Given parabolic subgroup $P, P=MN$ where $M$ is reductive and $N$ is the unipotent subgroup of P. All the books I read claim that $P\omega N$=$P\omega P$ where $\omega$ is an element of the weyl group correspoding to $P$.
When $P$ is just $B$ i.e. Borel subgroup, then since $M$ is just maximal torus $T$, and the weyl group now is just normalizer quotient by maximal torus, we have: $P\omega tN=P\omega t \omega^{-1} \omega N =Pt^{'} \omega N=P\omega N$. But for general parabolic, I don't think I can do the samething, so how do I show this maybe basic fact?
Thank you!