LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Distinct Verma weight spaces are Shapovalov-orthogonal

Statement

If $u$ and $v$ have distinct $h$ -weights, then $S_{λ} (u, v) = 0$ .

Facts & Assumptions

Given: The Shapovalov form and $τ (h) = h$ from Chevalley-contravariant forms.

Proof

technique · direct

1.1

Choose $h \in h$ separating the two weights $μ \neq ν$ . Contravariance gives $μ (h) S_{λ} (u, v) = S_{λ} (h u, v) = S_{λ} (u, h v) = ν (h) S_{λ} (u, v)$ .

givenalgebra

2.1

Since $μ (h) - ν (h) \neq 0$ , the displayed equality forces $S_{λ} (u, v) = 0$ .

givenalgebra∎

Depends on

Used by

The Shapovalov determinant on a weight space Definition

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

Mrudul Thatte, Category O: Verma's Thesis, Proposition 1.2(a) (standard reference, not scraped)