PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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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

1.1givenalgebra

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

2.1givenstep 1.1algebra∎

Such monomials span each displayed weight space. For fixed $β$ , only finitely many nonnegative tuples $(k_{α})$ have sum $β$ , since every positive root has nonnegative simple-root coordinates and each tuple coordinate is bounded by a coordinate of $β$ . Conversely, write $β = \sum_{i} n_{i} α_{i}$ with $n_{i} \geq 0$ . The ordered PBW monomial formed from $n_{i}$ copies of each simple negative-root vector is nonzero in $M (λ)$ and has weight $λ - β$ , so every claimed weight occurs. For $β = 0$ the zero tuple is the only one, giving $M (λ)_{λ} = C v_{λ}$ .

Depends on

The PBW model of a Verma module

Used by

Antidominant regular Verma modules are simple Corollary
The triangular restriction on projective Verma flags Corollary
Finite weight spaces and bounded support do not replace finite generation Counterexample
The central character does not determine the primitive ideal Counterexample
The full algebraic Verma dual is too large Counterexample
The Shapovalov determinant on a weight space Definition
The singular integral block and its dot stabilizer Definition
Truncation at a finite downward-closed ideal of a linkage class Definition
The regular integral sl2 block Example
Every nonzero Verma submodule contains a singular vector Lemma
Every Verma module contains a simple Verma submodule Lemma
Peeling a maximal-weight Verma from a standard filtration Lemma
Restricted self-duality of simple highest-weight modules Lemma
Standard-costandard Hom and Ext-one orthogonality Lemma
The generic radical on a Shapovalov factor hyperplane Lemma
The sum of all proper Verma submodules is proper Lemma
Verma self-extensions in O split Lemma
Weight-lambda vectors are singular at a maximal label Lemma
Simple and standard bases of K0(O) Proposition
The formal character of a Verma module Proposition
The support description of category O with finite generation Proposition
Verma and finite-dimensional weight modules belong to O Proposition
Verma composition multiplicities are finite Proposition
Parabolic category O is a highest weight category Theorem
Translation to the wall on Verma, simple and projective objects Theorem

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)