the normal bundle of $Z$ inside $G\times^P Z$. This normal bundle is the pullback of the normal bundle to the basepoint $P/P \in G/P$, hence its equivariant Euler class is the product $\prod_{\beta\in\Delta\setminus\Delta_P} \beta$ of the weights in the tangent space.

Applying $i^*$ to both sides of the formula, we therefore get $1 = \prod_{\beta\in\Delta\setminus\Delta_P} \beta/
  \prod_{\beta\in\Delta\setminus\Delta_P} \beta$. By the Weyl-invariance of both sides, the same confirmation holds for the pullback to each $wZ$. Now apply the injectivity above to conclude the formula on $G\times^P Z$ itself.

This has a well-known corollary due to Joseph:

[@Jo look in BBM]\[cor:Joseph\] Let $P_\alpha$ act on $Y$, and $Z$ be a $B$-invariant subscheme. Let $d$ be the degree of the map $P_\alpha \times^B Z \to P_\alpha\cdot Z$ unless $Z$ is $P_\alpha$-invariant, in which case let $d=0$. Let $\partial_\alpha$ denote the divided difference operator $\frac{1}{\alpha}(1 - r_\alpha)$, acting on ${\mathrm{Sym}}^\bullet(T^*)$. Then $$\partial_\alpha [Z]_Y = d\, [P_\alpha \cdot Z]_Y.$$

Let $L$ denote the Levi factor of $P_\alpha$ containing $T$, with semisimple part $L' {\mathop{\cong}}SL_2$. Then $P_\alpha = L B$, so $P_\alpha \cdot Z = L \cdot Z$. Applying Lemma \[lem:Hlocalization\], we learn $$\frac{[Z]_{L\times^B Z}}{\alpha}
  + \frac{r_\alpha\cdot [Z]_{L\times^B Z}}{-\alpha} = 1$$ as elements of $A_T(L\times^{B\cap L} 