LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-09-23 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Central elements lie in the zero-weight subspace of $U (g)$

Statement

Every central element of $U (g)$ lies in the zero-weight subspace for the adjoint action of the Cartan subalgebra. Equivalently, if $z \in Z (U (g))$ , then

$[h, z] = 0 (h \in h) .$

Facts & Assumptions

Given: A central element $z \in Z (U (g))$ and an element $h \in h$ .

Proof

technique · direct

1.1given

If $z$ is central, then it commutes with every element of $g$ , and in particular with every $h \in h$ . Thus $[h, z] = 0$ for all $h \in h$ .

2.1step 1.1∎

By definition of the adjoint weight decomposition that underlies The Harish-Chandra projection, the condition from step 1.1 is exactly the statement that $z$ has weight zero. Hence $z \in U (g)_{0}$ .

Depends on

The Harish-Chandra projection

Used by

Dependency tree · two levels

4 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 (standard reference, not scraped)

Central elements lie in the zero-weight subspace of U(g)U(g)