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A maximal-label Verma is projective in its truncation

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $Γ$ be a finite downward-closed ideal of a linkage class $C$ (Truncation at a finite downward-closed ideal of a linkage class) and let $λ \in Γ$ be maximal in $Γ$ . Then $Δ (λ) = M (λ)$ is a projective object of the truncation $O_{Γ}$ (Projective object).

More precisely, for every $X \in O_{Γ}$ evaluation at the highest-weight generator $v_{λ}$ is a natural isomorphism ${Hom}_{O_{Γ}} (M (λ), X) \to X_{λ}^{n^{+}} = X_{λ}$ , and $X \mapsto X_{λ}$ is exact, so ${Hom}_{O_{Γ}} (M (λ), -)$ is exact.

Under the fixed positive-Borel convention the essential hypothesis is maximality of $λ$ in the finite ideal $Γ$ : maximality, not any antidominance or sufficient-positivity condition, is what makes every $λ$ -weight vector singular. For a weight $λ$ that is not maximal in $Γ$ , $Δ (λ)$ need not be projective in $O_{Γ}$ .

Facts & Assumptions

Given: The Axiom of Choice, a finite downward-closed ideal $Γ$ of a linkage class, a maximal element $λ \in Γ$ , and an object $X \in O_{Γ}$ .

[F1]

For every $X \in O_{Γ}$ , every vector of weight $λ$ is annihilated by $n^{+}$ , so $X_{λ}^{n^{+}} = X_{λ}$ , and the weight functor $X \mapsto X_{λ}$ is exact on $O_{Γ}$ (Weight-lambda vectors are singular at a maximal label).

[F2]

Sending a homomorphism $M (λ) \to V$ to the image of $v_{λ}$ is a natural bijection onto the $n^{+}$ -fixed vectors of weight $λ$ in any $g$ -module $V$ (The universal property of Verma modules, Verma modules).

[F3]

An object $P$ of an abelian category is projective exactly when the functor $Hom (P, -)$ is exact (Projective object, Projective object characterisations).

Proof

technique · direct: identify the Hom functor with an exact weight functor through the universal property

1.1F1F2given

For $X \in O_{Γ}$ the universal property [F2] identifies ${Hom}_{O_{Γ}} (M (λ), X)$ with the space of $n^{+}$ -fixed vectors of weight $λ$ in $X$ , naturally in $X$ ; by [F1] this space is $X_{λ}^{n^{+}} = X_{λ}$ .

1.2F1given

The functor $X \mapsto X_{λ}$ is exact on $O_{Γ}$ by [F1].

2.1F3step 1.1step 1.2∎

Combining steps 1.1 and 1.2, ${Hom}_{O_{Γ}} (M (λ), -)$ is naturally isomorphic to the exact functor $X \mapsto X_{λ}$ , hence is exact; by the characterisation [F3] the Verma module $M (λ)$ is a projective object of $O_{Γ}$ .

Depends on

Used by

A Verma module need not be projective in its block Counterexample
The same Verma in two ambient categories Example
The two projectives in the principal sl2 block Example
Translation through the sl2 wall Example
Finite-dimensional tensoring reaches every simple of a linkage class Lemma

Dependency tree · two levels

28 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, Representations of Lie Groups (18.757, Fall 2023), Proposition 16.4 and its proof (standard reference, not scraped)
Lin Chen, lecture notes (Spring 2024), Lecture 8, Section 4 (standard reference, not scraped)