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The difference of the Weyl vector from its reflections is a sum of positive roots

Statement

Let $Φ$ be the root system with positive system $Φ^{+}$ , simple roots $α_{1}, \dots, α_{r}$ , Weyl group $W$ and Weyl vector $ρ = \frac{1}{2} \sum_{α \in Φ^{+}} α$ (Finite Weyl root system, lattice and chamber conventions, The Weyl vector rho for a chosen positive system, Root reflections and the Weyl group action). For every $w \in W$ , $ρ - w ρ = \sum_{\begin{array}{c} α \in Φ^{+} \\ w^{- 1} α \in Φ^{-} \end{array}} α,$ the sum running over the positive roots whose image under $w^{- 1}$ is a negative root. In particular $ρ - w ρ \in Q_{+}$ : it is a nonnegative integral combination of the simple roots.

Facts & Assumptions

Given: The finite root-system, positivity, length and lattice conventions of Finite Weyl root system, lattice and chamber conventions, a positive system $Φ^{+}$ with simple roots $α_{i}$ , the Weyl group $W$ , the Weyl vector $ρ$ , and an element $w \in W$ .

[F3]

The reflection $s_{α}$ acts by $s_{α} (λ) = λ - ⟨ λ, α^{\lor} ⟩ α$ and the Weyl vector is $ρ = \frac{1}{2} \sum_{α \in Φ^{+}} α$ (Root reflections and the Weyl group action, The Weyl vector rho for a chosen positive system).

[F4]

Every positive root is a nonnegative integral combination of the simple roots, so $Φ^{+} \subseteq Q_{+}$ (Simple roots form a signed integral basis, Finite Weyl root system, lattice and chamber conventions).

Proof

technique · direct

1.1F3givenalgebra

Put $S = {α \in Φ^{+} : w^{- 1} α \in Φ^{-}}$ . Since $w$ permutes the roots, $w Φ^{+}$ contains $α$ for each $α \in Φ^{+} ∖ S$ and $- α$ for each $α \in S$ , each exactly once: these assertions are respectively equivalent to $w^{- 1} α > 0$ and $w^{- 1} (- α) > 0$ . Therefore $2 w ρ = \sum_{α \in Φ^{+} ∖ S} α - \sum_{α \in S} α$ .

2.1F3F4step 1.1algebra∎

Subtracting the expression in step 1.1 from $2 ρ = \sum_{α \in Φ^{+}} α$ gives $ρ - w ρ = \sum_{α \in S} α$ . Every summand belongs to $Q_{+}$ by [F4], proving the claimed cone inclusion. For $w = 1$ or the empty root system the sum is empty and the same calculation gives zero.

Depends on

Used by

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Sources

P. Etingof, Lie Groups and Lie Algebras II (MIT 18.755, Spring 2024), complete lectures (standard reference, not scraped)
A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)