I am looking for a good (textbook) reference for the basic fact (due to Chevalley) that for every semisimple Lie group $G$ (without compact factors) with Weyl group $W$, the Bruhat order on $W$ coincides with the inclusion order between Schubert cycles in flag-manifolds of $G$. Sadly, the only reference I know is the original Chevalley's paper ("Sur les decompositions cellulaires des espaces $G/B$") published in 1994 (but written in 1950s). Note, I am not looking for a proof (I know how to prove it), just for a reference. Yes, I am looking at the groups which do not necessarily split (the proof I know covers such groups).
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$\begingroup$ Are you considering "split" groups (equivalently, is the Lie algebra split), or general semisimple groups (with their associated relative root systems in the sense of Borel-Tits)? $\endgroup$– user76758Commented Apr 14, 2014 at 21:24
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2$\begingroup$ @Misha: You need to clarify the precise set-up you have in mind for a real semisimple Lie group. In the extreme cases where this group is compact (with flag variety/manifold $G/T$) or is split, the complex case adapts well. But in general, there may be no Borel subgroup over $\mathbb{R}$ to use in the construction, and the Weyl group relative to a minimal parabolic is only a relative version of the usual Weyl group. It gets complicated. $\endgroup$– Jim HumphreysCommented Apr 14, 2014 at 21:34
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$\begingroup$ P.S. I'm still not sure what you mean by "flag manifold" for an arbitrary semisimple Lie group without compact factors. In the mostly equivalent algebraic setting, what Borel and Tits do seems to be optimal but deals with some $G/P$ and its points over the field. $\endgroup$– Jim HumphreysCommented Apr 15, 2014 at 13:14
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4 Answers
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The statement in Chevalley is for an algebraic group over an algebraically closed field, and Borel's fixed point theorem does hold in this context, but Chevalley's setup does not include real flag varieties. If you mean split real forms of semisimple groups, then in fact these come from $\Bbb{Z}$-forms, where the statement also holds. Jantzen's Representations of algebraic groups, Chapter 13, should have this general result.
The subtleties of attaching maps for odd-dimensional cells mean the (integral) cohomology of a $G_{\Bbb R}/B_{\Bbb R}$ is not straightforward to work out from the Bruhat order, but there is a thesis of Kocherlakota from the 1990's, and more recent papers of Casian and Kodama that do this.
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Does Hiller's "Geometry of Coxeter groups" fit?
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$\begingroup$ Thank you for the reference, but Hiller works over ${\mathbb C}$. Any reference which deals with real Lie groups? $\endgroup$– MishaCommented Apr 14, 2014 at 20:16
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The result is stated and proved as Corollary 2.2.2 of Michel Brion's Lectures on the Geometry of Flag Varieties.
answered Apr 14, 2014 at 19:56
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$\begingroup$ Thank you for the reference, but Brion (and his proof) works over ${\mathbb C}$. Any reference which deals with real Lie groups? $\endgroup$– MishaCommented Apr 14, 2014 at 20:11
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$\begingroup$ He uses Borel's fixed point theorem which fails over the reals. $\endgroup$– MishaCommented Apr 14, 2014 at 20:18
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1$\begingroup$ I thought Chevalley's paper also works over $\mathbb{C}$. Have you tried looking in Procesi's book Lie Groups: An Approach Through Invariants and Representations $\endgroup$ Commented Apr 14, 2014 at 20:23
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Concerning real semisimple Lie groups, the literature is somewhat scattered but does provide some help with the Bruhat decomposition and (Chevalley)-Bruhat ordering involved in closures of double cosets. Borel's lectures over many years gave considerable insight into the way his work with Tits on reductive algebraic groups (over fields) translates to the setting of Lie groups. There is also a useful section in the follow-up 1972 IHES paper he and Tits wrote: see section 3 and especially (3.13)-(3.15) here. They refer also to a discussion in section 8 of Steinberg's 1967-68 Yale lectures on page 107.
Note that the reference in Borel-Tits to the ordering is formulated only implicitly, but various papers by Deodhar bring out the equivalent conditions on the Bruhat ordering of a general Coxeter group including the Weyl groups or relative versions here.
If the group $G$ isn't split (or compact) over $\mathbb{R}$, all of the Borel-Tits machinery involving a minimal parabolic subgroup $P$ over $\mathbb{R}$ and the relative Weyl group comes into play. Fortunately, the real points of the homogeneous space $G/P$ behave similarly in the Zariski or real topology, as follows from the work of Borel and Tits. All of this involves some careful detail, however.
It's not clear to me that there is a single all-purpose reference covering everything asked for. For instance, comparisons of compact and split real groups (and their flag varieties) come up in older work of Bott and others but may not be covered by the standard algebraic group references.
answered Apr 14, 2014 at 22:53