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Rajkarov
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Parabolic subrgoups and BN-pairs

We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subrgoup containing $T$. We pute also, $N=N_{G}(T)$ the normalizer of $T$ in $G$. We know that $(B,N)$ is a BN-pair of $G$. How we prove that evrey parabolic subrgoup of $G$ containing $B$ is of the form $P=BW_{T}B$, where $W$ is the Weyl group of the BN-pair $(B,N)$, it's a same of a Weyl group of $G$, and $W_{T}=<T>$ a standard subgroup of $W$ ($W$ is a Coxeter groups with set of generators $S$, and $T$ is a subset of $S$ and finaly $W_{T}$ is a subgroup generated by $T$).

Rajkarov
  • 933
  • 4
  • 10