Cells in Coxeter GroupsPaul E. Gunnells

528 NOTICES OF THE AMS VOLUME 53, NUMBER 5

IntroductionCells—left, right, and two-sided—were introducedby D. Kazhdan and G. Lusztig in their study of therepresentation theory of Coxeter groups and Heckealgebras [22]. Cells are related to many disparateand deep topics in mathematics, including singu-larities of Schubert varieties [23], representationsof p-adic groups [24], characters of finite groupsof Lie type [25], the geometry of unipotent conju-gacy classes in simple complex algebraic groups[5,6], composition factors of Verma modules forsemisimple Lie algebras [21], representations ofLie algebras in characteristic p [19], and primitiveideals in universal enveloping algebras [32].1

In this article we hope to present a different andoften overlooked aspect of the cells: as geometricobjects in their own right, they possess an evoca-tive and complex beauty. We also want to draw at-tention to connections between cells and someideas from theoretical computer science.

Cells are subsets of Coxeter groups, and as suchcan be visualized using standard tools from the the-ory of the latter. How this is done, along with somebackground, is described in the next section. In themeantime we want to present a few examples, sothat the reader can quickly see how intriguingcells are.

Let p, q, r ∈ N∪ {∞} satisfy p−1 + q−1 + r−1

≤ 1, where we put 1/∞ = 0. Let ∆ = ∆pqr be a tri-angle with angles (π/p,π/q,π/r ). If p−1 + q−1

+r−1= 1, then ∆ is Euclidean and can be drawn inR2; otherwise ∆ lives in the hyperbolic plane. In ei-ther case, the edges of ∆ can be extended to lines,and reflections in these lines are isometries of theunderlying plane. The subgroup W = Wpqr of thegroup of isometries generated by these reflections

is an example of a Coxeter group. Under the actionof W , the images of ∆ become a tessellation of theplane, with tiles in bijection with W (Figure 1).Hence we can picture cells by coloring the tiles ofthis tessellation.

For example, the triangle ∆236 is Euclidean, andthe associated group W236 is also known as the

Paul Gunnells is assistant professor of mathematics at theUniversity of Massachusetts at Amherst. His email ad-dress is gunnells@math.umass.edu.

We thank M. Belolipetsky, W. Casselman, J. Humphreys, andE. Sommers for helpful conversations. Some computationsto generate the figures were done using software by W. Cas-selman, F. du Cloux, and D. Holt. In particular, the basicPostscript code to draw polygons in the Poincaré disk is dueto W. Casselman, as is the photo of G. Lusztig.

The author is partially supported by the U.S. National Sci-ence Foundation.1D. Vogan [32] also introduced cells for Weyl groups likethose of Kazhdan-Lusztig.

Figure 1. Generating a tessellation of thehyperbolic plane by reflections. The centralwhite tile is repeatedly reflected in the red,

green, and blue lines.

MAY 2006 NOTICES OF THE AMS 529

affine Weyl group G2. Figure 2 shows George Lusztigsporting a limited edition T-shirt emblazoned withthe two-sided cells of G2 [26], also reproduced inFigure 3. Figure 4 shows two hyperbolic examples,the groups W237 and W23∞ . The latter group is alsoknown as the modular group PSL2(Z). We invitethe reader to ponder how the three pictures are geo-metrically part of the same family.

Visualizing Coxeter GroupsBy definition, a Coxeter group2 W is a group gen-erated by a finite subset S ⊂ W where the definingrelations have the form (st)ms,t = 1 for pairs of gen-erators s, t ∈ S . The exponents ms,t are taken fromN∪ {∞} , and we require ms,s = 1. Hence each gen-erator s is an involution. Two generators s, t com-mute if and only if ms,t = 2.

The most familiar example of a Coxeter groupis the symmetric group Sn ; this is the group of allpermutations of an n-element set {1, . . . , n}. We cantake S to be the set of simple transpositions si ,where si is the permutation that interchanges iand i + 1 and fixes the rest. It’s not hard to see thatS generates Sn , and that the generators satisfy(sisi+1)3 = 1 and commute otherwise.

The triangle groups Wpqr from the introductionare also Coxeter groups, for which the generatorsare reflections through the lines spanned by theedges of the fixed triangle ∆pqr. The most important

Coxeter groups arecertainly the Weyland affine Weylgroups, which play avital role in geometryand algebra. In fact,the symmetric groupSn is also known tocognoscenti as theWeyl group An−1 ,while the three Eu-clidean trianglegroups W333 , W234 ,and W236 are exam-ples of affine Weylgroups.

The first step to-wards a geometricpicture of a Coxeter group is its standard geomet-ric realization. This is a way to exhibit W as a sub-group of GL(V ), where V is a real vector space ofdimension |S| . Suppose we have a basis∆ = {αs | s ∈ S} of the dual space V∗ . For eacht ∈ S , there is a unique point α∨s ∈ V such that〈αs,α∨t 〉 = −2 cos(π/ms,t ) for all s ∈ S , where thebrackets denote the canonical pairing between V∗and V . Each αs determines a hyperplane Hs, namelythe subspace of V on which αs vanishes. For eachs , let σs ∈ GL(V ) be the linear map σs (v) =v − 〈αs, v〉α∨s . Note that σs fixes Hs and takes α∨sto −α∨s (Figure 5(a)). One can show that the maps{σs | s ∈ S} satisfy (σsσt )ms,t = Id, which impliesthat the map s � σs extends to a representation ofW . It is known that this representation is faithful,and thus we can identify W with its image in GL(V ).3

Next we need the Tits cone C ⊂ V . Each hyper-plane Hs divides V into two halfspaces. We let H+

sbe the closed halfspace on which αs is nonnegative.The intersection Σ0 = ∩H+

s , where s ranges over S ,is a closed simplicial cone in V . The closure of theunion of all W-translates of Σ0 is a cone C in V ; thisis the Tits cone. It is known that C = V exactlywhen W is finite. Usually in fact C is much less thanall of V . Hence the Tits cone gives a better picturefor the action of W on V .

Under certain circumstances we can obtain amore succinct picture of the action of W on C. Forcertain groups W it is possible to take a nice “cross-section” of the simplicial cones tiling C to obtaina manifold M tessellated by simplices. An exam-ple can be seen in Figure 5(b) for the affine Weylgroup A2. This group has three generators r , s, t ,

2For more about Coxeter groups, we recommend [8, 20].

Figure 2. George Lusztig delivering anAisenstadt lecture at the Centre National de laRecherche Mathématique during the workshopComputational Lie Theory in spring 2002. Thisyear marks Lusztig’s 60th birthday; aconference in his honor will be held at MIT fromMay 30 to June 3, 2006.

Figure 3. G2 = W236.

3This construction allows us to define Weyl and affineWeyl groups. A Weyl group W is a finite Coxeter group gen-erated by a set S of real reflections and also preserving acertain Euclidean lattice L in its geometric realization.The associated affine Weyl group W is the extension of Wby L . As a Coxeter group W is generated by S and oneadditional affine reflection.

530 NOTICES OF THE AMS VOLUME 53, NUMBER 5

with the product of any two distinct generators hav-ing order three. Thus V = R3 , and the Tits cone Cis the upper halfspace {(x, y, z) ∈ R3 | z ≥ 0}. Itturns out that the action of A2 preserves the affinehyperplane M := {z = 1} , and moreover the inter-sections of M with translates of Σ0 are equilateraltriangles. This reveals that A2 is none other thanour triangle group W333. A similar picture works forany affine Weyl group, except that the trianglesmust be replaced by higher-dimensional simpliceswhose dihedral angles are determined by the ex-ponents ms,t.

For more examples we can consider the hyper-bolic triangle groups Wpqr , where p−1 + q−1

+r−1 < 1. In this case the Tits cone is a certainround cone in R3, and the manifold M is one sheetof a hyperboloid (Figure 6). Then M can be identi-fied with the hyperbolic plane; under this identifi-cation the intersections M ∩wΣ0 become the tri-angles of our tessellation.

W-graphs and CellsThere are two main ingredients needed to definecells: descent sets and Kazhdan-Lusztig

polynomials. To introduce them werequire a bit more notation.

The Coxeter group (W,S) comesequipped with a length function� : W → N∪ {0}, and a partial order≤ , the Chevalley-Bruhat order. Anyw ∈ W can be written as a finite prod-uct s1 · · · sN of the generators s ∈ S .Such an expression is called reducedif we cannot use the relations to pro-duce a shorter expression for w. Thenthe length �(w ) is the length N of a re-duced expression s1 · · · sN = w. Thepartial order ≤ can also be charac-terized via reduced expressions. Givenan expression s1 · · · sN , a subexpres-

sion is a (possibly empty) expression of the formsi1 · · · siM , where 1 ≤ i1 < · · · < iM ≤ N . Theny ≤ w if an expression for y appears as a subex-pression of a reduced expression for w . Althoughit is not obvious from this definition, this partialorder is well-defined.

The left descent set L(w ) ⊂ S of w ∈ W is sim-ply the set of all generators s such that�(sw ) < �(w ). There is an analogous definition forright descent set. The definition of the Kazhdan-Lusztig polynomials, on the other hand, is toolengthy to reproduce here, although it can bephrased in completely elementary terms. For eachpair y,w ∈ W satisfying y ≤ w, there is a Kazhdan-Lusztig polynomial Py,w ∈ Z[t] . By definitionPy,y = 1 ; otherwise Py,w has degree at mostd(y,w ) := (�(w )− �(y)− 1)/2 . These subtle poly-nomials are seemingly ubiquitous in representationtheory; they encode deep information about vari-ous algebraic structures attached to (W,S). More-over, computing these polynomials in practice isdaunting: memory is rapidly consumed in even thesimplest examples. In any case, for our purposeswe only need to know whether or not Py,w actuallyattains the maximum possible degree d(y,w ) for

Figure 4. (a) W237 (b) W23∞

Hs

αs

αs

∨

∨

Figure 5. (a) σs negates α∨s and fixesHs.

M

z = 0

Σ0

(b) Slicing the Tits cone for A2.

MAY 2006 NOTICES OF THE AMS 531

a given pair y < w. We write y—w if this is so;when w < y we write y—w if w—y holds.

We are finally ready to define cells. The leftW -graph ΓL of W is the directed graph withvertex set W , and with an arrow from y to wif and only if y—w and L(y) �⊂ L(w ). The leftcells are extracted from the left W -graph asfollows. Given any directed graph, we say twovertices are in the same strong connected com-ponent if there exist directed paths from eachvertex to the other. Then the left cells of W areexactly the strong connected components ofthe graph ΓL. The right cells are defined usingthe analogously constructed right W-graph ΓR,while y,w are in the same two-sided cell if theyare in the same left or right cell.

Figure 7 illustrates all the computationsnecessary to produce the cells for the sym-metric group S3 = 〈s, t | s2 = t2 = (st)3 = 1〉 . Fig-ure 7(a) shows S3 with its partial order and with theleft descent sets in boxes. For this group one cancompute that Py,w = 1 for all relevant pairs (y,w ) .Thus all the information needed to produce ΓL iscontained in the left descent sets. Figure 7(b) showsthe resulting graph ΓL, and Figure 7(c) shows thefour left cells. Computing right descent sets showsthat there are three two-sided cells, with the blueand green cells forming a single two-sided cell.

Now we can explain the coloring scheme used inFigures 3 and 4. All regions of a given color com-prise a two-sided cell. Moreover, the left cells areexactly the connected components of the two-sidedcells, in the following sense. Let us say two trian-gles are adjacent if they meet in an edge. Then bydefinition, a set T of triangles is connected if forany two triangles ∆,∆′ ∈ T it is possible within Tto build a sequence ∆∗ = ∆1,∆2, . . . of triangleswith each ∆i adjacent to ∆i+1, and such that the se-quence ∆∗ contains ∆ and ∆′. Note a significant dif-ference between the Euclidean group W236 and thetwo hyperbolic groups. For the former, each two-sided cell contains only finitely many left cells,whereas this is not necessarily the case in general.The latter phenomenon was first observed byR. Bédard [2], who also showed [3] that there are in-finitely many left cells for all rank 3 crystallographichyperbolic Coxeter groups (see the last section forthe definition of crystallographic). M. Belolipetskyproved that each Coxeter group in a certain infinitefamily has infinitely many left cells [4].

More ExamplesThere are two families of Coxeter groups for whichwe have a good combinatorial understanding oftheir cells: the symmetric groups Sn and the affineWeyl groups An . For the former, left cells appearnaturally in the combinatorics literature in thestudy of the Robinson-Schensted correspondence.A lucid exposition of this connection can be found

in Chapter 6 of the re-cently published [8].

The latter is the work ofJ.-Y. Shi [29]. To describesome of his results, recallthat we can associate tothe group An a tiling of Rn

by simplices. The sim-plices can be furthergrouped into certain con-vex sets called sign-typeregions. Figure 8(a) showsthe sixteen sign-type re-gions for A2; in general forAn there are nn+2 sign-type regions. One of Shi’smain results is that eachleft cell is a union of sign-type regions. Moreover,Shi also gave an explicitalgorithm that allows oneto determine to which leftcell a given region belongs.The algorithm requires toomuch notation to statehere, but it is completelyelementary and involvesno computation of Kazh-dan-Lusztig polynomials.Figure 8(b) shows the two-sided cells for A2 [26]; onecan clearly see how the re-gions are joined into cells.

Figures 9(a) and 9(b) de-pict the cells of A3.4 Theseimages were computed di-rectly from the data in

C

∆pqr

M

Σ0

Figure 6. Slicing the Tits cone for a hyperbolic triangle group.

Figure 7. (a) top, (b) center, (c)bottom.

4To keep the pictures unclut-tered, we have omitted theedges of the simplices.

(b) ΓL

(a) S3

(c) Left cells

532 NOTICES OF THE AMS VOLUME 53, NUMBER 5

[29, §7.3].5 In the ex-ploded view we haveomitted the red cells,which are all simplicialcones. Figure 10 showsthe left cells up to con-gruence. All left cells ina given two-sided cellare congruent, exceptfor the yellow two-sidedcell, which contains twodistinct types of leftcells up to congruence(an S and a U).

These figures also in-dicate relationships be-tween cells in differentrank groups. Perhaps

the most colorful way to describethem is through the permutahe-dron, which is a polytope ΠW at-tached to a Weyl group W as fol-lows. Let x ∈ V be a point in thestandard geometric realizationof W such that the W -orbit of xhas size |W | . Then ΠW is definedto be the closed convex hull ofthe points {w · x | w ∈ W} . Itturns out that the combinatorialtype of ΠW is independent of thechoice of x , and moreover thestructure of ΠW is easy to un-derstand: its faces are isomor-phic to lower-rank permutahe-dra ΠW ′ , where W ′ ⊂ W is thesubgroup generated by any sub-set S′ ⊂ S (such subgroups arecalled standard parabolic sub-groups). For example, the poly-tope underlying Figure 9(a) is the permutahedron for the symmetric group S4. The eighthexagonal (respectively, sixsquare) faces correspond to parabolic subgroups isomorphicto S3 (respectively, S2 × S2).

Now the relationship betweencells of affine groups of differentranks is conjectured to be as fol-lows. For any finite Weyl groupW , let W be the associated affineWeyl group. Then the intersec-tion of the cells of W with theface of ΠW corresponding to thestandard parabolic subgroup Pshould produce the picture for

the cells of the affine group P . This is clearly vis-ible in Figure 9(a): the cells for A2 (respectively,A1 × A1) appear when one slices the cells for A3

with hexagonal (respectively square) faces of ΠA3.Comparing the cells for C3 (Figure 11(b)), originallycomputed by R. Bédard [2], with the cells of C2

(Figure 11(a), [26]) shows another example of this.For more along these lines see [17].

Cells and AutomataSimple examples show that W -graphs can be quitecomplicated. However, despite this complexity lurk-ing in their construction, the cells themselves ap-pear to be very regular. In fact, for many groupsone can prove that the cells can be built using a rel-atively small set of rules, rules that involve noKazhdan-Lusztig polynomial computations at all[13], [14].

Computer scientists have a formal way to workwith this phenomenon, the theory of regular lan-guages and finite state automata [1]. One starts with

Figure 8. (a) A2 sign-type regions. (b) A2 cells.

Figure 10. A3 left cells up tocongruence.

Figure 9. Two views of the cellsof A3.

(a) (b)

5Unfortunately this data is incomplete due to a publisher error: four leftcells are missing.

MAY 2006 NOTICES OF THE AMS 533

a finite set A , called an alphabet.Words over the alphabet are se-quences of elements of A , and anyset L of words over A is called alanguage. Informally, a language isregular if its words can be recog-nized using a finite list of finitepatterns in the alphabet, patternsthat are familiar to anyone whohas ever used a Unix shell (e.g., ls*.tex). A finite state automaton Fover A is a finite directed graphwith edges labelled by elements ofA . The vertices of F are calledstates. All vertices are designatedas either accepting or nonaccept-ing, and one vertex is set to be theinitial state.

Such an automaton determines a language overA as follows. One starts at the initial state and fol-lows a directed path terminating at an acceptingstate. Such a path determines a word (one simplyconcatenates the labels of the edges along the pathto produce a word). We say that this word is rec-ognized by the automaton. The set of all words rec-ognized by an automaton is hence a language overA . A basic theorem is that a language is regular ex-actly when it can be recognized by a finite state au-tomaton.

For a Coxeter group W , the alphabet is the setof generators S , and the language is the setReducedW of all reduced expressions. By a resultof B. Brink and R. Howlett [10], the languageReducedW is regular. Any left cell C determines asublanguage ReducedW (C) := {w ∈ ReducedW | wis a word in C}. W. Casselman has conjecturedthat the language ReducedW (C) is always regular.

Figure 12 illustrates these ideas for one of theyellow left cells in A2 (Figure 8(b)). This cell has theproperty that every element in it has a unique re-duced expression; such cells were first consideredby G. Lusztig [24, Proposition 3.8]. The automatonhas edges labelled by elements of {r , s, t} . The

initial state is the encircled light purple vertex andis nonaccepting; all other vertices are accepting. Tomake the connection between the automaton andthe cell, start at the bottom grey triangle. Then ifwhile following a directed path we encounter an el-ement of S , we flip the indicated vertex to move toa new triangle in the cell. For another example fora cell in the hyperbolic group W343 , as well as moreinformation about the role of automata in the con-text of cells, we refer to [11], [12].

For W = An , the existence of automata forReducedW (C) follows easily from the work ofP. Headley [18] and Shi. Headley proved that onecan construct an automaton F recognizingReducedW in which the vertices are the sign-typeregions, and in which all vertices are accepting.Hence to recognize ReducedW (C) one merely takesF and makes a new automaton FC by designatingonly the vertices corresponding to regions in C asaccepting. In fact Headley’s automaton makes sensefor all Coxeter groups,6 although the examples ofC2 and G2 already show that the above argumentfor ReducedW (C) breaks down. However, for affine

6An exposition can be found in Chapter 4 of [8], where Fis called the canonical automaton.

Figure 11. (a) C2 (b) C3

s r

r

t s

s

r

t

t r

t

s

Figure 12 (a) (b)

534 NOTICES OF THE AMS VOLUME 53, NUMBER 5

Weyl groups, we have conjectured that a closely re-lated automaton works for ReducedW (C) [17].

Further QuestionsThe pictures in this paper certainly raise morequestions than they answer. For example, in thecase of affine Weyl groups, for all known examplesthe left cells are of “finite-type,” in the sense thatthey can be encoded by finitely much data. Herewe have in mind descriptions of the cells using suchtools as patterns among reduced expressions [2, 13,14], sign-types [29], or similar geometric struc-tures [2, 17].

The cells for general Coxeter groups, on theother hand, appear to be fractal in nature, andthus cannot be described in the same way. Au-tomata provide one convenient way to treat suchstructures, but they are not the only way. What areother techniques, and which are natural?

The situation becomes even more intriguingwhen one considers relationships between cellsand representation theory. For instance, Lusztigconjectured [24, 3.6] and proved [27] that an affineWeyl group W contains only finitely many two-sided cells. In fact, he proved much more: heshowed [28] that there is a remarkable bijection be-tween two-sided cells and the unipotent conjugacyclasses in the algebraic group dual to that of W .Moreover, each two-sided cell contains only finitelymany left cells. Lusztig also conjectured [24, 3.6]that the number of left cells in a two-sided cell canbe explicitly given in terms of the cohomology ofSpringer varieties [31].

For general Coxeter groups our knowledge ismuch more impoverished. First of all, it is notknown if there are always only finitely many two-sided cells, although in all known examples it is ev-idently true. Perhaps the only general result is dueto M. Belolipetsky, who showed that right-angledhyperbolic Coxeter groups have only 3 two-sidedcells [4]. Furthermore, in joint work withM. Belolipetsky we have conjectured that the Cox-eter group associated to a hyperbolic n-gon withn distinct angles has (n+ 2) two-sided cells.

The connection with geometry is even more ten-uous. If a Coxeter group W is crystallographic,which by definition means mst ∈ {2,3,4,6,∞} forall distinct generators s, t , then there is associatedto W an infinite-dimensional Lie group G called aKac-Moody group. In principle, G provides a set-ting to study geometric questions about cells, sincemany of the standard constructions (e.g., flag va-rieties, Schubert varieties) make sense there. Ofcourse, at the moment the connections with geom-etry are poorly understood. For instance, the factthat a two-sided cell can contain infinitely many leftcells [2–4] is somewhat sobering.

If W is not crystallographic, then there is nosuch group G . For such W we have no candidate

for an algebro-geometric picture. However, com-putations with many examples (cf. Figures 3 and4) indicate that certain structures vary “continu-ously” in families containing both crystallographicand non-crystallographic groups and that thesestructures are apparently insensitive to whether ornot the underlying group is crystallographic.

The situation is analogous to that of convexpolytopes. In the 1980s many difficult theoremsabout polytopes were first proven using the geom-etry of certain projective complex varieties—toricvarieties—built from the combinatorics of rationalpolytopes. Deep properties of the intersection co-homology of these varieties led to highly nontriv-ial theorems for rational polytopes; for some ofthese theorems no proofs avoiding geometry wereknown.

By definition rational polytopes are those whosevertices have rational coordinates. However, notevery polytope is rational, and for irrational poly-topes no toric variety exists. Yet irrational polytopesseem to share all the nice properties of their rationalcousins.

Today we have a much better understanding ofthis story. Recently several researchers have de-veloped purely combinatorial replacements for thetoric variety associated to a rational polytope andusing these replacements have extended variousdifficult results from the rational case to all poly-topes; see [9] for a recent survey of these results.

For Coxeter groups, the analogy suggests de-veloping combinatorial tools to take the role of thealgebro-geometric constructions that seem essen-tial in the study of crystallographic groups.7 Re-cently there has been significant progress in thiseffort [15, 16, 30]. Nevertheless, understandingthe geometry behind cells for general groups, if itexists, remains an intriguing and difficult problem.

References[1] A. V. AHO, J. E. HOPCROFT, and J. D. ULLMAN, The Design

and Analysis of Computer Algorithms, Addison-Wesley Publishing Co., Reading, MA-London-Amster-dam, 1975, Second printing.

[2] R. BÉDARD, Cells for two Coxeter groups, Comm. Alge-bra 14 (1986), no. 7, 1253–1286.

[3] ———, Left V-cells for hyperbolic Coxeter groups,Comm. Algebra 17 (1989), no. 12, 2971–2997.

[4] M. BELOLIPETSKY, Cells and representations of right-angled Coxeter groups, Selecta Math. (N.S.) 10 (2004),no. 3, 325–339.

[5] R. BEZRUKAVNIKOV, On tensor categories attached tocells in affine Weyl groups, Representation Theory ofAlgebraic Groups and Quantum Groups, Adv. Stud.Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004,pp. 69–90.

7In fact, the analogies between convex polytopes and Cox-eter groups go much further than what is suggested inthese paragraphs [7] and deserves a lengthy exposition ofits own.

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[6] R. BEZRUKAVNIKOV and V. OSTRIK, On tensor categoriesattached to cells in affine Weyl groups II, Represen-tation Theory of Algebraic Groups and QuantumGroups, Adv. Stud. Pure Math., vol. 40, Math. Soc.Japan, Tokyo, 2004, pp. 101–119.

[7] A. BJÖRNER, Topological combinatorics, lecture deliveredat IAS conference in honor of Robert MacPherson, Oc-tober 2004. Notes available at www.math.kth.se/~bjorner/files/MacPh60.pdf.

[8] A. BJÖRNER and F. BRENTI, Combinatorics of CoxeterGroups, Springer-Verlag, 2005.

[9] T. BRADEN, Remarks on the combinatorial intersectioncohomology of fans, arXiv:math.CO/0511488.

[10] B. BRINK and R. B. HOWLETT, A finiteness property andan automatic structure for Coxeter groups, Math. Ann.296 (1993), no. 1, 179–190.

[11] W. A. CASSELMAN, Automata to perform basic calcu-lations in Coxeter groups, Representations of Groups(Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math.Soc., Providence, RI, 1995, pp. 35–58.

[12] ——— , Regular patterns in Coxeter groups, talk at CRM,notes available from www.math.ubc.ca/~cass/crm.talk/toc.html, January 2002.

[13] C. D. CHEN, The decomposition into left cells of theaffine Weyl group of type D4, J. Algebra 163 (1994),no. 3, 692–728.

[14] J. DU, The decomposition into cells of the affine Weylgroup of type B3, Comm. Algebra 16 (1988), no. 7,1383–1409.

[15] P. FIEBIG, Kazhdan-Lusztig combinatorics via sheaveson Bruhat graphs, arXiv:math.RT/0512311.

[16] ——— , The combinatorics of Coxeter categories,arXiv:math.RT/0512176.

[17] P. E. GUNNELLS, On automata and cells in affine Weylgroups, in preparation.

[18] P. HEADLEY, Reduced expressions in infinite Coxetergroups, Ph.D. thesis, University of Michigan, 1994.

[19] J. E. HUMPHREYS, Representations of reduced envelopingalgebras and cells in the affine Weyl group,arXiv:math.RT/0502100.

[20] ——— , Reflection Groups and Coxeter Groups, Cam-bridge Stud. Adv. Math., vol. 29, Cambridge Univ.Press, Cambridge, 1990.

[21] J. C. JANTZEN, Moduln mit einem höchsten Gewicht, Lec-ture Notes in Math., vol. 750, Springer-Verlag, Berlin,1979.

[22] D. KAZHDAN and G. LUSZTIG, Representations of Cox-eter groups and Hecke algebras, Invent. Math. 53(1979), no. 2, 165–184.

[23] ——— , Schubert varieties and Poincaré duality, Geom-etry of the Laplace Operator (Univ. Hawaii, Honolulu,Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer.Math. Soc., Providence, RI, 1980, pp. 185–203.

[24] G. LUSZTIG, Some examples of square-integrable rep-resentations of p -adic semisimple groups, Trans.Amer. Math. Soc. 277 (1983), 623–653.

[25] ——— , Characters of finite groups of Lie type, Ann.of Math. Stud., vol. 107, Princeton Univ. Press, 1984.

[26] ——— , Cells in affine Weyl groups, Algebraic Groupsand Related Topics (Kyoto/Nagoya, 1983), Adv. Stud.Pure Math., vol. 6, North-Holland, Amsterdam, 1985,pp. 255–287.

[27] ——— , Cells in affine Weyl groups II, J. Algebra 109(1987), no. 2, 536–548.

[28] ——— , Cells in affine Weyl groups IV, J. Fac. Sci. Univ.Tokyo Sect. IA Math. 36 (1989), no. 2, 297–328.

[29] J. Y. SHI, The Kazhdan-Lusztig Cells in Certain AffineWeyl Groups, Lecture Notes in Math., vol. 1179,Springer-Verlag, Berlin, 1986.

[30] W. SOERGEL, Kazhdan-Lusztig-Polynome und unzer-legbare Bimoduln über Polynomringen, arXiv:math.RT/0403496.

[31] T. A. SPRINGER, Trigonometric sums, Green functionsof finite groups and representations of Weyl groups,Invent. Math. 36 (1976), 173–207.

[32] D. A. VOGAN JR., A generalized τ-invariant for the prim-itive spectrum of a semisimple Lie algebra, Math. Ann.242 (1979), no. 3, 209–224.