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

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A Verma embedding implies strong linkage

Statement

An embedding $M (μ) ↪ M (λ)$ implies $μ ↑ λ$ .

Facts & Assumptions

Given: Injectivity A nonzero homomorphism between Verma modules is injective and strong linkage The strong linkage principle for Verma modules.

Proof

technique · direct

1.1

The image of the embedding is a submodule isomorphic to $M (μ)$ , so its simple quotient $L (μ)$ is a subquotient, hence a composition factor, of $M (λ)$ .

given

2.1

Applying the strong linkage principle to that factor gives $μ ↑ λ$ .

step 1.1∎

Depends on

Used by

Antidominant regular Verma modules are simple Corollary
The BGG criterion for homomorphisms between Verma modules Theorem

Dependency tree · two levels

8 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, Corollary 20.14 (standard reference, not scraped)