and assume $Z$ and $G\cdot Z$ have rational singularities.

Let $m_0 = [Z]^K_Y$, and construct a sequence of Laurent polynomials $m_1, m_2,\ldots$ by applying Demazure operators $d_\alpha := (1 - \exp(-\alpha))^{-1} (1 - \exp(-\alpha) r_\alpha)$ to $m_0$, where $\alpha$ varies over the set of simple roots of $G$. Don’t bother applying any $d_\alpha$ that acts as the identity, and only stop when all $d_\alpha$ act as the identity. The sequence of simple roots can be taken to give a reduced expression for $w_0 w_0^P$. This process terminates after finitely many steps (namely, $\dim G\times^P Z - \dim G\cdot Z$). The last Laurent polynomial in this sequence is the $K$-polynomial $[G\cdot Z]^K_Y$. Moreover $$[G\cdot Z]^K_Y = \sum_{w\in W^P} w\cdot
  \frac{[Z]^K_Y}{\prod_{\beta\in \Delta\setminus\Delta_P} (1-\exp(-\beta))}.$$

In the cases where Theorem \[thm:Kpoly\] applies, it implies Theorem \[thm:multidegree\], by viewing the multidegree as the lowest-order homogeneous component of the $K$-polynomial.

Kempf worked only with the case that $Z \subseteq Y$ is a linear subspace (in which case its $K$-polynomial $[Z]^K_Y$ is again a very simple product over the weights in $Y/Z$), which will also suffice for our main application. It is frequently the case that the weights of $T$ on $Y$ are distinct, which implies that 