LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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Weights of a finite-dimensional simple module lie in the norm ball

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $ν$ be a dominant integral weight (Integral, dominant, and strictly dominant weights) and let $γ$ be a weight of the finite-dimensional simple module $L (ν)$ (Finite-dimensional simple modules are classified by dominant highest weights). Then, in the $W$ -invariant positive definite form on the real span $E$ of the roots (The roots form a reduced crystallographic Euclidean root system), $∣ γ ∣ \leq ∣ ν ∣,$ with equality if and only if $γ$ lies in the Weyl orbit $W ν$ ; moreover every weight in $W ν$ occurs in $L (ν)$ with multiplicity one.

Facts & Assumptions

Given: The Axiom of Choice, a dominant integral weight $ν$ , and a weight $γ$ of the finite-dimensional simple module $L (ν)$ .

[F1]

The module $L (ν)$ is finite-dimensional with highest weight $ν$ , its weights lie in $ν - Q^{+}$ , every weight $γ$ satisfies $ν - w^{- 1} γ \in Q^{+}$ for every $w \in W$ , and the weight $w ν$ occurs with multiplicity one for every $w$ (Extremal Weyl-orbit weights, Finite-dimensional simple modules are classified by dominant highest weights, Root order on weights).

[F2]

The form $(\cdot, \cdot)$ on $E$ is positive definite and $W$ -invariant; a dominant weight $ξ$ satisfies $(ξ, α_{i}) = ⟨ ξ, α_{i}^{\lor} ⟩ (α_{i}, α_{i}) / 2 \geq 0$ for every simple root $α_{i}$ , sums of dominant weights are dominant, and every $W$ -orbit in $E$ has exactly one dominant point (The roots form a reduced crystallographic Euclidean root system, Finite Weyl closed chambers and stabilizers, Integral, dominant, and strictly dominant weights).

Proof

technique · direct: pass to the dominant conjugate of the weight and compare squared lengths by a dominance computation

1.1F1F2given

The weight $γ$ lies in $E$ because it lies in $ν - Q^{+}$ , and the orbit $W γ$ has a unique dominant point, so there is $u \in W$ with $γ^{+} : = u γ$ dominant. Applying the extremal-weight bound of [F1] to $γ$ with the element $w = u^{- 1}$ gives $ν - u γ = ν - γ^{+} \in Q^{+}$ .

2.1F1F2step 1.1algebra

Put $β = ν - γ^{+} = \sum_{i} n_{i} α_{i} \in Q^{+}$ . Dominance gives $(γ^{+}, β) \geq 0$ , so $∣ ν ∣^{2} - ∣ γ^{+} ∣^{2} = 2 (γ^{+}, β) + ∣ β ∣^{2} \geq ∣ β ∣^{2} \geq 0$ . Since $∣ γ ∣ = ∣ γ^{+} ∣$ , this proves the norm bound. Equality forces $∣ β ∣^{2} = 0$ , hence $β = 0$ by positive definiteness and $γ^{+} = ν$ , so $γ \in W ν$ . Conversely $W$ -invariance gives equality for every $γ \in W ν$ .

3.1F1step 2.1∎

The multiplicity-one statement is the last assertion of [F1], and by step 2.1 the equality case is exactly $γ \in W ν$ ; this completes the proof of all three claims.

Depends on

Used by

Dependency tree · two levels

44 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

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Theorem 5.5(d)-(e) and remarks (standard reference, not scraped)
Pavel Etingof, Representations of Lie Groups (18.757, Fall 2023), Sec. 23.2 (standard reference, not scraped)