Affine Weyl groups as Coxeter groups

Question

If G is a reductive algebraic group (say over ℂ), T a maximal torus, then we can consider its Weyl group W which acts on the abelian group Y of one parameter subgroups of T. Thus we may form the semidirect product, which I will call the affine Weyl group.

In the semisimple simply connected case, this affine Weyl group is a Coxeter group. In the general setup I have here, the affine Weyl group is no longer a Coxeter group, but shares some properties of Coxeter groups. For example one can still define a length function, and considering an Iwahori subgroup of a loop group, we get something that looks like it wants to be a BN-pair/Tits system (but isn't, since we don't have a Coxeter group).

Since the axiomatic setup of Coxeter groups and BN-pairs is so convenient, I ask if there has been developed a generalisation of this setup to include the affine Weyl groups I mention above. I have some vague feeling that this should include some sort of 'disconnected' (I am thinking in the algebraic group sense when I use this word) Coxeter group, where the connected component is a genuine Coxeter group.

Answer 1

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).

Answer 2

I don't think I've ever seen it written down systematically, but it definitely could be. Many papers I've read use the almost-Coxeter theory either without any comment whatsoever, or they say something like "this generalized axiom we just used follows easily from the corresponding honest BN-pair axiom".