\dim G\cdot Z$), and the last polynomial in this sequence is $d$ times $[G\cdot Z]$.

In the case that $\kappa$ is generically finite, the sequence of simple roots can be taken to give a reduced expression for $w_0 w_0^P$, the product of the long elements of the Weyl groups $W$ of $G$ and $W_P$ of $P$ respectively. In this case there is an alternate formula $$d\,[G\cdot Z] = \sum_{w\in W^P} w\cdot
  \frac{[Z]}{\prod_{\beta \in \Delta\setminus \Delta_P} \beta}$$ where $W^P$ is the set of minimal coset representatives in $W/W_P$, and $\Delta$ and $\Delta_P$ are the sets of roots of $G$ and $P$ respectively.

In Section \[sec:mdegs\] we give an example in which $\kappa$ does not have connected general fiber.

When $\kappa$ does have connected fibers and $Z$ and $G\cdot Z$ have rational singularities, we can use the collapsing to compute a more precise invariant than the multidegree, which is the [*$K$-polynomial*]{} $[G\cdot Z]^K_Y$. Essentially, this is the numerator of the multigraded Hilbert series of the sheaf ${\mathcal{O}}_{G\cdot Z}$ on $Y$; we recall the precise definition in Section \[sec:mdegs\].

\[thm:Kpoly\] Let $\kappa : G\times^P Z\to Y$ be a Kempf collapsing whose general fiber is connected (so $d=1$ in the notation of Theorem \[thm:multidegree\]), 