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

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The first Jantzen filtration term is the maximal Verma submodule

Statement

The first Jantzen term satisfies $M^{1} (λ) = J (λ)$ , the maximal proper submodule of $M (λ)$ .

Facts & Assumptions

Given: The filtration The Jantzen deformation and filtration of a Verma module and the radical identification The Shapovalov radical is the maximal submodule.

Proof

technique · direct

1.1

Reducing the condition $S_{t} (v) \in t M_{t} (λ)^{\lor}$ modulo $t$ says exactly that the specialized Shapovalov form pairs $\overset{ˉ}{v}$ with every vector as zero. Thus $M^{1} (λ)$ is its radical.

given

2.1

The radical is $J (λ)$ by the radical theorem, so the two submodules agree.

step 1.1∎

Depends on

Used by

The strong linkage principle for Verma modules Theorem

Dependency tree · two levels

6 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, §20.5 (standard reference, not scraped)