Question

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!

Answer 1

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.