Lemma 6.2.4 (0P7K) · ScalingStacks

0P7K

Lemma 6.2.4. Let $w\in W_{|I|}$ , $m\in{\mathbf{Z}}$ and let $\sigma=F_{I}(wc^{m})$ be the element of $\operatorname{End}\nolimits_{{\mathcal{S}}_{n}}(I)$ corresponding to $wc^{m}$ . We have $\ell(\sigma)=\ell(w)$ , $\llbracket\sigma\rrbracket=m\cdot\delta$ and $m(\sigma)=2m\varepsilon_{I}$ .

0P7L

Proof. The first statement follows from the fact that $F_{I}$ preserves lengths (cf the discussion before Lemma 6.2.2).

Note that $\llbracket s_{i,j}\rrbracket=0$ for $i,j\in\tilde{I}$ with $i-j{\not\in}n{\mathbf{Z}}$ , while $\llbracket F_{I}(c)\rrbracket=\delta$ . We deduce that $\llbracket\sigma\rrbracket=m\cdot\delta$ .

The last statement of the lemma is immediate. ∎