LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Generalized central-character summands

Statement

Every $M \in O$ decomposes canonically into finitely many nonzero generalized central-character submodules:

$M = ⨁_{χ} M_{χ} .$

For each summand there is a single $N \geq 1$ such that $m_{χ}^{N} M_{χ} = 0$ . The decomposition of zero is empty.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

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 $M \in O$ and $Z = Z (U (g))$ , the image algebra $Z / {Ann}_{Z} (M)$ is finite dimensional over $C$ . (The center has finite-dimensional image on each O-object)

[F2]

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 $w \cdot λ = w (λ + ρ) - ρ$ . Let $Z = Z (U (g))$ and let $χ : Z \to C$ be a unital complex-algebra character, as in def-central-character-of-a-lie-algebra-module. Put $m_{χ} = \ker χ$ and, for $M$ in def-bgg-category-o, define $M_{χ} = {v \in M : m_{χ}^{N} v = 0 for some integer N \geq 1} .$ Here $m_{χ}^{N} v = 0$ means every element of that ideal kills $v$ ; the exponent may initially depend on $v$ . The full subcategory $O_{χ}$ consists of the objects with $M = M_{χ}$ . This is a generalized central-character condition, weaker than scalar central action. It does not by definition assert that $O_{χ}$ is an indecomposable block. (Generalized central-character subcategories)

[F3]

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 $w \cdot λ = w (λ + ρ) - ρ$ . The category $O$ is closed under submodules, quotients and finite direct sums and is an abelian category. If $0 \to A \to E \to B \to 0$ is exact, $A, B \in O$ , and $E$ is $h$ -semisimple, then $E \in O$ . The middle-term weight hypothesis is essential. (Category O is abelian and extension closed among weight modules)

Proof

1.1

If $M = 0$ there is nothing to split. Otherwise let $R = Z / {Ann}_{Z} (M)$ , a nonzero finite-dimensional commutative algebra. Choose a vector-space basis $a_{1}, \dots, a_{d}$ of $R$ and consider commuting multiplication operators on its regular representation $R$ .

F1construct

2.1

For each $a_{i}$ , its minimal polynomial splits over $C$ into powers of distinct linear factors. Bezout identities for these relatively prime factors produce polynomial projections onto their generalized eigenspaces: the polynomials are 1 modulo one factor and 0 modulo every other, hence are orthogonal idempotents summing to 1 when evaluated at $a_{i}$ . Multiply these commuting projections for all $i$ and omit zero products. We obtain nonzero orthogonal elements $e_{t} \in R$ summing to 1 and ideals $R_{t} = e_{t} R$ .

algebrastep 1.1

3.1

On $R_{t}$ , multiplication by each $a_{i}$ has one eigenvalue $c_{i t}$ and $a_{i} e_{t} - c_{i t} e_{t}$ is nilpotent. The ideal $J_{t}$ generated by these finitely many commuting nilpotents is nilpotent: if their nilpotence exponents are $n_{i}$ , any product of more than $\sum_{i} (n_{i} - 1)$ generators vanishes. Since the $a_{i}$ span $R$ , $R_{t} / J_{t}$ is spanned by $e_{t}$ ; it is nonzero because a nilpotent ideal cannot contain its nonzero unit. Thus this quotient is $C$ and gives a unique character $χ_{t} : R \to C$ on that factor.

algebrastep 2.1

4.1

The $e_{t}$ act centrally on $M$ , so $M = ⨁_{t} e_{t} M$ as $g$ -modules, with each summand in $O$ . A fixed power of $\ker χ_{t}$ kills $e_{t} M$ by the nilpotence just proved, where $χ_{t}$ is composed with $Z \to R$ . For a different character $ψ$ of $Z$ , choose $z$ with $ψ (z) \neq χ_{t} (z)$ . On $e_{t} M$ , $z - ψ (z)$ is a nonzero scalar plus a nilpotent operator, hence is invertible by a finite geometric series. It cannot kill a nonzero vector to any power.

F2F3algebrastep 3.1

5.1

It follows that the intrinsic submodule $M_{χ_{t}}$ is exactly $e_{t} M$ and all other $M_{χ}$ vanish. This proves independence from the chosen basis and projections, as well as the common annihilating power on each summand.

F2algebrastep 4.1∎

Depends on

Used by

Dependency tree · two levels

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Sources

§15.1 Corollary 15.7 and proof, p.80 (standard reference, not scraped)