -stable, $h$ -semisimple generating subspace $E$ . There is a flag $0 = E_{0} \subset E_{1} \subset \dots \subset E_{r} = E$ of $b$ -submodules whose quotients are one dimensional and annihilated by $n^{+}$ . For $M = 0$ take $E = 0$ and the empty flag.

Facts & Assumptions

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

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of thm-pbw-ordered-monomial-basis-for-the-enveloping-algebra and thm-triangular-decomposition-from-a-chosen-positive-root-system. (The classical BGG category O)

Proof

1.1F1construct

Choose finitely many generators and replace them by their finitely many weight components $v_{1}, \dots, v_{s}$ . They still generate $M$ . Set $E = \sum_{i} U (n^{+}) v_{i}$ . Local finiteness makes $E$ finite dimensional. For a weight vector $v_{i}$ , root monomials applied to $v_{i}$ are weight vectors; hence $E$ is $h$ -stable and $h$ -semisimple as well as $n^{+}$ -stable. For $M = 0$ choose no generators.

2.1choosestep 1.1

If $E \neq 0$ , its finite weight set has a maximal element for the positive-root order. A nonzero vector at that weight spans a $b$ -stable line: every positive-root operator raises the weight and therefore kills it. The quotient by that line remains finite dimensional and a weight module, since weight components descend.

3.1algebrastep 2.1∎

Repeat the preceding construction in the quotient and take inverse images of the resulting flag. Dimension drops by one each time, so the process ends at zero and gives exactly $r = \dim E$ one-dimensional quotients. When $E = 0$ there is no step to perform.

Depends on

The classical BGG category O

Used by

Dependency tree · two levels

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Sources

Lecture 2 §3 Proposition 3.6 and Lemma 3.7, p.5 (standard reference, not scraped)