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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A nonzero O-object has a highest-weight vector

Statement

Every nonzero $M \in O$ contains a nonzero weight vector killed by $n^{+}$ .

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 $w \cdot λ = w (λ + ρ) - ρ$ . Let $M$ be a finitely generated $h$ -semisimple $g$ -module. Then $M \in O$ if and only if $supp M \subset ⋃_{i = 1}^{r} (λ_{i} - Q^{+})$ for some finite list of weights. In either case every $M_{μ}$ is finite dimensional. The list may be empty for $M = 0$ ; finite generation is an independent hypothesis. (The support description of category O with finite generation)

Proof

1.1

Choose a weight $μ$ in the nonempty support. Above $μ$ , each containing cone $λ_{i} - Q^{+}$ has only finitely many possibilities: $μ \leq ν \leq λ_{i}$ implies $ν - μ, λ_{i} - ν \in Q^{+}$ and their sum is fixed. Bounding simple-root coefficients makes this set finite. Thus the support above $μ$ is a nonempty finite poset.

F1choose

2.1

Choose a maximal element $ν$ of that poset and $0 \neq v \in M_{ν}$ . Every positive-root operator sends $v$ to weight $ν + α$ , which would still be above $μ$ but is absent by maximality. All such operators kill $v$ , hence $n^{+} v = 0$ .

choosestep 1.1∎

Depends on

The support description of category O with finite generation

Used by

Finite weight-space detection of subquotients Lemma
Every object of O has finite length Theorem
The simple objects of O Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

§15.1 p.79, paragraph after Definition 15.1 (standard reference, not scraped)