\dim Z_k = \dim Z + k$, i.e. $k = \dim G\cdot Z-\dim Z$.

The map $BS_{\vec\alpha}\iota: BS_{\vec\alpha} Z \to Z_k = G\cdot Z$ is surjective. By dimension count it is generically finite-to-one.

To construct the sequence $(\beta_j)$, consider the $B$-equivariant map $\{pt\} \to G/B$ taking a point to the identity coset, and apply $BS_{\vec\alpha}$ to that. The result $BS_{\vec\alpha} \cdot pt \to G/B$ is a [[****]{}Demazure-Hansen resolution]{} [@De; @Ha] of a Schubert variety in $G/B$, where the source is a Bott-Samelson manifold. Now select $(\beta_i)$ following the same procedure as was used above, to construct a finite-to-one map $BS_{\vec\beta} BS_{\vec\alpha}\cdot pt \to G/B$. In fact the resulting map is generically $1$:$1$ [@BS]. This obviously extends to a map of $B$-bundles, and our map $BS_{\vec\beta} BS_{\vec\alpha} Z \to G \times^B Z$ is the corresponding map of associated $Z$-bundles. Consequently it too is generically $1$:$1$.

To finish setting up the diagram, define $$\begin{array}{rclrcl}
    \mu : G\times^B Z &{\mathop{\twoheadrightarrow}}& G\cdot Z
    &\qquad [g,z] &\mapsto& g\cdot z \\
    BS_{\vec\beta} BS_{\vec\alpha} Z &{\mathop{\twoheadrightarrow}}& G\times^B Z
    &\qquad [g_1,\ldots,g_{j+k},z] &\mapsto& [g_1 g_2\cdots g_{k+j},z] \\
    BS_{\vec\beta} BS_{\vec\alpha} Z &{\mathop{\twoheadrightarrow}}& BS_{\vec\beta} Z_k
    =  BS_{\vec\beta} (G\cdot Z)
    &\qquad [g_1,\ldots,g_{j+k},z]
    &\mapsto& [g_1 g_2\cdots g_j,g_{j+1}\cdots g_{j+k} \cdot z]
  \end{array}$$ which are 