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

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Weights of a Verma module lie below lambda

Statement

The weights of $M (λ)$ are exactly $λ - β$ for $β \in Q^{+}$ ; every weight space is finite dimensional, and $M (λ)_{λ} = C v_{λ}$ .

Facts & Assumptions

Given: The negative-root PBW basis of The PBW model of a Verma module.

Proof

technique · direct

1.1

A monomial with negative roots $- α$ occurring $k_{α}$ times has $h$ -weight $λ - \sum k_{α} α$ , by commuting $h$ past its factors.

givenalgebra

2.1

Such monomials span each displayed weight space. For fixed $β$ , only finitely many nonnegative tuples $(k_{α})$ have sum $β$ , and the zero tuple is the only tuple for $β = 0$ .

givenalgebra∎

Depends on

The PBW model of a Verma module

Used by

The Shapovalov determinant on a weight space Definition
Every nonzero Verma submodule contains a singular vector Lemma
The sum of all proper Verma submodules is proper Lemma
The formal character of a Verma module Proposition

Dependency tree · two levels

4 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, Lie Groups and Lie Algebras I & II, Corollary 25.9 (standard reference, not scraped)