is also easily derived from the above properties.) The third is a technical result in equivariant Chow theory:

\[lem:Hlocalization\] Let $Z$ be a $P$-variety and let $A_T(pt)_{frac}$ denote the field of fractions of the polynomial ring $A_T(pt)$. Then we have a formula in the localization $A_T(G\times^P Z){{\otimes}}_{A_T(pt)} A_T(pt)_{frac}$ of the equivariant Chow ring $A_T(G\times^P Z)$: $$1 = \sum_{w\in W^P} w\cdot
  \frac{[Z]_{G\times^P Z}}{\prod_{\beta\in\Delta\setminus\Delta_P} \beta}$$ where $[Z]_{G\times^P Z} \in A_T(G\times^P Z)$ is the class induced by the regularly embedded subvariety $Z$.

As the map $G\times^P Z {\mathop{\twoheadrightarrow}}G/P$ is $T$-equivariant (indeed, $G$-equivariant), all the $T$-fixed points in $G\times^P Z$ lie over the $T$-fixed points $\{wP : w\in W^P\}$ in $G/P$. So we get inclusions $(G\times^P Z)^T {\operatorname*{\hookrightarrow}}\bigcup_{w\in W^P} wZ {\operatorname*{\hookrightarrow}}G\times^P Z$.

Then we use the fact, proven in [@Br section 3.2], that the inclusion of fixed points (the composite of the two above) induces an injective pullback $A_T(G\times^P Z) {\operatorname*{\hookrightarrow}}A_T((G\times^P Z)^T)$. Hence the map $A_T(G\times^P Z) {\operatorname*{\hookrightarrow}}A_T(\cup_{w\in W^P} wZ) {\mathop{\cong}}\bigoplus_{w\in W^P} A_T(wZ)$ is injective, and to prove the two sides of the formula agree it will suffice to check their images.

Let $i:Z{\operatorname*{\hookrightarrow}}G\times^P Z$ take $z \mapsto [1,z]$. Then $i^* [Z]_{G\times^P Z} = i^* i_* 1 = $ the equivariant Euler class of 