# Reference request: expository text on the structure of reductive groups over non-archimedean local fields

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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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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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

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

Paul Garrett's book and lecture notes provide a reasonable approach to the subject, for which there are few textbook options. There are of course other lecture notes, usually slanted in some way which you might or might not be interested in. As Vishal indicates, the slender 1971 paperback volume by I.G. Macdonald (also referenced in Paul's book), based on his 1970 lectures in Madras, provides a nice shorter exposition with emphasis on spherical functions. Macdonald bases his account on an axiomatic version of the Bruhat-Tits theory (emphasizing properties of the BN-pair), in order to avoid assuming too much about algebraic groups or algebraic geometry.

There's also something to be said for going back to the historical origins in the 1965 IHES paper by Iwahori and Matsumoto here. They give the first systematic development of affine (and extended affine) Weyl groups, in a separate first section, followed by a detailed treatment of the BN-pair in a split semisimple $p$-adic group (obtained via Chevalley's reduction mod $p$ process from a semisimple Lie algebra over $\mathbb{C}$). Though some of their notation is obsolete, their approach to the structure theory is concrete and doesn't yet involve directly the theory of Bruhat-Tits buildings which came into play soon afterward. But they do emphasize the (Iwahori)-Hecke algebra, which originated in Iwahori's earlier work on finite Chevalley groups.

To get started with just the rank 1 case, there is a short exposition (with details) in my old lecture notes Arithmetic Groups (Lecture Notes in Math. 789, Springer, 1980), $\S15$. A description of the building is also given, which is just a tree in this case. Anyway, it's definitely best to start with the split groups, before the notation thickens.

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A very nice exposition is given in I.G. Macdonalds Spherical functions on a group of p-adic type

Another nice exposition is given in the section 3 of the these notes.

Garret's book is perfect for learning about buildings in general, while these notes summarize exactly how they one constructs a building for a reductive group.

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