The folding algorithm and the intersections U-vI∩IwI

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 April 2010

The folding algorithm and the intersections U-vI∩IwI

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 𝔤0 is finite dimensional. In this case, the extended loop Lie algebra 𝔤 defined in (5.1) is also a Kac–Moody Lie algebra. If G0 is the Tits group of 𝔤0 and G=G0 ℂt is the corresponding loop group then the subgroup I defined in (6.3) differs from the Borel subgroup of the Kac–Moody group GKM for 𝔤 only by elements of T, and the affine flag variety of G coincides with the flag variety of GKM Thus, in this case, Theorem 4.1 provides a labeling of the points of the affine flag variety.

Suppose that 𝔤0 is a finite dimensional complex semisimple Lie algebra presented as a Kac–Moody Lie algebra with generators e1,…,en,f1,…,fn,h1,…,hn and Cartan matrix A=αihj1≤i,j≤n. Let ϕ be the highest root of R (the highest weight of the adjoint representation), fix eϕ∈𝔤ϕ,fϕ∈𝔤-ϕsuch that eϕfϕ0=1, and let e0=e-ϕ+δ=tfϕ,f0=f-ϕ+δ=t-1eϕ,h0=e0f0=tx-ϕtxϕ-1=-hϕ+c, as in (5.9). The magical fact is that, in this case, 𝔤=𝔤0tt-1⊕ℂc⊕ℂd is a Kac–Moody Lie algebra with generators e1,…,en,f1,…,fn,h1,…,hn,d and Cartan matrix A1=αihj0≤i,j≤n,whereα0=-ϕ+δ  and  h0=-hϕ+c where δ is as in (5.6) (see [Kac, Theorem 7.4]).

The alcoves are the open connected components of 𝔥ℝ\∪-α+jδ∈R~relH-α+jδ,whereH-α+jδ=x∨∈𝔥ℝ |x∨α=j. Under the map in (5.16) the chambers wC of the Tits cone X (see (2.20) and (2.21)) become the alcoves. Each alcove is a fundamental region for the action of waff on 𝔥ℝ given by (5.17) and Waff acts simply transitively on the set of alcoves (see [Kac, Proposition 6.6]). Identify 1∈Waff with the fundamental alcove A0 =x∨∈𝔥ℝ|x∨αi>0  for all  0≤i≤n to make a bijection Waff↔alcoves.

For example, when 𝔤0=𝔰𝔩3,

Diagram 1

The alcoves are the triangles and the (centers of) hexagons are the elements of Q∨.

Let w∈Waff. Following the discussion in (4.4)–(4.6), a reduced expression w→=si1…sil is a walk starting at 1 and ending at w,