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

Question

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.

Answer 1

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

Answer 2

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.

Answer 3

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.

Answer 4

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.