ExampleConstruction: AI-generatedVerification: AI-generatedverified 2026-09-23 (gpt-6-sol)

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Dot-conjugate type- $A_{2}$ weights have the same central character

Example

In type $A_{2}$ , let $λ = 0$ and let $μ = s_{1} \cdot λ = - α_{1}$ . Then $λ$ and $μ$ have the same central character. Concretely,

$λ + ρ = (1, 0, - 1), μ + ρ = s_{1} (1, 0, - 1) = (0, 1, - 1),$

and the basic symmetric invariants take the same values on those two triples.

Facts & Assumptions

Given: Type $A_{2}$ in the realization ${(x_{1}, x_{2}, x_{3}) \in C^{3} : x_{1} + x_{2} + x_{3} = 0}$ , the simple reflection $s_{1}$ , and the weight $λ = 0$ .

Verification

technique · direct

1.1givenalgebra

In the $A_{2}$ realization, $ρ = (1, 0, - 1)$ and $s_{1}$ swaps the first two coordinates. Hence $μ = s_{1} \cdot 0 = s_{1} (ρ) - ρ = - α_{1}$ , and $μ + ρ = s_{1} (ρ) = (0, 1, - 1)$ .

2.1step 1.1∎

Here $⟨ ρ, α_{1}^{\lor} ⟩ = 1$ . By The simple-root singular vector in a Verma module, $f_{1} v_{0}$ is a nonzero highest vector of weight $μ = - α_{1}$ in $M (0)$ . The universal property The universal property of Verma modules gives a nonzero map $M (μ) \to M (0)$ . Every central element acts by a scalar on either cyclic highest-weight module by The Harish-Chandra projection computes the highest-weight scalar, and the map intertwines these actions. Hence $χ_{μ} = χ_{λ}$ . The displayed degree-two and degree-three symmetric invariants also agree on $ρ$ and $s_{1} (ρ)$ , as the coordinates show.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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)

Dot-conjugate type-A2A2​ weights have the same central character