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

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The Jantzen sum formula for a Verma module

Statement

For every $λ$ , $\sum_{i > 0} ch M^{i} (λ) = \sum_{α \in Φ^{+}} \sum_{\begin{array}{c} n > 0 \\ ⟨ λ + ρ, α^{\lor} ⟩ = n \end{array}} ch M (s_{α} \cdot λ) .$

Facts & Assumptions

Given: The filtration The Jantzen deformation and filtration of a Verma module, the Shapovalov determinant formula The Shapovalov determinant formula, and the Verma character The formal character of a Verma module.

Proof

technique · direct

1.1

On a finite free weight block, Smith normal form has diagonal entries $t^{a_{j}} u_{j} (t)$ . Both the order of its determinant and $\sum_{i > 0} \dim M^{i} (λ)_{λ - β}$ equal $\sum_{j} a_{j}$ : each $a_{j}$ contributes once for each $1 \leq i \leq a_{j}$ .

givenalgebra

2.1

Substitute the determinant formula along $λ + t ρ$ . Its order at $t = 0$ in the $λ - β$ block is $\sum_{α, n : ⟨ λ + ρ, α^{\lor} ⟩ = n} K (β - n α)$ , exactly the coefficient of that weight in the right-hand character sum. Equality coefficientwise for every $β$ proves the formula.

step 1.1algebra∎

Depends on

Used by

The strong linkage principle for Verma modules Theorem

Dependency tree · two levels

13 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, Exercises 20.11–20.12 (standard reference, not scraped)