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Finite-dimensional irreducible sl2 modules have exact integral central characters

Example

Let $G = {S L}_{2}$ and $n \geq 0$ . The finite-dimensional irreducible $V (n ρ)$ of highest weight $n ρ$ lies in BGG category $O$ and has exact central character $χ_{(n + 1) ρ}$ . Since the Weyl group sends $(n + 1) ρ$ to $- (n + 1) ρ$ , the same character has regular antidominant representative $κ = - (n + 1) ρ .$ The corollary The integral regular localization equivalence restricts to Category O therefore localizes this object to a coherent strongly B-equivariant $D_{κ}$ -module. The claim concerns the exact central-character subcategory $\ker χ_{κ} \cdot V = 0$ . The generalized-character condition $\forall v \exists N : (\ker χ_{κ})^{N} v = 0$ is a different condition; no equivalence for the entire generalized block is asserted.

Facts & Assumptions

Given: $G = {S L}_{2}$ , its Weyl group $W = {1, s}$ , $ρ (h) = 1$ , and an integer $n \geq 0$ .

[F1]

The simple finite-dimensional representation of highest weight $n ρ$ exists and is irreducible (Highest-weight classification).

[F2]

Finite-dimensional $h$ -semisimple modules belong to BGG category $O$ (Verma and finite-dimensional weight modules belong to O via The classical BGG category O).

[F3]

A central element acts on a highest-weight module of highest weight $λ$ by $pr (z) (λ)$ , and the shifted Harish–Chandra map is W-invariant (The Harish-Chandra projection computes the highest-weight scalar, Harish-Chandra isomorphism for the center).

[F4]

For integral regular antidominant $κ$ , the exact central-character part of BGG $O$ localizes to coherent strongly B-equivariant modules (The integral regular localization equivalence restricts to Category O).

Verification

1.1F1F2F3

By [F1], $V (n ρ)$ is finite-dimensional and irreducible. Its Cartan action is diagonalizable, so [F2] places it in $O$ . Schur's lemma makes every central element act by a scalar; evaluating on a highest vector and using [F3], its central character is $χ_{n ρ + ρ} = χ_{(n + 1) ρ}$ .

2.1F3step 1.1algebra

In type $A_{1}$ , $s ρ = - ρ$ , so $s ((n + 1) ρ) = - (n + 1) ρ = κ$ . W-invariance in [F3] gives $χ_{κ} = χ_{(n + 1) ρ}$ . The pairing $⟨ κ, α^{\lor} ⟩ = - (n + 1)$ is nonzero and nonpositive, so $κ$ is regular antidominant.

3.1F4step 2.1∎

The central ideal $\ker χ_{κ}$ annihilates $V (n ρ)$ exactly, not merely by a power, because its central character is $χ_{κ}$ by step 2.1. Thus it belongs to the exact subcategory in [F4], and that corollary gives the asserted coherent strongly B-equivariant localization. The generalized-character condition permits nilpotent action of the central ideal and is not used in this example.

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Sources

Masaki Kashiwara, Representation theory and D-modules on flag varieties, Astérisque 173–174 (1989) (standard reference, not scraped)