$BS_{\vec\alpha} Z {\mathop{\twoheadrightarrow}}G\cdot Z$. That degree is in turn the product of the degrees $d_i$ of the maps $P_{\alpha_i} \times^B Z_{i-1} \to Z_i$, since $BS_{\vec\alpha} Z {\mathop{\twoheadrightarrow}}G\cdot Z$ factors as $$\left(\prod_{i=1}^k P_\alpha \times^B\right) Z {\mathop{\twoheadrightarrow}}\left(\prod_{i=1}^{k-1} P_\alpha \times^B\right) Z_1 {\mathop{\twoheadrightarrow}}\left(\prod_{i=1}^{k-2} P_\alpha \times^B\right) Z_2 {\mathop{\twoheadrightarrow}}\cdots {\mathop{\twoheadrightarrow}}Z_k = G\cdot Z$$ where the $\{Z_i\}$ are as in the proof of Proposition \[prop:BSshrinkdim\], and the $j$th map is the associated map of bundles over $\left(\prod_{i=1}^j P_\alpha \times^B\right) \cdot pt$ to the $B$-equivariant map $P_{\alpha_i} \times^B Z_{i-1} \to Z_i$.

Hence by $k$ applications of Corollary \[cor:Joseph\], $$d\, [G\cdot Z]_Y = \left(\prod_{i=1}^k d_i\right) [G\cdot Z]_Y
  = \left(\prod_{i=1}^k \partial_{\alpha_i}\right) [Z]_Y.$$

In the case $\kappa:G\times^P Z \to G\cdot Z$ is generically a finite map, we can use Lemma \[lem:BSforGmodP\] to know that for $\vec\alpha$ giving a reduced word for $w_0 w_0^P$, the map $BS_{\vec\alpha}\cdot Z \to G\cdot Z$ is also generically finite (with the same degree).

To see the alternate formula, we apply (as in the proof of Joseph’s Lemma) the pushforward $\kappa_*$ to the equation from Lemma \[lem:Hlocalization\]: $$\kappa_*(1) =
  \sum_{w\in W^P} w\cdot
  \frac{[Z]_Y}{\prod_{\beta\in\Delta\setminus\Delta_P} \beta}.$$ Since $\kappa$ is generically finite of degree $d$, the left-hand side is $\kappa_*(1) = d\, [{\mathop{\mathrm{Im}}}\kappa]_Y$.

The first part of this theorem only 