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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!

random123
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user330928
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  • What is $U$? You never defined it. – Alex Youcis Sep 17 '19 at 23:14
  • @AlexYoucis Opps! Sorry, I think I mean N actually. – user330928 Sep 18 '19 at 01:09
  • @random123 Thank you very much! The definition of $W_P$ that I am more familiar with is double quotient of $W$ by $W_I$ where $W_I$ is the weyl group generated by $s_\alpha , \alpha \in I$ and $I$ is the subset of simple roots correspond to $P$. So how can I show these two definition are the same? Or could you please give a reference for that? – user330928 Sep 18 '19 at 16:40
  • @user330928 What I wrote above was incorrect as can be seen for the case of the group GL(n) itself. I shall delete my comment. – random123 Sep 19 '19 at 09:03

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