## The folding algorithm and the intersections ${U}^{-}vI\cap IwI$

Last update: 19 February 2012

## The folding algorithm and the intersections ${U}^{-}vI\cap IwI$

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 ${𝔤}_{0}$ defined in (5.1) is also a Kac-Moody Lie algebra. If ${G}_{0}$ is the Tits group of ${𝔤}_{0}$ and $G={G}_{0}\left(ℂ\left(\left(t\right)\right)\right)$ is the corresponding loop group then the subgroup $I$ defined in (6.3) differs from the Borel subgroup of the Kac-Moody group ${G}_{KM}.$ Thus, in this case, Theorem 4.1 provides a labeling of the points of the affine flag variety.

Suppose that ${𝔤}_{0}$ is a finite dimensional complex semisimple Lie algebra presented as a Kac-Moody Lie algebra with generators ${e}_{1},...,{e}_{n},{f}_{1},...,{f}_{n},{h}_{1},...,{h}_{n}$ and Cartan matrix $A={\left({\alpha }_{i}\left({h}_{j}\right)\right)}_{1\le i,j\le n}.$ Let $\phi$ be the highest root of $R$ (the highest weight of the adjoint representation), fix and let $e0= e-φ+δ =tfφ, f0= f-φ+δ = t - 1 eφ, h0= [e0, f0]= [tx-φ,txφ -1] =-hφ+c$ where $\delta$ is as in (5.6) (see [Kac, Theorem 7.4]).

The alcoves are the open connected components of $𝔥ℝ∖ ⋃ -α+jδ∈R˜rel H-α+jδ, where H-α+jδ= {x∨∈ 𝔥ℝ| ⟨x∨,α⟩=j}.$ 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 ${W}_{\mathrm{aff}}$ on ${𝔥}_{ℝ}$ given by (5.17) and ${W}_{\mathrm{aff}}$ acts simply transitively on the set of alcoves (see [Kac, Proposition 6.6]). Identify $1\in {W}_{\mathrm{aff}}$ with the fundamental alcove to make a bijection $Waff↔ {alcoves}.$

For example, when ${𝔤}_{0}={\mathrm{𝔰𝔩}}_{3},$

Diagram 1

The alcoves are the triangles and the (centers of) hexagons are the elements of ${Q}^{\vee }.$

Let $w\in {W}_{\mathrm{aff}}.$ Following the discussion in (4.4)-(4.6), a reduced expression $\stackrel{\to }{w}={s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ is a walk starting at $1$ and ending at $w,$

Diagram 2
and the points of $IwI= { xi1(c1) ni1 - 1 xi2(c2) ni2 - 1 ⋯ xil(cl) nil - 1 I| c1...cl∈ℂ } (1.1)$ are in bijection with labelings of the edges of the walk by complex numbers ${c}_{1},...,{c}_{l} .$ The elements of $R\left(w\right)=\left\{{\beta }_{1},...,{\beta }_{l}\right\}$ are the elements of ${\stackrel{˜}{R}}_{\mathrm{re}}^{I}$ corresponding to the sequence of hyperplanes crossed by the walk.

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 ${H}_{\alpha +k\delta }$ such that

1. $1$ is on the positive side of ${H}_{\alpha }$ for $\alpha \in {R}_{\mathrm{re}}^{+},$
2. ${H}_{\alpha +k\delta }$ and ${H}_{\alpha }$ have parallel orientations.

This orientation is such that

Together, (1.2) and (1.3) provide a powerful combinatorics for analyzing the intersections ${U}^{-}vI\cap IwI.$ 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 $v\in {W}_{\mathrm{aff}}$ and ${n}_{v}={{n}_{{j}_{1}}}^{-1}\cdots {{n}_{{j}_{k}}}^{-1}$ if $v={s}_{{i}_{1}}\cdots {s}_{{i}_{k}}$ is a reduced word, and ${\gamma }_{1},...,{\gamma }_{l},\in ,{\stackrel{˜}{R}}_{\mathrm{re}}^{U}$ so that ${x}_{{\gamma }_{1}}\left({c}_{1}\text{'}\right)\cdots {x}_{{\gamma }_{l}}\left({c}_{l}\text{'}\right)\in {U}^{-}.$ Then the procedure described in (1.5)-(1.7) will compute and ${\gamma }_{l+1}\in {\stackrel{˜}{R}}_{\mathrm{re}}^{U}$ 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 $b{x}_{j}\left(c\right){{n}_{j}}^{-1}\in I{s}_{j}I$ there are unique $\stackrel{˜}{c}\in ℂ$ and $b\text{'}\in I$ such that $b{x}_{j}\left(c\right){{n}_{j}}^{-1}={x}_{j}\left(\stackrel{˜}{c}\right){{n}_{j}}^{-1}b\text{'}$ 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 $v{\alpha }_{j}\in {\stackrel{˜}{R}}_{\mathrm{re}}^{U},$

then ${x}_{{\gamma }_{1}}\left({c}_{1}\text{'}\right)\cdots {x}_{{\gamma }_{l}}\left({c}_{l}\text{'}\right){n}_{v}{x}_{j}\left(\stackrel{˜}{c}\right){{n}_{j}}^{-1}b\text{'}$ is equal to $xγ1 (c1')⋯ xγl (cl') xvαj (±c˜) nvsj b'∈ U-vsj I∩IwsjI.$ In this case, and

Case 2. If $v{\alpha }_{j}\notin {\stackrel{˜}{R}}_{\mathrm{re}}^{U}$ and $\stackrel{˜}{c}\ne 0,$

where ${\gamma }_{l+1}=-v{\alpha }_{j}$ and $b\text{'}\text{'}={x}_{{\alpha }_{j}}\left(-\stackrel{˜}{c}\right){h}_{{\alpha }_{j}^{\vee }}\left(\stackrel{˜}{c}\right)b\text{'}.$ So

Case 3. If $v{\alpha }_{j}\in {\stackrel{˜}{R}}_{\mathrm{re}}^{U}$ and $\stackrel{˜}{c}=0,$

then $xγ1 (c1')⋯ xγl (cl') nvxαj (0) nj - 1 b' = xγ1 (c1')⋯ xγl (cl') nv x-αj(0) nj - 1 b' = xγ1 (c1')⋯ xγl (cl') xγl+1(0) nvsj b''∈U-v sjI∩IwsjI,$ where ${\gamma }_{l+1}=-v{\alpha }_{j}.$ So
We have proved the following theorem:

If $w\in {W}_{\mathrm{aff}}$ and $\stackrel{\to }{w}={s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ is a minimal length walk to $w$, define where a labeled folded path of type $\stackrel{\to }{w}$ is a sequence of steps of the form

where the kth step has $j={i}_{k}.$

Viewing ${U}^{-}vI\cap IwI$ as a subset of $G/I$, there is a bijection $𝒫(w→)v↔ U-vI∩IwI.$ 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 ${\left(\stackrel{\to }{w}\right)}_{v}$ indicate a decomposition of ${U}^{-}vI\cap IwI$ into "cells", where the cell associated to a nonlabeled path $p$ is the set of points of ${U}^{-}vI\cap IwI$ 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?