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

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

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 $G=G_0\left(ℂ\right(\left(t\right)\left)\right)$ 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 $$e_{\phi }\in {𝔤}_{\phi },f_{\phi }\in {𝔤}_{-\phi }such\; that\; {⟨e_{\phi },f_{\phi }⟩}_0=1,$$ and let $$e_0=e_{-\phi +\delta }=t{f}_{\phi },f_0=f_{-\phi +\delta }=t^{-1}{e}_{\phi },h_0=[e_0,f_0]=[t{x}_{-\phi },t{x_{\phi }}^{-1}]=-h_{\phi }+c$$ where $\delta$ is as in (5.6) (see [Kac, Theorem 7.4]).

The alcoves are the open connected components of $${𝔥}_{ℝ}\setminus \bigcup _{-\alpha +j\delta \in {\tilde{R}}_{\mathrm{re}}^l}{H}_{-\alpha +j\delta },where{H}_{-\alpha +j\delta }=\{x^{\vee }\in {𝔥}_{ℝ}|⟨x^{\vee },\alpha ⟩=j\}.$$ 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 $$A_0=\{x^{\vee }\in {𝔥}_{ℝ}|⟨x^{\vee },{\alpha }_i⟩>0 \; for\; all\; 0\le i\le n\}$$ to make a bijection $$W_{\mathrm{aff}}\leftrightarrow \left\{alcoves\right\}.$$

For example, when ${𝔤}_0={\mathrm{𝔰𝔩}}_3,$

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 $\overrightarrow{w}=s_{i_1}\cdots s_{i_l}$ is a walk starting at $1$ and ending at $w,$

and the points of $$IwI=\left\{x_{i_1}\right(c_1\left){n_{i_1}}^{-1}{x}_{i_2}\right(c_2){n_{i_2}}^{-1}\cdots x_{i_l}(c_l\left){n_{i_l}}^{-1}I\right| c_1,...,c_l \in ℂ\}(1.1)$$ are in bijection with labelings of the edges of the walk by complex numbers $c_1,...,c_l .$ The elements of $R\left(w\right)=\{{\beta }_1,...,{\beta }_l\}$ are the elements of ${\tilde{R}}_{\mathrm{re}}^I$ 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 $$x_{\alpha }\left(c\right){n_{\alpha }}^{-1}=x_{-\alpha }\left(c^{-1}\right)x_{\alpha }(-c)h_{{\alpha }^{\vee }}\left(c\right)(main\; folding\; law),$$ to rewrite the points of $IwI$ given in (1.1) as elements of $U^{-}vI.$ Suppose that $$x_{i_1}\left(c_1\right){n_{i_1}}^{-1}\cdots x_{i_l}\left(c_l\right){n_{i_l}}^{-1}=x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)n_{v}b,where\; b\in I,(1.4)$$ $v\in W_{\mathrm{aff}}$ 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 ${\gamma }_1,...,{\gamma }_l,\in ,{\tilde{R}}_{\mathrm{re}}^U$ 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 $c_{l+1}\text{'}\in ℂ, b\text{'}\in I, v\text{'}\in W_{\mathrm{aff}}$ and ${\gamma }_{l+1}\in {\tilde{R}}_{\mathrm{re}}^U$ so that $$x_{i_1}\left(c_1\right){n_{i_1}}^{-1}\cdots x_{i_l}\left(c_l\right){n_{i_l}}^{-1}{x}_j\left(c_l\right){n_j}^{-1}=x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)x_{{\gamma }_{l+1}}\left(c_{l+1}\right)n_v\text{'}b\text{'}.$$

Keep the notations in (1.4). Since $b{x}_j\left(c\right){n_j}^{-1}\in I{s}_{j}I$ there are unique $\tilde{c}\in ℂ$ and $b\text{'}\in I$ such that $b{x}_j\left(c\right){n_j}^{-1}=x_j\left(\tilde{c}\right){n_j}^{-1}b\text{'}$ and $$\begin{array}{rcl}{x}_{i_1}\left(c_1\right){n_{i_1}}^{-1}\cdots x_{i_l}\left(c_l\right){n_{i_l}}^{-1}{x}_j\left(c\right){n_j}^{-1}& =& x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)n_{v}b{x}_j(c,{n_j}^{-1})\\ & =& x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)n_v{x}_j\left(\tilde{c}\right){n_j}^{-1}b\text{'}.\end{array}$$

Case 1. If $v{\alpha }_j\in {\tilde{R}}_{\mathrm{re}}^U,$

then $x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)n_v{x}_j\left(\tilde{c}\right){n_j}^{-1}b\text{'}$ is equal to $$x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)x_{v{\alpha }_j}(\pm \tilde{c})n_{v{s}_j}b\text{'}\in U^{-}v{s}_{j}I\cap Iw{s}_{j}I.$$ In this case, ${\gamma }_{l+1}=v{\alpha }_j, v\text{'}=v{s}_j,$ and

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

where ${\gamma }_{l+1}=-v{\alpha }_j$ and $b\text{'}\text{'}=x_{{\alpha }_j}(-\tilde{c})h_{{\alpha }_j^{\vee }}\left(\tilde{c}\right)b\text{'}.$ So

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

then $$\begin{array}{rcl}{x}_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)n_v{x}_{{\alpha }_j}\left(0\right){n_j}^{-1}b\text{'}& =& x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)n_v{x}_{-{\alpha }_j}\left(0\right){n_j}^{-1}b\text{'}\\ & =& x_{{\gamma }_1}\left(c_1\text{'}\right)\cdots x_{{\gamma }_l}\left(c_l\text{'}\right)x_{{\gamma }_{l+1}}\left(0\right)nv{s}_{j}b\text{'}\text{'}\in U^{-}v{s}_{j}I\cap Iw{s}_{j}I,\end{array}$$ where ${\gamma }_{l+1}=-v{\alpha }_j.$ So We have proved the following theorem:

If $w\in W_{\mathrm{aff}}$ and $\overrightarrow{w}=s_{i_1}\cdots s_{i_l}$ is a minimal length walk to $w$, define $$𝒫{\left(\overrightarrow{w}\right)}_v=\left\{ \; labeled\; folded\; paths\; p \; of\; type\; \overrightarrow{w} \; which\; end\; in\; v\right\}forv\in W_{\mathrm{aff}},$$ where a labeled folded path of type $\overrightarrow{w}$ is a sequence of steps of the form

where the k^th step has $j=i_k.$

Viewing $U^{-}vI\cap IwI$ as a subset of $G/I$, there is a bijection $$𝒫{\left(\overrightarrow{w}\right)}_v\leftrightarrow U^{-}vI\cap IwI.$$ 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 ${\left(\overrightarrow{w}\right)}_v$ 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?

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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