DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-09-23 (gpt-6-sol)

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Standard and costandard objects

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ .

For $λ \in h^{*}$ , define the standard and costandard weight modules by

$Δ (λ) = M (λ), \nabla (λ) = D (M (λ)) .$

The Verma module is defined in Verma modules, and its restricted dual is defined in Restricted Chevalley dual. Under the Axiom of Choice (The Axiom of Choice), Restricted duality is exact and involutive on O places $\nabla (λ)$ in $O$ , so both displayed modules are objects of that category. These symbols impose no projectivity or highest-weight-category axiom.

Depends on

Used by

Injectives have costandard filtrations Corollary
Finite Verma flags and their multiplicities Definition
Standard- and costandard-filtered objects and tilting objects Definition
The integral parabolic category-O blocks for higher-level Schur-Weyl duality Definition
The regular integral block and its antidominant weight Definition
The singular integral block and its dot stabilizer Definition
The regular integral sl2 block Example
Standard-costandard Hom and Ext-one orthogonality Lemma
The simple socle of a costandard object Proposition
BGG reciprocity Theorem
Duflo–Joseph g-finite endomorphisms of every Verma module Theorem
Parabolic category O is a highest weight category Theorem
Simple Verma composition multiplicities are Kazhdan–Lusztig values Theorem
Singular category O multiplicities are parabolic Kazhdan–Lusztig values Theorem
Translation to the wall on Verma, simple and projective objects Theorem
Type-A parabolic category-O Ringel duality via the opposite parabolic Theorem

Dependency tree · two levels

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Sources

Lecture 8 §3 Definition 3.12, p.5 (standard reference, not scraped)