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We note $G$ a connected algebraic group, $T$ a maximal torus in $G$, $B$ a Borel subgroup containing $T$. We put 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 every parabolic subgroup 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 the same as a Weyl group of $G$, and $W_{T}=<T>$ a standard subgroup of $W$ ($W$ is a Coxeter group with set of generators $S$, and $T$ is a subset of $S$ and finally $W_{T}$ is a subgroup generated by $T$).

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2 Answers

up vote 6 down vote accepted

Combine Theorem 6.43 (1) pg. 315 in Abramenko-Brown "Buildings" together with Proposition 6.36 (6) on pg. 310.

These are two steps: As you mention, $BW_T B$ is a subgroup (Proposition 6.36).

$BW_TB$ contains $B$, any subgroup containing the Borel subgroup is parabolic, and any parabolic is conjugate to a subgroup containing a specific Borel $B$. (Theorem 6.43)

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Thanks pm, now it's clear. –  Rajkarov Feb 20 '12 at 10:23
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To provide a more balanced context for the question (and the answer by pm), it's useful to separate the elementary notion of BN-pair from the far more sophisticated structure theory of algebraic groups. Following Chevalley's 1955 Tohoku paper on finite simple groups of Lie type and his 1956-58 Paris seminar on classification of semisimple algebraic groups, Jacques Tits formulated an axiomatic framework to cover these and further examples. In a concise 1962 Comptes Rendus note, he defined BN-pairs and derived their main properties including the role of "Borel" and "parabolic" subgroups involved in answering Rajkarov's question. Chapter 4 of the Bourbaki treatise Groupes et algebres de Lie (1968) adopted the name "systemes de Tits" for BN-pairs and included in the exercises much of the contemporary development by Tits of the theory of buildings.

By now there are many treatments of BN-pairs in books, papers, and lecture notes, including the self-contained Section 29 of my 1975 Springer text on linear algebraic groups. But the theory of buildings lies beyond this elementary axiomatic stage and is not directly relevant to the question asked here. Similarly, the theory of BN-pairs is independent of algebraic groups but flexible enough to fit more advanced developments such as the Iwahori-Matsumoto study of groups over $p$-adic fields. (The latter IHES paper is now freely available online, but I'm not sure whether any other online sources contain complete details about BN-pairs.)

Maybe it's also worth mentioning that quite a few MO questions involve BN-pairs, also known as (B,N)-pairs or Tits systems, as well as buildings. So it may be worthwhile to browse through these.

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Thank you very much Mr Jim for details. –  Rajkarov Feb 20 '12 at 20:19
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