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

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Verma composition multiplicities are finite

Statement

For any weights $λ, μ$ , the composition multiplicity $[M (λ) : L (μ)]$ is finite.

Facts & Assumptions

Given: Central-character preservation A Verma composition factor has the same central character, dot-orbit classification Central characters are dot-Weyl orbits, Casimir scalars The quadratic Casimir eigenvalue on a highest-weight module is $(λ, λ + 2 ρ)$ , and the Verma weight cone Weights of a Verma module lie below lambda.

Proof

technique · direct

1.1

A factor $L (μ)$ can occur only when $μ$ is both a weight below $λ$ and in the dot orbit fixed by its central character. The Casimir equality restricts the possible lattice differences in each bounded weight cone to a finite set.

given

2.1

In particular, the $μ$ -weight space of $M (λ)$ is finite-dimensional and every copy of $L (μ)$ contributes its one-dimensional highest-weight line there. Therefore the multiplicity is bounded by $\dim M (λ)_{μ} < \infty$ .

step 1.1algebra∎

Depends on

Used by

The strong linkage principle for Verma modules Theorem

Dependency tree · two levels

18 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, Lemma 15.9 (standard reference, not scraped)