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The augmentation kernel is the sum of the simple-reflection Verma submodules

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $λ \in Λ^{+}$ and let $d_{1} ⁣ : ⨁_{i} M (s_{i} \circ λ) \to M (λ)$ be the first BGG differential of The BGG differential from signed Verma maps, indexed by the simple reflections $s_{i}$ . Then $im d_{1} = \sum_{i} M (s_{i} \circ λ) = \ker π$ , where $π ⁣ : M (λ) ↠ L (λ)$ is the canonical surjection and $M (s_{i} \circ λ) \subseteq M (λ)$ is the canonical submodule generated by the singular vector $f_{i}^{⟨ λ, α_{i}^{\lor} ⟩ + 1} v_{λ}$ of The simple-root singular vector in a Verma module. Consequently the augmented complex is exact at $C_{0}$ and $L (λ) = coker (d_{1})$ .

Facts & Assumptions

Given: The Axiom of Choice, a dominant integral weight $λ \in Λ^{+}$ , the Verma module $M (λ)$ with highest weight vector $v_{λ}$ , its canonical simple quotient $π ⁣ : M (λ) ↠ L (λ)$ , and the first BGG differential $d_{1} = ⨁_{i} (d_{1})_{s_{i}, e}$ .

[F1]

The elements of $W$ of length $1$ are exactly the simple reflections $s_{i}$ , and the arrows into the identity are the covers $s_{i} ⊳ e$ with label $α_{i}$ ; for every $i$ the component $(d_{1})_{s_{i}, e} = ε (s_{i}, e) ι_{s_{i} \to e}$ with $ε (s_{i}, e) = \pm 1$ is a nonzero scalar multiple of the canonical injective embedding $ι_{s_{i} \to e} ⁣ : M (s_{i} \circ λ) ↪ M (λ)$ , and all other components vanish (The BGG differential from signed Verma maps, Bruhat covers give canonical Verma embeddings, and composites are inclusions, Finite Weyl positive roots and simple reflections, The Bruhat graph and the BGG Verma sum in degree k).

[F2]

$M (s_{i} \circ λ)$ is the $g$ -submodule of $M (λ)$ generated by the singular vector $f_{i}^{m_{i} + 1} v_{λ}$ , where $m_{i} = ⟨ λ, α_{i}^{\lor} ⟩ \geq 0$ ; this vector has weight $s_{i} \circ λ \neq λ$ , and every weight of $M (s_{i} \circ λ)$ is $\leq s_{i} \circ λ$ in the root order, so $λ$ is not a weight of $M (s_{i} \circ λ)$ (Dominant integral dot translates embed canonically in the Verma module, The simple-root singular vector in a Verma module, Highest weight modules lie below the top weight).

[F3]

$M (λ) = U (g) / K$ as a left $U (g)$ -module, where $K$ is the left ideal generated by $n^{+}$ and by ${H - λ (H) : H \in h}$ , with $v_{λ} = 1 + K$ ; equivalently $M (λ) = U (g) \otimes_{U (b)} C_{λ}$ with the induced universal property (Verma modules, The universal property of Verma modules, The PBW model of a Verma module).

[F4]

The cyclic module $M_{i n t} (λ) = U (g) / I_{λ}$ of Dominant cyclic highest-weight presentation is generated by $v_{λ} = 1 + I_{λ}$ with $n^{+} v_{λ} = 0$ , $H v_{λ} = λ (H) v_{λ}$ , $f_{i}^{m_{i} + 1} v_{λ} = 0$ , and it is finite-dimensional (Simple-root integrability bounds the dominant cyclic module).

[F5]

The sum $J (λ)$ of all proper submodules of $M (λ)$ is the unique maximal submodule and $\ker π = J (λ)$ ; $L (λ) = M (λ) / J (λ)$ is simple, and $L (λ)$ is the unique simple quotient of $M (λ)$ (A Verma module has a unique simple quotient, The sum of all proper Verma submodules is proper, A proper Verma submodule misses the highest-weight line).

[F6]

Finite-dimensional $g$ -modules are completely reducible, and the finite-dimensional simple $g$ -modules are exactly the $L (μ)$ for $μ \in Λ^{+}$ , with $L (μ) ≅ L (μ^{'})$ only for $μ = μ^{'}$ ; every weight of $L (μ)$ is $\leq μ$ (Every finite-dimensional module is a direct sum of highest-weight modules, Finite-dimensional simple modules are classified by dominant highest weights, Highest weight modules lie below the top weight).

[F7]

For every simple root $α_{i}$ , $⟨ λ + ρ, α_{i}^{\lor} ⟩ = ⟨ λ, α_{i}^{\lor} ⟩ + 1 \geq 1$ , and $s_{i} \circ λ = λ - ⟨ λ + ρ, α_{i}^{\lor} ⟩ α_{i} \neq λ$ (Positive coroot pairings of a dominant integral weight, The Weyl vector rho for a chosen positive system).

Proof

1.1F1

The image of $d_{1}$ is the sum of the images of its components. By [F1] the only nonzero components are the $(s_{i}, e)$ -components, each of which is a nonzero scalar multiple of the injective embedding $ι_{s_{i} \to e}$ ; hence the image of the $i$ -th summand is exactly $M (s_{i} \circ λ) \subseteq M (λ)$ , and $im d_{1} = \sum_{i} M (s_{i} \circ λ)$ .

1.2F2F5F7

Each $M (s_{i} \circ λ)$ is a proper submodule: $s_{i} \circ λ \neq λ$ by [F7], and a submodule of $M (λ)$ containing $v_{λ}$ would be all of $M (λ)$ and would contain the weight $λ$ , which by [F2] is not a weight of $M (s_{i} \circ λ)$ . Hence $\sum_{i} M (s_{i} \circ λ) \subseteq J (λ) = \ker π$ .

1.3F2F3F4

The quotient $Q : = M (λ) / \sum_{i} M (s_{i} \circ λ)$ is isomorphic to $M_{i n t} (λ)$ . Indeed, by [F3] the preimage in $U (g)$ of $\sum_{i} M (s_{i} \circ λ)$ is the left ideal generated by $K$ together with the elements $f_{i}^{m_{i} + 1}$ , because the submodule generated by the vectors $f_{i}^{m_{i} + 1} v_{λ}$ has preimage $K + \sum_{i} U (g) f_{i}^{m_{i} + 1}$ and $M (s_{i} \circ λ)$ is exactly that submodule by [F2]; this preimage is precisely the left ideal $I_{λ}$ of [F4].

2.1step 1.3F2F4

$Q$ is finite-dimensional by [F4] and step 1.3, and its generator $\overset{ˉ}{v} = v_{λ} + \sum_{i} M (s_{i} \circ λ)$ is nonzero of weight $λ$ : it is nonzero because the sum is proper by step 1.2, and $Q = U (g) \overset{ˉ}{v}$ with $Q_{λ} = C \overset{ˉ}{v}$ because every weight of $Q$ is $\leq λ$ and the weight- $λ$ space of the quotient is the image of $M (λ)_{λ} = C v_{λ}$ .

3.1F4F5F6step 2.1

We identify $Q$ with $L (λ)$ . As a finite-dimensional $g$ -module, $Q$ is completely reducible, $Q ≅ ⨁_{j} L (μ_{j})$ , and the multiplicity of $L (λ)$ in $Q$ equals $\dim Q_{λ} = 1$ : a summand $L (μ)$ has a weight- $λ$ vector only if $λ \leq μ$ , and all weights $μ_{j}$ of $Q$ satisfy $μ_{j} \leq λ$ because $Q$ is a quotient of $M (λ)$ ; hence $μ = λ$ , and $L (λ)$ occurs with multiplicity $\dim Q_{λ} = 1$ . Since $Q$ is generated by $\overset{ˉ}{v} \in Q_{λ}$ , which lies in the unique $L (λ)$ -summand, $Q$ equals that summand: $Q ≅ L (λ)$ .

4.1F5step 1.1step 1.2step 3.1∎

Since $Q = M (λ) / \sum_{i} M (s_{i} \circ λ)$ is simple by step 3.1, the submodule $\sum_{i} M (s_{i} \circ λ)$ is maximal in $M (λ)$ . It is contained in $J (λ)$ by step 1.2, and $J (λ)$ is the unique maximal submodule by [F5], so $\sum_{i} M (s_{i} \circ λ) = J (λ) = \ker π$ . Combining with step 1.1 gives $im d_{1} = \ker π = \ker d_{0}$ : the augmented complex is exact at $C_{0}$ , and $coker (d_{1}) = M (λ) / im d_{1} ≅ L (λ)$ .

Depends on

Used by

The BGG resolution for sl2 Example
The BGG resolution of a finite-dimensional simple module Theorem

Dependency tree · two levels

74 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

Fan Zhou, The classical and the functorial BGG resolutions (Columbia thesis 2021), Part I Sec. 4.1 Step 2, pp. 15-17 (K8.27, K8.28a, K8.28b) (standard reference, not scraped)
A. Rocha-Caridi, Splitting criteria for modules induced from a subalgebra of a semisimple Lie algebra, Trans. AMS 262 (1980), Sec. 10, Lemma 10.1, p. 353 (standard reference, not scraped)