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Dominant integral weights are maxima of their Weyl orbits

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $ζ \in h^{*}$ be a dominant integral weight, that is, $⟨ ζ, α_{i}^{\lor} ⟩ \in Z_{\geq 0}$ for every simple root $α_{i}$ (equivalently, when $ζ$ lies in the real span of the roots, $⟨ ζ, α^{\lor} ⟩ \in Z_{\geq 0}$ for every positive root $α$ ). Then $ζ - w ζ \in Q^{+} for every w \in W,$ where $W$ is the Weyl group of Root reflections and the Weyl group action and $Q^{+}$ is the cone of Root order on weights. In particular $ζ$ is the maximum of its Weyl orbit for the order $μ \leq ν$ defined by $ν - μ \in Q^{+}$ , and if $w ζ \geq ζ$ then $w ζ = ζ$ .

Consequently, if $λ$ is a weight with $λ + ρ$ dominant integral ( $ρ$ the Weyl vector of The Weyl vector rho for a chosen positive system), then $λ - w \cdot λ \in Q^{+}$ for every $w \in W$ , and $λ$ is the maximum of its linkage class $W_{λ} \cdot λ$ of The integral Weyl group of a weight; here $W_{λ} = W$ because every simple reflection pairs integrally with $λ + ρ$ .

Integrality is used with its full strength: for a dominant $ζ$ that is not integral the conclusion $ζ - w ζ \in Q^{+}$ fails for reflections pairing non-integrally with $ζ$ , and only the reflections integral at $ζ$ can be used.

Facts & Assumptions

Given: The Axiom of Choice and a dominant integral weight $ζ$ in $h^{*}$ , and the Weyl group $W$ generated by the root reflections $s_{α}$ of Root reflections and the Weyl group action.

[F1]

The reflection is $s_{α} (λ) = λ - ⟨ λ, α^{\lor} ⟩ α$ , and under the identification of The roots form a reduced crystallographic Euclidean root system (parts (iii) and (iv)) these reflections are the reflections of the reduced crystallographic root system $Φ$ with $W$ -invariant positive definite form on the real span $E$ of the roots (The root set is a reduced crystallographic root system, Finite Weyl positive roots and simple reflections, Finite Weyl closed chambers and stabilizers).

[F2]

Simple reflections generate $W$ ; word length satisfies $ℓ (w s_{i}) = ℓ (w) - 1$ exactly when $w α_{i} < 0$ , and in a reduced word $w = s_{i_{1}} \dots s_{i_{k}}$ every prefix is reduced (Finite Weyl strong exchange and deletion, Finite Weyl positive roots and simple reflections).

[F3]

The relation $μ \leq λ$ defined by $λ - μ \in Q^{+}$ is a partial order on $h^{*}$ (Root order on weights).

Proof

technique · direct, through a reduced-word telescoping identity whose terms are nonnegative integer multiples of positive roots

1.1F1F2givenalgebra

By [F2] the simple reflections generate $W$ , so choose a reduced expression $w = s_{i_{1}} \dots s_{i_{k}}$ with $k = ℓ (w)$ and put $w_{j} = s_{i_{1}} \dots s_{i_{j}}$ . Since $w_{j} = w_{j - 1} s_{i_{j}}$ , the telescoping sum gives $ζ - w ζ = \sum_{j = 1}^{k} (w_{j - 1} ζ - w_{j} ζ) = \sum_{j = 1}^{k} w_{j - 1} (ζ - s_{i_{j}} ζ) = \sum_{j = 1}^{k} ⟨ ζ, α_{i_{j}}^{\lor} ⟩ w_{j - 1} α_{i_{j}}$ , using [F1] for the last equality.

2.1F1F2step 1.1algebra

Each coefficient is $⟨ ζ, α_{i_{j}}^{\lor} ⟩ \in Z_{\geq 0}$ because $α_{i_{j}}$ is simple and $ζ$ is dominant integral, and each vector $w_{j - 1} α_{i_{j}}$ is a positive root: otherwise $ℓ (w_{j - 1} s_{i_{j}}) = ℓ (w_{j - 1}) - 1$ by [F2], contradicting that the prefix $w_{j}$ of the reduced word $w$ is reduced and that $w = w_{j} s_{i_{j + 1}} \dots s_{i_{k}}$ has length $k$ . Hence every term of the sum of step 1.1 lies in $Z_{\geq 0} Φ^{+} \subseteq Q^{+}$ , and $ζ - w ζ \in Q^{+}$ for every $w \in W$ .

3.1F3step 2.1

Since $ζ - w ζ \in Q^{+}$ means $w ζ \leq ζ$ , every Weyl conjugate of $ζ$ lies below $ζ$ in the root order, so $ζ$ is the maximum of its Weyl orbit; and if in addition $ζ \leq w ζ$ , then $ζ = w ζ$ by antisymmetry of the partial order [F3].

3.2F1F2step 2.1algebra

For the dot-action statement let $λ$ be a weight with $λ + ρ$ dominant integral, and apply step 2.1 to $ζ = λ + ρ$ : then $(λ + ρ) - w (λ + ρ) \in Q^{+}$ , that is, $λ - w \cdot λ \in Q^{+}$ for every $w \in W$ ; moreover $⟨ λ + ρ, α_{i}^{\lor} ⟩ \in Z$ for every simple root $α_{i}$ , so every simple reflection lies in $W_{λ}$ and $W_{λ} = W$ by [F2]; hence every element of the linkage class $W_{λ} \cdot λ$ is a Weyl conjugate of $λ$ and lies below $λ$ .

4.1step 3.1step 3.2∎

Combining steps 3.1 and 3.2: for every $w \in W$ one has $ζ - w ζ \in Q^{+}$ and $λ - w \cdot λ \in Q^{+}$ in the dot setting, so $ζ$ and $λ$ are the maxima of their orbits and linkage classes respectively, and $w ζ \geq ζ$ forces $w ζ = ζ$ .

Depends on

Used by

A Verma module need not be projective in its block Counterexample
Finite-dimensional tensoring reaches every simple of a linkage class Lemma

Dependency tree · two levels

36 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 (18.757, Fall 2023), Corollary 16.1 and its proof (standard reference, not scraped)
Dennis Gaitsgory, Geometric Representation Theory (Fall 2005), proof of Theorem 4.26 (standard reference, not scraped)