Double coset isomorphism - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:42:27Z http://mathoverflow.net/feeds/question/98359 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98359/double-coset-isomorphism Double coset isomorphism th.ng 2012-05-30T10:58:05Z 2012-05-30T13:21:57Z <p>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 $N_G(T)$.</p> <p>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 $(x,y) \mapsto xy$ ?</p> <p>Thank you!</p> http://mathoverflow.net/questions/98359/double-coset-isomorphism/98377#98377 Answer by Tom De Medts for Double coset isomorphism Tom De Medts 2012-05-30T13:21:57Z 2012-05-30T13:21:57Z <p>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 <em>Bruhat decomposition</em>) can be found, for instance, in Borel's "Linear Algebraic Groups", Second Enlarged Edition, p. 195 at the bottom.</p>