MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer
I really meant $U\dot{v}U$, but I think what you mention will do the trick. I will read it then :) thank you! – May 30 '12 at 16:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.