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I am interested in an expository text in English, which summarizes the main results and aspects of the structure theory of reductive groups over local fields, in a hopefully not very technical manner (full proofs are not necessary, though sketches of the main arguments would be nice). Specifically covering the following topics:

  1. The apropriete BN-pair (with the Iwahori subgroup)

  2. The affine root system

  3. The affine Weyl group

  4. the Bruhat decomposition of the affine flag variety.

  5. Classification and structure of standard parabolic (or parahoric?) subgroups.

The emphasis is really being on readability and not thoroughness, so well written notes of some course or seminar on the subject would be great. I could not find any textbook on the subject, and the standard reference everywhere seems to be the original french papers of Bruhat and Tits from the 70's, but if there is such a textbook It would be optimal.

Another point is that I really don't know much about buildings, and even though It seems a fundamental part of the theory, It would be much easier for me to approach this at first from a direction not relaying heavily on the theory of buildings (if it is at all possible).

Finally, I am mostly interested in the case of the field $\mathbb{C}((t))$ (and not, say, $\mathbb{Q}_p$), which is not a local field in the strict sense (not locally compact), but is a complete non-archimedean DVR (which some people still call a local field), so I am looking for a source that applies for this case.

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2 Answers 2

up vote 5 down vote accepted

http://www.math.umn.edu/~garrett/m/buildings/book.pdf

was derived from seminar notes on structure of split classical p-adic groups, intending to circumvent the larger apparatus of algebraic groups and buildings. It became clear in the original project that it was necessary to develop some aspects of buildings, since they encapsulated and packaged-up some otherwise-clumsy (if not intractable) issues.

For split classical groups, it is possible to develop the building-theory "directly" (as J. Tits did, too, before the general development) in terms of flags of subspaces and flags of lattices (with additional structure...)

Edit: also, a smaller, newer treatment of buildings without Coxeter group stuff intervening is at http://www.math.umn.edu/~garrett/m/v/bldgs.pdf

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Dear @paul, The link seems to have a problem. It goes to a page that says "oops" at the top. –  Keenan Kidwell Jul 14 '13 at 15:45
    
Dear @KeenanKidwell, oop, thanks, fixed it. –  paul garrett Jul 14 '13 at 16:04
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Thank you for the links. It does seem very building-oriented, but I understand you are saying it is inevitable. Is it possible to reduce dependence on buildings, if one is willing to assume more classical algebraic group theory and/or algebraic geometry? –  KotelKanim Jul 14 '13 at 17:17
    
Once-upon-a-time, I had hoped to reduce dependence on "building theory" by assuming more geometric algebra and such, and this can be made to succeed for spherical buildings. However, it seems much harder to avoid "building theory" to address the affine case directly, even for $SL_n(\mathbb Q_p)$. As the second link suggests, it is possible to pare things down and avoid much discussion of Coxeter groups, but the geometry of buildings seems to be an essential aid. –  paul garrett Jul 14 '13 at 17:53
    
I accept this answer, as the best given yet. Thanks again for the links. Both are written in a very clear and accessible style. –  KotelKanim Jul 16 '13 at 7:35
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The standard (classical) survey is:

Tits, J. Reductive groups over local fields. Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, pp. 29--69, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

But it is still quite difficult. The following lectures are a helpful complement:

Yu, Jiu-Kang Bruhat-Tits theory and buildings. Ottawa lectures on admissible representations of reductive $p$-adic groups, 53--77, Fields Inst. Monogr., 26, Amer. Math. Soc., Providence, RI, 2009.

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I am aware of "Reductive groups over local fields" by Tits, but I will check the other one. Thanks. –  KotelKanim Jul 14 '13 at 17:45
    
I just want to confirm that Tits's notes are very challenging for the novice. At least they were for me: as a PhD student, I wanted but didn't strictly need to learn about Bruhat-Tits Theory. I was lucky enough to find hardcover copies of the Corvallis texts in a used bookstore early in my graduate career. But I never managed to make much of a dent in Tits's article. More accessible sources -- like the ones you and Paul Garrett provide -- are most welcome. –  Pete L. Clark Jul 15 '13 at 0:05
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