Kazhdan Lusztig Cells in Infinite Coxeter Groups

Groups defined by presentations of the form , 2 1, , 1, ( ) 1 ( , 1, , ) | 〈 ... = = = ... 〉 i j m n i i j s s s s s i j n are called Coxeter groups. The exponents , { } ∈ ∪ ∞  i j m form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group Sn is isomorphic to the Coxeter group with presentation 2 3 1 1 , 1 ( 1, , ), ( ) 1 ( 1, ) | , 1 + 〈 ... = = ... = = ... − 〉 n i i i s s s i n s s i n , and is also known as the Weyl group of type An−1.


Introduction
The notion of cells was introduced by Kazhdan and Lusztig [1,2] to study representations of Coxeter groups and their Hecke algebras. Later it was realized that cells arise in many different branches of mathematics and have many interesting properties. Some examples and references for the related results can be found in Gunnells et al. [3]. We are interested in the combinatorial structure of the cells in infinite Coxeter groups. Examples of such groups include the affine Weyl groups, as well as Weyl groups of hyperbolic Kac-Moody algebras. In contrast with the affine case, there are very few results on cells in hyperbolic Coxeter groups available so far. We refer to Bedard and Belolipetsky [4,5] for some work in this direction. Based on numerous computational experiments we introduce two conjectures that describe the structure of the cells in infinite Coxeter groups. Our conjectures use combinatorial rigidity for elements in an infinite Coxeter group ( §4). We expect that this notion may be of independent interest. We were able to check the conjectures for affine groups of small rank [6] ( §5), and to prove them in some special cases [7] ( §4). The latter include, in particular, the right-angled Coxeter groups previously studied in Belolipetsky [5]. The proof of the conjectures in their general form remains an open problem.

Visualization of cells
Let W be a Coxeter group with a fixed system of generators S. Consider a real vector space V of dimension |S| with a basis { | } α ∈ s s S . We define a symmetric bilinear form on V by Now for every s ∈ S we can define a linear map This map sends α s to −α s and fixes the hyperplane H s orthogonal to α s with respect to B in V. Therefore, σ s is a reflection of the space V. One can show that the map σ → s s extends to a faithful linear representation of the group W in GL(V), called the standard geometric realization of the group.
Next let us introduce the Tits cone ⊂  V of W. Every hyperplane H s divides V into two halfspaces. Let + s H denote the closed halfspace on which the element * α s dual to α s is nonnegative. The intersection of these halfspaces 0 + Σ = ∩ s H for s ∈ S is a closed simplicial cone in V. The closure of the union of all W -translates of 0 Σ is again a cone in V. This cone  is called the Tits cone. It is known that  = V if and only if W is finite. For infinite groups W the Tits cone is significantly smaller than the whole space and hence is more convenient for the geometric realization of the group.
Under some additional assumptions we can define the action of W on a section of the Tits cone. In particular, this way we can describe the action of affine or hyperbolic Coxeter groups of rank 3 on a Euclidean or Lobachevsky plane, respectively. For example, consider the affine group W of type 2  A . This group is generated by three involutions s 1 , s 2 , s 3 satisfying the relations The space V is isomorphic to 3  , and via this the Tits cone  can be identified with the upper halfspace . It is easy to check that the action of W preserves the affine plane := { = 1} M z , and that the images of Σ 0 intersect M in equilateral triangles ( Figure 1). This construction implies that the group W of type 2  A can be realized as the discrete subgroup of the affine isometrics of a plane generated by reflections in the sides of an equilateral triangle.
A generalization of this construction leads to triangle groups. Let  ∆ be a triangle with angles π/p, π/q, π/r, where p, q, r ∈  ∪ {∞}.

Journal of Generalized Lie Theory and Applications
The triangle ∆ lives on a sphere, or in an affine or hyperbolic plane, depending on whether 1/p + 1/q + 1/r is > 1, = 1, or < 1. The group W pqr of isometries of the corresponding space generated by reflections in the sides of ∆ is respectively a finite, affine, or hyperbolic Coxeter group. For example W 333 is the affine group of type 2 A  . In Figure 2 we show the tessellations of the hyperbolic plane corresponding to W 237 (the Hurwitz group, Figure 2a and W 23∞ (the modular group, Figure  2b). The coloring of the triangles indicates the partition of W into cells, which we will define in the next section.

Main Definitions
Consider a Coxeter group W with a system of generators S. Any element w ∈W can be written as a product, or word, in the generators: w s s s S . Such an expression is called reduced if we cannot use the relations in W to produce a shorter expression for w.
An element can have different reduced expressions but it is not hard to check that all of them have the same length. Therefore, we can define the length function : , which assigns to an element w ∈W the length of a reduced expression with respect to the generators S [1]. Another important notion which can be defined using the reduced expressions is the partial order ≤ of Chevalley-Bruhat. Let 1 … N s s be a word in the generators. We define a subexpression as to be any (possibly empty) product of the form 1 We say that y ≤ w if an expression for y appears as a subexpression of a reduced expression for w. It can be shown that the relation ≤ is a partial order on the group W [1].
Let  denote the Hecke algebra of W over the ring of Laurent polynomials in q 1/2 . This algebra is a free -module with a basis T w , w ∈ W and with multiplication defined by . This new basis, introduced by Kazhdan and Lusztig [2], has a number of important properties and has proven to be very convenient for describing the representations of W and . The elements C w can be expressed in terms of T w by the formulae  (y, w), we denote the leading coefficient by µ(y, w) = µ(w, y), and in all other cases (including when y and w are not comparable in the partial order) we put µ(y, w) = µ(w, y) = 0. We indicate that µ(y, w)≠ 0 by y−−w.
Using the polynomials P y,w we can define the partial orders ≤ L , ≤ R , ≤ LR on W. First, for w ∈W we define the left and right descent sets: Next, we say that ≤ L y w if there exists a sequence The relation R ≤ can be defined using : means that there exists a sequence 0 1 , , , = … = n y y y y wsuch that for all i<n, we have either . It is easy to check that ≤ L , ≤ R and ≤ LR are partial orders on W; we denote the corresponding equivalence relations by ∼ L , ∼ R and ∼ LR . The equivalence classes of ∼ L (respectively, ∼ R , ∼ LR ) are called the left cells (resp. right cells, two-sided cells) of W. It follows from the definitions that the left and right cells have very similar properties, thus we will mostly deal with only one of these two types of cells.
The colors in Figure 2 indicate the left cells of W 237 and the 2-sided cells of W 23∞ . Furthermore, the left cells of W 23∞ can be seen in Figure  2b also: they are the connected unions of triangles of a given color. Note that both of these groups apparently have infinitely many left cells. Now consider the multiplication of the C-basis element in . We can write , , , , Let a(z) be the smallest integer such that Lusztig proved that in affine groups there is a bijection between the set  and the left (or right) cells of W [8]. This deep result has many important corollaries and applications. One of the main goals of our work is to make this correspondence between distinguished involutions and cells explicit, so that one can describe the structure of the cells using the distinguished involutions of the group. Another goal is to find an algorithm that produces distinguished involutions of a given group. In the next section we will formulate two conjectures that answer these questions and present some results to support the conjectures.

Conjectures and Results
We will need some more notations and definitions. Given w ∈ W, we write w = x.y if w = xy and l(w) = l(x) + l(y). Denote by Z(w) the set of all v ∈ W such that w = x.v.y for some x, y ∈ W and v ∈ W I for some I ⊂ S with W I finite. (We recall that for a subset I ⊂ S, W I denotes the standard parabolic subgroup of W generated by s ∈ I.) We call v ∈ Z (w) maximal in w if it is not a proper sub word of any other ( ) ′ ∈ v Z w such that . .
l . This notion of combinatorial rigidity plays an important role in our considerations. Figure 3 helps to understand its meaning using the Cayley graph of the group W. Maximal distinguished involutions of the finite parabolic subgroups correspond to the "long cycles" in the graph. Combinatorial rigidity means that such a cycle cannot be shifted along the presentation of w in any direction. For example, in the triangle group W 333 of type 2  A , the element w = s 3 s 1 s 2 s 1 s 3 is rigid at v = s 1 s 2 s 1 , but Our conjectures can be formulated as follows:  w . Conjecture 1 can be used to inductively construct distinguished involutions in an infinite Coxeter group W starting from the involutions of its finite standard parabolic subgroups. Conjecture 2, in turn, allows one to obtain equivalences in the group using its distinguished involutions. Let us note that the usual method for obtaining results of this kind is based on computing Kazhdan-Lusztig polynomials. This requires many computations and, moreover, does not give any a priori information about the elements that are distinguished involutions or satisfy the cell equivalences in the group. For infinite Coxeter groups in which an exhaustive search is not possible this latter disadvantage becomes critical. Our Conjecture 2 does not necessarily give all equivalences in the group, but still one can expect that the equivalences provided by the conjecture suffice for describing the cells. More precisely, we have the following theorem.

Theorem 1:
If an infinite Coxeter group satisfies Conjectures 1 and 2, and also two conjectures of Lusztig, then [7].
(1) The set  of distinguished involutions consists of the union of v ∈  f and the elements of W obtained from them using Conjecture 1.
(2) The relations described in Conjecture 2 determine the partition of W into right cells.
(3) The relations described in Conjecture 2 together with its ∼ L -analogue determine the partition of W into two-sided cells.
The conjectures of Lusztig to which we refer in the theorem are socalled "positivity conjecture" and a conjecture about a combinatorial description of the function a(z). The positivity conjecture is now proved for a wide class of infinite Coxeter groups that includes affine Weyl groups. The precise statements of the conjectures and the references for the known results can be found in Belolipetsky et al. [7].
We were able to prove Conjectures 1 and 2 under certain additional assumptions. The results are given in the following two theorems.  [7].

Assume that
(1) For any and j tv t is rigid at v 0 .
(2) For any The additional assumption ( ) ( ) ≠ ∅  L vs R vs in Theorem 2 may seem minor, but unfortunately this is not the case. In particular, conditions (1) and (2) in Theorem 3 appear as a consequence of this assumption. The proof of the theorems 2 and 3 in Belolipetsky et al. [7] essentially uses the results from two unpublished letters of Springer and Lusztig [9]. A possible approach to the proof of our conjectures in general requires developing further the ideas of this correspondence.
Although the conditions of Theorems 2 and 3 are not always met for all W, the theorems can still be used to produce interesting results. We will give some examples in the next section, other applications of the theorems are considered in Belolipetsky et al. [7].

Cells in Affine Groups of Rank 3
Affine Weyl groups of rank 3 have type  The cells in these groups were first described by Lusztig [10]. In this section we will show how the same results can be relatively easily obtained using conjectures from §4. ∈  v s vs . Note that after this the inductive procedure terminates as the elements s 1 s 2 v and s 3 s 2 v which would come out on the next step are both non-rigid at v. We can apply the same procedure to the other two involutions from •  f . As a result we get Therefore, the group W of type 2  A has 10 left (right) cells. Using Conjecture 2 and the geometric realization of the group it is easy to show that the partition of W into cells is the one shown on Figure 4a, where two-sided cells correspond to the regions of the same color and left cells correspond to the connected components of the two-sided cells. This coincides with the result of Lusztig et al., [10]. Note that in order to produce the cells we only need Theorems 2 and 3, and thus our results for this case are unconditional.

=  s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s
The partition of W into cells, which we get using Conjecture 2, is shown in Figure 4b. One can quite easily obtain this partition by following the cycles which correspond to the distinguished involutions on the tessellation of the plane. The result is again in agreement with Lusztig et al. [10]. Note that in this case one can check that the assumptions of Theorem 2 hold, but not the assumptions of Theorem 3. Thus we can compute the distinguished involutions, but we cannot prove that our relations suffice to generate the cells, and therefore cannot be sure that we actually have all the distinguished involutions without using Lusztig's computations in Lusztig et al. [10] or referring to our conjectures.