Z)$. Let $\kappa$ denote the action map $L\times^{B\cap L} Z \to Y$, and apply $\kappa_*$ to both sides: $$\frac{[Z]_Y - r_\alpha\cdot [Z]_Y}{\alpha} = \kappa_*(1).$$ If $\kappa$ is generically finite of degree $d$, the right-hand side is $d\, [L\cdot Z]$, and otherwise $0$.

(In this corollary we see the reason for $Lie(P)$ to contain all the [*negative*]{} root spaces rather than the positive ones; divided difference operators are usually defined for application to Schubert polynomials, which come from Schubert varieties that are $B_-$-invariant not $B$-invariant.)

Use Proposition \[prop:BSshrinkdim\] to create a sequence $(\alpha_i)$. The condition in Proposition \[prop:BSshrinkdim\] on $(\alpha_i)$ is that $Z_i$ should grow in dimension at each step, which is the condition that the $d$ from Corollary \[cor:Joseph\] is nonzero. By the assumption that all the weights of $Y$ lie in a half-space, $[P_\alpha \cdot Z]_Y \neq 0$. Hence the dimension grows if and only if $\partial_\alpha$ does not act as zero. So the conditions on $(\alpha_i)$ in the theorem’s statement match those used in Proposition \[prop:BSshrinkdim\].

By Proposition \[prop:BSshrinkdim\], the map $BS_{\vec\alpha} Z \to Y$ has image $G\cdot Z$. The number of components in a general fiber of $G\times^B Z {\mathop{\twoheadrightarrow}}G\cdot Z$ is the degree of the map 