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

1.1

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.

F1construct

2.1

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.

choosestep 1.1

3.1

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.

algebrastep 2.1∎

Depends on

The classical BGG category O

Used by

Finite filtrations by highest-weight quotients Lemma
The support description of category O with finite generation Proposition

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)