Balanced representations, the asymptotic Plancherel formula, and Lusztig’s conjectures for C2

We prove Lusztig’s conjectures P1 – P15 for the aﬃne Weyl group of type ˜ C 2 for all choices of positive weight function. Our approach to computing Lusztig’s a -function is based on the notion of a “balanced system of cell representations”. Once this system is established roughly half of the conjectures P1 – P15 follow. Next we establish an “asymptotic Plancherel Theorem” for type ˜ C 2 , from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig’s conjectures for all rank 1 and 2 aﬃne Weyl groups for all choices of parameters.


Introduction
The theory of Kazhdan-Lusztig cells plays a fundamental role in the representation theory of Coxeter groups and Hecke algebras.In their celebrated paper [12] Kazhdan and Lusztig introduced the theory in the equal parameter case, and in [14] Lusztig generalised the construction to the case of arbitrary parameters.A very specific feature in the equal parameter case is the geometric interpretation of Kazhdan-Lusztig theory, which implies certain "positivity properties" (such as the positivity of the structure constants with respect to the Kazhdan-Lusztig basis).This was proved in the finite and affine cases by Kazhdan and Lusztig in [13], and the case of arbitrary Coxeter groups was settled only very recently by Elias and Williamson in [4].However, simple examples show that these positivity properties no longer hold for unequal parameters, hence the need to develop new methods to deal with the general case.
A major step in this direction was achieved by Lusztig in his book on Hecke algebras with unequal parameters [15,Chapter 14] where he introduced 15 conjectures P1-P15 which capture essential properties of cells for all choices of parameters.In the case of equal parameters these conjectures can be proved for finite and affine types using the above mentioned geometric interpretation (see [15]).For arbitrary parameters the existing state of knowledge is much less complete.
Recently in [11] we developed an approach to proving P1-P15 and applied it to the case G2 with arbitrary parameters.This provided the first infinite Coxeter group, apart from the infinite dihedral group, where Lusztig's conjectures have been established for arbitrary parameters.Indeed outside of the equal parameter case P1-P15 are only known to hold in the following very limited number of cases: • the quasisplit case where a geometric interpretation is available [15,Chapter 16]; • finite dihedral type [7] and infinite dihedral type [15,Chapter 17] for arbitrary parameters; • finite type Bn in the "asymptotic" parameter case [2,7]; • finite type F4 for arbitrary parameters [7]; • affine type G2 for arbitrary parameters [11].Our approach in [11] hinges on two main ideas: (a) the notion of a balanced system of cell representations for the Hecke algebra, and (b) the asymptotic Plancherel formula.In the present paper we develop these ideas in type C2.This three parameter case turns out to be considerably more complicated than the two parameter G2 case, and this additional complexity requires us to take a somewhat more conceptual approach here.
We now briefly describe the ideas (a) and (b) above.Let (W, S) be a Coxeter system with weight function L : W → N>0 and associated multi-parameter Hecke algebra H defined over Z[q, q −1 ].Let Λ be the set of two-sided cells of W with respect to L, and recall that there is a natural partial order ≤LR on the set Λ. Let (Cw)w∈W denote the Kazhdan-Lusztig basis of H.
One of the main challenges in proving Lusztig's conjectures is to compute Lusztig's a-function since, in principle, it requires us to have information on all the structure constants with respect to the Kazhdan-Lusztig basis.In [11] we showed that the existence of a balanced system of cell representations is sufficient to compute the a-function.Such a system is a family (πΓ) Γ∈Λ of representations of H, each equipped with a distinguished basis, satisfying various axioms including (1) πΓ(Cw) = 0 for all w ∈ Γ ′ with Γ ′ ≥LR Γ, (2) the maximal degree of the coefficients that appear in the matrix πΓ(Cw) is bounded by a constant aπ Γ , (3) this bound is attained if and only if w ∈ Γ.This concept is inspired by the work of Geck [7] in the finite dimensional case.
Thus a main part of the present paper is devoted to establishing a balanced system of cell representations in type C2 for each choice of parameters.For this purpose we use the explicit partition of W into Kazhdan-Lusztig cells that was obtained by the first author in [10].It turns out that the representations associated to finite cells naturally give rise to balanced representations and so most of our work is concerned with the infinite cells.In type C2 there are either 3 or 4 such two-sided cells depending on the choice of parameters.To each of these two-sided cells we associate a natural finite dimensional representation admitting an elegant combinatorial description in terms of alcove paths.Using this description we are able to give a combinatorial proof of the balancedness of these representations.In fact we study these representations as representations of the "generic" affine Hecke algebra of type C2, thereby effectively analysing all possible choices of parameters simultaneously.Once a balanced system of cell representations is established for each choice or parameters we are able to compute Lusztig's a-function for type C2, and combined with the explicit partition of W into cells the conjectures P4, P8, P9, P10, P11, P12, and P14 readily follow.
The second main part of this paper is establishing an "asymptotic" Plancherel formula for type C2, with our starting point being the explicit formulation of the Plancherel Theorem in type C2 obtained by the second author in [19] (this is in turn a very special case of Opdam's general Plancherel Theorem [18]).In particular we show that in type C2 there is a natural correspondence, in each parameter range, between two-sided cells appearing in the cell decomposition and the representations appearing in the Plancherel Theorem (these are the tempered representations of H).Moreover we define a q-valuation on the Plancherel measure, and show that in type C2 the q-valuation of the mass of a tempered representation is twice the value of Lusztig's a-function on the associated cell.This observation allows us to introduce an asymptotic Plancherel measure, giving a descent of the Plancherel formula to Lusztig's asymptotic algebra J .In particular we obtain an inner product on J , giving a satisfying conceptual proof of P1 and P7.Moreover we are able to determine the set D of Duflo involutions, and conjectures P2, P3, P5, P6, and P13 follow naturally.
The remaining conjecture P15 is of a slightly different flavour.In [22] Xie has proved this conjecture under an assumption on Lusztig's a-function.We are able to verify this assumption using our calculation of the a-function and the asymptotic Plancherel formula, hence proving P15 and completing the proof of all conjectures P1-P15.
We conclude this introduction with an outline of the structure of the paper.In Section 1 we recall the basics of Kazhdan-Lusztig theory, and we recall the axioms of a balanced system of cell representations from [11].Section 2 provides background on affine Weyl groups, root systems, the affine Hecke algebra, and the combinatorics of alcove paths.In Section 3 we recall the partition of C2 into cells for all choices of parameters from [10], and introduce some notions such as the generating set of a two-sided cell, cell factorisation and the ã-function.In Section 4 we define various representations of the affine Hecke algebra in preparation for the important Sections 5 and 6 where we establish the existence of the a balanced system of cell representations for each choice of parameters.The main work here is in Section 6, where we conduct a detailed combinatorial analysis of certain representations associated to the infinite two-sided cells.In Section 7 we establish connections between the Plancherel Theorem and the decomposition into cells, hence establishing the asymptotic Plancherel Theorem for type C2.The proofs of P1-P15 are given progressively throughout the paper (see Corollaries 3.1, 6.2, 6.23, 7.9, 7.11, and Theorems 7.7 and 7.13).

Kazhdan-Lusztig theory and balanced cell representations
In this section we recall the definition of the generic Hecke algebra and the setup of Kazhdan-Lusztig theory, including the Kazhdan-Lusztig basis, Kazhdan-Lusztig cells, and the Lusztig's conjectures P1-P15.In this section (W, S) denotes an arbitrary Coxeter system (with |S| < ∞) with length function ℓ : W → N.For I ⊆ S let WI be the standard parabolic subgroup generated by I.

Generic Hecke algebras and their specialisations
Let (qs)s∈S be a family of commuting invertible indeterminants with the property that qs = q s ′ whenever s and s ′ are conjugate in W .Let Rg = Z[(q ±1 s )s∈S].The generic Hecke algebra of type (W, S) is the Rg-algebra Hg with basis {Tw | w ∈ W } and multiplication given by (for w ∈ W and s ∈ S) TwTs = Tws if ℓ(ws) = ℓ(w) + 1 Tws + (qs − q −1 s )Tw if ℓ(ws) = ℓ(w) − 1. (1.1) We set qw := qs 1 • • • qs n where w = s1 . . .sn ∈ W is a reduced expression of w.This can easily be seen to be independent of the choice of reduced expression (using Tits' solution to the Word Problem).
Let L : W → N be a positive weight function on W . Thus L : W −→ N satisfies L(ww ′ ) = L(w) + L(w ′ ) whenever ℓ(ww ′ ) = ℓ(w) + ℓ(w ′ ).Let q be an invertible indeterminate and let R = Z[q, q −1 ] be the ring of Laurent polynomials in q.The Hecke algebra of type (W, S, L) is the R-algebra H = HL with basis {Tw | w ∈ W } and multiplication given by (for w ∈ W and s ∈ S) (1.2) We refer to (Tw)w∈W as the "standard basis" of H.Of course H is obtained from Hg via the specialisation qs → q L(s) , with the multiplicative property of weight functions ensuring that this specialisation compatible with the fact that qs = q s ′ whenever s and s ′ are conjugate in W .For a given weight function L, we denote the above specialisation by ΘL : Hg → H.

3
While Kazhdan-Lusztig theory is setup in terms of the specialised algebra H = HL, we will also need the generic algebra Hg at times in this paper (particularly in Section 6).We sometimes write Qs = qs − q −1 s , or Qs = q L(s) − q −L(s) depending on context (particularly in matrices for typesetting purposes).If S = {s0, . . ., sn} we will also often write, for example, 0121 as shorthand for s0s1s2s1, and thus in the Hecke algebra T0121 = Ts 0 s 1 s 2 s 1 .In particular, note that 1 is shorthand for s1, and therefore to avoid confusion we denote the identity of W by e.

The Kazhdan-Lusztig basis
Let L be a positive weight function and let H = HL.The involution ¯on R which sends q to q −1 can be extended to an involution on H by setting In [12], Kazhdan and Lusztig proved that there exists a unique basis {Cw | w ∈ W } of H such that, for all w ∈ W , Cw = Cw and Cw = Tw + y<w Py,wTy where Py,w This basis is called the Kazhdan-Lusztig basis (KL basis for short) of H.The polynomials Py,w are called the Kazhdan-Lusztig polynomials, and to complete the definition we set Py,w = 0 whenever y < w (here ≤ denotes Bruhat order on W ) and Pw,w = 1 for all w ∈ W .We note that the Kazhdan-Lusztig polynomials, and hence the elements Cw, depend on the the weight function L (see the following example).
Example 1.1.Let (W, S, L) be a Coxeter group and let J ⊆ S be such that the group WJ generated by J is finite.
Let wJ be the longest element of WJ .The Kazhdan-Lusztig element Cw J is equal to w∈W J q L(w)−L(w J ) Tw. Indeed, this element has the required triangularity with respect to the standard basis and it is stable under the bar involution.Further, if we set Cw J := w∈W qwq −1 w J Tw ∈ Hg then we have ΘL(Cw J ) = Cw J for all positive weight functions L on W . Now assume that S contains two elements s1, s2 such that (s1s2) 4 = e.If we set a = L(s1) and b = L(s2) then we have Indeed, the expressions on the right-hand side are stable under the bar involution and since they have the required triangularity property, they have to be the Kazhdan-Lusztig element associated to 212.Unlike the case where w = wJ , there is no generic element in Hg that specialises to C212 ∈ H(W, S, L) for all positive weight functions L. We also note that when b > a we have P2,212 = q −b−a − q −b+a , showing that the Kazhdan-Lusztig polynomials can have negative coefficients in the unequal parameter case.
Let x, y ∈ W .We denote by hx,y,z ∈ R the structure constants associated to the Kazhdan-Lusztig basis: 15,Chapter 13]).The Lusztig a-function is the function a : W → N defined by When W is infinite it is, in general, unknown whether the a-function is well-defined.However in the case of affine Weyl groups it is known that a is well-defined, and that a(z) ≤ L(w0) where w0 is the longest element of an underlying finite Weyl group W0 (see [15]).The a-function is a very important tool in the representation theory of Hecke algebras, and plays a crucial role in the work of Lusztig on the unipotent characters of reductive groups.
Definition 1.3.For x, y, z ∈ W let γ x,y,z −1 denote the constant term of q −a(z) hx,y,z.
The coefficients γ x,y,z −1 are the structure constants of the asymptotic algebra J introduced by Lusztig in [15, Chapter 18].

Kazhdan-Lusztig cells and associated representations
Define preorders ≤L, ≤R, ≤LR on W extending the following by transitivity: x ≤L y ⇐⇒ there exists h ∈ H such that Cx appears in the decomposition in the KL basis of hCy, x ≤R y ⇐⇒ there exists h ∈ H such that Cx appears in the decomposition in the KL basis of Cyh, x ≤LR y ⇐⇒ there exists h, h ′ ∈ H such that Cx appears in the decomposition in the KL basis of hCyh ′ .
We associate to these preorders equivalence relations ∼L, ∼R, and ∼LR by setting (for * ∈ {L, R, LR}) x ∼ * y if and only if x ≤ * y and y ≤ * x.
The equivalence classes of ∼L, ∼R, and ∼LR are called left cells, right cells, and two-sided cells.
Example 1.4.For y, w ∈ W we write y w if and only if there exists x, z ∈ W such that w = xyz and ℓ(w) = ℓ(x) + ℓ(y) + ℓ(y).In this case it is not hard to see, using the unitriangularity of the change of basis matrix from the standard basis to the Kazhdan-Lusztig basis, that TxCyTz ∈ Cw + z<w azCz and therefore w ≤LR y.
We denote by Λ the set of all two-sided cells (note that of course Λ depends on the choice of weight function).Given any cell Γ (left, right, or two-sided) we set Γ ≤ * := {w ∈ W | there exists x ∈ Γ such that w ≤ * x} and we define Γ ≥ * , Γ> * and Γ< * similarly.
To each right cell Υ of W there is a natural right H-module HΥ constructed as follows.The R-modules are right H-modules by definition and therefore the quotient is a right H-module with basis {Cw | w ∈ Υ} where Cw is the class of Cw in HΥ.Given a left cell (respectively a two-sided cell) we can follow a similar construction to produce left H-modules (respectively H-bimodules).
This is well defined because Px,y ∈ q −1 Z[q −1 ] for all x, y ∈ W .Let The elements of D are called Duflo elements (or, somewhat prematurely, Duflo involutions; see P6 below).
P12.If I ⊆ S then the a-function of WI is the restriction to WI of the a-function of W .
P13.Each right cell Υ of W contains a unique element d ∈ D, and we have γ x,x −1 ,d = 0 for all x ∈ Υ.

Balanced system of cell representations
In [11] we introduced the notion of a balanced system of cell representations, inspired by the work of Geck [5,7] in the finite case.We recall this theory here.If (π, M) is a (right) representation of H, and if B is a basis of M, we write (for h ∈ H and u, v ∈ B) π(h; B) and [π(h; B)]u,v for the matrix of π(h) with respect to the basis B, and the (u, v) th entry of π(h; B).
We define the degree of a Laurent polynomial f (q) ∈ R[q, q −1 ] to be the greatest integer n ∈ Z such that q n appears in f (q) with nonzero coefficient (with deg(0) = −∞).For example deg(3q −1 + q −2 ) = −1 and deg(3q Definition 1.5.We say that H admits a balanced system of cell representations if for each two-sided cell Γ ∈ Λ there exists a representation (πΓ, MΓ) defined over an R-polynomial ring RΓ (where we could have RΓ = R) and a basis BΓ of MΓ such that the following 6 properties hold: B2.There exist bounds aπ Γ ∈ N such that deg[πΓ(Cw; BΓ)]u,v ≤ aπ Γ for all w ∈ W and all u, v ∈ BΓ.
B5. Let Γ ∈ Λ.For each z ∈ Γ, there exists (x, y) ∈ Γ 2 such that γx,y,z −1 = 0, where γx,y,z −1 ∈ Z is defined by the equation Remark 1.6.We make the following remarks: 1) We note that B1 does not depend on the basis BΓ.A representation with property B1 is called a cell representation for the two-sided cell Γ.It is clear that the representations associated to cells that we introduced in Section 1.3 are cell representations (see [11,Section 2.1]).
2) If the basis BΓ of MΓ is clear from context we will sometimes write cπ Γ (w) in place of cπ Γ (w; BΓ).
3) By [11,Corollary 2.4] the axioms B1-B4 and B6 alone imply that the Z-span JΓ of the matrices {cπ Γ (w; BΓ) | w ∈ Γ} is a Z-algebra.Hence the definition of γx,y,z −1 in B5 is not itself a separate axiom; these integers are the structure constants of the algebra JΓ.

4)
We note that in B2 and B3 it is equivalent to replace Cw by Tw, because Cw = Tw + v<w pv,wTv with pv,w ∈ q −1 Z[q −1 ].However in B1 one cannot replace Cw by Tw.
5) Finally we note that we have slightly changed the numbering from [11], where B5 was denoted B4 ′ , and B6 was denoted B5.
In [11] we showed that the existence of a balanced system of cell representations is sufficient to compute Lusztig's a-function.In particular, we have: Theorem 1.7 ([11, Theorem 2.5 and Corollary 2.6]).Suppose that H admits a balanced system of cell representations.Then a(w) = aπ Γ for all w ∈ Γ.Moreover, for each Γ ∈ Λ the Z-algebra JΓ spanned by the matrices {cπ Γ (w; BΓ) | w ∈ Γ} is isomorphic to Lusztig's asymptotic algebra associated to Γ, and γx,y,z = γx,y,z.
Note that the first part of this theorem implies that the bounds aπ Γ are in Definition 1.5 are in fact unique.That is, if there exist two balanced systems of cell representations then their bounds coincide.
2 Affine Weyl groups, affine Hecke algebras, and alcove paths We begin this section with some basic facts about root systems and Weyl groups.We then recall the combinatorial language of alcove paths from [20], and the concept of alcove paths confined to strips from [11].We also discuss the combinatorics of the affine Hecke algebra (and extended affine Hecke algebra) of type C2.
For each α ∈ Φ let sα be the orthogonal reflection in the hyperplane Hα = {x ∈ R 2 | x, α = 0} orthogonal to α, and for i ∈ {1, 2} let si = sα i .The Weyl group of Φ is the subgroup W0 of GL(V ) generated by the reflections s1 and s2 (this is a Coxeter group of type B2 = C2).The Weyl group W0 acts on Q and the affine Weyl group is W = Q ⋊ W0 where we identify λ ∈ Q with the translation t λ (x) = x + λ.The affine Weyl group is a Coxeter group with generating set S = {s0, s1, s2}, where s0 = t ϕ ∨ sϕ, with ϕ = 2α1 + 2α2 the highest root of Φ.
If w ∈ W we define the final direction θ(w) ∈ W0 and the translation weight wt(w) ∈ Q by the equation Let F denote the union of the hyperplanes H α,k with α ∈ Φ and k ∈ Z.The closures of the open connected components of R 2 \F are called alcoves (these are the closed triangles in Figure 1).The fundamental alcove is given by The hyperplanes bounding A0 are called the walls of A0.Explicitly these walls are Hα i ,0 with i = 1, 2 and Hϕ,1.We say that a face of A0 (that is, a codimension 1 facet) has type si for i = 1, 2 if it lies on the wall Hα i ,0 and of type s0 if it lies on the wall Hϕ,1.
The affine Weyl group W acts simply transitively on the set of alcoves, and we use this action to identify the set of alcoves with W via w ↔ wA0.Moreover, we use the action of W to transfer the notions of walls, faces, and types of faces to arbitrary alcoves.Alcoves A and A ′ are called s-adjacent, written A ∼s A ′ , if A = A ′ and A and A ′ share a common type s face.Thus under the identification of alcoves with elements of W , the alcoves w and ws are s-adjacent.
Let w = si 1 si 2 • • • si ℓ be an expression for w ∈ W , and let v ∈ W .A positively folded alcove path of type w starting at v is a sequence p = (v0, v1, . . ., v ℓ ) with v0, . . ., v ℓ ∈ W such that The end of p is end(p) = v ℓ .Let P( w, v) = {all positively folded alcove paths of type w starting at v}.
Less formally, a positively folded alcove path of type w starting at v is a sequence of steps from alcove to alcove in W , starting at v, and made up of the symbols (where the kth step has s = si k for k = 1, . . ., ℓ): If p has no folds we say that p is straight.Note that, by definition, there are no "negative" folds.
If p is a positively folded alcove path we define, for each sj ∈ S, fj(p) = #(positive sj-folds in p).
be the region between the hyperplanes H α ′ i ,0 and We refer to the two symbols in (b) as "s-bounces" rather than folds, since they play a different role in the theory.It turns out that there is no need to distinguish between "positive" and "negative" s-bounces.We note that bounces only occur on the hyperplanes H α ′ i ,0 and H α ′ i ,1 .Moreover, note that there are no folds or crossings on the walls H α ′ i ,0 and H α ′ i ,1 -the only interactions with these walls are bounces.In the case i = 1 every bounce has type 1.In the case i = 2, 3 the bounces on H α ′ 2 ,0 have type 2, and those on H α ′ 2 ,1 have type 0 (see Figures 1 and 3).Let p be an i-folded alcove path.For each j ∈ {0, 1, 2} let fj (p) = #(sj -folds in p) and gj(p) = #(sj-bounces in p).where ψi : W0 → W i 0 is the natural projection map taking u ∈ W0 to the minimal length representative of Wiu, and ω1, ω2 are as defined in Section 2.1.For later use, we also set and let τ3 = τ2.Observe that τi preserves Ui.It is not hard to see that for each p ∈ Pi( w, u) the path τi(p) obtained by applying τi to each part of p is again a valid i-folded alcove path starting at τiu (the main point here is that in the case i = 1 the reflection part of τ1 is in the simple root direction α1, and thus sends Φ + \{α1} to itself; in the cases i = 2, 3 the element τ2 = τ3 is a pure translation, and so the result is obvious).Moreover θ i (p) is preserved under the application of τi, and a direct calculation shows that wt i (τ k i (p)) = k + wt i (p).Note that W i 0 is a fundamental domain for the action of τi on Ui.Let B be any other fundamental domain for this action.For w ∈ Ui we define wt i B (w) ∈ Z and θ i B (w) ∈ B by the equation and for i-folded alcove paths p we define It is easy to see that in the case B = W i 0 these definitions agree with those for wt i (p) and θ i (p) made above.
Example 2.1.Figure 3 shows three examples of i-folded alcove paths, with i = 1 in the first two cases, and i = 2 or i = 3 in the third case.In each case the identity alcove is shaded in dark green.The first and second paths have type w = 210121012120 and start at u = e, and the third path has type w = 121021210120120 and starts at u = 12.

Fig. 3: i-folded alcove paths
The first and second figures illustrate two choices of fundamental domain B for the action of τ1 on U1 (indicated by green and red shading).In the first example B = W 1 0 , and we have wt

The affine Hecke algebra of type C2
Let (W, S) be the C2 Coxeter system and let Hg be the associated generic affine Hecke algebra, as in (1.1).The algebra Hg is generated by T0 = Ts 0 , T1 = Ts 1 and T2 = Ts 2 subject to the relations (for i = 0, 1, 2) where qi = qs i and Qi = qi − q −1 i .Let v ∈ W and choose any expression v = si 1 • • • si ℓ (not necessarily reduced).Consider the associated straight alcove path (v0, v1 . . ., v ℓ ), where v0 = e and v k = si 1 • • • si k .Let ε1, . . ., ε ℓ be defined using the periodic orientation on hyperplanes as follows: It is easy to check (using Tits' solution to the Word Problem) that the element does not depend on the particular expression v = si 1 • • • si ℓ we have chosen (see [9]).If λ ∈ Q we write and it follows from the above definitions that (the second equality follows since t wt(v) is on the positive side of every hyperplane through wt(v), and the third equality follows since Xu = T −1 u −1 for all u ∈ W0).Moreover since Xv = Tv + (lower terms) the set {Xv | v ∈ W } is a basis of Hg, called the Bernstein-Lusztig basis.
Let Rg[Q] be the free Rg-module with basis {X λ | λ ∈ Q}.We have a natural action of W0 on Rg[Q] given by wX λ = X wλ .We set It is a well-known result that the centre of Hg is The combinatorics of positively folded alcove paths encode the change of basis from the standard basis (Tw)w∈W of Hg to the Bernstein-Lusztig basis (Xv)v∈W .This is seen by taking u = e in the following proposition (see [20,Theorem 3.3], or [11,Proposition 3.2]).
Proposition 2.2.(c.f.[20, Theorem 3.3]) Let w, u ∈ W , and let w be any reduced expression for w.Then We have For later reference we record the following complete set of relations for Hg in the Bernstein-Lusztig presentation.Let Y1 = X ω 1 and Y2 = X ω 2 .Then Remark 2.3.Let L : W → N>0 be the weight function with L(s1) = a, L(s2) = b, and L(s0) = c, and let H be the associated affine Hecke algebra, as in (1.2).The results of the above section of course apply equally well to H after applying the specialisation ΘL.For example, Proposition 2.2 applies with the obvious modification

The extended affine Hecke algebra
If q0 = q2 (or, in the specialisation, c = b) one can slightly enlarge the affine Hecke algebra as follows.Let The Weyl group W0 acts on P , and the extended affine Weyl group is Note that P/Q ∼ = Z2.Let σ ∈ W be the nontrivial element of P/Q.Then σsiσ −1 = s σ(i) for each i = 0, 1, 2, where σ(i) denotes the nontrivial diagram automorphism of (W, S).
The length function on W is extended to W by setting ℓ(wσ) = ℓ(w) for all w ∈ W . Thus the length 0 elements of W are precisely the elements e and σ.
Under the assumption q0 = q2 we have . The extended affine Hecke algebra is the algebra Hg over Rg with basis {Tw | w ∈ W } and multiplication (for u, v, w ∈ W and s ∈ S) given by The definition of the Bernstein-Lusztig basis {Xv | v ∈ W } can be extended to Hg by considering W as 2 sheets of W , and an alcove path of type w = si 1 • • • si k σ consists of an ordinary alcove path of type si 1 • • • si k followed by a jump to the σ-sheet of W (see [20]).The centre of Hg is Rg[P ] W 0 .
The Hecke algebra Hg (with q0 = q2) is naturally a subalgebra of Hg.Indeed Hg is generated by T0, T1, T2, and the additional element

Schur functions
The following Schur functions will play a role later.Let In particular we have

Kazhdan-Lusztig cells in type C2
Let W be a Coxeter group of type C2 with weight diagram That is, L(s1) = a, L(s2) = b, and L(s0) = c.In this section we recall the decomposition of W = C2 into cells for all choices of parameters (a, b, c) ∈ N 3 .We then study the properties of this partition and introduce various notions such as the generating set of a two-sided cell, cell factorisations and the ã-function.The ã-function is defined using the values of Lusztig a-function in finite parabolic subgroups of W and as a consequence of the main result of this paper, it turns out that a = ã, and thus the table listed in Section 3.5 in fact records the values of Lusztig's a-function (however, of course, this cannot be assumed at this stage).• L ∈ D for a weight function L to mean (L(s2)/L(s1), L(s0)/L(s1)) ∈ D. In a similar spirit, when considering a statistic F that depends on the choice of parameters (for instance the partition into cells), we will write F (L), or F (a, b, c) or F (r1, r2) to mean that we consider the statistic F with respect to the weight function L, the triplet (a, b, c) ∈ N 3 or the pair

Partition of C2 into cells
, we will also write F (D) to denote the common value of F on D.
The partition of W into cells has been obtained by the first author in [10].Even though there are an infinite number of positive weight functions on W , there are only a finite number of partitions of W into cells (as conjectured by Bonnafé in [1]).In order to describe these partitions we first need to define a set R of subsets of Q 2 >0 on which the partition into cells will be constant.
We define open subsets A1, . . ., A10 of Q 2 >0 in Figure 4. Write A i ′ = A ′ i for the region Ai reflected in the line r1 = r2 (we call this the "dual" region).For "adjacent" regions Ai and Aj (respectively Ai and A ′ i ), let Ai,j (respectively A i,i ′ ) be the line segment Ai∩Aj (respectively Ai ∩A i ′ ) with the endpoints removed.This partitions the set A1,2, A2,3, A3,4, A4,5, A3,6, A6,7, A4,7, A7,8, A5,8, A7,9, A9,10, A8,10 (open intervals), For any region D ∈ R, the decomposition of W into right cells and two-sided cells is the same for all choices of parameters (r1, r2) ∈ D. In Figure 5, we represent Λ(D) for all The alcoves with the same colour lie in the same two-sided cell and the right cells in a given two-sided cell are the connected components.The Hasse diagram on the right of each partition describes the two-sided order on the two-sided cells, going from the highest cell at the top to the lowest one at the bottom.Finally to obtain the decomposition and the two-sided order for a region included in one simply applies the diagram automorphism σ to the partition for the dual region.Hence the partition of C2 into two-sided cells and right cells is known for all choices of parameters.

Semicontinuity conjecture
The parameters (r1, r2) ∈ Q 2 >0 are called generic if there exists an open subset O of R 2 that contains (r1, r2) and such that for all . According to Figure 4, we see that the generic parameters for W are exactly those that lie in some Ai or In [1], Bonnafé has conjectured that the partition of an arbitrary Coxeter group into cells satisfies certain "semicontinuity properties".The basic idea of his conjecture is that the partition for all parameters can be determined from the knowledge of the partition for generic parameters.More precisely the partition Λ(D) for D ∈ R is the finest partition of W that satisfies the following property: For all A ∈ RD, and for all Γ ∈ Λ(A), there exists a cell Γ ′ ∈ Λ(D) such that Γ ⊆ Γ ′ .
In the case of C2 the conjecture is known to hold (by direct inspection using Figure 5).Thus it is (retrospectively) sufficient to know Λ(A) for all A ∈ R• to determine Λ(D) for all D ∈ R (in fact, using the diagram automorphism σ it is enough to know Λ(Ai) for all 1 ≤ i ≤ 10).The most striking example of the semicontinuity phenomenon is when D = P2 (the equal parameter case) where one has to look at the partition of W into cells for parameters in the regions A2, A 2 ′ , A3, A 3 ′ , A4, A 4 ′ , A5 and A 5 ′ to determine the partition into cells.As a result, all finite cells get absorbed into the infinite cells.

Generating sets of two-sided cell
Recall the definition of in Example 1.4.Given a subset C of W we denote by C + the set that consists of all elements w ∈ W that satisfy u w for some u ∈ C. By inspection of Figure 5 we see that for all D ∈ R and all Γ ∈ Λ(D) there exists a minimal subset JΓ(D) of W such that We call this set the generating set of Γ.We have for all D ∈ R and all Γ ∈ Λ(D) (1) JΓ(D) ⊆ I S WI; (2) the elements of JΓ are involutions; where the inclusion can be strict (see the example D = P2 below); (5) the set {Cw | w ∈ JΓ(D)} generates the module Of course, it is also possible to have we will denote by wΓ the element of this set (or simply wi if Γ = Γi).In the table below, we give the elements wi for all Aj ∈ R and Γi ∈ Λ(Aj).

Cell factorisations
When the set JΓ(D) contains a unique element then the two-sided cell Γ admits a cell factorisation.We refer to [11, §4] for a detailed description of this concept in type G2.To illustrate cell factorisation here, consider the lowest two-sided cell Γ0 in the regime r2 < r1.In this case we see that JΓ 0 (r1, r2) = {w0} where w0 = s1s2s1s2.By direct inspection of Figure 5 we have the following representation of elements of Γ0: • Each right cell Υ ⊆ Γ0 contains a unique element wΥ of minimal length.
• The element w0 is a suffix of each wΥ.
In the infinite cells Γ = Γi with i = 1, 2, 3 cell factorisation (if it exists) takes a similar form: • Each right cell Υ ⊆ Γ contains a unique element wΥ of minimal length.
We represent this factorisation in Figure 6.The set of grey alcoves together with the black alcove A0 on the left hand side is B −1 Γ , and the small diagram on the right hand side illustrates BΓ.The connected sets of dark blue (respectively light blue) alcoves are the sets of the form {u −1 wΓt n Γ v | u, v ∈ BΓ} where n is odd (respectively even).
There are also cases where there is a kind of "generalised" cell factorisation that involves the extended affine Weyl group.Specifically, these cases are Γ0 with r2 = r1, the cell Γ2 in the case r2 = r1 and r2 < 1, and the cell Γ2 in the case r2 = r1 and r2 > 1.We will discuss these factorisations at the appropriate time.
Suppose that Γ is a cell admitting a cell factorisation.If w ∈ Γ is written as w = u −1 wΓt n Γ v with u, v ∈ BΓ and n ∈ N we write uw = u, vw = v, and τw = n (and in the case of Γ0 we have w = u −1 wΓt λ v and τw = λ).Let x, y ∈ Γ.With these notations, we have for all generic parameters: x ∼L y ⇐⇒ vx = vy and x ∼R y ⇐⇒ ux = uy.

The ã-function
A useful auxiliary notion is the ã-function, defined as follows.The values of the a-function are explicitely known for finite dihedral groups (see, for example, [11,Table 1]) and Lusztig's conjectures have been verified in this case (see [7,Proposition 5.1]).Therefore, for all choices of parameters, we can define a-functions a k : WI k → N (k = 0, 1, 2) where I k := S\{k}, however we emphasise that it is not clear that a k is the restriction of a to WI k ; this is the content of P12.
It turns out, by direct observation, that if u, v ∈ Γ lie in a common two-sided cell, with u ∈ WI j and v ∈ WI k for j, k ∈ {0, 1, 2}, then aj(u) = a k (v).These observations, together with the fact that every two-sided cell intersects a finite parabolic subgroup, allows us to define a function ã : W → N (for each choice of parameters) by By definition ã is constant on each two-sided cell Γ, and therefore we write ã(Γ) for the value of ã on any element of Γ, thereby considering ã as a function ã : Λ(r1, r2) → N. We remark that ã is a deacreasing function on the set Λ. Indeed it is not hard to check that ã(Γ) ≥ ã(Γ ′ ) whenever Γ ≤LR Γ ′ .Finally, the values of ã are "generically invariant" on the regions D ∈ R as shown in the following proposition.
Proof.This can deduced from the values of the a-function in dihedral groups: see, for example, [11,Table 1].
Since the values of ã-function will play a crucial role in the reminder of the paper, we record these values in the table below.
Tab. 2: The values of ã(Γ i ) for (b/a, c/a) ∈ A j Table 2 only lists the values of ã(Γ k ) for (a, b, c) such that (r1, r2) ∈ Ai for some 1 ≤ i ≤ 10.The remaining cases can also be computed using Proposition 3.2.However we now explain another method to deduce these values (essentially due to semicontinuity).

Representations of H
Let (W, S) be the Coxeter group of type C2 and let L : W → N be a positive weight function.In this section we construct representations of H that will ultimately be used to produce a balanced system of cell representations for each parameter regime.In fact it is convenient to define representations of the generic Hecke algebra Hg of type C2, from which representations of H are obtained by the specialisation ΘL.In what follows we will use the same notations (eg, πi) for the representations of Hg and H.

The diagram automorphism
Let σ be the nontrivial diagram automorphism of (W, S).Then σ induces a ring automorphism of Rg by swapping q0 and q2, and it is easy to check that the formula

The principal series representation
Let ζ1 and ζ2 be commuting indeterminants, and let M0 be the with generator ξ0 and Rg[Q]-action given by linearly extending ξ0 We sometimes write π0 = π ζ 0 when the dependence on ζ = (ζ1, ζ2) requires emphasis.Note that {ξ0 ⊗ Xu | u ∈ W0} is a basis of M0.More generally, if B is a fundamental domain for the action of Q on W then it is clear that is a basis of M0.We will often write π0(Tw; B) in place of π0(Tw; B), even though strictly speaking B is not a basis of M0 (c.f.notation in Section 1.5).
We have the following important alcove path interpretation of the matrix coefficients [π0(Tw; B)]u,v, as in [11].
Theorem 4.1.Let B be a fundamental domain for the action of Q on W .For u, v ∈ B we have and where w is any reduced expression for w.

Induced representations
Let Hi (i = 1, 2) be the subalgebra of Hg generated by Ti, X1 and X2.Let ζ be an invertible indeterminant.Let M1 be the 1-dimensional (right) H1-module over the ring Rg[ζ, ζ −1 ] generated by ξ1 with and for j ∈ {2, 3} let Mj be the 1-dimensional (right) H2-module over the ring Rg[ζ, ζ −1 ] generated by ξj with ).One uses the formulae in Section 2.4 to check that the above formulae do indeed define representations of H1 and H2.
Let (πj , Mj) with j = 1, 2, 3 be the representations M1 = Ind Hg H 1 (M1) and Mj = Ind Hg H 2 (Mj) for j = 2, 3. Then each Mj is a 4-dimensional (right) Hg-module.Indeed {ξi ⊗ Xu | u ∈ W i 0 } is a basis of Mi (where we set W 3 0 = W 2 0 ).More generally, if B is a fundamental domain for the action of τi on Ui (see Section 2.3) then If p is an i-folded alcove path we define We note that the action of τi on the set of i-folded alcove paths preserves Qi.
We have the following analogue of Theorem 4.1, giving a combinatorial formula for the matrix entries of πi(Tw; B) (i = 1, 2, 3) in terms of i-folded alcove paths.
Theorem 4.3.Let w ∈ W , i ∈ {1, 2, 3}, and let B be a fundamental domain for the action of τi on Ui.Then where w is any choice of reduced expression for w.
For example, using the "standard basis" B = W i 0 we have

Square integrable representations
The representations in this section will play a role in the analysis of the finite cells.It turns out that they are also "square integrable representations" (of certain natural C-algebra specialisations of Hg), although this fact will not be particularly important in this paper.
Define 1-dimensional representations πi, 4 ≤ i ≤ 9, of Hg by We now define 3-dimensional representations π10 and π11.These representations were constructed as modules HΥ for some right cell Υ, however since we now consider them as representations of the generic Hecke algebra Hg we will simply provide explicit matrices, from which the defining relations are easily checked.In the case π10 we require two choices of basis for our applications, and we write the resulting matrices as π10( • ; A) and π10( • ; B).In the case π11 we require three choices of basis, and we write the resulting matrices as π11( • ; A), π11( • ; B), and π11( • ; C).The third case only occurs for the specialised algebras with q0 = q1, and indeed the matrices provided for this case below only give a representation of Hg under the specialisation q0 = q1.
where µi,j = qiq −1 j + q −1 i qj, and Similarly we define a 2-dimensional representation π12, equipped with two choices of basis, by We will some times write π A i in place of πi( • ; A), and similarly for π B i and π C i .

A generic version of axiom B1
The aim of this section is to show that the representations πi defined above "generically" satisfy B1 for the cell Γi.Our first task is to define some specific elements in Hg that specialise to Kazhdan-Lusztig elements.As we have seen in Example 1.1, this can easily be done when w is the longest element of some parabolic subgroup.In this section, we extend this construction to all elements in the sets JΓ.
Let D ∈ R and w ∈ JΓ(D) where Γ ∈ Λ(D).Then either w is the longest element of some parabolic subgroup WI or it is of the form w = sts where L(s) > L(t) for all weight functions L ∈ D. In the first case we set and in the second case we set Here, the element Cw on the right-hand side is computed with respect to the parameters (r1, r2).
Proof.This is a consequence of Example 2.12 in [6].
From the properties (5) and ( 6) of the sets JΓ, we see that this theorem can be interpreted as a generic version of B1.

Finite cells
In this section we construct balanced representations for each finite cell.Recall that constructing such a system requires us to associate not only a representation to each two-sided cell, but also a distinguished basis of that representation.Theorem 5.1.Each finite two-sided cell Γ admits a representation πΓ equipped with a basis B satisfying B1-B5 with aπ Γ = ã(Γ).Moreover, in all cases where the finite cell Γ admits a cell factorisation we have cπ Γ (w; B) = ±Eu w ,vw for all w ∈ Γ. (5.1) Proof.For the moment exclude the cell Γ13 from consideration.For all other finite cells we take πΓ to be the cell module HΥ where Υ is any right cell contained in Γ, equipped with the natural Kazhdan-Lusztig basis.The matrices for πΓ have been computed using the CHEVIE package [8,17] in GAP3 [21].For 4 ≤ i ≤ 9 we have πΓ i = πi (these representations are 1-dimensional, and hence have unique representing matrices).For i ∈ {10, 11, 12} we have the following explicit matrices: 11 is indeed a representation).It is then immediate that B1 is satisfied.However we note that B1 also follows from Theorem 4.6 (without needing to know that the representations above are the cell modules).
More interestingly, sometimes (Γi) ≥LR contains an infinite cell.These cases are outlined below (we note that this situation did not occur in type G2; see [11]).
Let us consider one case in detail (the remaining cases are similar).Suppose that (r1, r2) ∈ A7,8 (thus c = a and 2a < b < 3a).Let z = q 4a−2b .By diagonalising π C 11 (t2) we obtain where It is then a straightforward (although somewhat tedious) exercise to show that the degrees of the matrix entries of π C 11 (w) are strictly bounded by a + b for all elements w = u −1 w2t n 2 v ∈ Γ2.
It is then easy to compute π A 12 (Tw) for all w = u −1 w1t n 1 v with n ∈ N and u, v ∈ BΓ 1 , and the result follows.Thus B1, B2, and B3 hold for all cells Γi with 4 ≤ i ≤ 12.Moreover, these cells admit cell factorisations, and the leading matrices are easily computed directly, verifying that (5.1) holds.For the cells Γi with 4 ≤ i ≤ 9 the sign in (5.1) is easily computed (since the associated representations are 1-dimensional).In the remaining cases we have the + sign except for the case π12 with (r1, r2) ∈ A1 ∪ A2 ∪ A1,2 in which case we have the − sign.
It is thus clear, from (5.1), that B4 holds.To verify B5 for the cell Γ = Γi we note that if w = u −1 wΓv then This completes the analysis for the finite cells Γi with 4 ≤ i ≤ 12.
Next we note that B2 and B3, with aπ Γ = a, hold by an easy direct calculation (note that Γ ≥LR = Γ4 ∪ Γ13 is finite).Moreover the leading matrices are computed directly as and hence B4 holds.Let d1, d2 ∈ Γ13 be the elements d1 = 0 and d2 = 1 (these turn out to be the Duflo involutions; see Theorem 7.8).Then the formulae above give and hence B5 holds.Once again, Theorem 4.6 yields that πΓ satisfies B1.Moreover B2 and B3 hold by direct calculation with aπ Γ = a, and the leading matrices are computed as and B4 follows.Let d1 = 1, d2 = 2, and d3 = 01 (again, these turn out to be the Duflo involutions; see Theorem 7.8).Then the formulae above give and hence B5 holds, completing the proof.

Infinite cells
In this section we construct balanced representations for the infinite cells Γi with i ∈ {0, 1, 2, 3} for all choices of parameters.The results of this section, along with Theorem 5.1, give the following: Theorem 6.1.For each choice of parameters (a, b, c) ∈ Z 3 >0 there exists a balanced system of cell representations (πΓ) Γ∈Λ for H with bounds aπ Γ = ã(Γ).
Proof.It follows from Theorems 1.7 and 6.1 that Lusztig's a-function is given by Table 2. Conjectures P4, P9, P10, P11 and P12 are then easily checked using the explicit values of the a-function.In fact, due to the logical dependencies amongst the conjectures established in [15,Chapter 14] it is sufficient to prove P4, P10, and P12, which are obvious from the explicit values of the a-function and the explicit decomposition of W into right cells given in Figure 5. Then P10 ⇒ P9 and P4 + P9 + P10 ⇒ P11.
Of course it remains to exhibit balanced systems for the infinite cells.We undertake this rather intricate task in the present section.Let us begin by noting the following immediate consequence of Theorem 4.6.Corollary 6.3.Let i ∈ {0, 1, 2, 3}.The representation πi satisfies B1 for the cell Γi.

The lowest two-sided cell
Suppose first that c = b.It is sufficient to consider the case c < b, for if c > b one can apply the diagram automorphism σ.In the case c < b the lowest two-sided cell Γ0 admits a cell factorisation where B0 = {e, 0, 01, 012, 010, 0102, 01021, 010210}, and if w = u −1 w0t λ v is written in this form we define uw = u, vw = v, and τw = λ.
Since B0 is a fundamental domain for the action of Q on W the set B0 = {ξ0 ⊗ Xu | u ∈ B0} is a basis of M0.The proof of the following theorem is very similar to [11,Section 6] with only some minor adjustments, and so we will only sketch the argument.Proof.We have already verified B1 in Corollary 6.3.To verify B2 we note that deg Q(p) ≤ max{2a + 2b, 2a + 2c} for all positively folded alcove paths, and so for c < b we have deg Q(p) ≤ 2a + 2b (see [11,Lemma 6.2]).Thus B2 follows from Theorem 4.1.
Axiom B3 is verified as in [11,Theorem 6.6], with one additional ingredient: If deg(Q(p)) = 2a + 2b then necessarily p has no folds on type 0-walls (for otherwise the degree is bounded by 2a + b + c < 2a + 2b).The only simple hyperplane direction available in the "box" B0 is a type 0-wall, and thus if p is a maximal path of type u −1 w0t λ v with u, v ∈ B0 then by the above observation there is no fold on this wall in the final v-part of the path (see [11,Remark 6.4]).With this observation in hand the proof of [11,Theorem 6.6] applies verbatim, including the calculation of the leading matrices.Linear independence of the Schur functions gives B4, and to verify B5 we note that if w ∈ Γ0 then and the proof is complete.Now suppose that b = c.In this case we will work in the extended affine Weyl group W and the extended affine Hecke algebra H. See Remark 4.2 for the definition of the principal series representation (π0, M0) in this case.
Let B 1/2 = {e, 0, 01, 012} be the "half box".Each element w ∈ W of the (non-extended) affine Weyl group can be written uniquely as We will work with the basis of the module M0.Then, as in Theorem 4.1, with respect to this basis we have where, if w = t λ v as in (6.1), then wt(w) = λ and θ(w) = v.
We have the following generalised cell factorisation: Each w ∈ Γ0 has a unique expression as If w ∈ Γ0 is written in the form (6.3) then we define uw = u, vw = v, and τw = λ.Proof.The proof is again very similar to [11,Theorem 6.6].The choice of "box" B ′ 0 = B 1/2 ∪ B 1/2 σ again implies that if p is a maximal path of type u −1 w0t λ v with u, v ∈ B ′ 0 then there are no folds in the final v-part of the path.Moreover, a slight generalisation of [11,Theorem 3.4] gives X wt(p) for λ ∈ P + , and the proof of [11,Theorem 6.6] now applies verbatim.

Slices of the induced representations and folding tables
In the following sections we analyse the remaining infinite cells Γi with i ∈ {1, 2, 3}.The basic idea is to use the combinatorial description of the matrix entries from Theorem 4.3 to show that the representation πi is balanced for the cell Γi.Thus we are primarily interested in the i-folded alcove paths that attain the maximal value of deg(Qi(p)), as these are the terms that contribute to the leading matrices.However the situation is complicated by the large number of distinct parameter regimes for the cells Γi as the i-folded alcove paths that attain the maximal value of deg(Qi(p)) vary with the parameter regimes.
Therefore it is desirable to be able to work with all parameter regimes simultaneously.To achieve this we work in the generic Hecke algebra Hg.In this setting the degree of the multivariate polynomial Qi(p) (see (4.1)) is too crude for our purposes, and so we introduce a more refined statistic, which we call the exponent of Qi(p), defined as follows.Firstly, if x = (x, y, z) ∈ Z 3 then the exponent of the monomial q x := q x 1 q y 2 q z 0 is exp(q x ) = (x, y, z) ∈ Z 3 .Let denote the partial order on Z 3 with (x ′ , y ′ , z ′ ) (x, y, z) if and only if x − x ′ ≥ 0, y − y ′ ≥ 0, and z − z ′ ≥ 0. Definition 6.6.Let p be an i-folded alcove path.Then Qi(p) has a unique monomial with exponent maximal with respect to .We denote this maximal exponent by exp(Qi(p)).Explicitly, Note that if exp(Qi(p)) = (x, y, z) then on specialising q0 → q c , q1 → q a , q2 → q b we have Proof.We may write each u ∈ B as u = τ k i u ′ for some k ∈ Z and u ′ ∈ B ′ .We claim that Consider the case i = 2, 3. Then by (2.1) we have Xu = X kω 1 X u ′ , and the result follows since ξi this follows from the fact that τ1 preserves the orientation of all hyperplanes except for the hyperplanes in the α1 parallelism class, and that this class is not encountered in U1).Since Thus the change of basis matrix from the B basis to the B ′ basis is a monomial matrix with entries in Z[ζ], and the lemma follows.
Thus we can define E(πi) = E(πi; B) for any fundamental domain B.
We will show below (in the course of the proof of Theorem 6.18) that the elements of E(πi) are bounded above in each component -we will assume this fact for the moment.Let M(πi) = {maximal elements of the partially ordered set (E(πi), )}.Definition 6.9.Let B be a fundamental domain for the action of τi on Ui.For x = (x, y, z) ∈ Z 3 the x-slice of πi(Tw; B) is the matrix c x π i (w; B) whose (u, v) th entry is the coefficient of The following key theorem shows that the slices c x π i (w; B) with x ∈ M(πi) are sufficient to compute leading matrices in all parameter ranges.Theorem 6.10.Let (a, b, c) be a fixed choice of parameters, and suppose that property B2 holds for πi(•, B) with bound aπ i .Suppose that xa + yb + zc ≤ aπ i for all (x, y, z) ∈ M(πi).Then where the sum is over those x = (x, y, z) ∈ M(πi) with xa + yb + zc = aπ i .
Proof.By Theorem 4.3 the entry [cπ i (w; B)]u,v of the leading matrix cπ i (w; B) is given as a sum over paths p ∈ Pi( w; u) with deg(Qi(p)) = aπ i .Thus it suffices to show that if exp(Qi(p)) / ∈ M(πi) then, after specialising, deg(Qi(p)) < aπ i .
Thus our approach in the following sections is to compute M(πi) and the slices corresponding to these maximal exponents.In fact, the cell Γ2 turns out to be the most complicated, in part due the intricate equal parameter regime.Thus we give complete details for Γ2, and we will only state the results for the easier cells Γ1 and Γ3.Remark 6.11.The hypothesis xa + yb + zc ≤ aπ i for all (x, y, z) ∈ M(πi) in Theorem 6.10 is required because it is a priori possible that there exists and i-folded alcove path p with exp(Qi(p)) = (x, y, z) and xa + yb + zc > aπ i .The leading contributions from all such paths in Theorem 4.3 must cancel (after specialisation) for otherwise B2 is violated.While indeed cancellations can (and do) occur, it turns out that the condition xa + yb + zc > aπ i in fact never occurs.We will see this in the course of the calculations in the following sections.
We will use "folding tables" to analyse i-folded alcove paths (i ∈ {1, 2, 3}).We give a brief outline below, and we refer to [11, §7.2] for further details.Let v ∈ W i 0 and x ∈ W with reduced expression x = si 1 . . .si n .We denote by p( x, v) ∈ Pi( x, v) the unique i-folded alcove path of type x starting at v with no folds.Of course p( x, v) may still have bounces, because i-folded alcove paths are required to say in the strip Ui.Nonetheless, we refer to p( x, v) as the straight path of type x starting at v. Let I − ( x, v) = {k ∈ {1, . . ., n} | p( x, v) makes a negative crossing at the kth step} ) makes a positive crossing at the kth step} Note that I − ∪ I + ∪ I * = {1, . . ., n}.We define a function as follows.For k ∈ I − ( x, v) let p k be the i-folded alcove path obtained from the straight path p0 = p( x, v) by folding at the kth step (note that after performing this fold one may need to include bounces at places where the folded path p k attempts to exit the strip Ui).Let ) and p k agree after the kth step.
The i-folding table of x is the 4 × ℓ(x) array Fi( x) with (j, k) th entry equal to f j,k ( x).Remark 6.13.If y is a prefix of y then Fi( y) is the subarray of F( x) consisting of the first ℓ(y) columns.Also note that of course any other enumeration of W i 0 can be used in the definition.
Example 6.14.Let ti = tω i for i = 1, 2. The 2-folding tables for the elements t1 = 0121 and t2 = 010212 are shown in Table 3, where the rows are indexed by W 2 0 in the order e, 1, 12, 121, and the t2 table excludes the final column.Note that we have appended a 0-row and 0-column to the table for convenience.The 0-row is called the "header" of the table.The folding tables for the elements v ∈ B0 are also given by these tables, because the reduced expressions for the elements of B0 are the strict prefixes of t2, along with 010210 (which is given in the t2 table by removing the penultimate column), along with 012 (which is a subexpression of t1).The folding table Fi( w) can be used to compute Qi(p) for all p ∈ Pi( w, u) with u ∈ W i 0 as follows (see [11] for more details).We begin an excursion through the table Fi( w) starting at the first cell on row ℓ(u) + 1 (the row corresponding to u ∈ W i 0 ) with a counter Z starting at Z = 1.At each step we move to a cell strictly to the right of the current cell and modify Z according to the following rules.Suppose we are currently at the N th cell of row r, and this cell contains the symbol x ∈ {−, * , 1, 2, 3, 4}.Let j ∈ {0, 1, 2} denote the header entry of the N th column.
1) If x = − then we move to the (N + 1) st cell of row r and Z remains unchanged.
2) If x = * then we move to the (N + 1) st cell of row r and replace Z by Z ′ where if i = 3 and j = 0.
3) If x = k ∈ {1, 2, 3, 4} then we have two options: (a) we can move to the (N + 1) st cell of row r and leave Z unchanged, or (b) we can move to the (N + 1) st cell of row k and replace Z by Z × (qj − q −1 j ).The set of all such excursions through the table is naturally in bijection with the set of i-folded alcove paths Pi( w, u), and the final value of the counter Z at the end of the excursion is Qi(p).Moreover, the final exiting row gives the value of θ i (p).It may help to note that cases (1), ( 2), (3)(a) and (3)(b) correspond to a positive crossing, bounce, negative crossing, and fold respectively.
Suppose that w = t m ω 1 • t n ω 2 where m, n ∈ N, and let u ∈ W i 0 .Then Fi( w) is the concatenation of m copies of the i-folding table of tω 1 followed by n copies of the i-folding table of tω 2 (for this observation to hold it is important that tω 1 and tω 2 are translations).Thus the process described above may be regarded as "m passes through the tω 1 table, followed by n passes through the tω 2 table" in an obvious way.

The cell Γ 2
The cell Γ2 is stable on each of the following regions: To explain this notation, notice that R1 and R2 are open regions, and R1,2 is the boarder between these regions.Moreover, R 1,1 ′ is the boarder between the regions R1 and R 1 ′ , where R j ′ denotes the σ-dual of Rj .Similarly R 2,2 ′ if the boarder between the regions R2 and R 2 ′ .We begin by describing the cell Γ2 in each of the above regions and setting up notation for the statement of the main theorems.The cases (r1, r2) ∈ Rj with j = 1, 2 are "generic", and admit cell factorisations where and Bj = (e, 2, 21, 210) if j = 1 (e, 1, 10, 101) if j = 2.
For each j = 1, 2 let zj ∈ Bj be such that B ′ j = {z −1 j u | u ∈ Bj} is a fundamental domain for the action of τ2 on U2 with z −1 j e on the negative side of each hyperplane separating z −1 j e from z −1 j u with u ∈ Bj .Specifically, z1 = 21 and z2 = 1.Define an ordered basis Bj of M2 by Bj = (ξ2 The proof of the above theorems will be given towards the end of this section.We first analyse the slices of the matrices π2(Tw; W 2 0 ).This in turn requires, by Theorem 4.3, a rather detailed analysis of 2-folded alcove paths.Each w ∈ W can be written uniquely as w = ut m 2 t n 1 v with u ∈ W0, v ∈ B0, and m, n ∈ Z (where we write t1 = tω 1 and t2 = tω 2 ) and necessarily ℓ(w) = ℓ(u) + nℓ(t1) + mℓ(t2) + ℓ(v).We choose and fix the reduced expressions for each u ∈ W0, v ∈ B0, and t1, t2, which are lexicographically minimal.Thus w0 = 1212, t1 = 0121, and t2 = 010212, and the expressions v for v ∈ B0 are the prefixes of b0 = 010210, along with the element 012 (see Example 6.14).These choices give a distinguished reduced expression for each w ∈ W , namely with the reduced expressions for each component chosen as above.We fix this choice throughout this section.If p is a path of type To efficiently record 2-folded alcove paths we will use the notation î to denote an i-fold, and ǐ to denote an i-bounce.Thus, for example, p = 2 101 21 is a 2-folded alcove path whose second and fourth steps are 1-folds, and whose third step is a 0-bounce (for example, this is a valid 2-folded alcove path starting at 1).
The main theorems of this section will follow from the following combinatorial theorem.
The folding tables for the elements t1 = 0121 and t2 = 010212 are shown in Table 3.The following observation will be used frequently: If a pass of either the t1 or t2 table is completed on a row containing at least one * , and if no folds are made in this pass, then where p ′ is the path obtained from p 0 by removing this copy of t1 or t2.Thus such paths necessarily have strictly dominated exponents, and can therefore be discarded in the following analysis.
The claim follows from the following four points.
3) Suppose that start(p 0 ) = 12.We first claim that if ℓ > 0 then exp(Q2(p)) ≺ x for some x ∈ E. By the observation made in (6.6) it suffices to assume that if ℓ > 0 then a fold is made in the first pass of the t2 table.Thus the first part of the path is one of the following: Thus we may assume that ℓ = 0, and so p 0 has type t k 1 • v for some k ≥ 0 and some v ∈ B0.If k > 0 then by the observation above we may assume that a fold is made in the first pass of the t1 table.Thus the first part of the path is necessarily 01 21, which has exponent (1, 0, −1) and exits on row 1 of the folding table.Any further t1 factors will have no effect on the exponent, and the final v factor can have contribution at most (0, 0, 0), and this occurs if and only if v ∈ {e, 0, 01, 012}.Thus the paths Tab.5: Optimal p 0 parts (starting at 21) all have exponent precisely (1, 0, −1), and when composed with an optimal p0 path we have exp(Q2(p)) = (2, 0, −1) ∈ E. These paths are listed on rows 2, 9, 16 and 25 of Table 5.
4) Suppose that start(p 0 ) = 121.A very similar argument to the case start(p 0 ) = 12 shows that if ℓ > 0 then exp(Q2(p)) ≺ x for some x ∈ E. Thus we may assume that ℓ = 0. Thus p 0 has type t k 1 • v for some k ≥ 0 and v ∈ B0.Since the 4th row of the 2-folding table of t1 contains no bounces, one may begin by making any number k1 ≤ k passes through the folding table with no folds.
The theorem now follows by combining Tables 4 and 5.
It is then clear that B4 holds (by linear independence of Schur characters), and the formula The case (r1, r2) ∈ R2 is very similar -one first identifies the paths with exponent (0, 1, 0), and then rewrites these paths in the cell factorisation u −1 w2t N 2 v with u, v ∈ B2.Next one adjusts the start of the paths according to the fundamental domain B ′ 2 = z −1 2 B2 = {e, 1, 0, 01} (paths starting at 121 now start at 0, and those starting at 12 now start at 01).Since Q2(p) = q2 − q −1 2 has leading term +q2 for all such paths we finally obtain +sN (ζ).
We omit the details for the "generic" cases in Theorem 6.16 which involve the extended affine Weyl group -the general approach is similar to the above.Thus consider the most intricate case of all -the equal parameter case of Theorem 6.17.
In this case, quite remarkably, the maximum value of xa + yb + zc is a, attained at all x ∈ M(πi).One checks directly from Theorem 6.18 that if p is of type w with exp(Q2(p)) ∈ M(π2) then either w ∈ Γ2(R6) or p 0 is on row 32 * or 33 * .However, as explained in Remark 6.19, the paths on rows 32 * and 33 * may be discarded (as their leading terms cancel one another).We now compute the sum of all slices: x∈M(π 2 ) c x π 2 (w; W 2 0 ) with respect to the standard basis.A rather miraculous calculation (with many cancellations occurring) shows that this sum of slices is precisely as stated in Theorem 6.17.This computation can be read immediately off the tables in Theorem 6.18, because we work in the standard basis and thus no modifications or conversions are required; however one must be rather careful with signs.For example, let us compute the sum of slices for w = w k 3,3 .We look through the tables to find all paths of type w k 3,3 = 21  For each j = 1, 2, 3 let zj ∈ Bj be such that {z −1 j u | u ∈ Bj } is a fundamental domain for the action of τ1 on U1 with z −1 j e on the negative side of each hyperplane separating z −1 j e from z −1 j u with u ∈ Bj.Specifically, zj = 2, 12, e in the cases j = 1, 2, 3. Define an ordered basis Bj of M1 by Bj = (ξ1 ⊗ X z −1 j u | u ∈ Bj ).Thus (ξ1 ⊗ X2, ξ1 ⊗ Xe, ξ1 ⊗ X20, ξ1 ⊗ X0) if j = 1 (ξ1 ⊗ X21, ξ1 ⊗ X2, ξ1 ⊗ X20, ξ1 ⊗ Xe) if j = 2 (ξ1 ⊗ Xe, ξ1 ⊗ X0, ξ1 ⊗ X01, ξ1 ⊗ X010) if j = 3.
Next one classifies the paths p for which exp(Q1(p)) = x for some x ∈ M(π1).Theorem 6.21 now follows as in the Γ2 case.
b) v k−1 is on the positive side of the hyperplane separating v k−1 and v k−1 si k .We note that condition 2)(a) can only occur if v k−1 and v k−1 si k are separated by either H α ′ i ,0 or H α ′ i ,1 .The end of the i-folded alcove path p = (v0, . . ., v ℓ ) is end(p) = v ℓ .Let Pi( w, v) = {all i-folded alcove paths of type w starting at v}. Less formally, i-folded alcove paths are made up of the following symbols, where x ∈ Ui and s ∈ S: When the alcoves x and xs both belong to Ui When xs lies outside of Ui

A
positive weight function L on W is completly determined by its values L(s1) = a, L(s2) = b and L(s0) = c on the set S of generators.If the triplet (a, b, c) ∈ N>0 admits a common divisor d then the algebra H defined with respect to (a, b, c) is easily seen to be isomorphic to the one defined with respect to (a/d, b/d, c/d).Therefore the Hecke algebra H defined with respect to (a, b, c) only depends on the ratios b/a and c/a, and hence also the decomposition into cells depends only on these ratios.Thus we set In this paper, many notions will depend on the choice of parameters and, as far as Kazhdan-Lusztig theory is concerned, it is equivalent to fix a weight function L, a triplet (a, b, c) ∈ N 3 or a pair (r1, r2) ∈ Q 2 .Given D ⊂ Q 2 >0 , we write • (a, b, c) ∈ D for (a, b, c) ∈ N 3 to mean (b/a, c/a) ∈ D; 1), and P5 = (3, 1) (points).The set Q 2 >0 is so partitioned into 55 regions (20 open subsets, 27 open intervals, and 8 points).Let R be the set of all such regions and let R
Twσ for aw ∈ Rg defines an involutive automorphism of Hg.Suppose that (π, M) be a right Hg-module over a ring S = Rg[ζ ±1 1 , . . ., ζ ±1 n ], where ζ1, . . ., ζn are invertible pairwise commuting indeterminants.The diagram automorphism σ of (W, S) gives rise to a "σ-dual" representation (π σ , M) of Hg by π σ (h) = π(h σ ) σ , where the outer σ is the homomorphism of End S (M) induced by σ.This construction will allow us to concentrate on the case c ≤ b for much of what follows, with the c > b case dealt with by replacing each representation with its σ-dual.

. 4 ) 6 . 7 .
deg(Qi(p)) = xa + yb + zc.(6Definition Let B be a fundamental domain for the action of τi on Ui.Let E(πi; B) = {x ∈ Z 3 | q x appears with nonzero coefficient in some matrix entry of πi(Tw; B) for some w ∈ W }, where B = {ξi ⊗ Xu | u ∈ B} is the basis of Mi associated to B. Lemma 6.8.If B and B ′ are fundamental domains for the action of τi on Ui then E(πi; B) = E(πi; B ′ ).

Table 4 (
the * in rows 17 and 18 will be explained later).Tab.4:Optimalp 0 partsTo establish the claim we note that the paths listed obviously have the stated exponents.One now constructs all paths p0 of type u with u ∈ W0 starting at some u0 ∈ {e, 1, 12, 121}, and verifies the claim directly.For example, the paths starting and ending at e are precisely the following: e, 1, 2, 12 , 21 , 1 21, 121 , 212 , 121 2, 1212 4 are as follows: 10, and t k 1 1 • 01 2. These paths are listed in rows 10/38, 17/41, 18/42, 32 * /46, and 26/45 of Table5(the * will be explained later in Remark 6.19, and again it is convenient to split the k1 = 0 and k1 > 0 cases).If k1 < k then we assume that the (k1 + 1)-st pass of the t1 table has a fold.The possibilities on this pass are • t k ω 1 • 01.These paths are listed in Table6.