Last update: 19 February 2012
In this section we prove our main theorem, which gives a precise connection between the alcove walks in [Ra] and the points in the affine flag variety. The algorithm here is essentially that which is found in [BD] and, with our setup from the earlier sections, it is the 'obvious one'. The same method has, of course, been used in other contexts, see, for example, [C].
A special situation in the loop group theory is when defined in (5.1) is also a Kac-Moody Lie algebra. If is the Tits group of and is the corresponding loop group then the subgroup defined in (6.3) differs from the Borel subgroup of the Kac-Moody group Thus, in this case, Theorem 4.1 provides a labeling of the points of the affine flag variety.
Suppose that is a finite dimensional complex semisimple Lie algebra presented as a Kac-Moody Lie algebra with generators and Cartan matrix Let be the highest root of (the highest weight of the adjoint representation), fix and let where is as in (5.6) (see [Kac, Theorem 7.4]).
The alcoves are the open connected components of Under the map in (5.16) the chambers of the Tits cone (see (2.20) and (2.21)) become the alcoves. Each alcove is a fundamental region for the action of on given by (5.17) and acts simply transitively on the set of alcoves (see [Kac, Proposition 6.6]). Identify with the fundamental alcove to make a bijection
For example, when
The alcoves are the triangles and the (centers of) hexagons are the elements of
Let Following the discussion in (4.4)-(4.6), a reduced expression is a walk starting at and ending at
The labeling of the hyperplanes in Diagram 1 is such that neighbouring alcoves have
The periodic orientation (illustrated in Diagram 1) is the orientation of the hyperplanes such that
This orientation is such that
Together, (1.2) and (1.3) provide a powerful combinatorics for analyzing the intersections We shall use the first identity in (3.3), in the form to rewrite the points of given in (1.1) as elements of Suppose that and if is a reduced word, and so that Then the procedure described in (1.5)-(1.7) will compute and so that
Keep the notations in (1.4). Since there are unique and such that and
Case 1. If
Case 2. If and
Case 3. If and
If and is a minimal length walk to , define
where a labeled folded path of type is a sequence of steps of the form
where the kth step has
Viewing as a subset of , there is a bijection Theorem 1.1 is a strengthening of the connection between the path model and the geometry of the affine flag variety as observed, in the case of the loop Grassmannian, in [GL] and, in terms of crystal bases, in [BG].
The paths in indicate a decomposition of into "cells", where the cell associated to a nonlabeled path is the set of points of which have the same underlying nonlabeled path. It would be very interesting to understand, combinatorially, the closure relations between these cells.