gr.group theory - Affine Weyl groups as Coxeter groups

In the abstract Bourbaki set-up, the affine Weyl group is defined to be a
semidirect product of an irreducible Weyl group with its coroot lattice.
This is naturally a Coxeter group, characterized in terms of its
positive semidefinite Coxeter matrix. The basic theory is developed
independently of applications in Lie theory, but is directly usable if you
start with a connected semisimple algebraic group (over an algebraically
closed field) and require its root system to be irreducible of type A, B,
etc. Most of the time this causes no trouble. While it is natural to work
with a connected reductive group, people often use the expression "affine
Weyl group" too loosely in this general context. For example, the standard
features of alcove geometry require irreducibility. Otherwise you have
to deal with products of simplexes, etc. In any case, the difference between
reductive and semisimple groups such as general linear and special linear
is sometimes significant.

In the Iwahori-Matsumoto (or Bruhat-Tits) setting over local fields, a
more intrinsic affine Weyl group occurs directly within the structure of
the group itself. Here one has to be cautious about applying
abstract Coxeter group theory or BN-pair theory, as I believe most authors
are. Already in the proceedings of the 1965 Boulder AMS summer institute,
Iwahori had to formulate a more complicated "generalized BN-pair" formalism
for this situation. I'm not sure what has become standard by now in the
literature.

In other situations (the classical study of compact Lie groups, or
the later application of affine Weyl groups in modular representation
theory starting with Verma) there is usually no difficulty about specializing
to the irreducible case. Here the affine Weyl group lives outside the
actual group under study. This is the situation I'm most comfortable with.

You need to make precise the setting in which you really want to study reductive groups, in order to adapt the Bourbaki language and results.
There are several distinct issues here: 1) Extra care is needed in treating
disconnected algebraic groups such as orthogonal groups, or in treating
reductive rather than semisimple groups. 2) Adjoint groups, simply
connected groups, and the occasional intermediate type: not all details of
structure are exactly the same. 3) Most important for working over
local fields is the natural use of an "extended affine Weyl group" (as in
much of Lusztig's work involving Hecke algebras, cells, etc.). Here you
start with the Bourbaki version of the affine Weyl group (a Coxeter group)
and form a semidirect product with a finite group $Omega$ isomorphic to the weight lattice mod root lattice (universal center). This amounts to
working with a semidirect product of the Weyl group and the full (co)weight
lattice rather than the (co)root lattice. Fortunately it's easy to extend
notions like length function to this extended group.

EDIT: Besides Bourbaki's treatment of Coxeter groups and root systems (1968),
foundational papers from that era include Iwahori-Matsumoto (IHES Publ. Math. 25, 1965), at http://www.numdam.org, and Iwahori's 1965 article in the AMS Boulder proceedings
http://www.ams.org/books/pspum/009/0215858/pspum0215858.pdf, followed by much more technical work by Bruhat-Tits.
Lusztig has written many technical papers on extended affine Weyl groups and corresponding affine Hecke algebras, including his four part series on cells in affine Weyl groups and later work on multiparameter cases. Much of this work is motivated by reductive groups over local fields, as well as the modular representation theory of reductive groups and their Lie algebras (where "linkage" of weights appears at first relative to an extended affine Weyl group).