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

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Verma embedding for an arbitrary positive root

Statement

For $α \in Φ^{+}$ , if $⟨ λ + ρ, α^{\lor} ⟩ \in Z_{> 0}$ , then $M (s_{α} \cdot λ) ↪ M (λ)$ .

Facts & Assumptions

Given: Simple-reflection embeddings Simple-reflection embeddings of Verma modules and the fixed reflection and dot-action conventions Root reflections and the Weyl group action and The Weyl vector rho for a chosen positive system.

[L1]

Etingof's Theorem 15.11: when two shifted weights differ by a positive-integral root reflection, the corresponding Verma module embeds uniquely in the other; its proof uses the Shapovalov determinant generically and then takes a limit.

Proof

technique · direct

1.1

Put $ν = λ + ρ$ and $n = ⟨ ν, α^{\lor} ⟩$ . Then $n \in Z_{> 0}$ and $s_{α} ν = ν - n α$ , so $s_{α} ν$ is related to $ν$ by one positive-integral root reflection.

givenalgebra

2.1

The source theorem [L1] applies to this one-reflection relation and gives a unique embedding $M (s_{α} ν - ρ) ↪ M (ν - ρ)$ . Since $s_{α} ν - ρ = s_{α} \cdot λ$ , this is the asserted embedding.

step 1.1L1∎

Depends on

Used by

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, Theorem 15.11 (standard reference, not scraped)