ScalingStacks

Assume $\zeta\in L(\theta)$. Since $\bar{\zeta}$ is negative, it follows that $\zeta$ is positive, then $(\theta^{\zeta})_{s}=\theta_{s}$ for $s{\not\in}\{s_{1},s_{2}\}$, while $(\theta^{\zeta})_{s_{1}}=\beta_{s_{2}}\circ\zeta$ and $(\theta^{\zeta})_{s_{2}}=\alpha_{s_{1}}\circ\zeta^{-1}$. We deduce that $(\beta\cdot u)\boxtimes v=\theta^{\zeta}$ if $\beta\cdot u\neq 0$. So, the assertion (8.1.3) is a consequence of the following:

(8.1.4)

$$\text{given }\zeta\in L(\theta)\text{ positive, we have }\beta\cdot u\neq 0\text{ if and only if }\zeta\in D(\theta).$$