LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Central elements act by scalars on cyclic highest-weight modules

Statement

Every central element acts on a cyclic highest-weight module by a scalar. In particular, each cyclic highest-weight module has a well-defined central character in the sense of Central character of a Lie algebra module.

Facts & Assumptions

Given: A cyclic highest-weight module $M = U (g) v$ of highest weight $λ$ with highest vector $v$ , and a central element $z \in Z (U (g))$ .

Proof

technique · direct

1.1

The highest-weight line is one-dimensional, so $z v$ must equal $c v$ for a unique scalar $c \in C$ . Indeed, $z$ commutes with $h$ and $n^{+}$ , so $z v$ has the same weight as $v$ and is again killed by $n^{+}$ .

givenalgebra

2.1

Every element of $M$ has the form $u v$ with $u \in U (g)$ , and centrality gives $z (u v) = u (z v) = u (c v) = c u v$ . Thus $z$ acts as $c {id}_{M}$ .

step 1.1algebra

3.1

Sending $z$ to the scalar $c$ from step 1.1 is an algebra homomorphism on the center, so $M$ has the central character promised in Central character of a Lie algebra module.

step 2.1∎

Depends on

Used by

The quadratic Casimir eigenvalue on a highest-weight module is (λ,λ+2ρ) Proposition

Dependency tree · two levels

5 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

Yiannis Sakellaridis, Verma Modules and the Category O (standard reference, not scraped)
Lin Chen, Geometric Representation Theory I, Lecture 4 (standard reference, not scraped)