Let $G = GL_n(\mathbb{C})$, $P \subseteq G$ a parabolic subgroup, and consider the decomposition $\frac{G}{P} = \bigsqcup_{w \in W^P} \frac{BwP}{P}$ where $W \cong S_n$ is the Weyl group of $G$, $W_P$ is the subset of $W$ associated to $P$ and $W^P = \frac{W}{W_P}$ and we consider $w \in W^P$ of minimal length in the above decomposition. Define $\phi: \frac{G}{P} = \bigsqcup_{w \in W^P} \frac{BwP}{P} \to G$ via $\phi(bwP) = bw$. My question is:
Is the map $\phi(bwP) = bw$ a morphism of algebraic varieties? Is it continuous in the Zariski topology?
$\textbf{Motivation}$: I am trying to show that the map $\frac{G}{B} \to \frac{G}{P}$ via $gB \mapsto gP$ is a fiber bundle. I have not been able to prove this directly, but a Theorem in Steenrod's book $\textit{Topology of Fibre Bundles}$ was suggested as helpful.
Let $G$ be a topological group and $H$ a closed subgroup. Let $x_0$ denote the coset $P \in \frac{G}{P}$, and suppose $V$ is an open neighborhood of $x_0$ in $\frac{G}{P}$. In Chapter 7, Steenrod defines a $\textit{local cross section}$ of $P$ in $G$ to be a continuous map $f: V \to G$ such that $p(f(xP)) = xP$ where $p: G \to \frac{G}{P}$ is the canonical quotient (I think this is called being locally constant?).
Theorem 3 in this chapter states that if we have closed subgroups $B \subset P \subset G$ and if $P$ admits a local cross section the the map from $\frac{G}{B} \to \frac{G}{P}$ via $gB \to gP$ is a fiber bundle.
Taking $V$ to be the whole space $G/P$ I believe my map above $\phi(bwP) = bw$ is a local cross section. I am not sure how to show that it is an morphism of algebraic varieties, or that it is continuous in the Zariski topology. Any and all help would be much appreciated.