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Roots of the infinite dihedral geometric representation

Statement

Use the real Coxeter representation of Real Coxeter representation and contragredient notation for $S = {s, t}$ and $m (s, t) = \infty$ . Write $α = α_{s}$ , $β = α_{t}$ and $δ = α + β$ . Its roots are exactly ${\pm (α + k δ) : k \in Z}$ . In dual coordinates $x = f (α)$ , $y = f (β)$ , $a = x + y$ and $b = x$ , its distinct root hyperplanes are $H_{k} = {b + k a = 0}$ , indexed by $k \in Z$ .

Facts & Assumptions

[F1]

Given: The geometric form has $B (α, β) = - 1$ and the roots are the orbits of the two simple roots, as defined in Real Coxeter representation and contragredient notation. Goal: Compute those orbits and kernels.

Proof

technique · direct

1.1F1algebra

The reflection formula gives $s α = - α$ , $s β = β + 2 α$ , $t α = α + 2 β$ , $t β = - β$ . Thus both generators fix $δ$ . If $r_{k} = α + k δ$ , then $s r_{k} = - r_{- k}$ and $t r_{k} = - r_{- k - 2}$ . The set ${\pm r_{k}}$ is invariant and contains $α = r_{0}$ and $β = - r_{- 1}$ , so it contains every root.

2.1step 1.1algebra

Composing the two displayed transformations changes the index by $2$ or $- 2$ , without changing the overall sign. Hence the orbit of $α$ contains all even-indexed $r_{k}$ , and the orbit of $β$ contains all odd-indexed $r_{k}$ , up to sign. Applying a reflection supplies their negatives as well. This proves equality with the root set.

3.1step 2.1algebra∎

Evaluation gives $f (r_{k}) = b + k a$ . Since $(a, b)$ are independent coordinates, the forms $b + k a$ and $b + ℓ a$ are proportional only if $k = ℓ$ . Their kernels are precisely the distinct root hyperplanes.

Depends on

Real Coxeter representation and contragredient notation

Used by · 1

The infinite dihedral Tits cone is an open half-plane with its vertex

Dependency tree · 2 levels

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