CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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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.
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Generic Verma modules are simple

Statement

If $⟨ λ + ρ, α^{\lor} ⟩ \notin Z_{> 0}$ for every $α \in Φ^{+}$ , then $M (λ)$ is simple. In particular this holds on the complement of the countable union of positive-integral reflection hyperplanes.

Facts & Assumptions

Given: The determinant irreducibility criterion The Verma irreducibility criterion from Shapovalov determinants.

Proof

technique · direct

1.1given

The hypothesis is precisely the absence of the positive-integral pairings in the criterion.

2.1step 1.1∎

The criterion therefore says that $M (λ)$ is simple.

Depends on

The Verma irreducibility criterion from Shapovalov determinants

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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 (standard reference, not scraped)