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The classical BGG category O

Definition

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Let ${α_{i}}_{i \in I}$ be the simple roots of the chosen positive system and write $Q^{+} = \sum_{i \in I} Z_{\geq 0} α_{i}$ and $μ \leq λ$ when $λ - μ \in Q^{+}$ . Let $W$ be the Weyl group of Root reflections and the Weyl group action and let $ρ$ be the Weyl vector of The Weyl vector rho for a chosen positive system; for $w \in W$ set $w \cdot λ = w (λ + ρ) - ρ$ .

The BGG category $O$ is the full subcategory of left $U (g)$ -modules $M$ satisfying all three conditions: $M$ is finitely generated; $M = ⨁_{μ \in h^{*}} M_{μ}$ where $M_{μ} = {v : h v = μ (h) v for all h \in h}$ ; and $U (n^{+}) v$ is finite dimensional for each $v \in M$ . Its morphisms are all $g$ -linear maps. The zero module is included, generated by the empty set. The enveloping and triangular conventions are those of PBW gives an ordered monomial basis for the enveloping algebra and Triangular decomposition from a chosen positive root system.

Depends on

Used by

The integral regular localization equivalence restricts to Category O Corollary
An ambient extension can leave category O Counterexample
Finite weight spaces alone do not give category O Counterexample
Finite weight spaces and bounded support do not replace finite generation Counterexample
O is not closed under arbitrary tensor products Counterexample
Unsigned Bruhat edge sums need not square to zero Counterexample
Category O, KLR, and Hecke route conventions Definition
Finite Verma flags and their multiplicities Definition
Generalized central-character subcategories Definition
Parabolic category O for a standard parabolic Definition
The Bruhat graph and the BGG Verma sum in degree k Definition
The deformed category O family over the Cartan symmetric algebra 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 standard induced resolution of the trivial module Definition
Truncation at a finite downward-closed ideal of a linkage class Definition
Type of a module with a Verma filtration Definition
Degree-one Kostant classes correspond to simple reflections Example
Finite-dimensional irreducible sl2 modules have exact integral central characters Example
Nonintegral sl2 central characters split into two simple blocks Example
The one-strand degenerate action on the trivial module has scalar dot Example
The two projectives in the principal sl2 block Example
The wall-crossing functor on Verma modules in rank one Example
Dimension of the kernel modulo n-minus equals the next term (BGG 10.7) Lemma
Finite Borel-stable generators and weight flags Lemma
Nonzero highest-weight images survive modulo n-minus (BGG 10.6b) Lemma
Surjectivity modulo n-minus for free weight-generated modules (BGG 10.5) Lemma
The BGG differential induces an injection into kernel coinvariants (BGG 10.6) Lemma
The center has finite-dimensional image on each O-object Lemma
Tor with the trivial module is computed by the weak BGG resolution Lemma
Weight-lambda vectors are singular at a maximal label Lemma
Finite-dimensional Hom spaces in O Proposition
Finite-dimensional tensoring preserves O Proposition
Kostant cohomology and BGG characters give the same Weyl numerator Proposition
The support description of category O with finite generation Proposition
Verma and finite-dimensional weight modules belong to O Proposition
Category O is abelian and extension closed among weight modules Theorem
Parabolic category O is a highest weight category Theorem
The BGG resolution of a finite-dimensional simple module Theorem

…and 1 more result.

Dependency tree · two levels

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Sources

Lecture 2 §3 Definition 3.1, pp.4–5 (standard reference, not scraped)