PropositionStatement: Literature-sourcedProof: AI-adaptedverified 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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The Harish-Chandra map on the center is injective

Statement

Assume the Axiom of Choice. For a finite-dimensional complex semisimple Lie algebra $g$ and any Cartan subalgebra $h$ in the nilpotent self-normalizing sense, if a central element $z \in Z (U (g))$ has zero Harish-Chandra projection relative to a chosen positive root system, then $z = 0$ . Equivalently, the Harish-Chandra map on the center is injective.

Facts & Assumptions

Given: The Axiom of Choice, a finite-dimensional complex semisimple Lie algebra $g$ , any Cartan subalgebra $h$ in the nilpotent self-normalizing sense, chosen positive roots, and a central element $z \in Z (U (g))$ with $pr (z) = 0$ .

[A1]

The Axiom of Choice is The Axiom of Choice and is inherited through [F1].

[F1]

Under [A1], the shifted Harish-Chandra map for these data is an algebra isomorphism (Harish-Chandra isomorphism for the center).

Proof

technique · direct

1.1A1F1given

By [A1, F1], the shifted Harish-Chandra map ${HC}_{ρ} (w) (λ) = pr (w) (λ - ρ)$ is an algebra isomorphism from $Z (U (g))$ to $S (h)^{W}$ . Therefore it is injective.

2.1step 1.1∎

If $pr (z) = 0$ , then ${HC}_{ρ} (z) = 0$ as a polynomial function on $h^{*}$ . Injectivity from step 1.1 then forces $z = 0$ .

Depends on

Used by

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Dependency tree · two levels

13 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)
Yiannis Sakellaridis, Verma Modules and the Category O (standard reference, not scraped)