In a non-reduced root system, $s_{\alpha}(\beta)^{\vee} = s_{\alpha^{\vee}}(\beta^{\vee})$

Question

In Bill Casselman's notes on root systems (http://www.math.ubc.ca/~cass/courses/tata-07a/Roots.pdf), I am confused about the proof of the result $s_{\alpha}(\beta)^{\vee} = s_{\alpha^{\vee}}(\beta^{\vee})$ for any roots $\alpha,\beta$ (Corollary 3.10).

In these notes a root system is a quadruple $(V,\Sigma, V^{\vee}, \Sigma^{\vee})$, where $V, V^{\vee}$ are finite dimensional real vector spaces in a perfect pairing, and $\Sigma \subseteq V - 0, \Sigma^{\vee} \subseteq V^{\vee} - 0$ are finite subsets in bijection $\beta \mapsto \beta^{\vee}$. The usual axioms for the root system are given, except we do not require that $\Sigma$ spans $V$.

It is claimed that the result $s_{\alpha}(\beta)^{\vee} = s_{\alpha^{\vee}}(\beta^{\vee})$ can be proved in two ways: from Corollary 3.9, and from the orthogonal reflection formula (Corollary 3.3).

I can't see how this follows from Corollary 3.3, nor why Corollary 3.9 is true.

Method 1: Using Corollary 3.9, which states that the coroot $\alpha^{\vee}$ of a root $\alpha$ is the unique element of $V^{\vee}$ which lies in the span of $\Sigma^{\vee}$, satisfies $\langle \alpha, \alpha^{\vee} \rangle = 2$, and satisfies $$\sum\limits_{\gamma \in (\beta + \mathbb{Z}\alpha) \cap \Sigma} \langle \gamma, \alpha^{\vee} \rangle = 0 $$

for all roots $\beta$. It is claimed that if $l$ is another element of $V^{\vee}$ satisfying these properties, then $l - \alpha^{\vee}$ must be zero, but I don't see why this is.

Method 2: For $v, w \in V$, define

$$v \bullet w = \sum\limits_{\gamma \in \Sigma} \langle v, \gamma^{\vee} \rangle \langle w, \gamma^{\vee} \rangle$$

Then Corollary 3.3 says that for any $v \in V$ and any root $\beta$,

$$\langle v, \beta^{\vee} \rangle = 2 \frac{v \bullet \beta}{\beta \bullet \beta}$$

I don't see why Corollary 3.3 implies that $s_{\alpha^{\vee}}(\beta^{\vee}) = s_{\alpha}(\beta)^{\vee}$.

Answer 1

Proof of uniqueness in Method 1: it suffices to show that $\langle \beta, l - \alpha^{\vee} \rangle = 0$ for all roots $\beta$. Write

$$(\beta + \alpha\mathbb{Z}) \cap \Sigma = \{ \beta + n_i \alpha, i = 1, ... , r \}$$

this is nonempty.

Then

$$0 = \sum\limits_{i=1}^r \langle \beta + n_i \alpha, l \rangle = r \langle \beta, l \rangle + 2(n_1 + \cdots + n_r)$$

And also

$$0 = \sum\limits_{i=1}^r \langle \beta + n_i \alpha, \alpha^{\vee}\rangle = r \langle \beta, \alpha^{\vee} \rangle + 2(n_1 + \cdots + n_r)$$

so $\langle \beta, l \rangle = \langle \beta , \alpha^{\vee} \rangle$.