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LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-05
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The rho-shift intertwines the dot and ordinary Weyl actions

Statement

Let T be the translation operator on polynomial functions on h∗ defined by

(Tf)(λ):=f(λ−ρ).

Then f is invariant under the dot action of W if and only if Tf is invariant under the ordinary action of W.

Facts & Assumptions

Given: A Weyl-group element w∈W, the Weyl vector ρ, and the dot action w⋅λ:=w(λ+ρ)−ρ.

[F1]

The Weyl group acts linearly on h∗, and the dot action is defined by w⋅λ=w(λ+ρ)−ρ (Root reflections and the Weyl group action, The Weyl vector rho for a chosen positive system).

Proof

technique · direct
1.1

If f is dot-invariant, then for every w∈W and λ∈h∗ one has (Tf)(wλ)=f(wλ−ρ)=f(w⋅(λ−ρ))=f(λ−ρ)=(Tf)(λ), because [F1] gives w⋅(λ−ρ)=w((λ−ρ)+ρ)−ρ=wλ−ρ.

F1givenalgebra
1.2

Conversely, if Tf is ordinarily W-invariant, then for every w∈W and λ∈h∗, [F1] gives f(w⋅λ)=f(w(λ+ρ)−ρ)=(Tf)(w(λ+ρ))=(Tf)(λ+ρ)=f(λ). So f is dot-invariant.

F1givenalgebra
2.1

Thus translation by ρ converts the dot action into the ordinary Weyl action and intertwines their invariant polynomials.

step 1.1step 1.2∎

Depends on

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