LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Every nonzero Verma submodule contains a singular vector

Statement

Every nonzero submodule of $M (λ)$ contains a nonzero vector annihilated by $n^{+}$ .

Facts & Assumptions

Given: The weight support and finite weight spaces of Weights of a Verma module lie below lambda.

Proof

technique · direct

1.1

As in the weight-projection argument, a nonzero submodule has a nonzero weight vector. Choose one of weight $λ - β$ with $β \in Q^{+}$ of minimal height among its occurring weights.

givenchoose

2.1

If $x \in g_{α} \subseteq n^{+}$ , then $x v$ has weight $λ - (β - α)$ . If it were nonzero, $β - α \in Q^{+}$ would have smaller height, contradicting the choice; hence $n^{+} v = 0$ .

givencontradiction∎

Depends on

Used by

The Shapovalov determinant formula Theorem

Dependency tree · two levels

4 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

Pavel Etingof, Representations of Lie Groups, §15.1 (standard reference, not scraped)