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

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A Verma module has a unique simple quotient

Statement

The proper submodule $J (λ)$ which is the sum of all proper submodules is the unique maximal submodule of $M (λ)$ . The quotient $L (λ) : = M (λ) / J (λ)$ is simple and is its unique simple quotient.

Facts & Assumptions

Given: Properness of $J (λ)$ from The sum of all proper Verma submodules is proper.

Proof

technique · direct

1.1givenalgebra

Every proper submodule is contained in $J (λ)$ by its definition, so $J (λ)$ is maximal and unique among proper maximal submodules.

2.1givenalgebra∎

A submodule of $M (λ) / J (λ)$ lifts to a submodule containing $J (λ)$ ; it is either $J (λ)$ or all of $M (λ)$ . Thus the quotient is simple, and the kernel of any simple quotient is a maximal submodule, necessarily $J (λ)$ .

Depends on

The sum of all proper Verma submodules is proper

Used by

The triangular restriction on projective Verma flags Corollary
The BGG complex cannot be used unchanged at a singular weight Counterexample
Parabolic category O for a standard parabolic Definition
The BGG differential from signed Verma maps Definition
The regular integral block and its antidominant weight Definition
Truncation at a finite downward-closed ideal of a linkage class Definition
A singular integral A2 central-character summand Example
The finite-dimensional sl2 quotient of a Verma module Example
The regular integral sl2 block Example
A Verma composition factor has the same central character Lemma
A Verma embedding implies strong linkage Lemma
Dimension of the kernel modulo n-minus equals the next term (BGG 10.7) Lemma
Finite-dimensional tensoring reaches every simple of a linkage class Lemma
Jordan-Holder factors of Verma modules dominate the head (BGG 8.12) Lemma
Nonzero highest-weight images survive modulo n-minus (BGG 10.6b) Lemma
Restricted self-duality of simple highest-weight modules Lemma
Simple extensions cannot cross linkage classes Lemma
The augmentation kernel is the sum of the simple-reflection Verma submodules Lemma
Weight-lambda vectors are singular at a maximal label Lemma
Annihilators of simple highest-weight modules are primitive Proposition
Simple and standard bases of K0(O) Proposition
The BGG differential squares to zero Proposition
The Shapovalov radical is the maximal submodule Proposition
The simple socle of a costandard object Proposition
The Verma annihilator contains the central-character ideal Proposition
Verma and finite-dimensional weight modules belong to O Proposition
Central-character summands refine into linkage blocks Theorem
The simple objects of O Theorem
Translation to the wall on Verma, simple and projective objects Theorem

Dependency tree · two levels

3 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, Proposition 25.12 (standard reference, not scraped)