Let $G$ be a connected reductive group (although I don't think this is relevant here), $B$ a Borel subgroup containing a maximal torus $T$ and $U$ the associated unipotent radical ($U^-$ the opposite unipotent subgroup). Finally, let $v \in W$ and $\dot{v}$ a lift of $v$ in $N\_G(T)$.


How do you prove that there is an isomophism of varieties 
$$ (U\dot{v} \cap \dot{v}U^-) \times U \to U\dot{v}U, $$
which is the multiplication $(x,y) \mapsto xy$ ?


Thank you!



Double coset isomorphism
I assume you mean that there is an isomorphism of varieties
$$ (U \dot v \cap \dot v U^-) \times B \to B \dot v B : (x,y) \mapsto xy.$$
Note that it is actually somewhat more natural to write the isomorphism as
$$ (U \cap \dot v U^- \dot v^{-1}) \times B \to B \dot v B : (x,y) \mapsto x \dot v y.$$
The proof of this fact (which is part of what is known as the *Bruhat decomposition*) can be found, for instance, in Borel's "Linear Algebraic Groups", Second Enlarged Edition, p. 195 at the bottom.


