The folding algorithm and the intersections U-vI IwI

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 19 February 2012

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 defined in (5.1) is also a Kac-Moody Lie algebra. If G0 is the Tits group of 𝔤0 and G=G0(((t))) is the corresponding loop group then the subgroup I defined in (6.3) differs from the Borel subgroup of the Kac-Moody group 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 and Cartan matrix A= (αi(hj)) 1i,jn . Let φ be the highest root of R (the highest weight of the adjoint representation), fix eφ 𝔤φ, fφ 𝔤-φ such that   eφ,fφ0 =1, and let e0= e-φ+δ =tfφ, f0= f-φ+δ = t - 1 eφ, h0= [e0, f0]= [tx-φ,txφ -1] =-hφ+c where δ is as in (5.6) (see [Kac, Theorem 7.4]).

The alcoves are the open connected components of 𝔥 -α+jδR˜rel H-α+jδ, where H-α+jδ= {x 𝔥| 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 𝔥 given by (5.17) and Waff acts simply transitively on the set of alcoves (see [Kac, Proposition 6.6]). Identify 1Waff with the fundamental alcove A0={ x 𝔥| x, αi >0   for all   0in} to make a bijection Waff {alcoves}.

For example, when 𝔤0=𝔰𝔩3,

Hα2+δ Hα2 H-α2+δ H-α2+3δ H-α2+5δ H-φ+4δ H-φ+3δ H-φ+2δ Hα0 Hφ Hφ+δ Hφ+2δ Hφ+3δ Hφ+4δ H-α1+δ Hα1 Hα1+δ Hα1+3δ Hα1+5δ - + - + - + - + - + - + - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - 1 s0 w0 w0s0 s1 s2 s0s1 s0s2 s1s0 s2s0
Diagram 1

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

Let wWaff. Following the discussion in (4.4)-(4.6), a reduced expression w= si1 sil is a walk starting at 1 and ending at w,

1 w Hβ2 Hβ4 Hβ1 Hβ3 Hβ5
Diagram 2
and the points of IwI= { xi1(c1) ni1 - 1 xi2(c2) ni2 - 1 xil(cl) nil - 1 I| } (1.1) are in bijection with labelings of the edges of the walk by complex numbers The elements of R(w)= {β1, ...,βl} are the elements of R˜reI corresponding to the sequence of hyperplanes crossed by the walk.

The labeling of the hyperplanes in Diagram 1 is such that neighbouring alcoves have

Hvαj v vsj with vαjR˜reI if v is closer to 1 that vsj. (1.2)

The periodic orientation (illustrated in Diagram 1) is the orientation of the hyperplanes Hα+kδ such that

  1. 1 is on the positive side of Hα for αRre+,
  2. Hα+kδ and Hα have parallel orientations.

This orientation is such that

Hvαj v vsj - + vαjR˜reU iff (1.3)

Together, (1.2) and (1.3) provide a powerful combinatorics for analyzing the intersections U-vIIwI. We shall use the first identity in (3.3), in the form xα(c) nα - 1 = x-α( c - 1 )xα (-c)hα(c) (main folding law), to rewrite the points of IwI given in (1.1) as elements of U-vI. Suppose that xi1 (c1) ni1 - 1 xil (cl) nil - 1 = xγ1 (c1') xγl (cl') nvb, where   bI, (1.4) vWaff and nv= nj1 - 1 njk - 1 if v= si1 sik is a reduced word, and γ1...γl R˜reU so that xγ1 (c1') xγl (cl') U-. Then the procedure described in (1.5)-(1.7) will compute cl+1',  b'I,  v'Waff and γl+1R˜reU so that xi1 (c1) ni1 - 1 xil (cl) nil - 1 xj (cl) nj - 1 = xγ1 (c1') xγl (cl') xγl+1 (cl+1) nv'b'.

Keep the notations in (1.4). Since bxj (c) nj - 1 IsjI there are unique c˜ and b'I such that bxj(c) nj - 1 = xj(c˜) nj - 1 b' and xi1 (c1) ni1 - 1 xil (cl) nil - 1 xj (c) nj - 1 = xγ1 (c1') xγl (cl') nvbxj (c, nj - 1 ) = xγ1 (c1') xγl (cl') nvxj (c˜) nj - 1 b'.

Case 1. If vαjR˜reU,

Hvαj v vsj c˜
then xγ1 (c1') xγl (cl') nvxj (c˜) nj - 1 b' is equal to xγ1 (c1') xγl (cl') xvαj (±c˜) nvsj b' U-vsj IIwsjI. In this case, γl+1 =vαj,  v'=vsj, and
Hvαj v vsj c˜ becomes (1.5) Hvαj v vsj ±c˜

Case 2. If vαjR˜reU and c˜0,

Hvαj v vsj - + c˜
where γl+1=-vαj and b''=xαj (-c˜)hαj(c˜) b'. So
Hvαj v vsj c˜ - + becomes (1.6) Hvαj v ±c˜-1 - +

Case 3. If vαj R˜reU and c˜=0,

Hvαj v vsj - + 0
then xγ1 (c1') xγl (cl') nvxαj (0) nj - 1 b' = xγ1 (c1') xγl (cl') nv x-αj(0) nj - 1 b' = xγ1 (c1') xγl (cl') xγl+1(0) nvsj b''U-v sjIIwsjI, where γl+1=-vαj. So
Hvαj v vsj 0 - + becomes (1.7) Hvαj v vsj 0 - +
We have proved the following theorem:

If wWaff and w=si1sil is a minimal length walk to w, define 𝒫(w)v = {  labeled folded paths   p   of type   w   which end in   v} for vWaff, where a labeled folded path of type w is a sequence of steps of the form
Hvαj v vsj c , Hvαj v vsj 0 , - + (1.8) Hvαj v c-1 - + ,
where the kth step has j=ik.

Viewing U-vIIwI as a subset of G/I, there is a bijection 𝒫(w)v U-vIIwI. 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 (w)v indicate a decomposition of U-vIIwI into "cells", where the cell associated to a nonlabeled path p is the set of points of U-vIIwI which have the same underlying nonlabeled path. It would be very interesting to understand, combinatorially, the closure relations between these cells.

Notes and References

Where are these from?

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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