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

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Simple-reflection embeddings of Verma modules

Statement

If $⟨ λ + ρ, α_{i}^{\lor} ⟩ \in Z_{> 0}$ , there is an embedding $M (s_{i} \cdot λ) ↪ M (λ)$ .

Facts & Assumptions

Given: The singular vector The simple-root singular vector in a Verma module and the universal property The universal property of Verma modules.

Proof

technique · direct

1.1

The singular vector $f_{i}^{m} v_{λ}$ has weight $s_{i} \cdot λ$ , so the universal property gives a homomorphism $M (s_{i} \cdot λ) \to M (λ)$ taking its highest vector to it.

givenconstruct

2.1

In the PBW model the vector $f_{i}^{m} v_{λ}$ is nonzero. If $u v_{s_{i} \cdot λ}$ lay in the kernel, then $u f_{i}^{m} = 0$ in $U (n^{-})$ . The PBW-degree associated graded is the domain $S (n^{-})$ , so $u = 0$ . Thus the nonzero homomorphism in step 1.1 is injective and is the asserted embedding.

step 1.1algebra∎

Depends on

Used by

The sl2 Verma embedding chain Example
Verma embedding for an arbitrary positive root 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

Yiannis Sakellaridis, Verma Modules and the Category O, Proposition 3.2 (standard reference, not scraped)