Jérémie Guilhot

James Parkinson

A proof of Lusztig's conjectures for affine type G 2 with arbitrary parameters

We prove Lusztig's conjectures P1-P15 for the affine Weyl group of type G2 for all choices of parameters. Our approach to compute Lusztig's a-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the a-function and we explicitly construct such a system in type G2 for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type G2, allowing us to prove P1 and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types G2 and A1, along with some explicit computations for the finite cells.

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 [START_REF] Kazhdan | Representations of Coxeter groups and Hecke algebras[END_REF] Kazhdan and Lusztig introduced the theory in the equal parameter case, and in [START_REF] Lusztig | Left cells in Weyl groups[END_REF] 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 [START_REF] Kazhdan | Schubert varieties and Poincaré duality[END_REF], and the case of arbitrary Coxeter groups was settled only very recently by Elias and Williamson in [START_REF] Elias | The Hodge theory of Soergel bimodules[END_REF]. 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 [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]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 using the above mentioned geometric interpretation. In the case of more general parameters P1-P15 are only known to hold in the following situations:

• the quasisplit case where a geometric interpretation is also available [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]Chapter 16];

• finite dihedral type [START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF] and infinite dihedral type [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]Chapter 17] for arbitrary parameters;

• type Bn in the "asymptotic" case [START_REF] Bonnafé | Left cells in type Bn with unequal parameters[END_REF][START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF];

• F4 for arbitrary parameters [START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF]. Note that the only example of an infinite Weyl group where P1-P15 are known for arbitrary parameters is the infinite dihedral group, where the proof proceeds by explicit computations. In this paper we prove Lusztig's conjectures in type G2 for arbitrary parameters. This provides the very first example of an affine Weyl group of rank greater than 1 in which the conjectures have been proved. Furthermore, our methods provide a theoretical framework that one may hope to apply to other types of affine Weyl groups. For instance, the approach outlined in this paper could be applied to the case B2.

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. Our approach to this problem is based on the notion of a balanced system of cell representations inspired by the work of Geck [START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF] in the finite case. This notion can be defined for an arbitrary Coxeter group (W, S) with weight function L : W → N 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 let Γ ∈ Λ. We say that a representation π is Γ-balanced if it admits a basis such that (1) the maximal degree of the coefficients that appear in the matrix π(Tw) is bounded by a constant aπ (here Tw denotes the standard basis of H) and [START_REF] Bonnafé | Semicontinuity properties of Kazhdan-Lusztig cells[END_REF] this bound is attained if and only if w ∈ Γ. A balanced systems of cell representations is a family (πΓ) Γ∈Λ of Γ-balanced representations that satisfy some extra axioms (see Section 2). We show that the existence of such a system is sufficient to compute Lusztig's a-function, and as a byproduct we obtain an explicit construction of Lusztig's asymptotic algebra J [25, Chapter 18].

In the case of G2, we construct a balanced system of cell representations for each parameter regime. Our starting point is the partition of W into Kazhdan-Lusztig cells that was proved by the first author in [START_REF] Guilhot | Kazhdan-Lusztig cells in affine Weyl groups of rank 2[END_REF]. It turns out that

The Kazhdan-Lusztig basis

The Hecke algebra H associated to (W, S, L) is the algebra over R with basis {Tw | w ∈ W } and multiplication given by TwTs = Tws if ℓ(ws) = ℓ(w) + 1 Tws + (q L(s) -q -L(s) )Tw if ℓ(ws) = ℓ(w) -1.

The basis {Tw | w ∈ W } is called the standard basis of H. We set qs = q L(s) for s ∈ S.

The involution ¯on R which sends q to q -1 can be extended to an involution on H by setting

w∈W awTw = w∈W aw T -1 w -1 .
In [START_REF] Kazhdan | Representations of Coxeter groups and Hecke algebras[END_REF], 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 ∈ q -1 Z[q -1 ].

This basis is called the Kazhdan-Lusztig basis 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 ). We note that the Kazhdan-Lusztig polynomials, and hence the elements Cw, depend on the the weight function L. For example, in the dihedral group I2(2m) with m ≥ 2, L(s1) = a, and L(s2) = b, we have

Ps 1 ,s 1 s 2 s 1 =      q -(b-a) + q -a-b if a < b q -2a if a = b q -(a+b) -q -(a-b) if a > b.
In particular, this example shows that the positivity properties enjoyed by Py,z in the equal parameter case (that is, L = ℓ) do not transfer across to the general case.

Let x, y ∈ W . We denote by hx,y,z ∈ R the structure constants associated to the Kazhdan-Lusztig basis:

CxCy = z∈W hx,y,zCz.

Definition 1.1 ([25, Chapter 13]). The Lusztig a-function is the function a : W → N defined by a(z) := min{n ∈ N | q -n hx,y,z ∈ Z[q -1 ] for all x, y ∈ W }.

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 the underlying finite Weyl group W0 (see [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]). 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 1.2. 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 [25, 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.

We denote by Λ the set of all two-sided cells (note that Λ depends on the choice of parameters). 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.

Example 1.3. The following table records the decomposition of the dihedral group W = I2(m) = s1, s2 into two-sided cells for all choices of weight function L(s1) = a and L(s2) = b (up to duality). Lusztig's conjectures are known to hold for dihedral groups. In particular the a-function is constant on two-sided cells, and we list these values below. This example turns out to be particularly useful -for all affine rank 3 (dimension 2) Weyl groups every two-sided cell intersects a finite parabolic subgroup (hence a dihedral group), and so assuming the Lusztig conjectures P4 and P12 the table below gives conjectural values of the a-function on all cells. These 'conjectures' become 'theorems' for type G2 due to the results of this paper.

W two-sided cells values of the a-function I 2 (2), a ≥ b {e}, {s 1 }, {s 2 }, {w 0 } 0, a, b, a + b I 2 (m), a = b, m ≥ 2 {e}, W \{e, w 0 }, {w 0 } 0, a, ma I 2 (2m), a > b, m ≥ 2 {e}, W \{e, s 2 , w 0 s 2 , w 0 }, {s 2 }, {w 0 s 2 }, {w 0 } 0, a, b, ma -(m -1)b, ma + mb I 2 (∞), a = b {e}, W \{e} 0, a I 2 (∞), a > b {e}, W \{e, s 2 }, {s 2 } 0, a, b

Tab. 1: Cells and the a-function for dihedral groups

Lusztig conjectures

Define ∆ : W → N and nz ∈ R\{0} by the relation Pe,z = nzq -∆(z) + strictly smaller powers of q.

This is well defined because Px,y ∈ q -1 Z[q -1 ] for all x, y ∈ W . Let

D = {w ∈ W | ∆(w) = a(w)}.
The elements of D are called Duflo elements (or, somewhat prematurely, Duflo involutions; see P6 below).

In [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]Chapter 13], Lusztig has formulated the following 15 conjectures, now known as P1-P15.

P1. For any z ∈ W we have a(z) ≤ ∆(z).

P2. If d ∈ D and x, y ∈ W satisfy γ x,y,d = 0, then x = y -1 .

P3. If y ∈ W then there exists a unique d ∈ D such that γ y -1 ,y,d = 0.

P4. If z ′ ≤LR z then a(z ′ ) ≥ a(z). In particular the a-function is constant on two-sided cells. P7. For any x, y, z ∈ W , we have γx,y,z = γy,z,x.

P8. Let x, y, z ∈ W be such that γx,y,z = 0. Then x ∼L y -1 , y ∼L z -1 , and z ∼L x -1 .

P9. If z ′ ≤L z and a(z ′ ) = a(z), then z ′ ∼L z.

P10. If z ′ ≤R z and a(z ′ ) = a(z), then z ′ ∼R z.

P11. If z ′ ≤LR z and a(z ′ ) = a(z), then z ′ ∼LR z.

P12. If I ⊆ S then the a-function of WI is the restriction of the a-function of W .

P13. Each right cell Γ of W contains a unique element d ∈ D. We have γ x -1 ,x,d = 0 for all x ∈ Γ.

P14. For each z ∈ W we have z ∼LR z -1 .

P15. If x, x ′ , y, w ∈ W are such that a(w) = a(y) then

y ′ ∈W h w,x ′ ,y ′ ⊗ h x,y ′ ,y = y ′ ∈W h y ′ ,x ′ ,y ⊗ h x,w,y ′ in R ⊗ Z R.
As noted in the introduction, these conjectures have been established in the following cases: (1) when W is a Weyl group or an affine Weyl group with equal parameters (see [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF] and the updated version available on ArXiv), (2) in the "quasisplit case" [START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]Chapter 16] where W is obtained by "twisting" a larger Coxeter group W , and L is the restriction of the length function on W to W , (3) when W is a dihedral group (finite or infinite) for all choices of parameters (see [START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF][START_REF] Lusztig | Hecke algebras with unequal parameters[END_REF]), (4) when W = Bn in the "asymptotic case" (see [START_REF] Bonnafé | Left cells in type Bn with unequal parameters[END_REF][START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF]), [START_REF] Braverman | Remarks on the asymptotic Hecke algebra[END_REF] when W = F4 for any choices of parameters (see [START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF]). We note that in case (1) and (2) the proof relies on deep results including the positivity of the Kazhdan-Lusztig polynomials in equal parameters. This approach cannot work for the general case, since we have seen that the positivity no longer holds in this case.

In this paper we prove all conjectures P1-P15 for G2 for all choices of parameters. Our approach extends naturally to all rank 2 affine Weyl groups, although the details in the three parameter case B2 becomes rather involved due to the large number of distinct regimes of cell decompositions.

Systems of balanced representations and Lusztig a-function

In this section we define a balanced system of cell representations, inspired by the work of Geck [START_REF] Geck | Constructible characters, leading coefficients and left cells for finite coxeter groups with unequal parameters[END_REF][START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF] in the finite case. We show that the existence of such a system, plus one additional axiom, is sufficient for the computation of Lusztig's a-function. This gives us our primary strategy for resolving Lusztig's conjectures in type G2.

System of balanced representation

We now introduce the central notion of this paper, based on the work of Geck [START_REF] Geck | Constructible characters, leading coefficients and left cells for finite coxeter groups with unequal parameters[END_REF][START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF] in the finite case.

Definition 2.1. We say that H admits a balanced system of cell representations if for each Γ ∈ Λ there exists a representation (πΓ, VΓ) defined over an R-polynomial ring RΓ (where we could have RΓ = R) and a basis (ej) of VΓ such that the following holds, where [πΓ(Cw)]i,j denotes the matrix entries of πΓ(Cw) with respect to the basis (ej):

B1. If w / ∈ Γ ≥ LR then πΓ(Cw) = 0.
B2. There exists bounds aπ Γ ∈ N such that deg[πΓ(Cw)]i,j ≤ aπ Γ for all w ∈ W and all 1 ≤ i, j ≤ dim(VΓ).

B3. We have maxi,j deg[πΓ(Cw)]i,j = aπ Γ if and only w ∈ Γ. We define the leading matrices by

cπ Γ ,w = sp | q -1 =0 q -aπ Γ [πΓ(Cw)] .
B4. The set {cπ Γ ,w | w ∈ Γ} of leading matrices is free over Z.

B5. If Γ ′ ≤LR Γ then aπ Γ ′ ≥ aπ Γ .
The natural numbers aπ Γ are called the bounds of the balanced system of cell representations. The main approach of this paper hinges on the construction of a balanced system of cell representations for the Hecke algebra of type G2 in each parameter regime.

Note that B1 above does not depend on the special choice of basis. 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 1.2 are cell representations. To see this, let Υ be a right cell, and let HΥ be as in Section 1.2. If Cw acts nontrivially on HΥ then there exist u, v ∈ Υ such that Cu • Cw = z hu,w,zCz with hu,w,v = 0. Thus v ≤LR w.

We say that a representation (π, V ) is Γ-balanced for the two-sided cell Γ if V admits a basis such that B2, B3 and B4 hold. 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. With this notation ρ ∅ is the sign representation and ρS is the trivial representation. It is easy to see that (1) ρ ∅ is Γe-balanced where Γe is the two-sided cell that contains precisely the identity and (2) ρS cannot be balanced for any two-sided cell Γ.

Consider the representation ρI where I = {s0, s2} . We will see in Section 4 that Γ5 := {s0s2s0} is a two-sided cell in W when a/b > 2. For w ∈ W we have ρI (w) = q bℓ 2 (w) (-q) -aℓ 1 (w) where ℓ2(w) is the number of s2 and s0 generators in any reduced expression of w and ℓ1(w) is the number of s1 generators. Saying that the representation ρI is Γ5-balanced for a/b > 2 means that bℓ2(w) -aℓ1(w) ≤ 3b for all w and that there is equality if and only if w = s0s2s0. This can be done by studying reduced expressions in W , and we will see another proof using Kazhdan-Lusztig theory in Section 5.

Proceeding as above, one can show that the representation ρI where I = {s1} is Γ7-balanced whenever a/b < 1. Once again we will see in Section 4 that Γ7 := {s1} is a two-sided cell in W for this parameter range.

Computing the a-function

In this section we show that axioms B1-B5, along with an additional axiom B4 ′ introduced below, are sufficient to show that Lusztig's a-function is constant on two-sided cells, and moreover we are able to compute the value of the a-function in terms of the bounds (aπ Γ ) Γ∈Λ from B2.

Let (πΓ) Γ∈Λ be a balanced system of cell representations for H with bounds aπ Γ for all Γ ∈ Λ. We have CxCy = Γ∈Λ z∈Γ hx,y,zCz.

(2.1)

Proposition 2.3. Let x, y ∈ W and w ∈ Γ where Γ ∈ Λ. We have deg(hx,y,w) ≤ aπ Γ .

Proof. We proceed by downwards induction. Let Γ ∈ Λ and suppose that deg(hx,y,z) ≤ aπ Γ ′ for all z ∈ Γ ′ where Γ ′ >LR Γ. Then applying πΓ to (2.1) using the fact that πΓ is a cell representation gives

πΓ(CxCy) = z∈Γ hx,y,zπΓ(Cz) + Γ ′ ∈Λ, Γ ′ > LR Γ z∈Γ ′ hx,y,zπΓ(Cz).
(2.2)

By B2 the degree of the matrix coefficients of πΓ(CxCy) = πΓ(Cx)πΓ(Cy) is bounded by 2aπ Γ . By the induction hypothesis and properties of balanced representations the degree of the matrix coefficients for each term in the double sum on the right is strictly bounded by aπ Γ ′ + aπ Γ ≤ 2aπ Γ . Indeed the maximal degree that can appear in πΓ(Cz) is stricly less than aπ Γ since z / ∈ Γ and the bounds of the balanced system are decreasing with respect to ≤LR. We now show that deg(hx,y,z) ≤ aπ Γ for all z ∈ Γ.

Let m = max{deg(hx,y,z) | z ∈ Γ} and let Z = {z ′ ∈ Γ | deg(h x,y,z ′ ) = m} = ∅.
For z ∈ Z define γx,y,z -1 ∈ Z by hx,y,z = q m γx,y,z -1 + lower terms. By B3 we have πΓ(Cz) = q aπ Γ cπ Γ ,z + lower terms, with cπ Γ ,z = 0 the leading matrix.

Then the right hand side of (2.2) is of the form

q m+aπ Γ z∈Z γx,y,z -1 cπ Γ ,z + lower terms,
and by B4 the expression in the sum (that is, the coefficient of q m+aπ Γ ) is nonzero. By comparing with the lefthand side in (2.2) it follows that m + aπ Γ ≤ 2aπ Γ that is m ≤ aπ Γ as required.

Corollary 2.4. Let Γ ∈ Λ. The subset JΓ of M dim(π Γ ) (RΓ) generated by {cπ Γ ,w | w ∈ Γ} is a Z-subalgebra.
Proof. Let Γ be the two-sided cell an let x, y ∈ Γ. Applying πΓ to CxCy, using B1, and multiplying by q -2aπ Γ we get

[q -aπ Γ πΓ(Cx)][q -aπ Γ πΓ(Cy)] = z∈Γ [q -aπ Γ hx,y,z][q -aπ Γ πΓ(Cz)] + Γ ′ ∈Λ, Γ ′ > LR Γ z∈Γ ′ [q -aπ Γ hx,y,z][q -aπ Γ πΓ(Cz)].
Specialising at q -1 = 0 will annihilate all the terms in the double sum. Indeed for z ∈ Γ ′ with Γ ′ >LR Γ we have deg(hx,y,z) ≤ aπ Γ ′ ≤ aπ Γ and the maximal degree that can appear in πΓ(Cz) is strictly less that aπ Γ . Thus we obtain

cπ Γ ,xcπ Γ ,y = z∈Γ γx,y,z -1 cπ Γ ,z
where γx,y,z -1 ∈ Z is the coefficient of degree aπ Γ of hx,y,z.

We introduce the following additional axiom, where γx,y,z -1 ∈ Z is the coefficient of degree aπ Γ of hx,y,z. B4 ′ . Let Γ ∈ Λ. For each z ∈ Γ, there exists (x, y) ∈ Γ 2 such that γx,y,z -1 = 0.

We can now show that if all axioms B1-B5 and B4 ′ are satisfied, then we can compute Lusztig's a-function in terms of the bounds aπ Γ . Theorem 2.5. Suppose that B1-B5 and B4 ′ are satisfied. Then a(w) = aπ Γ for all w ∈ Γ.

Proof. According to Proposition 2.3, we only need to show that aπ Γ ≥ a(w). To do so it is enough to find x, y ∈ W such that deg(hx,y,w) = aπ Γ or equivalently, to find x, y ∈ W such that γx,y,w = 0. Hence the result using B4 ′ . Corollary 2.6. Assuming B1-B5 and B4 ′ , the ring JΓ is isomorphic to Lusztig's asymptotic algebra associated to Γ.

Proof. The elements γx,y,z -1 are the coefficients of hx,y,z of degree aπ Γ , and are the structure constants of JΓ with respect to the basis {cπ Γ ,w | w ∈ Γ}. Indeed once we know that aπ Γ = a(Γ) we know that the structure constants of JΓ are γ x,y,z -1 .

We note that our construction above parallels Geck's construction from [12, §1.5]. Another construction of the asymptotic algebra has been obtained by Koenig and Xi [19] in the case that H is affine cellular.

Affine Weyl groups, affine Hecke algebras, and alcove walks

We begin this section by recalling basic definitions and terminology concerning affine Weyl groups. While we are primarily interested in G2 in this paper, some of our results apply in arbitrary type, and in any case the general language turns out to be more appropriate for the formulation of our results and their proofs. Next we recall the Bernstein-Lusztig basis of the affine Hecke algebra, and its combinatorial interpretation using alcove walks, following [START_REF] Ram | Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux[END_REF]. Finally we present a combinatorial formula for the Weyl character that will be used in Section 6.

Root systems, Weyl groups, and affine Weyl groups

Let Φ be a reduced, irreducible, finite, crystallographic root system with simple roots α1, . . . , αn in an n-dimensional real vector space V with inner product •, • . Let Φ + be the set of positive roots relative to the simple roots α1, . . . , αn. Let W0 be the Weyl group; the subgroup of GL(V ) generated by the reflections sα, α ∈ Φ, where

sαλ = λ -λ, α α ∨ with α ∨ = 2α/ α, α .
The group W0 is a finite Coxeter group with distinguished generators s1, . . . , sn, where si = sα i . Let w0 be the longest element of W0.

Let F0 denote the union of the hyperplanes Hα = {x ∈ V | x, α = 0} with α ∈ Φ. The closures of the open connected components of V \F are geometric cones, called Weyl chambers. The fundamental Weyl chamber is given by

C0 = {x ∈ V | x, α ≥ 0 for all α ∈ Φ + }.
The Weyl group W0 acts simply transitively on the set of Weyl chambers, and we sometimes use this action to identify the set of Weyl chambers with W0 via w ↔ wC0.

The dual root system is Φ ∨ = {α ∨ | α ∈ Φ} and the coroot lattice of Φ is Q = Z-span of Φ ∨ .
The fundamental coweights of Φ are the vectors ω1, . . . , ωn where ωi, αj = δij . The coweight lattice is P = Zω1 + • • • + Zωn, and the cone of dominant coweights is

P + = P ∩ C0 = Nω1 + • • • + Nωn. Note that Q ⊆ P .
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. We have the following standard facts: 1) W is generated by the orthogonal reflections s α,k in the affine hyperplanes

H α,k = {x ∈ V | x, α = k} with α ∈ Φ and k ∈ Z. Explicitly, s α,k (x) = x -( x, α -k)α ∨ , so that s α,k = t kα ∨ sα.
2) The affine Weyl group W is a Coxeter group with generating set S = {s0, s1, . . . , sn}, where s0 = t ϕ ∨ sϕ, with ϕ the highest root of Φ.

Each hyperplane H α,k with α ∈ Φ + and k ∈ Z divides V into two half spaces, denoted

H + α,k = {x ∈ V | x, α ≥ k} and H - α,k = {x ∈ V | x, α ≤ k}.
This "orientation" of the hyperplanes is called the periodic orientation, since it is invariant under translation by λ ∈ Q.

If w ∈ W we define the final direction θ(w) ∈ W0 and the translation weight wt(w) ∈ Q by the equation w = t wt(w) θ(w).

Here we use the fact that each element w ∈ W can be written uniquely as w = t λ v with v ∈ W0 and λ ∈ Q.

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 V \F are called alcoves. The fundamental alcove is given by

A0 = {x ∈ V | 0 ≤ x, α ≤ 1 for all α ∈ Φ + }.
The hyperplanes bounding A0 are called the walls of A0. Explicitly these walls are Hα i ,0 with i = 1, . . . , n and Hϕ,1.

We say that a face of A0 (that is, a codimension 1 facet) has type si for i = 1, . . . , n 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. Under the identification of alcoves with elements of W , the alcoves w and ws are s-adjacent.

For any sequence w = (si 1 , si 2 , . . . , si ℓ ) of elements of S we have

e ∼s i 1 si 1 ∼s i 2 si 1 si 2 ∼s i 3 • • • ∼s i ℓ si 1 si 2 • • • si ℓ .
In this way, sequences w of elements of S determine alcove walks of type w starting at the fundamental alcove e = A0. We will typically abuse notation and refer to alcove walks of type w = si 1 si 2 • • • si ℓ rather than w = (si 1 , si 2 , . . . , si ℓ ). Thus "the alcove walk of type w = si 1 si 2 • • • si ℓ " is the sequence (v0, v1, . . . , v ℓ ) of alcoves, where v0 = e and

v k = si 1 • • • si k for k = 1, . . . , ℓ.
We are, of course, primarily interested in the case where Φ is a root system of type G2. We outline this example below.

Example 3.1. Let Φ be a root system of type G2 with simple roots α1 and α2. We have P = Q, and the dual root system is

Φ ∨ := ±{α ∨ 1 , α ∨ 2 , α ∨ 1 + α ∨ 2 , α ∨ 1 + 2α ∨ 2 , α ∨ 1 + 3α ∨ 2 , 2α ∨ 1 + 3α ∨ 2 }.
The fundamental alcove is shaded in Figure 1, and the periodic orientation on some hyperplanes is shown.

α ∨ 2 α ∨ 1 ω 1 ω 2 H 2α 1 +3α 2 ,0 H 2α 1 +3α 2 ,1 H 2α 1 +3α 2 ,2 H 2α 1 +3α 2 ,-1 H 2α 1 +3α 2 ,-2 H α 2 ,0 H α 2 ,-2 H α 2 ,2 + - + - + - + - + - -+ -+ -+
Fig. 1: The root system of type G 2

Alcove walks and the Bernstein-Lusztig basis of H

Let W be an affine Weyl group as in the previous section. Let L be a weight function on W . The standard basis of H is well adapted to the Coxeter structure of the affine Weyl group. We now describe another basis of H, due to Bernstein and Lusztig, that is well adapted to the semi-direct product structure of W . Our approach here follows Ram's alcove walk model [START_REF] Ram | Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux[END_REF].

Let w = si 1 si 2 • • • si ℓ be an expression for w ∈ W , and let v ∈ W . A positively folded alcove walk of type w starting at v is a sequence p = (v0, v1, . . . , v ℓ ) with v0, . . . , v ℓ ∈ W such that

1) v0 = v, 2) v k ∈ {v k-1 , v k-1 si k } for each k = 1, . . . , ℓ, and
3) if v k-1 = v k then v k-1 is
(p) = s∈S (qs -q -1 s ) fs(p) .
Let v ∈ W and choose any expression v = si 1 • • • si ℓ (not necessarily reduced). Consider the associated straight alcove walk (v0, v1 . . . , v ℓ ), where v0 = 1 and v k = si 1 • • • si k . Let ε1, . . . , ε ℓ be defined using the periodic orientation on hyperplanes as follows:

ε k = +1 if v k-1 -| + v k (that is, a positive crossing) -1 if v k -| + v k-1 (that is, a negative crossing).
It turns out that the element

Xv = T ε 1 s i 1 . . . T ε ℓ s i ℓ
of H does not depend on the particular expression v = si 1 • • • si ℓ we have chosen (see [START_REF] Görtz | Alcove walks and nearby cycles on affine flag manifolds[END_REF]). If λ ∈ Q we write

X λ = Xt λ .
It follows from the above definitions that

Xv = X t wt(v) θ(v) = X wt(p) X θ(v) = X wt(v) T -1 θ(v) -1
(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 H, called the Bernstein-Lusztig basis.

Let R[Q] be the free R-module with basis {X λ | λ ∈ Q}. We have a natural action of W0 given by wX λ = X wλ . We set

R[Q] W 0 = {p ∈ R[Q] | w • p = p for all w ∈ W0}. It is a well-known result that the centre of H is Z(H) = R[Q] W 0 .
The combinatorics of positively folded alcove walks encodes the change of basis from the standard basis (Tw)w∈W of H to the Bernstein-Lusztig basis (Xv)v∈W . This is seen by taking u = e in the following proposition. Proof. Suppose that ℓ(ws) = ℓ(w) + 1. Then

XuTws = XuTwTs = p∈P( w,u) Q(p)X end(p) Ts.

Now, using the formula

Ts = T -1 s + (qs -q -1
s ) in the second case below, we have

X end(p) Ts = X end(p•ǫ + s ) if end(p) -| + end(p)s X end(p•ǫ - s ) + (qs -q -1 s )X end(p•f + s ) if end(p)s -| + end(p)
where p • ǫ + s , p • ǫ - s , and p • f + s denote, respectively, the path p followed by a positive s-crossing, a negative s-crossing, and a positive s-fold. The result follows by induction.

Example 3.3. Let (W, S) be the affine Weyl group of type G2 with diagram and weight function as in Example 2.2. Write q1 = q L(s 1 ) and q2 = q L(s 2 ) = q L(s 0 ) . The coroot system Φ ∨ is as in Example 3.1. Writing X1 = X α ∨ 1 and X2 = X α ∨ 2 , the Hecke algebra H asscociated to W has generators T1 = Ts 1 , T2 = Ts 2 , X1 and X2 with relations

T 2 1 = 1 + (q1 -q -1 1 )T1 T1X1 = X -1 1 T1 + (q1 -q -1 1 )(1 + X1) T 2 2 = 1 + (q2 -q -1 2 )T2 T2X2 = X -1 2 T2 + (q2 -q -1 2 )(1 + X2) (T1T2) 3 = (T2T1) 3 T2X1 = X1X 3 2 T -1 2 -(q2 -q -1 2 )X1X2(1 + X2) X1X2 = X2X1 T1X2 = X1X2T -1 1 .

A formula for the Weyl character

In this subsection we use the Hecke algebra as a tool to establish a combinatorial formula for the Weyl character s λ (X).

It is sufficient for this purpose to consider the Hecke algebra H with weight function L = ℓ (that is, the equal parameter case). Let 10 = w∈W 0 q ℓ(w) Tw.

We have Tw10 = 10Tw = q ℓ(w) 10 for all w ∈ W0. For dominant λ, the Macdonald spherical function is the unique element P λ (X, q -1 ) of R[Q] such that P λ (X, q -1 )10 = q -2ℓ(w 0 ) 10X λ 10.

The well known explicit formula for P λ (X, q -1 ), due to Macdonald (see [START_REF] Macdonald | Spherical functions on a group of p-adic type[END_REF], and also [START_REF] Ram | Kostka-Foulkes polynomials and Macdonald spherical functions[END_REF] for a proof in the Hecke algebra context) is

P λ (X, q -1 ) = w∈W 0 w X λ α∈Φ + 1 -q -2 X -α ∨ 1 -X -α ∨ ,
from which we see that P λ (X, q -1 ) ∈ R[Q] W 0 and that on specialising q -1 = 0 we have P λ (X, 0) = s λ (X).

Let w, u ∈ W and let w be any reduced expression for w. Let

P( w, u) = {p ∈ P( w, u) | f (p) = ℓ(w0)}. (3.1)
The following theorem is well known, however we sketch the proof for completeness.

Theorem 3.4. If λ ∈ Q ∩ P + then s λ (X) = p∈P( w 0 • t λ ,e)
X wt(p) ,

Proof. Let H be the Hecke algebra with L = ℓ. Since TuTt λ = Tut λ for all u ∈ W0 we have, by Proposition 3.2, P λ (X, q -1 )10 = q -2ℓ(w 0 ) u∈W 0 q ℓ(u) TuTt λ 10 = q -2ℓ(w 0 ) u∈W 0 p∈P( u• t λ ,1)

q ℓ(u) (qq -1 ) f (p) X end(p) 10.

Since X end(p) 10 = X wt(p) T -1 θ(p) -1 10 = q -ℓ(θ(p)) X wt(p) 10 it follows that P λ (X, q -1 ) = u∈W 0 p∈P( u• t λ ,1)

(q -1 ) 2ℓ(w 0 )-ℓ(u)-f (p)+ℓ(θ(p)) (1q -2 ) f (p) X wt(p) .

For each positively folded alcove walk p ∈ P( u • t λ , e), let p0, . . . , p f (p) be the partial folding sequence of p, where pj is the positively folded alcove walk that agrees with p up to (and including) the jth fold of p, and is straight thereafter.

It is simple to see (either using the technique of Lemma 6.2 in this paper, or see [START_REF] Ram | Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux[END_REF]) that ℓ(θ(pj+1)) < ℓ(θ(pj)) for all j = 0, . . . , f (p) -1. Thus ℓ(θ(pj))ℓ(θ(pj+1)) -1 ≥ 0, and it follows by summing that ℓ(θ(p0

)) -ℓ(θ(p f (p) )) -f (p) ≥ 0, and hence f (p) ≤ ℓ(u) -ℓ(θ(p)),
with equality if and only if ℓ(θ(pj))ℓ(θ(pj+1)) -1 = 0 for each j = 0, . . . , f (p) -1. Thus the exponent of q -1 in the above formula for P λ (X, q -1 ) is

2ℓ(w0) -ℓ(u) -f (p) + ℓ(θ(p)) ≥ 2(ℓ(w0) -ℓ(u) + ℓ(θ(p))) = 2(ℓ(w0u -1 ) + ℓ(θ(p))) ≥ 0,
with equality if and only if f (p) = ℓ(u)ℓ(θ(p)), ℓ(w0u -1 ) = 0, and ℓ(θ(p)) = 0. Thus equality occurs if and only if u = w0, θ(p) = e, and f (p) = ℓ(w0). Therefore, upon specialising at q -1 = 0 only the terms with u = w0 and f (p) = ℓ(w0) survive, hence the result.

Kazhdan-Lusztig cells in type G2

In this section we recall the decomposition of G2 into right cells and two-sided cells for all choices of parameters (a, b) ∈ N 2 from [START_REF] Guilhot | Kazhdan-Lusztig cells in affine Weyl groups of rank 2[END_REF]. We also recall some "cell factorisation" properties for the infinite two-sided cells from [START_REF] Guilhot | Affine cellularity of affine hecke algebras of rank two[END_REF].

The partition of G2 into cells

Let W be an affine Weyl group of type G2 with diagram and weight function L(s1) = a and L(s2) = L(s0) = b as in Example 2.2. The partition of W into two-sided cells depends only on the ratio r = a/b of the parameters, and it turns out that there are precisely 7 distinct regimes. We recall these decompositions in the diagrams below where

• w and w ′ are in the same two-sided cell if and only if they have the same colour;

• w and w ′ are in the same right cell if and only if they have the same colour and lie in a common connected component;

• the graphs represent the two-sided order on two-sided cells for all regimes from r > 2 on the left to r < 1 on the right. Let Γ be a two-sided cell for the parameter r ∈ Q. We say that r is generic for Γ if there exists η > 0 such that Γ is a cell for all parameters r ′ ∈ Q such that r ∈ ]rη, r + η[. Looking at the decomposition of G2 into cells, it is easy to see that the only pairs (Γ, r) such that r is non-generic for Γ are (Γ2, 2), (Γ1, 3/2) and (Γ3, 1).

r > 2 r = 2 2 > r > 3/2 r = 3/2 3/2 > r > 1 r = 1

Cell factorisation for the lowest two-sided cell Γ 0

Note that the yellow two-sided cell is constant for all choices of r (see Figure 2). This cell is called the lowest two-sided cell, and is denoted Γ0. By direct inspection we have the following representation of elements of Γ0 (see Figure 3):

• Each right cell Γ k ⊆ Γ (1 ≤ k ≤ 12) contains a unique element w Γ k of minimal length. • The longest element w0 of G2 is a suffix of each w Γ k , 1 ≤ k ≤ 12. Let u k = w0w -1 Γ k for 1 ≤ k ≤ 12 (these elements are the inverses of the grey elements on the left). Let B0 = {u k | 1 ≤ k ≤ 12} (this "box" is illustrated on the right). • We have Γ0 = {u -1 w0t λ v | u, v ∈ B0, λ ∈ P + }.
Moreover, each w ∈ Γ0 has a unique expression in the form w = u -1 w0t λ v with u, v ∈ B0 and λ ∈ P + , and this expression is reduced (that is, We use the third property to define functions u, v : Γ0 -→ BΓ 0 and τ : Γ0 → {t λ | λ ∈ P + } by the equation w = u(w) -1 w0τ (w)v(w). We will usually write uw, vw and τw in place of u(w), v(w) and τ (w). Thus the cell factorisation of w ∈ Γ0 is the expression w = u -1 w w0τwvw.

ℓ(w) = ℓ(u -1 ) + ℓ(w0) + ℓ(t λ ) + ℓ(v)).

Cell factorisation for the cells Γ 1 and Γ 2 with generic parameters

Note that for each value of r = a/b there are precisely three infinite two-sided cells (including the lowest two-sided cell Γ0). With reference to Figure 2, let Γ1 be the green cell, and let Γ2 be the blue cell. Note that the two-sided cells Γ1 and Γ2 are dependent on the choice of r. It turns out that for most parameters (a, b) the infinite two-sided cells Γ1 and Γ2 admit analogous cell factorisations to Γ0. Recall from above that:

Convention: When speaking about the cell Γi in the "non-generic case", we will mean either the cell Γ1 in the case r1 = a/b = 2 or the cell Γ2 in the case r2 = a/b = 3/2. All other parameter values are generic for these cells.

With this convention, if Γ ∈ {Γ1, Γ2} and r is generic for Γ then we have the following cell factorisation properties: Let Γ 1 , . . . , Γ 6 be the right cells contained in Γ. Then Explicitly, in each case the elements wΓ and tΓ, and the "box" BΓ are as follows. For Γ1 there are 2 distinct generic regimes, given by r > 2, and r < 2 (see Figure 4). We have

wΓ 1 = 01 if r > 2 020 if r < 2 tΓ 1 = 210 if r > 2 120 if r < 2 and BΓ 1 = {e, 2, 20, 21, 212, 2120} if r > 2 {e, 1, 12, 121, 1212, 12120} if r < 2.
Note that the translates of BΓ 1 by tΓ 1 tessellate a "strip" in W . For Γ2 there are 2 distinct generic regimes, given by r > 3/2, and r < 3/2 (see Figure 5). We have

wΓ 2 = 12121 if r > 3/2 01 if r < 3/2 tΓ 2 = 02121 if r > 3/2 21210 if r < 3/2 and BΓ 2 = {e, 0, 02, 021, 0212, 02120} if r > 3/2 {e, 2, 21, 212, 2121, 2120} if r < 3/2.
Note that the translates of BΓ 2 by tΓ 2 tessellate a "strip" in W . We will typically write wi and Bi in place of wΓ i and BΓ i (for i = 0, 1, 2) and ti in place of tΓ i (for i = 1, 2).

For i ∈ {1, 2} we use the third property of cell factorisation to define functions u, v : Γi -→ Bi and τ : Γi → {t n i | n ∈ N} by the equation w = u(w) -1 wiτ (w)v(w). We will usually write uw, vw and τw in place of u(w), v(w) and τ (w). Thus the cell factorisation of w ∈ Γi is the expression w = u -1 w wiτwvw.

Remark 4.1. It is possible to have similar decompositions for most finite cells Γ when r is generic for Γ:

• For Γ3 there are 2 distinct generic regimes, given by r > 1 and r < 1. When r > 1, if we set wΓ 3 := s1, tΓ 3 := s2s1 and BΓ 3 := {e, s2, s2s0} then we have

Γ3 = {u -1 wΓ 3 t k Γ 3 v | u, v ∈ BΓ 3 , k ∈ {0, 1}}.
• For Γ4 there are 2 distinct generic regimes, given by r > 1. When r < 1 if we set wΓ 4 := s2s1s2s1s2 and BΓ 4 := {e, s0} then we have

Γ4 = {u -1 wΓ 4 v | u, v ∈ BΓ 4 }.
• For Γ6 there is only one regime given by 2 > r > 3/2. If we set wΓ 6 := s1s0 and BΓ 6 := {e, s2, s2s1, s2s1s2, s2s1s2s0} then we have Γ6 = {u -1 wΓ 6 v | u, v ∈ BΓ 3 }.

• All other finite cells in generic parameters contain a unique element wΓ and we set BΓ = {e}. It would also possible to have a similar description for Γ3 when r < 1 and Γ4 when r > 1 but notation becomes more technical since we have to take into account that there is a graph automorphism of the parabolic subrgoup W {0,2} .

We will typically write wi and Bi in place of wΓ i and BΓ i and t3 in place of tΓ 3 . As above, when there is a cell factorisation for the finite cell Γ, we obtain functions u, v on Γ. For the two-sided cells Γ3 when r > 1, we also have a function

τ3 : Γ3 -→ {t k 3 | k = 0, 1}.
Remark 4.2. Let w, w ′ ∈ Γi where Γi for which there is a cell factorisation. Then we have

w ∼L w ′ ⇐⇒ vw = v w ′ and w ∼R w ′ ⇐⇒ uw = u w ′ .
Furthermore we note that τ (w

-1 ) = τ (w). Indeed if w ∈ Γi where i = 0, 1, 2, 3 then w -1 = v -1 τ -1 w wiu = v -1 wiτwu.

Cell factorisation for the cells Γ 1 and Γ 2 with non-generic parameters

Let r1 = 2 and r2 = 3/2. The behaviour of the cell Γ1 when r = r1 is similar to the behaviour of the cell Γ2 when r = r2.

The two-sided cell Γi is the union of Γ + i , the two-sided cell in the generic case a/b > ri and Γ - i , the two-sided cell in the generic case a/b < ri (in line with the semicontinuity conjecture of Bonnafé [START_REF] Bonnafé | Semicontinuity properties of Kazhdan-Lusztig cells[END_REF]). More precisely we have

Γ + 1 \Γ - 1 = {u -1 w + 1 v | u, v ∈ B + 1 ∩ s1B - 1 }, Γ - 1 \Γ + 1 = {w - 1 }, Γ - 2 \Γ + 2 = {u -1 w + 1 v | u, v ∈ B - 2 ∩ s0B + 2 }, Γ + 2 \Γ - 2 = {w + 1 }.
Furthermore, each right cell Υ ⊂ Γi is either

• equal to a right cell in the case a/b > ri, in which case we say Υ is of positive type;

• equal to a right cell in the case a/b < ri, in which case we say Υ is of negative type; Definition 4.3. Let w ∈ Γi. We say that w is of type (ε1, ε2) where ε k = ± if w belongs to a right cell of type ε1 and w -1 belongs to a right cell of type ε2.

It is easy to see from the definition that if w is of type (ε1, ε2) then w -1 will be of type (ε2, ε1). We represent the types of the elements of Γi in Figure 6: the dark blue, light blue, light red, dark red alcoves are respectively of type (-, -), (-, +), (+, -) and (+, +).

Fig. 6:

(ε 1 , ε 2 )-type in Γ i .
We denote by w ε i , B ε i , and P ε i the data associated to Γ ε i where ε = ±. We define uε, vε :

Γ ε i -→ B ε i and τ ε : Γ ε i → {t n i,ε | n ∈ N} by the equation w = uε(w) -1 w ε i τε(w)vε(w).
In the case where i = 1, we extend the definition of uε, vε and τε by setting for all u, v ∈ B + 1 ∩ s1B -

1 u+(w - 1 ) := s2s0 u-(u -1 w + 1 v) = s1u, v+(w - 1 ) := s2s0 u-(u -1 w + 1 v) = s1v, τ+(w - 1 ) := -1 τ-(u -1 w + 1 v) = -1.
Similarly when i = 2, we extend the definition of uε, vε and τε by setting for all u, v ∈ B -

2 ∩ s0B + 2 u-(w + 2 ) := s2s1s2s1 u+(u -1 w - 2 v) = s0u, v-(w + 2 ) := s2s1s2s1 u+(u -1 w - 2 v) = s0v, τ-(w + 2 ) := -1 τ+(u -1 w - 2 v) = -1.
These definitions are coherent since we have

• for all w ∈ Γi and ε = ± we have w = u -1 ε (w)w ε i τε(w)vε(w);

• for all w, w ′ ∈ Γi, w ∼L w ′ if and only if vε(w) = vε(w ′ );

• for all w, w ′ ∈ Γi, w ∼R w ′ if and only if uε(w) = uε(w ′ ). The relation between those two expressions when i = 1 are as follows

• if w is of type (+, +) then u-(w) = s1u+(w), v-(w) = s1v+(w) and τ-(w) = τ+(w) -1; • if w is of type (-, -) then u+(w) = s0s2 = v+(w) and τ+(w) = τ-(w) -1; • if w is of type (+, -) then u-(w) = s1u+(w), v-(w) = s0s2 and τ-(w) = τ+(w); • if w is of type (-, +) then u-(w) = s0s2, v-(w) = s1v+(w) and τ-(w) = τ+(w).
There are similar formulas for i = 2.

Cell representations in type G 2

In this section we prove that each finite cell admits a finite dimensional representation satisfying B1-B4 and B4 ′ . Moreover, we show that each infinite cell admits a finite dimensional representation satisfying B1.

We will use the following notation. We write Ei,j for the square matrix with 1 in the (i, j) place, and zeros elsewhere (the dimension of the matrix will be clear from context). For i, j ∈ Z we write µi,j = q ia-jb + q -ia+jb .

Finite cells

Let Γ be a finite two-sided cell and let Υ be a right cell lying in Γ. By Table 2, Γ intersects a dihedral parabolic subgroup WI , and we set ãΓ = aI (z) for any z ∈ Γ ∩ WI (here aI is Lusztig's a-function on WI). It is easily verified, using Table 1, that this is well defined.

We write ρ ∼ Υ to indicate that ρ is the cell module over R associated to Υ equipped with the natural Kazhdan-Lusztig basis as in Section 1.2. From the data in Figure 2, we see that Υ ≥ LR and Υ> LR are also finite subsets of W . Proof. We have already noted in Section 2 that ρ satisfies B1. To check B2, note that the set Υ ≥ LR is finite, and hence it is clear that there exists

M ≥ 0 such that deg([ρ(Cw)]i,j ) ≤ M for all w ∈ Υ ≥ LR . Since ρ satisfies B1 we have ρ(Cw) = 0 if w / ∈ Υ ≥ LR , so B2 holds.
We now verify B3, B4 and B4 ′ . Since ρ(Cw) = 0 if w / ∈ Υ ≥ LR it is sufficient to look at the matrices ρ(Cw) where w lies in the finite set Υ ≥ LR .

We start by treating the 1-dimensional cells. There are 4 such two-sided cells, Γe = {e} (in all parameter regimes) and the cells Γ5 = {s0s2s0} (for r > 2), Γ7 = {s1s2s1s2s1} (for 3/2 > r > 1) and Γ7 = {s1} (for r < 1). The associated cell modules are ρI where I = ∅, {s1} or {s0, s2} (see Example 2.2). We now verify B3 for each of these cells. Then B4 and B4 ′ are obvious since there is only one leading matrix, and it is just a nonzero element of R.

• We have ρ ∅ ∼ Γe and since max deg(ρ ∅ (Γe)) = 0 = ãΓe the result is clear. This shows that aρ = 3a -2b = ãΓ 7 and hence B3 holds.

• When r < 1, we have ρI ∼ Γ7 where I = {1}. We have Γ7 ≤ LR = Γ7 ∪ Γe and max deg(ρ(Γ7)) = a. This shows that aρ = a = ãΓ 7 and hence B3 holds.

We now consider the remaining finite cells. Consider Γ6, which occurs in the regime 2 > r > 3/2 only. Let ρ ∼ Υ where Υ is a right cell included in Γ6. Thus ρ is a 5-dimensional representation with basis indexed by the elements of Υ. To be concrete we will take Υ = {s1s0, s1s0s2, s1s0s2s1, s1s0s2s1s2, s1s0s2s1s2s0}, however it turns out that the representations for the right cells are pairwise isomorphic. Then the matrices of Ts 1 , Ts 2 , and Ts 0 are, respectively,

   q a µ 1,1 0 1 -µ 2,3 0 -q -a 0 0 0 0 1 q a µ 1,1 0 0 0 0 -q -a 0 0 0 0 0 -q -a    ,    -q -b 0 0 0 0 1 q b 0 0 0 0 0 -q -b 0 0 0 0 1 q b 1 0 0 0 0 -q -b    ,     q b 1 -µ 1,2 0 0 0 -q -b 0 0 0 0 0 -q -b 0 0 0 0 0 -q -b 0 0 0 0 1 q b     .
We have This shows that aρ = a + b = ãΓ 6 and B3 holds. To verify B4 requires further computation. Recall that any w ∈ Γ6 can be written in a unique way in the form u -1 s1s0v where u, v ∈ B6, see Remark 4.1. Again by direct computation we see that cρ,w = Es 1 s 0 u,s 1 s 0 v if w = u -1 s1s0v with u, v ∈ B6 (recall that the rows and columns of the matrices for ρ(Tw) are indexed by the elements of

Γ6 ≤ LR = Γ6 ∪ Γ3 ∪ Γ4 ∪
Υ = {s1s0v | v ∈ B6}). Thus B4 holds. To verify B4 ′ we note that if w = u -1 s1s0v ∈ Γ6 with u, v ∈ B6 then writing x = u -1 s1s0 ∈ Γ6 and y = s1s0v ∈ Γ6 we have cρ,xcρ y = Es 1 s 0 u,s 1 s 0 Es 1 s 0 ,s 1 s 0 v = Es 1 s 0 u,s 1 s 0 v = cρ,w.
Consider Γ4, which occurs in for all r = 1. The matrices for ρΥ(Tj) are easily computed, and we find

ρΥ(Ts 0 ) = q b 1 0 -q -b ρΥ(Ts 1 ) = -q -a 0 0 -q -a ρΥ(Ts 2 ) = -q -b 0 1 q b If r > 1 then Γ 4≤ LR = Γ4 ∪ Γe,
and by direct computation we see that max deg(ρ(Γ4)) = b, hence B3 holds. We compute

cρ,s 0 = E1,1, cρ,s 0 s 2 = E1,2, cρ,s 2 = E2,2, cρ,s 2 s 0 = E2,1, from which B4 and B4 ′ follow. If r < 1 then Γ 4≤ LR = Γ4 ∪ Γ7 ∪ Γ3 ∪ Γe.
By direct calculation we have max deg(ρ(Γ4)) = 3b -2a, max deg(ρ(Γ7)) = -a, and max deg(ρ(Γ3)) = 2ba, and hence B3 holds. We have

cρ,s 0 s 2 s 1 s 2 s 1 s 2 = E1,2, cρ,s 0 s 2 s 1 s 2 s 1 s 2 s 0 = E1,1, cρ,s 2 s 1 s 2 s 1 s 2 = E2,2, cρ,s 2 s 1 s 2 s 1 s 2 s 0 = E2,1,
and hence B4 and B4 ′ hold.

We are left with the red cells Γ3. When r > 1 all the representations afforded by the right cells are isomorphic and the matrices of Ts 1 , Ts 2 and Ts 0 are given by

     q a µ 1,1 0 0 1 0 0 -q -a 0 0 0 0 0 0 -q -a 0 0 0 0 1 0 q a µ 1,1 0 0 0 0 0 -q -a 0 0 0 0 0 0 -q -a      ,      -q -b 0 0 0 0 0 1 q b 1 0 0 0 0 0 -q -b 0 0 0 0 0 0 -q -b 0 0 0 0 0 1 q b 1 0 0 0 0 0 -q -b      ,      -q -b 0 0 0 0 0 0 -q -b 0 0 0 0 0 1 q b 0 0 0 0 0 0 -q -b 0 0 0 0 0 0 -q -b 0 0 0 0 0 1 q b      .
A direct check shows that deg([ρ(Tw)]i,j ) is bounded by ãΓ 3 = a and that B3 holds. Moreover,

{cρ,w | w ∈ Γ3} = {E1+i,1+j + E4+i,4+j, E1+i,4+j + E4+i,1+j | 0 ≤ i, j ≤ 2},
from which B4 and B4 ′ follow. The case r < 1 can be treated similarly.

The case (Γ, r) = (Γ3, 1) is slightly different since the right cells contained in Γ do not give rise to isomorphic cell representations (there are two right cells with 8 elements, and one with 7). However in this case it turns out, by calculation, that the direct sum of these representations is bounded by ãΓ 3 = 1 and B3, B4 and B4 ′ hold. Explicit matrices for all finite cells can be found on the authors' webpage, and are provided below.

Remark 5.2. When Γ = Γ3 and r > 1, it is possible to use the cell factorisation described in Remark 4.1 to construct a 3 dimensional balanced representation over a quotient of an R-polynomial ring (this is a slight generalisation of our definition of balanced representations). The construction is based on the induction process introduced by Geck in [START_REF] Geck | On the induction of Kazhdan-Lusztig cells[END_REF].

Recall that B3 = {e, s2, s2s0} and t3 = s2s1. For all x ∈ B3 ∪ {t3}, there exist hx ∈ H such that Cs 1 hx ≡ Cs 1 x mod HΓ 3 .

Then

HΓ 3 = h ♭ u Cs 1 h k t 3 hv | u, v ∈ B3, k ∈ {0, 1} R
where ♭ denotes the anti-involution defined [25, §3.4]. This allows us to define a 3 dimensional representation

ρ over R[ε]/(ε 2 -1) with basis {es 1 v | v ∈ B3} by setting es 1 v • Tw = v∈B 3 ,k∈{0,1} λ k,v ′ v,w ε k e s 1 v ′ whenever Cs 1 v • Tw ≡ v ′ ∈B 3 ,k∈{0,1} λ k,v ′ v,w Cs 1 h k t 3 h v ′ mod HΓ 3 .
We obtain the following matrices for Ts 1 , Ts 2 and Ts 0 :

q a ε+µ 1,1 0 0 -q -a 0 0 0 -q -a , -q -b 0 0 1 q b 1 0 0 -q -b and -q -b 0 0 0 -q -b 0 0 1 q b .
Then it can be checked that ρ is Γ3-balanced. More precisely we have cρ,w

= ε k Es 1 u,s 1 v if w = u -1 s1t k 3 v.
It is useful for later results to understand the decomposition of cell modules of finite cells into irreducible components. We summarise this in the following proposition.

Proposition 5.3. Let Γ be a finite two-sided cell and let Υ be a right cell in Γ. Let ρΥ ∼ Υ.

1) If Γ = Γ3 then the representations ρΥ are irreducible and pairwise isomorphic.

2) If Γ = Γ3 and r = 1 then the representations ρΥ are pairwise isomorphic and decompose into a direct sum ρΥ = ρ + 3 ⊕ ρ - 3 where ρ ± 3 are irreducible 3 dimensional representations with ρ + 3 ∼ = ρ - 3 . 3) Suppose that Γ = Γ3 and r = 1. Let Υ1, Υ2, and Υ3 be the right cells containing s1, s0, and s2 (respectively).

Then

ρΥ 1 ∼ = ρ + 3 ⊕ ρ - 3 ⊕ ρ ′ 3 and ρΥ 2 ∼ = ρΥ 3 ∼ = ρ + 3 ⊕ ρ - 3 ⊕ ρ ′′ 3
where ρ + 3 , ρ - 3 , ρ ′ 3 , and ρ ′′ 3 are pairwise non-isomorphic irreducible representations of dimension 3, 3, 1, and 2.

Proof. Statement (1) follows by direct calculation, and we omit the details.

Suppose that Γ = Γ3 and r = 1. Again we verify that each right cell gives rise to an isomorphic representation by direct calculation. Let us discuss the decomposition into irreducible components. If r > 1 then the cell Γ3 admits a cell factorisation, and it follows from Remark 5.2 that ρ decomposes as ρ + 3 ⊕ ρ - 3 , where the matrices for ρ ε 3 (Tj ) are as in Remark 5.2 (with ε now considered to be ±1, and so these representations are over R). If r < 1 then we compute directly that ρΥ ∼ = ρ + 3 ⊕ ρ - 3 with matrices

ρ ε 3 (T0) = -q -b 0 0 1 q b 0 0 0 -q -b , ρ ε 3 (T1) = -q -a 0 0 0 -q -a 0 1 0 q a and ρ ε 3 (T2) = q b 1 ε+µ 1,1 0 -q -b 0 0 0 -q -b
In each case it is easy to see that ρ ε 3 is irreducible, and that ρ + 3 ∼ = ρ - 3 . Hence (2). Finally, consider Γ = Γ3 with r = 1. In this case the result follows from [24, (3.13.1)]. Indeed, if ρΥ 1 is constructed using the basis of residues (e1, e2, e3, e4, e5, e6, e7) = (C1, C12, C120, C121, C1212, C12120, C12121) then the submodules giving the claimed decomposition are e1 + 2e4 + e7, e2 + e5, e3 + e6 , e1 -e7, e2 -e5, e3 -e6 , and e1 -e4 + e7 . If ρΥ 2 is constructed using the basis of residues (e1, e2, e3, e4, e5, e6, e7, e8) = (C0, C02, C021, C0212, C02120, C02121, C021212, C0212120) then the submodules are e3 +e6, e2 +2e4 +e7, e1 +2e5 +e8 , e3 -e6, e2 -e7, e1 -e8 , and e2 -e4 +e7, e1 -e5 +e8 . The same submodule structure works for ρΥ 3 using the basis of residues (C20, C2, C21, C212, C0212, C2121, C21212, C212120).

Remark 5.4. We note the following for later use. In the case Γ = Γ3 the representations ρ + 3 , ρ - 3 , ρ ′ 3 , and ρ ′′ 3 , equipped with the bases from the above proposition, satisfy B1 and B2 (in the respective parameter regimes). It is clear that B1 holds (because if π is semisimple and π(Cw) = 0 then π ′ (Cw) = 0 for all submodules). To see that B2 holds we note that the change of basis matrix that converts the cell representation into block form is independent of q.

The principal series representation π 0

We now associate a representation π0 to the lowest two-sided cell Γ0. It is convenient to set this section up in arbitrary type, and so H is an affine Hecke algebra of rank n. Recall that R[Q] denotes the subalgebra of H spanned by the elements {X λ | λ ∈ Q}. We use this large commutative subalgebra to construct finite dimensional representations of H as follows. Let ζ1, . . . , ζn be commuting indeterminants, and let M0 be the

1-dimensional right R[Q]-module over the ring R[ζ1, . . . , ζn, ζ -1 1 , . . . , ζ -1 n ],
with generator ξ0 and R[Q]-action given by linearly extending Since Γ0 is the lowest two-sided cell, the representation π0 trivially satisfies B1 with respect to Γ = Γ0

ξ0 • X µ = ξ0 ζ µ where ζ µ = ζ k 1 1 • • • ζ kn n if µ = k1α ∨ 1 + • • • + knα ∨ n ∈ Q. Now let (π0, M0) be the induced right H-module. That is, M0 = Ind H R[Q] (M0) = M0 ⊗ R[Q] H. Since {X µ T -1 u -1 | µ ∈ Q, u ∈ W0} is a basis of H, and since ξ0 ⊗ X µ = (ξ0 ⊗ 1)ζ µ , we see that {ξ0 ⊗ Xu | u ∈ W0} is a basis of M0. Thus M0 is a |W0|-dimensional

The induced representations π 1 and π 2

For each i ∈ {1, 2} let Hi be the subalgebra of H generated by Ti, X1, X2 (where Xj = X α ∨ j ). Let ζ be an indeterminant, and for each i ∈ {1, 2} let Mi be the 1-dimensional right Hi-module over the ring R[ζ, ζ -1 ] with generator ξi and Hi-action given by

ξ1 • T1 = ξ1(-q -a ) ξ1 • X1 = ξ1 q -2a ξ1 • X2 = ξ1 (-q a ζ) ξ2 • T2 = ξ2(-q -b ) ξ2 • X1 = ξ2 (-q 3b ζ) ξ2 • X2 = ξ2 q -2b
One checks directly using the formulae in Example 3.3 that these are representations.

For i ∈ {1, 2}, let (πi, Mi) be the induced right H-module. Thus Mi = Mi ⊗H i H. For i ∈ {1, 2} let Wi = si and let W i 0 denote the set of minimal length coset representatives for cosets in Wi\W0. Note that the module Mi has basis {ξi ⊗ Xv | v ∈ W i 0 } for i = 1, 2.

Theorem 5.5. Let i ∈ {1, 2}. The representation πi satisfies B1 with respect to Γ = Γi.

Proof. We need to show that πi(Cw) = 0 for all w ∈ Γ with Γ ≤LR Γi. The set of such Γ is determined by the Hasse diagrams in Figure 2. It suffices to show that πi(Cw j ) = 0 whenever Γj ≤LR Γi (here j ∈ {0, 1, 2}), plus in the regime r < 1 we need to show that π2(Cw) = 0 for all w in the finite cell Γ4. For example, in the parameter regime 2 > r > 3/2 we need to check that π1(Cw 0 ) = π1(Cw 2 ) = 0 and π2(Cw 0 ) = π2(Cw 1 ) = 0.

In the cases that wi is the longest element of some dihedral parabolic subgroup WJ we have the formula

Cw i = q -L(w i )
w∈W J q L(w) Tw.

The only case required when wi is not the longest element of a dihedral parabolic subgroup is w2 in the parameter regime 2 > r > 3/2. In this case

Cw 2 =(q -3a-2b -q -3a + q -3a+2b )Te + (q -3a-b -q -3a+b )Ts 2 + (-q -2a-2b + q -2a + q -2a+2b )Ts 1 + (q -2a-b + q -2a+b )(Ts 2 s 1 + Ts 1 s 2 ) + q -3a Ts 2 s 1 s 2 + (q -a-b + q -a+b )Ts 1 s 2 s 1 + +q -a (Ts 2 s 1 s 2 s 1 + Ts 1 s 2 s 1 s 2 ) + Ts 1 s 2 s 1 s 2 s 1
For the case r < 1, to show that π2(Cw) = 0 for w ∈ Γ4 it is sufficient to show that π2(Cs 2 s 1 s 2 s 1 s 2 ) = 0. The formula for Cs 2 s 1 s 2 s 1 s 2 in the Tw basis is as in the Cw 2 formula above with the roles of s1 and s2 interchanged. The result now follows by direct computation.

6 The lowest two-sided cell Γ 0

In this section we show that the principal series representation π0, equipped with certain natural bases, satisfies B2-B4 and B4 ′ for the cell Γ0, with bound aπ 0 = L(w0). It is convenient to work more generally than G2. However since ultimately we are interested in G2, and in this case Q = P , we will work in this setting here. Thus the analysis below applies verbatim to the cases G2, F4, and Ẽ8, however we note that by slight modifications (in particular to the definition of B0) the analysis below applies to all (extended) affine Weyl groups.

We first show that the degree of the matrix coefficients of π0(Tw) are bounded by L(w0) for all w ∈ W (verifying B2), and then we determine explicitly the set of w ∈ W for which this bound is attained: it turns out to be precisely the lowest two-sided cell Γ0 (hence B3). Finally we will compute the leading matrices cπ 0 ,w in terms of Schur functions, verifying B4 and B4 ′ . The following theorem generalises the formula presented in [START_REF] Parkinson | A local limit theorem for random walks on the chambers of Ã2 buildings[END_REF]Theorem 5.16].

Theorem 6.1. Let B be a fundamental domain for Q. The set {ξ0 ⊗ Xu | u ∈ B} is a basis for M0, and with respect to this basis the matrix entries of π0(Tw), w ∈ W , are given by

[π0(Tw)]u,v = {p∈P B ( w,u)|θ B (p)=v} Q(p)ζ wt B (p)
where w is any reduced expression for w.

Proof. Since W0 is a fundamental domain for Q, each u ∈ B can be written as b = tµ u u ′ for some µu ∈ Q and some

u ′ ∈ W0. Then ξ0 ⊗ Xu = ξ0 ⊗ X µu X u ′ = (ξ0 ⊗ X u ′ )ζ µu . The first claim follows since {ξ0 ⊗ X u ′ | u ′ ∈ W0} is clearly a basis of M0.
Let w be any reduced expression for w. Using Proposition 3.2 we have

(ξ0 ⊗ Xu) • Tw = p∈P( w,u) (ξ0 ⊗ X wt B (p) X θ B (p) )Q(p) = v∈W 0 {p∈P B ( w,u)|θ B (p)=v} (ξ0 ⊗ Xv)Q(p)ζ wt B (p) ,
hence the result.

Leading matrices for π 0

We begin with some definitions in preparation for the following lemma. Let u, w ∈ W , let w be a reduced expression, and let p ∈ P( w, u). The partial foldings of p are the positively folded alcove walks p0, p1, . . . , p f (p) , where pj is the positively folded alcove walk of type w starting at u that agrees with p up to (and including) the jth fold of p, and is straight thereafter. Thus p0 is the straight path of type w starting at u, and p f (p) = p. The pivots of p are the alcoves u0, . . . , u f (p) in which the folds occur, with u0 = u. More formally, if the folds of p occur at positions k1 < . . . < k f (p) in the reduced expression w = r1 • • • r ℓ (with rj ∈ S) then the pivots of p are the alcoves u0 = u, u1 = ur1 • • • r k 1 -1 , and

uj+1 = ujr k j +1 • • • r k j+1 -1 for j = 1, . . . , f (p) -1.
Lemma 6.2. Let u, w ∈ W and let wt(u) = µ. Let v ∈ W0 be such that uw ∈ tµvC0. Let p ∈ P( w, u), and suppose that the folds of p occur on the hyperplanes

H β 1 ,k 1 , . . . , H β f (p) ,k f (p)
, where β1, . . . , β f (p) ∈ Φ + . Let v0 = v, and let vj+1 = s β j+1 vj for j = 0, 1, . . . , f (p) -1. Then:

1) ℓ(vj+1) < ℓ(vj) for j = 0, 1, . . . , f (p) -1.

2)

f (p) ≤ ℓ(v) -ℓ(v f (p)
) with equality if and only if ℓ(vj+1) = ℓ(vj ) -1 for all j = 0, 1, . . . , f (p) -1.

3 Proof. 1) We may assume that µ = 0 (if not, translate the entire proof by t-µ, and then translate back at the end). Thus u ∈ W0. Let p ∈ P( w, u), and let f = f (p). Let p0, . . . , p f be the partial foldings of p. Let p ∞ 0 be an "infinite continuation" of p0 such that each finite segment of p ∞ 0 is reduced, and p ∞ 0 moves into the "interior" of the Weyl chamber vC0 (that is, away from all walls). More formally, p ∞ 0 can be constructed by first extending p0 to y = t wt(uw) v (the longest element of uwW0 ∩ vC0) and then appending infinitely many copies of a fixed reduced expression for tρ, where ρ = ω1 + • • • + ωn (or any other choice of strictly dominant coweight). Verifying that any finite segment of the resulting infinite path p ∞ 0 is reduced is a straightforward exercise in computing separating hyperplanes. Let p ∞ 1 , . . . , p ∞ f be the infinite extensions of p1, . . . , p f induced from p ∞ 0 . In other words, p ∞ 1 , . . . , p ∞ f are generated by successively performing the folds of p to p ∞ 0 . The hyperplane H β j+1 ,k j+1 separates the pivot uj from all alcoves of p ∞ j occurring after uj , and uj is on the positive side of this hyperplane. Thus the linear hyperplane H β j+1 ,0 separates the identity alcove e from all alcoves sufficiently far along p ∞ j (this is because the former is on the positive side of this hyperplane, and the latter are on the negative side). It is clear that all alcoves sufficiently far along p ∞ j lie in vj C0 (here it is important that ρ is strictly dominant). Thus H β j+1 ,0 separates the Weyl chamber C0 from the Weyl chamber vjC0. By the strong exchange condition s β j+1 vj is obtained from a reduced expression of vj by deleting a generator, and thus ℓ(s β j+1 vj) < ℓ(vj). Therefore ℓ(vj+1) < ℓ(vj ) for all j = 0, 1, . . . , f -1.

) If f (p) = ℓ(w0) then v = w0, v f (p) = e,
2) By the above we have ℓ(vj+1)ℓ(vj) + 1 ≤ 0 for all j = 0, . . . , f -1, and hence

0 ≥ f -1 j=0 (ℓ(vj+1) -ℓ(vj) + 1) = ℓ(v f ) -ℓ(v) + f (p),
with equality if and only if ℓ(vj+1) = ℓ(vj) -1 for all j = 0, . . . , f -1.

3) If f (p) = ℓ(w0) then by 2) we have v = w0 and v f = e. Applying the equality ℓ(vj+1) = ℓ(vj ) -1 in the cases j = 0 and j = f -1 gives ℓ(s β 1 w0) = ℓ(w0) -1 and ℓ(e) = ℓ(s β f ) -1, which forces β1 and β f to be simple roots.

4) The conditions ℓ(vj+1) < ℓ(vj) and vj+1 = s β j+1 vj imply, by the strong exchange condition, that vj+1 is obtained from a reduced expression of vj by deleting a single generator. Moreover, by the proof of the strong exchange condition this deleted generator is conjugate to s β j+1 , and thus L(vj+1) ≤ L(vj ) -L(s β j+1 ). It follows that

deg(Q(p)) = f -1 j=0 L(s β j ) ≤ f (p)-1 j=0 (L(vj ) -L(vj+1)) = L(v) -L(θ(p)).
Hence deg(Q(p)) ≤ L(w0), and the condition for equality is clear. Proof. This follows immediately from Theorem 6.1 and Lemma 6.2. Remark 6.4. Note that part 3) of Lemma 6.2 says that if p ∈ P( w, u) with f (p) = ℓ(w0) then the first and last folds of p occur on simple root directions. Here we mean 'simple direction' when p is drawn, as usual, in 'folded form'. One can also draw p in 'unfolded form' by drawing the unfolded path p0 and marking the positions on this path where the folds of p occur. We may then ask if f (p) = ℓ(w0) forces the first and last folds in the unfolded form to also be on simple root directions. Indeed this is the case. The first fold is on the same hyperplane in both the folded and unfolded forms. We note that in the notation of Lemma 6.2 the last fold in unfolded form occurs on a hyperplane whose linear root is

s β f (p)-1 • • • s β 1 β f (p) = s β f (p) v f (p) β f (p) = s β f (p) β f (p) = -β f (p)
, which is a negative simple root. Corollary 6.5. Let p be a positively folded alcove walk of reduced type w starting at u ∈ W . If f (p) = ℓ(w0) then the straight path from u to uw of type w crosses at least one hyperplane of each direction.

Proof. In the notation of the lemma, we see that the set of hyperplanes on which the infinite extensions p ∞ j make negative crossings has strictly decreasing cardinality as j increases. It follows that if f (p) = ℓ(w0) then p0 crosses at least one hyperplane of each of the ℓ(w0) directions.

The main result of this section is the following. Recall that Γ0 = {u -1 w0t λ v | u, v ∈ B0, λ ∈ P + }, and for w ∈ Γ0 we define uw, vw ∈ B0 and τw ∈ P + by w = u -1 w w0τwvw.

Theorem 6.6. The representation π0, equipped with the basis {ξ0 ⊗ Xu | u ∈ B0}, satisfies B3, B4 and B4 ′ . Moreover, cπ 0 ,w = sτ w (ζ)Eu w ,vw for all w ∈ Γ0.

Proof. Suppose that w ∈ W is such that [π0(Tw)]u,v has degree L(w0) for some u, v ∈ B0. Thus by Theorem 6.1 we see that for every reduced expression w there exists a path p ∈ P( w, u) such that deg(Q(p)) = L(w0) and f (p) = ℓ(w0). By Corollary 6.5 the straight path from u to uw of type w crosses every hyperplane direction. It follows that uw lies in the anti-dominant sector based at 0. To see this, recall that there are no simple directions available in B0 (as Q = P ), and thus if all hyperplane directions are crossed then the hyperplanes Hα i are crossed for each 1 ≤ i ≤ n. Thus we may choose a reduced expression for w such that the straight path from u to uw of type w passes through the alcoves 1 and w0. It follows that w admits a reduced expression of the form w = u -1 • w0 • t λ • v for some λ ∈ P + and v ∈ B0, and hence w ∈ Γ0.

We now consider the converse. Let w ∈ Γ0 and write w = uw • w0 • τw • vw. If there exists p ∈ P( w, uw) with f (p) = ℓ(w0) then p has no folds in the initial u -1 w part (since the first fold must be on a simple direction by Lemma 6.2). Thus in the notation of (3.1) we have P( w, uw) = P( w0 • τw • vw, e). Moreover there are no folds in the final vw part (from Lemma 6.2 and Remark 6.4) and thus {p ∈ P( w0

• τw • vw, e) | θ B 0 (p) = vw} = P( w0 • τw • vw, e).
Finally, there is a bijection from P( w0 • τw • vw, e) to P( w0 • τw, e) by simply removing the final vw part, and it follows from Theorem 6.1, Theorem 3.4, and the above observations, that

[cπ 0 ,w ]u w ,vw = p∈P( w 0 • τw • vw ,e) ζ wt B 0 (p) = p∈P( w 0 • τw ,e) ζ wt(p) = sτ w (ζ).
From this formula it follows, in particular, that P( w, uw) = ∅, and hence B3 holds. Moreover, if uw = u then by the first paragraph of the proof that f (p) < L(w0) for all p ∈ P( w, u) and hence [cπ 0 ,w ]u,v = 0. If u = uw and v = vw then by an observation above we have {p ∈ P( w0 • τw • vw, e) | θ B 0 (p) = v} = ∅, and so again [cπ 0 ,w]u,v = 0. This proves that cπ 0 ,w = sτ w (ζ)Eu w ,vw for all w ∈ Γ0.

We also see that B4 holds, because the set of matrices {s λ (ζ)Eu,v | λ ∈ P + , u, v ∈ B0} is free over Z (using linear independence of the Schur characters). 7 The infinite cells Γ 1 and Γ 2

In this section we carry out an analogue of the work of Section 6 for the other infinite cells Γi with i = 1, 2. We begin by introducing and developing a combinatorial model of "αi-folded alcove walks". We then show that this model encodes the matrix coefficients of πi(Tw), and we use this model to prove that our representations are balanced for the cells Γ1 and Γ2, compute the bounds for the degree of matrix coefficients in each parameter regime, and compute the leading matrices in terms of Schur functions of type A1. This section is necessarily more involved that the previous section, since we need to pay careful attention to the non-generic parameter regimes.

α i -folded alcove walks

The following definitions apply to any affine Coxeter group. Let αi be a fixed simple root, and let Ui = {x ∈ V | 0 ≤ x, αi ≤ 1} be the region between the hyperplanes Hα i ,0 and Hα i ,1. Let w ∈ W and write w = si 1 • • • si ℓ . An αi-folded alcove walk of type w starting at v ∈ Ui is a sequence p = (v0, v1, . . . , v ℓ ) with v0, . . . , v ℓ ∈ Ui such that 1) v0 = v, and We refer to the two symbols in (b) as "s-bounces" rather than folds, since they play a different role in the theory. Note that bounces only occur on the hyperplanes Hα i ,0 and Hα i ,1. Moreover, note that there are no folds on the walls Hα i ,0 and Hα i ,1 -the only interactions with these walls are bounces. We note that in all cases except for Ã1 and Cn every s-bounce necessarily has qs = qs i (although it is not necessarily true that s = si). In type Ã1 and Cn this property holds under the assumption that either L(s0) = L(sn), or by modifying the definition of Ui. In any case, here we are interested in G2, and in this case we have qs = qs i for all s-bounces. Thus we will typically simply say bounces.

v k ∈ {v k-1 , v k-1 si k } for each k = 1, . . . , ℓ, and
2) if v k-1 = v k then either: (a) v k-1 si k / ∈ Ui, or (b) v k-1 is
Let p be an αi-folded alcove walk. Let fs(p) = #(s-folds in p) and b(p) = #(bounces in p).

Define a modified q-weight for p by

Qi(p) = (-q -1 s i ) b(p) s∈S (qs -q -1 s ) fs(p) .
Finally, for each 1 ≤ i ≤ n define θ i (p) = ψi(θ(p)) and wt i (p) = wt(p), ωi ,

where ψi : W0 → W i 0 is the natural projection map taking u ∈ W0 to the minimal length representative of Wiu, and ω1, . . . , ωn are the fundamental coweights of Φ.

Thus if wt(p) = m1α ∨ 1 + • • • + mnα ∨ n then wt i (p) = mi.
We refer to θ i (p) as the final direction of p, and wt i (p) as the weight of p (with respect to αi).

We now specialise to the case G2. Let

σ1 = sα 1 ,1tα ∨ 1 +α ∨ 2 = t α ∨ 1 +α ∨ 2 s1 and σ2 = sα 2 ,1tα ∨ 1 +2α ∨ 2 = t α ∨ 1 +2α ∨ 2 s2.
Observe that for each i ∈ {1, 2} the "glide reflection" σi preserves Ui, and that W i 0 is a fundamental domain for the action of σi 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

w = σ wt i B (w) i θ i
B (w), and for αi-folded alcove walks p we define

wt i B (p) = wt i B (end(p)) and θ i B (p) = θ i B (end(p)).
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 7.1. Let i = 1. Let w = 121021210212102120212102120 (a reduced expression). Figure 8 illustrates an α1-folded alcove walk of type w, with two choices of fundamental domain B (the gray shaded regions). The tessellation of U1 by B is shown. The alcove walk has 2 folds and 3 bounces, and Q1(p) = -q -3a (q aq -a )(q bq -b ). The weight of p is 4 with respect to the first fundamental domain, and 2 with respect to the second fundamental domain.

(a) B = W 1 0 , wt 1 (p) = 4, θ 1 (p) = 21212 (b) B = {e, 0, 2, 21, 212, 2121, 2120}, wt 1 B (p) = 2, θ 1 B (p) = s0
Fig. 8: An α 1 -folded alcove walk p, with two choices of fundamental domain B

We now prove an analogue of Theorem 6.1, giving a combinatorial formula for the matrix entries of πi(Tw) in terms of αi-folded alcove walks. We first consider the fundamental domain W i 0 , and then deduce the general case in Corollary 7.3 below.

Theorem 7.2. Let i ∈ {1, 2} and let w ∈ W . With respect to the basis {ξi ⊗ Xu | u ∈ W i 0 } of Mi, the matrix entries of πi(Tw) are given by

[πi(Tw)]u,v = {p∈P i ( w,u)|θ i (p)=v} Qi(p)ζ wt i (p)
where w is any reduced expression for w.

Proof. We will prove the case i = 1. The case i = 2 is completely analogous. We first prove the following formula by induction on ℓ(w):

(ξ1 ⊗ Xu) = p∈P 1 ( w,u) (ξ1 ⊗ X end(p) )Q1(p). (7.1)
Suppose that ℓ(ws) = ℓ(w) + 1. Then by the induction hypothesis

(ξ1 ⊗ Xu) • Tws = p∈P 1 ( w,u) (ξ1 ⊗ X end(p) )TsQ1(p).
Let p ∈ P1( w, u). Consider the following cases:

1) If end(p) -| + end(p)s with end(p)s ∈ U1 then (ξ1 ⊗ X end(p) )TsQ1(p) = (ξ1 ⊗ X end(p•ǫ + s ) )Q1(p • ǫ + s ),
where p • ǫ + s denotes the path obtained from p by appending a positive s-crossing.

2) If end(p) + | -end(p)s with end(p)s ∈ U1 then using Ts = T -1 s -(qs -q -1 s ) gives (ξ1 ⊗ X end(p) )TsQ1(p) = (ξ1 ⊗ X end(p•ǫ - s ) )Q1(p • ǫ - s ) + (ξ1 ⊗ X end(p•fs) )Q1(p • fs),
where p • fs denotes the path obtained from p by appending an s-fold.

3) If end(p) -| + end(p)s with end(p)s / ∈ U1 then necessarily end(p) ∩ end(p)s is a face of Hα 1 ,1 (since the crossing is positive). Then end(p)s = sα 1 ,1end(p • bs) where p • bs denotes the path obtained from p by appending an s-bounce, and since sα 1 ,1 = t α ∨ 1 s1 and

X s α 1 ,1 end(p•bs) = X α ∨ 1 T -1 s 1 X end(p•bs) , we have (ξ1 ⊗ X end(p) )TsQ1(p) = (ξ1 ⊗ X end(p)s )Q1(p) = (ξ1 • X α ∨ 1 T -1 s 1 ⊗ X end(p•bs) )Q(p) = (ξ1 ⊗ X end(p•bs ) )Q1(p)(-q1)(q -2 1 ) = (ξ1 ⊗ X end(p•bs ) )Q1(p • bs).
+ (q1 -q -1
1 ), and the fact that end(p)s = s1end(p) = s1end(p • bs), we have

(ξ1 ⊗ X end(p) )TsQ1(p) = (ξ1 ⊗ X end(p)s )Q1(p) + (ξ1 ⊗ X end(p) )(q1 -q -1 1 )Q1(p) = (ξ1 ⊗ X s 1 end(p•bs) )Q1(p) + (ξ1 ⊗ X end(p•bs) )(q1 -q -1 1 )Q1(p) = (ξ1 • T -1 s 1 ⊗ X end(p•bs) )Q1(p) + (ξ1 ⊗ X end(p•bs) )(q1 -q -1 1 )Q1(p) = (ξ1 ⊗ X end(p•bs) )(-q1)Q1(p) + (ξ1 ⊗ X end(p•bs) )(q1 -q -1 1 )Q1(p) = (ξ1 ⊗ X end(p•bs) )Q1(p • bs).

Equation (7.1) follows.

Let p ∈ P1( w, u) and write end(p) = tµv with µ ∈ Q and v ∈ W0.

Then µ ∈ Hα 1 ,0 ∪ Hα 1 ,1 (since end(p) ∈ U1). If µ ∈ Hα 1 ,0 then µ = kα ∨ 1 + 2kα ∨ 2 for some k ∈ Z and v ∈ W 1 0 . Thus ξ1 ⊗ X end(p) = ξ1 • X µ ⊗ Xv = (ξ1 ⊗ Xv)ζ 2k = (ξ1 ⊗ X θ 1 (p) )ζ wt 1 (p) . If µ ∈ Hα 1 ,1 then µ = kα ∨ 1 + (2k -1)α ∨ 2 for some k ∈ Z, and v / ∈ W 1 0 . Thus θ 1 (p) = s1v, and hence ξ1 ⊗ X end(p) = ξ1 • X µ ⊗ Xv = (ξ1 ⊗ X s 1 θ 1 (p) )q -1 1 (-ζ 2k-1 ) = (ξ1 • T -1 s 1 ⊗ X θ 1 (p) )q -1 1 (-ζ 2k-1 ) = (ξ1 ⊗ X θ 1 (p) )ζ 2k-1 = (ξ1 ⊗ X θ 1 (p) )ζ wt 1 (p) ,
and the theorem follows.

It is convenient to have a version of Theorem 7.2 for other choices of fundamental domain. It is not hard to see that for each p ∈ Pi( w, u) the path σi(p) obtained by applying σi to each part of p is again a valid αi-folded alcove walk starting at σiu (the main point here is that the reflection part of σi is in the simple root direction αi, and thus sends Φ + \{αi} to itself). Moreover, Qi(p) and θ i (p) are preserved under the application of σi, and a direct calculation shows that wt i (σ k i (p)) = k + wt i (p). Corollary 7.3. Let w ∈ W , i ∈ {1, 2}, and let B be a fundamental domain for the action of σi on Ui. Then the matrix entries of πi(Tw) with respect to the basis {ξi ⊗ Xu | u ∈ B} are

[πi(Tw)]u,v = {p∈P i ( w,u)|θ i B (p)=v} Qi(p)ζ wt i B (p) ,
where w is any choice of reduced expression for w.

Proof. We will prove the result for i = 1, with the case i = 2 being similar. For each u ∈ B define k(u) ∈ Z and u ′ ∈ W 1 0 by the formula u = σ

k(u) 1 u ′ . A direct calculation, using the formulae σ 2k 1 = t kα ∨ 1 +2kα ∨ 2 and σ 2k-1 1 = t kα ∨ 1 +(2k-1)α ∨ 2 s1 shows that ξ1 ⊗ Xu = ξ1 ⊗ X σ k(u) 1 u ′ = (ξ1 ⊗ X u ′ ) ζ k(u) .
It follows from Theorem 7.2 (by applying change of basis) that v) .

[π1(Tw)]u,v = {p∈P 1 ( w,u ′ )|θ 1 (p)=v ′ } Qi(p)ζ wt 1 (p)+k(u)-k(

By definition we have

θ 1 (p) = v ′ if and only if θ 1 B (p) = v. Recall that σ k(u) 1
(P1( w, u ′ )) = P1( w, u) and that for each p ∈ P1( w, u ′ ) the value of Q1(p) is preserved under this transformation. Thus

[π1(Tw)]u,v = {p∈P 1 ( w,u)|θ 1 B (p)=v} Qi(p)ζ wt 1 (p)-k(v) ,
and the result follows since wt 1

B (p) = wt 1 (p) -k(v) if θ 1 B (p) = v.

Folding tables and admissible sequences

Our next task is to show that the representations π1 and π2 satisfy B2. By our combinatorial formula for the matrix coefficients of πi(Tw) in terms αi-folded alcove walks it is equivalent to show that deg(Qi(p)) is bounded by some numbers aπ i for all αi-folded alcove walks p. In this subsection we explain our approach to bounding the degree of αi-folded alcove walks.

Note that every w ∈ W admits a reduced expression of the form

w = v • t m ω 1 • t n ω 2 • b with v ∈ W0, m, n ∈ N, and b ∈ B0, (7.2)
and each walk p ∈ Pi( w, u) with u ∈ W i 0 and w as above can naturally be decomposed as p = p0 • p 0 where p0 ∈ Pi( v, u) and p 0 ∈ Pi( w1, end(p0)) where

w1 = t m ω 1 • t n ω 2 • b.
Since Qi(p) = Qi(p0)Qi(p 0 ) it is sufficient to bound the degrees of Qi(p0) and Qi(p 0 ). The former is straight forward (since v is in the dihedral group G2). Thus the main effort is involved in bounding the degree of Qi(p 0 ). For this purpose we will fix reduced expressions for tω 1 and tω 2 , and construct folding tables that record the possible degrees of Qi(p 0 ).

We now explain the construction of our folding tables, via an analogue of the admissible sets of Lenart and Postnikov [START_REF] Lenart | Affine weyl groups in k-theory and representation theory[END_REF][START_REF] Lenart | A combinatorial model for crystals of Kac-Moody algebras[END_REF]. 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 αi-folded alcove walk of type x starting at v with no folds. Of course p( x, v) may still have bounces, because αi-folded alcove walks are required to say in the strip Ui. Nonetheless, we refer to p( x, v) as the straight walk of type x starting at v. Let Note that I -∪ I + ∪ I * = {1, . . . , n}. We define a function

I -( x, v) = {k ∈ {1, . . . , n} | p( x,
ϕ v x : I -( x, v) → W i 0 × Z as follows. For k ∈ I -( x, v
) let p k be the αi-folded alcove walk obtained from the straight walk 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 walk p k attempts to exit the strip Ui; also note that this notation differs from the partial foldings defined earlier). Let

ϕ v x (k) = the unique (u, n) ∈ W i 0 × Z such that p( x, σ n i u
) and p k agree after the kth step.

Equivalently, (u, n) is the unique pair such that end(p( x, σ n i u)) = end(p k ), and thus σ n i u is simply the end of the straight alcove walk p(rev( x), end(p k )), where rev( x) is the expression x read backwards. Definition 7.4 (Folding table). Fix the enumeration y1, . . . , y6 of W i 0 with ℓ(yj) = j -1 for j = 1, . . . , 6. For each (j, k) with 1 ≤ j ≤ 6 and 1 ≤ k ≤ ℓ(x) define f j,k ( x) ∈ {-, * , 1, 2, 3, 4, 5, 6} by

f j,k ( x) =      -if k ∈ I + ( x, yj) * if k ∈ I * ( x, yj ) j ′ if k ∈ I -( x, yj) and ϕ y j x = (y j ′ , n) for some n ∈ Z.
The αi-folding table of x is the 6 × ℓ(x) array F( x) with (j, k) th entry equal to f j,k ( x).

Remark 7.5. If y is a prefix of y then F( 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 7.6. We will need the αi-folding tables of tω 1 = 0212012121, tω 2 = 021212, and each element b in B0.

For efficiency, we note that 10 of the 12 elements of B0 are prefixes of tω 1 , and one of the remaining elements of B0 is a prefix of tω 2 . Thus the folding tables of these 11 elements of B0 are 'contained' in the folding tables F( tω 1 ) and F( tω 2 ) (see Remark 7.5). The final element of B0 (namely the longest element B0) is b0 = 0212012120 and thus agrees with tω 1 except in the last step. Thus in the tables below we will record the folding tables of tω 1 and b0 simultaneously, with the table for tω 1 obtained by deleting the last column, and the table for b0 obtained by deleting the penultimate column.

The αi-folding tables of tω 1 , tω 2 , and elements of B0 are given in Tables 2 and3 below. This a direct calculation, and we have also automated the process in a computational algebra package.

0 2 1 2 0 1 2 1 2 1 0 1 --------- * - 2 ----- * ----- 3 -1 * -1 -- * 1 2 - 4 2 - * 2 -1 2 * --2 5 3 2 1 3 2 * 3 1 2 4 3 6 1 4 2 1 4 3 1 2 4 * 1 (a) α1-folding table of tω 1 and b0 0 2 1 2 1 2 1 ------ 2 ---- * 1 3 -1 * --- 4 2 - * 2 1 3 5 3 2 1 3 * - 6 1 4 2 1 3 5 (b) α1-folding table of tω 2
Tab. 2: α 1 -folding tables for B 0 ∪ { t ω1 , t ω2 }.

0 2 1 2 0 1 2 1 2 1 0 1 ----------- 2 - * -- * --- * 1 - 3 * -- * -1 * --- * 4 * 1 2 * 1 - * 2 1 3 * 5 1 * 3 1 * 2 1 3 * -1 6 2 3 1 2 3 4 2 1 3 5 2 (a) α2-folding table of tω 1 and b0 0 2 1 2 1 2 1 ----- * 2 - * ---- 3 * -- * 1 2 4 * 1 2 * -- 5 1 * 3 1 2 4 6 2 3 1 2 4 * (b) α2-folding table of tω 2 Tab. 3: α 2 -folding tables of B 0 ∪ { t ω1 , t ω2 }.
The connection between the αi-folding tables and the degree Qi(p) of an αi-folded alcove walk is understood through the notion of ( x, v)-admissible sequences defined below.

Definition 7.7. Let x ∈ W with reduced expression x = si 1 . . . si ℓ and let v ∈ W i 0 . We say that a sequence (k1, . . . , kr)

with 1 ≤ k1 < k2 < . . . < kr ≤ ℓ is ( x, v)-admissible if, for all 0 ≤ j ≤ r -1, kj+1 ∈ I -( x, σ n j i vj ) where (v0, n0) = (v, 0) and (vj , nj ) = ϕ v j-1 x (kj) for j > 0. Proposition 7.8. Let x ∈ W with reduced expression x = si 1 • • • si n and let v ∈ W i 0 .
There is a bijection between the set of all ( x, v)-admissible sequences and the set Pi( x, v).

Proof. It is clear that if p ∈ P( x, v) with v ∈ W i 0 , and if the folds of p occur at indices k1 < k2 < . . . < kr, then J = (k1, . . . , kr) is an ( x, v)-admissible sequence.

Consider the converse. If p = (wt) ℓ r=0 is an αi-folded alcove walk and j ≤ k we write p[j, k] = (wr) k t=j (this is the segment of p between the j th and k th steps). Let J = (k1, . . . , kr) be an ( x, v)-admissible sequence. Define (v0, n0) = (v, 0) and let (vj , nj ) = ϕ v j-1 x (kj). Induction shows that the concatenation of paths

pJ = p( x, v0)[0, k1 -1] • p( x, σ n 1 i v1)[k1, k2 -1] • • • • • p( x, σ nr i vr)[kr, ℓ]
is an αi-folded alcove walk, and that J is the set of indices where the walk pJ folds.

The above proposition encodes how one uses folding tables to compute Qi(p) for all p ∈ Pi( w, u) with u ∈ W i 0 . Let us explain this in an example. In fact we are mainly interested in deg(Qi(p)), and so we consider this below. Let w = t m ω 1 • t n ω 2 where m, n ∈ N, and let u ∈ W i 0 . Let T be the table obtained by concatenating the αi-folding tables of tω 1 and tω 2 with m copies of the tω 1 table followed by n copies of the tω 2 table. The elements of Pi( w, u) correspond to the excursions through T with the properties described below. We begin the excursion by entering the table T at the first cell on row ℓ(u) + 1, and at each step we move to a cell strictly to the right of the current cell 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, 5, 6}.

1) If x =then we move to the (N + 1) st cell of row r. These steps correspond to positive crossings, and have no contribution to deg(Qi(p)).

2) If x = * then we move to the (N + 1) st cell of row r, and we have a contribution of -L(si) to deg(Qi(p)). These steps correspond to bounces on either Hα i ,0 or Hα i ,1.

3) If x = j ∈ {1, 2, 3, 4, 5, 6} then we have two options.

(a) We can move to the (N +1) st cell of row r. These steps correspond to negative crossings, with no contribution to deg(Qi(p)).

(b) We can move to the (N + 1) st cell of row j. These steps correspond to folds, and give a contribution of L(s k ) to deg(Qi(p)), where k ∈ {0, 1, 2} is the entry in the N th cell of the "0-row" (the header) of T .

In the case that N is the last cell of the table, moving to the (N + 1) st cell should be interpreted as exiting the table and completing the excursion. We note that the above process can be regarded as m passes through the αi-folding table of tω 1 , followed by n passes through the αi-folding table of tω 2 , rather than concatenating the m + n tables into one table.

Remark 7.9. In the above explanation, concatenating the folding tables relied on the constituent pieces tω 1 and tω 2 being translations. If w = w1 • w2 is a reduced expression with w1 and w2 not necessarily translations, then one needs to make a correction when combining the individual tables for w1 and w2 into the table for w1 • w2. Specifically, one adds an extra column at the end of the w1 table with j th entry θ i (yjw1). This records the "exit orientation" of the path, and when concatenating the tables for w1 and w2, the rows of the w2 table are permuted so that they match with the exit column of w1. Alternatively, to interpret this process as one pass through w1 followed by one pass through w2 one should simply take the exit column entry of w2 to indicate the row on which to enter the w1 table.

Bounding the degree of matrix coefficients

We are now able to establish bounds on the degree of Qi(p) for all αi-folded alcove walks, from which B2 will follow.

Theorem 7.10. Let p be an αi-folded alcove walk of reduced type. Then deg(Qi(p)) ≤ aπ i where

aπ 1 = a + b if a ≥ 2b, 3b if a ≤ 2b and aπ 2 = 3a -2b if 2a ≥ 3b, a + b if 2a ≤ 3b. Moreover, if p ∈ Pi( w, u) with u ∈ W i 0 is such that deg(Qi(p)) = aπ i then uw ∈ Ui.
Proof. Using the action of σi on αi-folded paths we may assume that p starts at u ∈ W i 0 . We note that if w and w ′ are two reduced expressions for the same element w and if deg(Qi(p)) ≤ aπ i for all p ∈ Pi( w, u), then Theorem 7.2 implies that deg(Qi(p)) ≤ aπ i for all p ∈ Pi( w ′ , u). Thus we are free to choose any reduced expression for w. We choose a reduced expression for w as in (7.2). Let w1 = t m ω 1 • t n ω 2 • b, and decompose p ∈ Pi( w, u) as p = p0 • p 0 where p0 ∈ Pi( v, u) and p 0 ∈ Pi( w1, u0), where u0 = end(p0) ∈ W i 0 . The bounds for Qi(p0) in Table 4 are elementary (the left hand columns represent the elements of W i 0 in the natural order of increasing length). One can now use the folding tables from Example 7.6 to produce bounds for deg(Qi(p 0 )). The following observations make this possible. Firstly, all folding tables for tω 1 , tω 2 , and b with b ∈ B0 have the property that for 1 ≤ j ≤ 6, all entries in the j th row are either -, * , or are strictly smaller than j. This means that with each fold we move to a strictly lower row. Secondly, if one makes a full pass of a table without making any folds (that is, without changing row) then the contribution to deg(Qi(p)) is at most 0 and since the entry and exit rows are the same this pass can be ignored for the purpose of bounding deg(Qi(p)). Thus we may assume that at least one row change is made on each pass through a table, and therefore, by the above observation, we need only consider

w1 = t m ω 1 • t n ω 2 with m + n ≤ 6 and w1 = t m ω 1 • t n ω 2 • b with m + n ≤ 5.
This reduces the work to a finite problem. As a third observation, we note that every row in the α1-folding table of tω 1 , and every row in the α2-folding table of tω 2 , contains a * , and thus these tables tend to have a negative influence on deg(Q1(p)) and deg(Q2(p)), respectively.

u 0 = end(p 0 ) a ≥ b a < b 1 a 3b -2a 2 a 2b -a 3 a 2b -a 4 a b 5 b b 6 0 0 (a) i = 1 u 0 = end(p 0 ) b ≥ a b < a 1 b 3a -2b 2 b 2a -b 3 b 2a -b 4 b a 5 a a 6 0 0 (b) i = 2
Tab. 4: Bounds deg(Q i (p 0 )) where p 0 ∈ P i ( v, u) with u ∈ W i 0 and v ∈ W 0 .

With the above observations in mind we find the bounds on deg(Qi(p)) for p ∈ Pi( w1, u0) with u0 ∈ W i 0 and w1 = t m ω 1 • t n ω 2 • b listed in Table 5 below. We have checked these both by hand, and also implemented the process in MAGMA [START_REF] Bosma | The magma algebra system I: The user language[END_REF]. Moreover we see that if these bounds are attained then if i = 1 then m = 0, and if i = 2 then n = 0 (intuitively this is due to the third observation above).

u 0 a ≥ 2b a ≤ 2b 1 0 0 2 0 max{0, -a + b} 3 b b 4 b 2b 5 a 2b 6 a + b 3b (a) i = 1 u 0 2a ≥ 3b 2a ≤ 3b 1 0 0 2 max{0, a -3b} 0 3 max{0, a -2b} 0 4 2a -3b max{0, a -b} 5 2a -2b b 6 3a -2b a + b (b) i = 2
Tab. 5:

Bounds deg(Q i (p 0 )) where p 0 ∈ P i ( w 1 , u 0 ) with u 0 ∈ W i 0 and w 1 = t m ω1 • t n ω2 • b.
The bounds aπ 1 and aπ 2 follow by combining the bounds in Tables 4 and5.

We now analyse paths such that deg(Qi(p)) = aπ i . We claim that in this case uw ∈ Ui. We have already shown that w = v • t n ω j • b for some v ∈ W0, n ∈ N, and b ∈ B0, where {j} = {1, 2}\{i}. In combining the bounds in Tables 4 and5 we see that if deg(Qi(p)) = aπ i then either:

1) i = 1, a ≥ 2b, and u0 ∈ {3, 4, 5, 6}, or a < 2b and u0 ∈ {4, 5, 6}, or 2) i = 2 and u0 ∈ {5, 6}.

Consider the case i = 1 and a ≥ 2b. If u0 = 6 (that is u6 = s2s1s2s1s2) then deg(Q1(p0)) = 0, and it follows that the walk p0 is straight with no bounces, and thus uv = s2s1s2s1s2 (with u and v as in Table 4). Therefore uw = s2s1s2s1s2t n ω 2 b for some b ∈ B0, and all such elements are obviously in Ui.

Suppose now that u0 = 5. In this case we see that for the bound in Table 4 to be attained we see, by direct observation, that (u, v) = (e, s2s1s2s1s2), (s2, s1s2s1s2), (s2s1, s2s1s2), (s2s1s2, s1s2), or (s2s1s2s1, s2) with the last step of v a fold. Thus uw = s2s1s2s1s2t n ω 2 b for some b ∈ B0, and so again uw ∈ Ui. Suppose now that u0 = 4. Since the bound deg(Q1(p0)) = a in Table 4 is attained we see that (u, v) = (e, s2s1s2s1), (s2, s1s2s1), (s2s1, s2s1), or (s2s1s2, s1) with the last term of v being a fold. Thus uw = s2s1s2s1t n ω 2 b for some b ∈ B0. However an easy check using the folding table shows that if n ≥ 1 then the maximum bound in deg(Q1(p 0 )) is not attained. Moreover, again by the folding tables, we see that b is such that uw = s2s1s2s1b ∈ U1.

The remaining cases are similar. Proof. This is immediate from Corollary 7.3 and Theorem 7.10.

Leading matrices for generic parameters

In this subsection we assume generic parameters. Thus, by our convention, if i = 1 then a = 2b and if i = 2 then 2a = 3b. If p ∈ Pi( w, u) with deg(Qi(p)) = aπ i then p is called a maximal path. In this section we determine all maximal paths, and show that πi(Tw) has a matrix coefficient of maximal degree if and only if w ∈ Γi, for i = 1, 2. Finally, we compute the leading matrices cπ i ,w in terms of Schur functions of type A1 and deduce that B3, B4, and B4 ′ hold.

To tighten the connection between πi and Γi it is convenient to work with the following fundamental domains in Corollary 7.3. Of course, using the action of σi on Ui, the choice of fundamental domain does not change the bounds on deg(Qi(p)). We define

g1 = s2s1s2 if a/b > 2 s2s1s2s1 if a/b < 2 and g2 = e if a/b > 3/2 s1s2s1s2 if a/b < 3/2, (7.3)
and set B ′ i = giBi, where Bi = BΓ i is as in Section 4.3. Then B ′ i is a fundamental domain for the action of σi on Ui, represented as the green region in Figure 9. The blue and red regions are translates of B ′ i by σi, and the "base alcove" gi of B ′ i is heavily shaded. We fix an indexing of B ′ i in Figure 9 in two cases for later use. Generally we write bu = giu for u ∈ Bi, and so Lemma 7.12. Let w ∈ W and u ∈ Bi with i ∈ {1, 2}. Let w be any reduced expression for w. If Pi( w, bu) contains a maximal path then w = u -1 wit N i v for some u, v ∈ Bi and N ∈ N, and hence w ∈ Γi.

B ′ i = {bu | u ∈ Bi}.
Proof. Let p be a maximal path. Thus buw ∈ Ui by Theorem 7.10. Note that the second sentence in the proof of Theorem 7.10 we may choose any reduced expression w for w. We first claim that there is a minimal length (straight) path from bu to buw passing through the element buu -1 wi (geometrically this element is the element "opposite" the base alcove of B ′ i , and is shaded yellow in Figure 10). If no minimal length path passes through buu -1 wi then buw lies in either the red, green, or blue region in Figure 10. It is clear that if buw lies in the red region then deg(Qi(p)) = 0 since there are no negative crossings in the straight path from bu to buw. Thus buw lies in either the green region (that is, B ′ i ) or in the blue region. Hence there are finitely many possibilities for w, and quick check shows that for these w there is no path attaining the degree bound aπ i .

(a) a > 2b (b) a < 2b (c) 3a > 2b (d) 3a < 2b
Thus w admits a reduced expression with u -1 • w1 as a prefix. Since buw lies in Ui it follows that w admits a reduced expression of the form w = u -1 • wi • t N i • v for some u, v ∈ Bi and N ∈ N, and thus w ∈ Γi by the cell factorisation of Section 4.3.

The following Theorem, along with Theorem 7.10 and Lemma 7.12, verifies that πi satisfies B3 for generic parameters. Recall that if w ∈ Γi with generic parameters then w = u -1 w wit τw i vw with uw, vw ∈ Bi and τw ∈ N (we sometimes write τw in place of t τw i by identifying N with {t k i | k ∈ N}).

Theorem 7.13. Let w ∈ Γi with reduced expression w = u -1 w • wi • t τw i • vw. For generic parameters we have:

1) There exist precisely τw + 1 maximal paths in Pi( w, bu w ).

2) For each 0 ≤ n ≤ τw there is a unique maximal path p ∈ Pi( w, bu w ) such that wt i

B ′ i (p) = τw -2n.
3) For all maximal paths we have θ i

B ′ i (p) = vw.
• There is one fold in the w2 part at position 5, followed by 2 folds in the subsequent t2 at positions 3 and 5.

• There are three folds distributed over two consecutive t2 cycles, at position 5 in the pass cycle, and then positions 3 and 5 in the next pass.

The theorem follows in this case in a similar way to the previous example. The two remaining cases are similar.

[cπ i ,w ]u,v = sp | q -1 =0 q -aπ i [πi(Tw)]u,v = {p∈P i ( w,u)|θ i B ′ i (p)=v} ζ wt i B ′ i (p)
(note that there are either no bounces, or precisely two bounces in maximal paths p, and thus Qi(p) is positive, and so q -aπ i Qi(p) specialises to +1). The verification of B4 and B4 ′ follows from cπ i ,w = sτ w (ζ)Eu w,vw in an analogous way to Theorem 6.6.

Leading matrices for non-generic parameters

In this final subsection we compute the leading matrix coefficients for πi with non-generic parameters r = ri. This if i = 1 and r = 2, and if i = 2 and r = 3/2. In fact most of the work has been done in the previous sections, and all that remains is to piece together the paths from the generic regimes on either side of the generic parameter.

Recall the notation of Section 4.4. For ε ∈ {±} we define g ε i as in (7.3) with g - i corresponding to r < ri and g + i corresponding to r > ri. We set

B ′ε i = g ε i B ε i and write B ′ε i = {b ε u | u ∈ W i 0 }.
When working in the case where i = 1 all the matrices will be written in the basis {ξ1 ⊗ X b + u | u ∈ W 1 0 }, and for i = 2 we use the ε =basis.

Theorem 7.15. Let w ∈ Γ1 and r = 2 and let u -1 w + 1 t N 1,+ v be the positive cell factorisation of w. We have

cπ 1 ,w =          (sN (ζ) + sN-1(ζ)) Eu,v if w is of type (+, +); (sN (ζ) + sN+1(ζ)) Eu,v if w is of type (-, -); (1 + ζ -1 )sN (ζ)Eu,v if w is of type (+, -); (1 + ζ)sN (ζ)Eu,v if w is of type (-, +)
where by definition we set s-1(ζ) := 0. If i = 2 the corresponding result applies with all signs reversed.

Proof. The case i = 2 is completely analogous to the case i = 1, and so we only present the i = 1 case. First assume that w is of type (+, +). Then there exists (u, v)

∈ B + 1 ∩ s1B - 1 such that w = u -1 w + 1 t N 1,+ v = u -1 s1w - 1 t N-1 1,-s1v.
According to Theorem 7.13, we see that there will be two families of maximal paths starting at b + u , one with endpoints of the form b + e t N-2r

1,+ v for all 0 ≤ r ≤ N and one with endpoints of the form b - e t N-1-2r

1,-s1v for all 0 ≤ r ≤ N -1. We have wt 1

B ′+ 1 (b + e t N-2r 1,-v) = N -2r, θ 1 B ′+ 1 (b + e t N-2r 1,+ v) = v, wt 1 B ′+ 1 (b - e t N-1-2r 1,- s1v) = N -2r -1 and θ 1 B ′+ 1 (b - e t N-1-2r 1,- s1v) = v.
It follows that cπ 1 ,w = (sN (ζ) + sN-1(ζ)) Eu,v in this case.

Assume that w of type (-, -). Then we have

w = s0s2w + 1 t N 1,+ s2s0 = w - 1 t N+1 1,-.
We see that there will be two families of maximal paths starting at b + s 2 s 0 = b - e t1,-, one with endpoints of the form b + e t N-2r 1,+ s2s0 for all 0 ≤ r ≤ N and one with endpoints of the form b - e t N+2-2r

1,-for all 0 ≤ r ≤ N + 1. We have

wt 1 B ′+ 1 (b + e t N-2r 1,+ s2s0) = N -2r, θ 1 B ′+ 1 (b + e t N-2r 1,+ s2s0) = s2s0, wt 1 B ′+ 1 (b - e t N+2-2r 1,- ) = N + 1 -2r and θ 1 B ′+ 1 (b - e t N+2-2r 1,- ) = s2s0 since b - e t N+2-2r 1,- = b + s 2 s 0 t N+1-2r 1,- . It follows that cπ 1 ,w = (sN (ζ) + sN+1(ζ)) Eu,v in this case.
Assume that w is of type (+, -). Then there exists u ∈ B + 1 ∩ s1B - 1 such that

w = u -1 w + 1 t N 1,+ s2s0 = u -1 s1w - 1 t N 1,-.
We see that there will be two families of maximal paths starting at b + u , one with endpoints of the form b + e t N-2r 1,+ s2s0 and one with endpoints of the form b - e t N-2r

1,-for all 0 ≤ r ≤ N . We have

wt 1 B ′+ 1 (b + e t N-2r 1,+ s2s0) = N -2r, θ 1 B ′+ 1 (b + e t N-2r 1,+ s2s0) = s2s0, wt 1 B ′+ 1 (b - e t N-2r 1,-) = N -2r -1 and θ 1 B ′+ 1 (b - e t N-2r 1,-) = s2s0.
It follows that cπ 1 ,w = (1 + ζ -1 )sN (ζ) in this case.

Assume that w is of type (-, +). Then there exists v ∈ B + 1 ∩ s1B - 1 such that

w = s0s2w + 1 t N 1,+ v = w - 1 t N 1,-s1v.
We see that there will be two families of maximal paths starting at b + s 2 s 0 = b - e t1,-, one with endpoints of the form b + e t N-2r

1,+ v for all 0 ≤ r ≤ N and one with endpoints of the form b - e t N+1-2r

1,-s1v for all 0 ≤ r ≤ N . But we have wt 1

B ′+ 1 (b + e t N-2r 1,+ u) = N -2r, θ 1
B ′+ 1 (b + e t N-2r 1,+ u) = u, wt 1 B ′+ 1 (b - e t N+1-2r 1,- s1v) = N + 1 -2r and θ 1 B ′+ 1 (b - e t N+1-2r 1,- s1v) = v.
It follows that cπ 1 ,w = (1 + ζ)sN (ζ) in this case.

Remark 7.16. Note that the formulae in Theorem 7.15 show how the two leading matrices from the generic regimes on either side of the parameter r = r1 combine to give the leading matrix at r = r1. This suggests an approach to understanding the semicontinuity conjecture of Bonnafé [START_REF] Bonnafé | Semicontinuity properties of Kazhdan-Lusztig cells[END_REF].

We define the following sets which are the sets of non-zero leading matrix coefficients of the elements w of (ε, ε ′ )-type

Bε,ε = {sN (ζ) + sN-1(ζ) | N ≥ 0}, and Bε,-ε = {(1 + ζ ε )sN (ζ) | N ≥ 0}. We will write s ε,ε ′ N ∈ B ε,ε ′ to denote the element corresponding to N in B ε,ε ′ .
The following proposition is useful at a later stage.

Proposition 7.17. Let ε1, ε2, ε3 ∈ {-, +} and k, ℓ ∈ N.

1) We have s

(ε 1 ,ε 2 ) k • s (ε 2 ,ε 3 ) ℓ = m∈N µ m k,ℓ (ε1, ε2, ε3)s (ε 1 ,ε 3 ) m
for some integers µ m k,ℓ (ε1, ε2, ε3). 2) We have µ m k,ℓ (ε1, ε2, ε3) = µ ℓ m,k (ε3, ε1, ε2) for all k, ℓ, m ∈ N. 3) We have µ 0 k,ℓ (ε1, ε2, ε1) = 0 if and only if ε1 = ε2 and k = ℓ.

Proof. By obvious symmetry and commutativity it is sufficient to check the cases (ε1, ε2, ε3) = (+, +, +), (+, +, -), and (+, -, +). We first recall that the Schur functions s λ = s λ (ζ) form an orthonormal basis with respect to the Hall inner product •, • , and in type A1 they are self adjoint with respect to this inner product. Therefore if

s k s ℓ = m c m k,ℓ sm we have c m k,ℓ = s k s ℓ , sm = sms k , s ℓ = s ℓ sm, s k , and thus c m k,ℓ = c ℓ m,k = c k ℓ,m . Furthermore, if ℓ ≤ k we have s k s ℓ = ℓ j=0 s k-ℓ+2j
, and thus c 0 k,ℓ = δ k,ℓ (if k < ℓ then interchange the roles of k and ℓ).

Consider the case (ε1, ε2, ε3) = (+, +, +). Using the formula for Schur functions of type A1 we compute s

(+,+) k = s k (ζ) + s k-1 (ζ) = s 2k (ζ 1/2 ), where we introduce a new formal indeterminant ζ 1/2 with (ζ 1/2 ) 2 = ζ. It follows that s (+,+) k s (+,+) ℓ = s k (ζ 1/2 )s ℓ (ζ 1/2
) can be expressed as a linear combination of sm(ζ 1/2 ), and that the coefficients in this expansion are µ m k,ℓ (+, +, +) = c 2m 2k,2ℓ . Thus (1) and (2) hold, and the "if" part of (3). Consider the case (ε1, ε2, ε3) = (+, +, -). Then

s (+,+) k s (+,-) ℓ = (1 + ζ)(s k + s k-1 )s ℓ = m (c m k,ℓ + c m k-1,ℓ )(1 + ζ)sm, and so µ m k,ℓ (+, +, -) = c m k,ℓ + c m k-1,ℓ . Similarly, µ ℓ m,k (-, +, +) = c ℓ m,k + c ℓ m,k-1
, and thus µ ℓ m,k (-, +, +) = µ m k,ℓ (+, +, -). Consider the case (ε1, ε2, ε3) = (+, -, +). Then

s (+,-) k s (-,+) ℓ = (2s0 + s1)s k s ℓ = m c m k,ℓ (2s0 + s1)sm = m (c m k,ℓ + c m-1 k,ℓ )(sm + sm-1) = m (c m k,ℓ + c m-1 k,ℓ )s (+,+) m ,
and so µ m k,ℓ (+, -,

+) = c m k,ℓ + c m-1 k,ℓ
(here c -1 k,ℓ = 0 by definition). An easy calculation gives µ ℓ m,k (+, +, -) = c ℓ m,k + c ℓ m-1,k , and hence (2). The 'only if' part of (3) also follows.

Thus, finally we have: Theorem 7.18. For each choice of parameters there exists a balanced system of cell representations (πΓ) Γ∈Λ .

Proof. This follows from Theorem 5.1, Theorem 6.6, Corollary 7.11, Corollary 7.14 and Theorem 7.15. Property B5 is checked directly from our formulae for aπ Γ .

We can now explicitly compute Lusztig's a-function for G2.

Corollary 7.19. In type G2 we have a(w) = aπ Γ if w ∈ Γ.

Proof. This follows from Theorem 7.18 and Theorem 2.5.

8 The Plancherel Theorem, conjecture P1, and the Duflo involutions

In this section we prove P1 and compute the set D of Duflo "involutions" for all choices of parameters (and hence see that the elements of D are indeed involutions). The main piece of machinery is the Plancherel Theorem of Opdam [START_REF] Opdam | On the spectral decomposition of affine Hecke algebras[END_REF] and the explicit G2 formulation of this theorem computed by the second author in [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF].

Let us first briefly recall the situation for finite dimensional Hecke algebras. In this case the canonical trace Tr : H → R with Tr( awTw) = ae decomposes as

Tr(h) = π∈Irrep(H) mπχπ(h) for all h ∈ H, (8.1)
where the elements mπ are the generic degrees of H (see [START_REF] Pfeiffer | Characters of finite Coxeter groups and Iwahori-Hecke algebras[END_REF]Chapter 11]). This formula is a crucial ingredient in Geck's proof [START_REF] Geck | On Iwahori-Hecke algebras with unequal parameters and Lusztig's isomorphism theorem[END_REF] of Lusztig's conjectures for spherical type F4. In particular, the observation that the "q-valuation" νq(mπ) (see below) of mπ is equal to 2aπ played a central role in Geck's proof.

There is an analogue of (8.1) for affine Hecke algebras in the form of the remarkable Plancherel formula of Opdam [START_REF] Opdam | On the spectral decomposition of affine Hecke algebras[END_REF] (see also Opdam and Solleveld [START_REF] Opdam | Discrete series characters for affine Hecke algebras and their formal degrees[END_REF]). The summation in (8.1) becomes an integral over irreducible representations of a C * -algebra completion of H, and the generic degrees become the Plancherel measure dµ.

In this section we recall the explicit formulation of the Plancherel formula in type G2 computed by the second author in [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF], and show that in this case there is an analogue of the formula νq(mπ) = 2aπ in terms of the Plancherel measure. We will use this observation to prove P1, P7, and compute the set D. Along the way we will also introduce the asymptotic Plancherel measure (which we believe is a new, although it appears to be related to recent work of Braverman and Kazhdan [START_REF] Braverman | Remarks on the asymptotic Hecke algebra[END_REF]) and show that this measure induces an inner product on Lusztig's asymptotic algebra J (at least in type G2). We believe that these observations provide an intriguing connection between Kazhdan-Lusztig cells and the Plancherel formula -see also the conjectures listed at the end of Section 9.

The Plancherel formula

The main references for this section are [START_REF] Opdam | On the spectral decomposition of affine Hecke algebras[END_REF] and [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF]. The Plancherel Theorem is an analytic concept, and therefore we now take a slightly different view of the affine Hecke algebra. We extend the scalars to C, and specialise q to a real number q > 1. Thus H is now an algebra over C. We write Tw and Cw for the images of the standard basis and Kazhdan-Lusztig basis elements in H. Note also that the representations HΥ for any right cell Υ can naturally be regarded as representations of the Hecke algebra H defined over C by extending scalars and specialising q.

Let (π, V ) be a finite dimensional H-module (now over C). Recall that 2 ) defines a Hermitian inner product on H. Let h 2 = (h, h) be the ℓ 2 -norm. The algebra H acts on itself by left multiplication, and the corresponding operator norm is h = sup{ hx 2 : x ∈ H, x 2 ≤ 1}. Let H denote the completion of H with respect to this norm. Thus H is a non-commutative C * -algebra. The irreducible representations of H are the (unique) extensions of the irreducible representations of H that are continuous with respect to the ℓ 2 -operator norm, and it is known that these are the irreducible tempered representations of H (see [29, §2.7 and Corollary 6.2]). In particular, every irreducible 9.1 The conjectures P4, P7-P12, and P14 Knowing the value of Lusztig's a-function (from Corollary 7.19), and the partition of W into cells (from Figure 2), it is elementary that P4, P9-P12 and P14 hold. We prove P7 and P8 in the following theorem (note that we obtained a different proof of P7 in Corollary 8.15). Theorem 9.1. Let x, y, z ∈ W . 1) We have γ x,y,w -1 = 0 if and only if x ∼R w, y ∼L w and x ∼L y -1 .

V = ζ∈Hom(P,C × ) V gen ζ where V gen ζ = {v ∈ V | for each λ ∈ P we have (X λ -ζ λ ) k v = 0 for some k ∈ N} is the generalised ζ-weight space of V . Let supp(π) = {ζ ∈ Hom(P, C × ) | V gen ζ = {0}} be the support of (π, V ). A representation (π, V ) is tempered if |ζ λ | ≤ 1 for all ζ ∈ supp(π)
2) We have γx,y,w = γy,w,x = γw,x,y.

Proof. Let w ∈ Γ and x, y ∈ W be such that γ x,y,w -1 = 0. In the case where Γ ∈ Λ f , the result can be checked by explicit computations using the (finite) set of matrices BΓ := {cπ Γ ,w | w ∈ Γ}. For example to prove (1), we simply need to check that if cπ Γ ,w appears in the expansion of cπ Γ ,xcπ Γ ,y in the basis BΓ then we have x ∼R w, y ∼L w and x ∼L y -1 . This is easily checked using the explicit matrices provided on the authors' webpages.

Assume that Γ ∈ Λ∞ and that r is generic for Γ. By Theorem 6.6 and Corollary 7.14, the equality cπ Γ ,xcπ Γ ,y = z γ x,y,z -1 cπ Γ ,z becomes sτ x Eu x ,vx • sτ y Eu y ,vy = z∈Γ γ x,y,z -1 sτ z Eu z ,vz .

Since γ x,y,w -1 = 0, the term indexed by w on the righthand side is nonzero and this implies that the whole sum is nonzero by B4. It follows that the lefthand side is nonzero hence it is equal to sτ x sτ y Eu x,vy and we have vx = uy (or in other words x ∼L y -1 ). From there we see that if γ x,y,z -1 = 0 then we must have (a) uz = ux and vz = vy and (b) c τz τx ,τy = 0 where c τz τx,τy = sτ x sτ y , sτ z . In particular, since γ x,y,w -1 = 0 we have uw = ux and vw = vy or in other words x ∼R w and y ∼L w. This completes the proof of (1).

We now show that γx,y,w = γy,w,x = γw,x,y. We may assume that γx,y,w = 0 since if γx,y,w = 0 then γw,x,y = γy,w,x = 0 by (1). We know that γx,y,w is the coefficient of sτ w -1 in the product sτ x sτ y , which is equal to the coefficient of sτ w since by Remark 4.2 we have sτ w -1 = sτ w . Then using standard results on Weyl characters we get that γx,y,w = γw,x,y = γy,w,x. Consider the case where r is not generic for Γi, with i ∈ {1, 2}. Consider the case i = 1, and so r = 2 (the case i = 2 is similar). Recall the notation of Theorem 7.15. Let x be of (ε1, ε2) type, and let y be of (ε ′ 2 , ε3) type. If ε2 = ε ′ 2 then γx,y,z = 0 (this follows from Theorem 7.15 and the cell factorisation in Section 4.4). Thus suppose that ε2 = ε ′ 2 . Moreover, if γx,y,w = 0 then w -1 is of type (ε1, ε3). Then γx,y,w is the coefficient of s

(ε 1 ,ε 3 ) τ w -1
in the expansion of

s (ε 1 ,ε 2 ) τx s (ε 2 ,ε 3 ) τy in the (ε1, ε3) 'basis'. Similarly γw,x,y is the coefficient of s (ε 3 ,ε 2 ) τ y -1
in the expansion of s

(ε 3 ,ε 1 ) τw s (ε 1 ,ε 2 ) τx
. Hence by Proposition 7.17 we have γx,y,w = γw,x,y.

Hence P7 and P8 are proven. 2) If d ∈ D and x, y ∈ W are such that γ x,y,d = 0 then x = y -1 (hence P2 holds).

3) If y ∈ W , there exists a unique d ∈ D such that γ y -1 ,y,d = 0 (hence P3 holds). Proof. The first statement follows immediately from the explicit calculation of D given in Theorem 8.13. For the remaining statements, note that if Γ ∈ Λ f then the results can be proved by explicit matrix calculations, and thus we will focus here on the case where Γ ∈ Λ∞. Let d ∈ DΓ and assume that r is generic for Γ. Let x, y ∈ W be such that γ x,y,d = 0. We have the equality sτ x Eu x ,vx • sτ y Eu y ,vy = γ x,y,z -1 sτ z Eu z ,vz .

Arguing as in the proof of Theorem 9.1 we obtain:

• the lefthand side is equal to sτ x sτ y Eu d ,u d ;

• If γ x,y,z -1 = 0 then uz = u d = vz and c τz τx ,τy = 0. In particular since τ d = 0 we have c 0 τx ,τy = 0 which implies that τx = τy. Finally we have

x -1 = (u -1 x wiτxvx) -1 = v -1 x τ -1 x wiu -1 y = u -1 y wiτyvy = y as required in (2). In the case where r is not generic, we can argue in the same fashion using the result of Proposition 7.17 to get that τx = τy. Hence (2).

Let y ∈ W and let Γi ∈ Λ∞ be such that y ∈ Γi. If r is generic for Γi then setting d = v -1 y wivy we easily see arguing as above that γ y -1 ,y,d = 1 since c 0 τy ,τy = 1. In the case where r = 2 and y ∈ Γ1 then we have using Proposition 7.17

• if y is of type (ε, -) then γ y -1 ,y,d = 1 where d = s0s2s0;

• if y is of type (ε, +) then γ y -1 ,y,d = 1 where d = v -1 y s1s0vy. The case r = 3/2 and y ∈ Γ2 is similar. The statements (3), ( 4) and (5) follow readily.

The conjecture P15

We now prove P15. The technique here is somewhat different to the proofs of P2-P14 given above, and relies on the process of generalised induction introduced by the first author in [START_REF] Guilhot | Generalized induction of Kazhdan-Lusztig cells[END_REF]. An alternative proof of P15 can also be found in [START_REF] Xie | A decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2[END_REF]Theorem 6.2].

In order to present a uniform proof of Theorem 9.5, we will only consider the two-sided cells Γi for which there is a cell factorisation. Thus the proof below applies to the infinite cells Γi, i = 0, 1, 2, and also all finite cells except for Γ3 with r ≤ 1 and Γ4 with r ≥ 1 (see Remark 4.1). In these remaining cases we have checked P15 by explicit computations. We also present in Remark 9.6 an indication of how P15 can be proved in the interesting case Γ3 when r = 1 (where there is no cell factorisation).

Let Υ be the right cell in Γi that contains wi. In the case where r is not generic, we assume that wi = w + i or w - i and we choose the positive or negative cell factorisation in the following definitions. To lighten the notation, we will not write the superscript ± when it is clear from the context. Most of the equalities in this section will hold modulo H> LR Γ i and we sometimes write simply ≡ and omit mod H> LR Γ i .

We set T0 = {tω 1 , tω 2 }, Ti = {ti} for i = 1, 2, 3 and write Pi for the set of monomials in the variables Ti (see Remark 4.1 for the case i = 3). When i > 3 we simply set Pi = {e}. One can verify that Υ = {wiτ v | τ ∈ Pi, v ∈ Bi} where the set Bi and wi have been defined in Sections 4.2 and 4.3. For all x = τ v there exists an element H(x) ∈ H such that Cw i H(x) ≡ Cw i x and H(x) ∈ Tx + y<x,y∈X i q -1 Z[q -1 ]Ty.

These elements can be constructed using the induction process; see [START_REF] Geck | On the induction of Kazhdan-Lusztig cells[END_REF]Proposition 4.3] and the references therein. Using the anti-involution ♭, we easily see that H(x) ♭ Cw i ≡ C x -1 w i . For τ ∈ Pi, v ∈ Bi and x = τ v we define

hτ = H(tω 1 ) m H(tω 2 ) n if τ = mω1 + nω2 ∈ P0 H(ti) n if τ = t n i ∈ Pi and hx = hτ H(v).
It is important to notice here that we do not have hτ = H(τ ). Some basic properties of these elements are presented in Section [16, §4] where hx is denoted P(x). In particular, it is shown that the R-module of residues modulo HΓ i generated by {Cw i hτ hv | τ ∈ Pi, v ∈ Bi} is a right H-module. We set

Cw i hv • Tw ≡ τ ∈P i ,v ′ ∈B i ν τ,v ′ v,w Cw i hτ h v ′ and TwC u -1 w i ≡ τ ∈P i ,v ′ ∈B i λ τ,u ′ v,w h ♭ u ′ h ♭ τ Cw i .
Since Cw i hτ = h ♭ τ Cw i , we get that the R-module of residues modulo HΓ i generated by {h ♭ u Cw i hτ hv | τ ∈ Pi, u, v ∈ Bi} is a two-sided H-module. Further the coefficient λ and ν completely determined the structure of this module. Indeed we have

h ♭ u Cw i hτ hvTw = h ♭ u h ♭ τ Cw i v Tw = τ ′ ∈P i ,v ′ ∈B i ν τ ′ ,v ′ v,w h ♭ u h ♭ τ ′ Cw i hτ h v ′ = τ ∈P i ,v ′ ∈B i ν τ,v ′ v,w Cw i h τ +τ ′ h v ′ .
A similar formula holds for left multiplication.

Remark 9.3. In the case where r is generic, it is shown in [START_REF] Guilhot | Affine cellularity of affine hecke algebras of rank two[END_REF]Proposition 4.6] that the two-sided H-module defined above is in fact equal to the two-sided cell module HΓ i .

Proposition 9.4. Let u, v ∈ Bi and τ ∈ Pi. We have h ♭ u Cw i hτ hv ≡ z∈Γ i azCz where az ∈ Z.

Proof. We start by proving the result when u and v are equal to the identity. By a straightforward induction, it is enough to show that Cw i τ ht is Z-linear combination of Kazhdan-Lusztig elements. Here t = tω 1 or tω 2 if i = 0 and t = ti if i = 1, 2, 3. Since Cw i τ ht ≡ Cw i H(τ )ht ≡ H(τ ) ♭ h ♭ t Cw i mod HΓ i we obtain that Cw i τ ht ≡ τ ′ ∈P i b τ ′ C τ ′ w i where b τ ′ ∈ R; see [START_REF] Guilhot | Affine cellularity of affine hecke algebras of rank two[END_REF]Lemma 4.3]. We now show that the coefficients b τ ′ lie in Z. The following argument is inspired by [38, Proof of Theorem 6.2]. We have hw i ,w i ,w i Cw i τ ht ≡ H(τ ) ♭ Cw i Cw i ht ≡ C τ -1 w i Cw i t ≡ z∈Γ i h τ -1 w i ,w i t,z Cz ≡ τ ′ ∈P i b τ ′ hw i ,w i ,w i C τ ′ w i mod HΓ i which implies that bz ∈ Z since deg(hw i ,w i ,w i ) = a(wi) and deg(h τ -1 w i ,w i t,z ) ≤ a(z) = a(wi) by P11.

In order to prove the proposition, it is now enough to show that h ♭ u Cw i τ hv is a Z-linear combination of Kazhdan-Lusztig element. We start by proving the result in the generic case. By the generalised induction process [START_REF] Guilhot | Generalized induction of Kazhdan-Lusztig cells[END_REF] and explicit computations in G2 [START_REF] Guilhot | Kazhdan-Lusztig cells in affine Weyl groups of rank 2[END_REF] we have (1) h ♭ u Cw i τ hv ≡ C u -1 w i τ hv; (2) h ♭ u Cw i τ hv ≡ τ ′ ∈P i b τ ′ C u -1 w i τ ′ v where b τ ′ ∈ R. The first statement was the key fact in [START_REF] Guilhot | Kazhdan-Lusztig cells in affine Weyl groups of rank 2[END_REF] to determine the partition of G2 into cells. Then multiplying C u -1 w i τ hv by hw i ,w i ,w i we can conclude as above that b τ ′ ∈ Z.

The case where r is not generic for Γi is more delicate. We will only treat the case i = 1 and write t-for the translation t1,-and t+ for t1,+. In the case where u, v ∈ B + 1 ∩ s1B - 1 we can proceed exactly as above since (1) and (2) still hold. Next we can show by explicit computations that

h ♭ u C w + i hv ≡ C u -1 w + i v if u ∈ B + 1 ∩ s1B - 1 , C w - i + C w - i t - if u = v = s2s0
so that the result holds in this case. Assume that τ = t n + with n ≥ 1 and u = s2s0. We have

h ♭ s 2 s 0 C w + i t n + hv ≡ h ♭ s 2 s 0 C s 1 w - 1 t n-1 - s 1 hv ≡ h ♭ s 2 s 0 C s 1 w - 1 H(t n-1 -s1)hv ≡ C w - i + C w - i t -H(t n-1 -s1
)hv (by explicit computations)

≡ C w - i t n-1 - s 1 + C w - i t n -s 1 hv ≡    C s 0 s 2 w + i t n-1 + v + C s 0 s 2 w + i t n + v if v ∈ B + 1 ∩ s1B - 1 , C w - 1 t n-1 - + 2C w - 1 t n - + C w - 1 t n+1 - if u = v = s2s0.
The case where τ = t n + with n ≥ 1, v = s2s0 and u ∈ B + 1 ∩ s1B - 1 can be dealt with using the ♭ anti-involution. The case where i = 2 and r = 3/2 is completely analogous, based on the equalities

h ♭ s 1 s 2 s 1 s 2 C w - 2 hs 2 s 1 s 2 s 1 ≡ C w + 2 + C w + 2 t 2,+ and h ♭ s 1 s 2 s 1 s 2 C s 0 w - 2 ≡ C w - 2 + C w - 2 t 2,-.
Theorem 9.5. Let Γ be a two-sided cell of G2 and let x, y, w, w ′ ∈ W be such that x, y ∈ Γ. Then z∈W h x,w ′ ,z ⊗ hw,x,y = z∈W h x,w ′ ,y ⊗ hw,x,z.

In other words, Conjecture P15 holds.

Proof. First, we remark that the sum is in fact over z ∈ Γ. Indeed, if there is a non-zero term in the left sum, that is h x,w ′ ,z ⊗ hw,z,y = 0 then h x,w ′ ,z = 0 which implies that z ≤R x and hw,z,y = 0 which implies that y ≤L z. Then we have a(z) ≥ a(x) and a(y) ≥ a(z) so that a(y) = a(x) = a(y) since x, y lie in the same cell. In turn, using P9 and P10 we get that z ∼R x and y ∼L z and therefore z ∈ Γ.

Next, following [12, Remark 2. Then, P15 states that E is a two-sided (H1, H2)-bimodule.

We have seen the set of residues modulo H< LR Γ of the form h ♭ ux Cw i hτ x hv x is a two-sided submodule of the cell module HΓ i . The right action (respectively the left action) of H on this basis only depends on vx (respectively on ux) and is determined by the coefficient λ and ν. By Proposition 9.4, we can define a submodule E ′ of E with basis {e ′ uxw i τxvx | x ∈ Γi} and with action of H1 and H2 defined by x w i τxτ b .

In the non-generic case, we defined two-submodules E ε = {e ′ uε (x)w ε i τε(x)vε(x) | x ∈ Γ ε i } where ε = ±. From there it is easy to see that the submodule of E ′ is a two-sided (H1, H2)-module since the coefficent of e ′ z in Tw • (e ′ x • T w ′ ) and in (Tw • e ′

x ) • T w ′ are equal to τ,τ ′ ∈P i τ +τ ′ +τx=τz ν vz ,τ ′ vx ,w ⊗ λ uz ,τ ux,w .

When the parameter r is generic for Γi, this conclude the proof since the submodule E ′ is equal to E . When the parameter r is not generic, we also get the result since E + + E -= E .

P5.

  If d ∈ D, y ∈ W , and γ y -1 ,y,d = 0, then γ y -1 ,y,d = n d = ±1. P6. If d ∈ D then d 2 = 1.

Example 2 . 2 .

 22 Let W be an affine Weyl group of type G2 with diagram and weight function defined by a, b are positive integers. Let I ⊆ S be a union of conjugacy classes in S. We define the one dimensional representation ρI of W by ρI (Ts) = qs if s ∈ I, -q -1 s otherwise.

  on the positive side of the hyperplane separating v k-1 and v k-1 si k . The end of p is end(p) = v ℓ . Let P( w, v) = {all positively folded alcove walks of type w starting at v}. Less formally, a positively folded alcove walk 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): s-crossing) If p has no folds we say that p is straight. If p is a positively folded alcove walk we define, for each s ∈ S, fs(p) = #(positive s-folds in p) and Q

Proposition 3 . 2 .

 32 (c.f.[START_REF] Ram | Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux[END_REF] Theorem 3.3]) Let w, u ∈ W , and let w be any reduced expression for w. ThenXuTw = p∈P( w,u) Q(p)X end(p) .

Fig. 2 :

 2 Fig. 2: Partition of G2 into Kazhdan-Lusztig cells, r = a/b

  This expression is called the cell factorisation of w ∈ Γ0. It should be understood in the following way: The element u -1 indicates in which connected component (right cell) of Γ0 the alcove w lies. The element λ indicates in which translate of the box u -1 w0BΓ 0 the alcove w lies. The element v indicates location of w in the box u -1 w0t λ BΓ 0 . Note that the translates of BΓ 0 cover W . An analogue of the above cell factorisation applies to the lowest two-sided cell in arbitrary type, see [37, Proposition 4.3] and [1, Proposition 3.1].

Fig. 3 :

 3 Fig. 3: The lowest two-sided cell Γ 0

(a) r > 2 (b) r < 2 (c) The box BΓ 1 Fig. 4 :

 2214 Fig. 4: The green cell Γ 1 in generic regimes (r = 2)

( 2 Fig. 5 :

 25 Fig. 5: The blue cell Γ 2 in generic regimes (r = 3/2)

Theorem 5 . 1 .

 51 Let Γ be a finite two-sided cell. If (Γ, r) = (Γ3, 1) let Υ be any right cell contained in Γ and let ρ ∼ Υ. If (Γ, r) = (Γ3, 1) let ρ be the direct sum of the cell representations for each of the right cells contained in Γ. Then ρ satisfies B1-B4 and B4 ′ with aρ = ãΓ. Thus ρ is Γ-balanced over R.

•

  When r > 2, we have ρI ∼ Γ5 where I = {0, 2}. We have Γ5 ≤ LR = Γ5 ∪ Γ4 ∪ Γe and by direct calculation max deg(ρ(Γ5)) = 3b and max deg(ρ(Γ4)) = 2b. This shows that aρ = 3b = ãΓ 5 and hence B3. • When 3/2 > r > 1, we have ρI ∼ Γ7 where I = {1}. We have Γ7 ≤ LR = Γ7 ∪ Γ3 ∪ Γ4 ∪ Γe and max deg(ρ(Γ7)) = 3a -2b, max deg(ρ(Γ3)) = 2ab, and max deg(ρ(Γ4)) = -b.

  Γe and we check by direct computation that max deg(ρ(Γ6)) = a + b, max deg(ρ(Γ3)) = a, and max deg(ρ(Γ4)) = b.

  right H-module, called the principal series representation with central character ζ = (ζ1, . . . , ζn).

6. 1

 1 Path formula for the principal series representation π 0 Let B be any fundamental domain for the action of the group Q of translations on the set of alcoves (for example, both W0 and B0 are fundamental domains). Thus any w ∈ W can be written uniquely as w = tµu for some u ∈ B, and we set wt B (w) = µ and θ B (w) = u. If p is a positively folded alcove walk we write wt B (p) = wt B (end(p)) and θ B (p) = θ B (end(p)).

4 )

 4 and β1 and β f (p) are simple roots. We have deg(Q(p)) ≤ L(w0) with equality if and only if f (p) = ℓ(w0).

Corollary 6 . 3 .

 63 The representation π0, equipped with any basis of the form {ξ0 ⊗ Xu | u ∈ B} with B a fundamental domain for the action of Q on W , satisfies B2 with aπ 0 = L(w0).

Finally, to

  check B4 ′ , let w ∈ Γ0, and let x = u -1 w w0 ∈ Γ0 and y = w0τwvw ∈ Γ0. Then cπ 0 ,xcπ 0 ,y = s0(ζ)sτ w (ζ)Eu w ,eEe,vw = sτ w (ζ)Eu w,vw = cπ 0 ,w , completing the proof.We note that the above theorem recovers a result of Xie[START_REF] Xie | The based ring of the lowest generalized two-sided cell of an extended affine Weyl group[END_REF] Corollary 5.4].

  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 Hα i ,0 or Hα i ,1. The end of p = (v0, . .. , v ℓ ) is end(p) = v ℓ .Less formally, αi-folded alcove walks 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

4 )

 4 If end(p) + | -end(p)s with end(p)s / ∈ U1 then necessarily end(p) ∩ end(p)s is a face of Hα 1 ,0 (since the crossing is negative). Using the formula Ts = T -1 s

  v) makes a negative crossing at the kth step}I + ( x, v) = {k ∈ {1, . . . , n} | p( x,v) makes a positive crossing at the kth step} I * ( x, v) = {k ∈ {1, . . . , n} | p( x, v) bounces at the kth step}.

Corollary 7 . 11 .

 711 Let i ∈ {1, 2}. For generic parameters the representation πi, equipped with any basis of the form {ξi ⊗ Xu | u ∈ B} with B a fundamental domain for the action of σi on Ui, satisfies B2 with aπ i as in Theorem 7.10.

Fig. 9 :

 9 Fig. 9: The set B ′ i and translates by σ i .

Fig. 10 :

 10 Fig. 10: Configuration for Lemma 7.12

Corollary 7 . 14 .

 714 Let w ∈ Γi with generic parameters. Then cπ i ,w = sτ w (ζ)Eu w ,vw , where s k (ζ) is the Schur function of type A1. Thus πi satisfies B4 and B4 ′ . Proof. Let Pi( w, uu) = {p ∈ Pi( w, uu) | deg(Qi(p)) = aπ i } be the set of maximal paths. By Corollary 7.3 and the definition of cπ i ,w we have

Theorem 7 .

 7 13 gives {Pi( w, u) | θ i B ′ i (p) = v} = ∅ unless u = uw and v = vw, and thus [cπ i ,w ]u,v = 0 unless u = uw and v = vw. Moreover Theorem 7.13 gives [cπ i ,w ]u w ,vw = τw n=0 ζ 2τw -n = sτ w (ζ).

  and all λ ∈ P + , and it is square integrable if |ζ λ | < 1 for all ζ ∈ supp(π) and all λ ∈ P + \{0}. Define an involution * on H and the canonical trace functional Tr : H → C by w∈W awTw * = w∈W aw T w -1 and Tr w∈W awTw = ae where now aw denotes complex conjugation. An induction on ℓ(v) shows that Tr(TuT * v ) = δu,v for all u, v ∈ W , and hence Tr(h1h2) = Tr(h2h1) for all h1, h2 ∈ H. It follows that (h1, h2) = Tr(h1h *

9. 2

 2 The conjectures P2, P3, P5, P6, and P13 We now consider the conjectures involving the set D, computed in Theorem 8.13. Note that for the infinite cells Γ, if d ∈ DΓ then u d = v d and τ d = 0. Therefore we have c π Γ ,d = s0(ζ) for all d ∈ D ∩ (Γ0 ∪ Γ1 ∪ Γ2), this will be of crucial importance in the proof below. Theorem 9.2. We have the following. 1) If d ∈ D then d 2 = 1 (hence P6 holds).

4 )

 4 If d ∈ D, y ∈ W , γ y -1 ,y,d = 0, then γ y -1 ,y,d = n d = ±1 (hence P5 holds).5) Any right cell Υ of W contains a unique element d ∈ D. We have γ x -1 ,x,d = 0 for all x ∈ Υ (hence P13 holds).

( 1 ⊗

 1 3.7], we note that Conjecture P15 is in fact a statement of a certain bimodule structure. To see this, consider the ring A := R ⊗ Z R and let E be a free A-module with basis {ew | w ∈ Γ} where Γ is a two-sided cell of W . Let H1 := A ⊗ R H where R is embedded into A via a -→ 1 ⊗ a and H2 := H ⊗ R A where R is embedded into A via a -→ a ⊗ 1. We can define a left H1-action and a right H2-action by Cw • ex = z∈Γ hw,x,z)ez and ex • Cw = z∈Γ (hx,w,z ⊗ 1)ez.

Tw

  • e ′ x = (b,τ )∈B i ×P i (1 ⊗ λ b,τ ux,w )e ′ b -1 w i τxτ vx and e ′ x • Tw = (b,τ )∈B i ×P i (ν b,τ vx,w ⊗ 1)e ′ u -1

• Each right cell Γ k contains a unique element w Γ k of minimal length.• There exists a unique element wΓ ∈ Γ of maximal length subject to the conditions that wΓ lies in a finite parabolic subgroup of W and wΓ is a suffix of each w Γ k for all 1 ≤ k ≤ 6. The element wΓ is called the generating element of Γ. We setu k = wΓw -1 Γ k for all k and BΓ = {u k | 1 ≤ k ≤ 6}. • There exists tΓ ∈ Γ such that Γ = {u -1 wΓt n Γ v | u, v ∈ BΓ, n∈N}, and moreover each w ∈ Γ has a unique expression in the form w = u -1 wΓt n Γ v with u, v ∈ BΓ and n ∈ N.

This work was partially supported by the regional project MADACA. Moreover, both authors would like to thank the Université de Tours for its invited researcher program under which a significant part of this research was undertaken. 1

Proof. Write u = uw, v = vw, and N = τw. Let p be maximal. We claim that there are no folds in the initial u -1 segment. This is easily checked directly in each case. For example, consider i = 1 and a > 2b, and suppose that u = s2s1s2s0. Then bu = s0 is the "top right" element of B ′ 1 . Suppose that the path p of type u -1 • w1 • t N 1 • v folds in the initial u -1 part. If this fold occurs on the 4 th step, then the remainder of the path consists of positive crossings only, and hence has degree b. If the fold occurs on the 3 rd step, then the 4 th , 5 th , and 6 th steps (the last two coming from w1 = s0s1) are forced to be, respectively, a positive crossing, a positive crossing, and a bounce. After this the path consists of positive crossings (and perhaps bounces) and so the degree is bounded by ab. The remaining cases are similar.

Writing p = p0 •p 0 , with p0 corresponding to the initial u -1 segment, the previous paragraph shows that deg(Qi(p0)) = 0, and that p 0 starts at end(p0) = buu -1 = gi (the "base" alcove of B ′ i ). Note that p 0 has type w1 = wi • t N i • v. Consider the case i = 1 and a > 2b. We construct the α1-folding tables of the elements w1, t1, and v ∈ B ′ 1 in Table 6 below. We construct these tables with respect to the fundamental domain B ′ i (rather than W i 0 ), and thus we modify the definition of ϕ v x (k) given in the previous section (and in the notation of that section) to be ϕ v x (k) = the unique (u, n) ∈ B ′ i × Z such that p( x, σ n i u) and p k agree after the kth step.

Note that the elements of B ′ 1 are the prefixes of s2s1s2s0, along with the element v ′ = s2s0, and so it suffices to provide the tables for these two elements of B ′ 1 . These are given in Table 6 below. See Remark 7.9 for the meaning of the final "exit columns" in these tables, and note that we use the indexing of B ′ 1 as shown in Figure 9.

Tab. 6: α 1 -folding tables for

Note that a path p 0 of type w1 • t N 1 • v starting at g1 enters the w1 table on row 1, and it is then elementary to check that such a path is maximal if and only if it either folds at both places of the w1-part, or at both places of the s1s0 part of t1 in on of the passes of t1. That is,

v if the two folds occur in the n th pass of t1, where as usual ŝj indicates that the term is omitted. In the first case we have wt 1

In the second case the equality w1t n-1

s2t n 1 = e for all n shows that wt 1

This establishes the theorem in this case. Now consider the case i = 2 with 3a > 2b. The α2-folding tables of w2 = s1s2s1s2s1 and t2 = s0s2s1s2s1 and v ′ ∈ B ′ 2 with respect to B ′ 2 are given in Table 7 below (note that every element of

1 2 1 2 1

Tab. 7: α 2 -folding tables for w 2 , t 2 , and v ′ ∈ B ′ 2 with respect to B ′ 2 , in regime 3a > 2b.

Using these tables it is easy to check that a path p 0 of type w2 • t N 2 • v starting at g2 is maximal if and only if one of the following occur (recall we enter the w2 table on row 1):

• There are three folds in the w2 part, at positions 1, 3, 5 (and hence no further folds).

representation of H is finite dimensional (since every irreducible representation of H has degree at most |W0|), and it follows from the general theory of traces on "liminal" C * -algebras that there exists a unique positive Borel measure µ, called the Plancherel measure, such that (see [6, §8.8])

The Plancherel formula has been obtained in general by Opdam [START_REF] Opdam | On the spectral decomposition of affine Hecke algebras[END_REF]. We now recall the explicit formulation in type G2 from [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF]. We first describe the representations that appear in the Plancherel formula.

We define the representations π0, π1, and π2 as in Sections 5.2 and 5.3, however now the ring of scalars is C, and ζ ∈ Hom(P, C × ) in the case π0, and ζ ∈ Hom(Z, C × ) in the cases π1 and π2. To emphasise the dependence on the central character ζ we write πi = π ζ i for i = 0, 1, 2, and we write χ ζ i for the corresponding characters. These representations are tempered if and only if |ζ λ | = 1 for all λ ∈ P (in the case i = 0) and |ζ n | = 1 for all n ∈ Z (in the cases i = 1, 2). Therefore the tempered representations correspond to ζ ∈ T 2 (in the case i = 0) and ζ ∈ T (in the case i = 1, 2), where

Let π3 = ρ ∅ be the 1-dimensional representation of H with π3(Tj ) = -q -L(s j ) for j = 0, 1, 2 (using the notation of Example 2.2). Let π4 = ρ + 3 and π5 = ρ - 3 be the two three dimensional irreducible representations constructed in Proposition 5.3, and let π6 ∼ Γ4 in the case r = 1, and let π6 = ρ ′′ 3 in the case r = 1 (recall the ∼ notation from Section 5 and the definition of ρ ′′ 3 from Proposition 5.3). Finally, let π7 be the following representation, depending on the parameter regime (if r ∈ {3/2, 2} then π7 is not defined, and does not appear in the Plancherel Theorem below):

Let χ3, . . . , χ7 be the characters of the above representations.

We now describe the Plancherel measure. Let ω = e 2πi/3 and define functions cj (ζ), j = 0, 1, 2, by

) .

(We note that there is a change in the formulae for c1(ζ) and c2(ζ) from those in [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF] to reflect the fact that our representations π ζ 1 and π ζ 2 are related to the representations in [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF] by

) .

Theorem 8.1 (Plancherel Theorem for G2, [31, Theorem 4.7]). For each h ∈ H we have

where dζ the normalised Haar measure on the group T (thus

Remark 8.2. The representations π4, . . . , π7 were constructed differently in [START_REF] Parkinson | On calibrated representations and the Plancherel Theorem for affine Hecke algebras[END_REF], however it is an easy exercise to verify that they are isomorphic to the representations given above.

The Plancherel formula and cell decompositions

It is convenient to group the representations that appear under integral signs in the Plancherel formula (Theorem 8.1) into "classes"

The remaining representations (the "point masses" in the Plancherel formula) are taken to be in their own classes: Πj = {πj } for 3 ≤ j ≤ 7. We make the following observation comparing the cell decomposition and the Plancherel formula in type G2. Proposition 8.3. For each parameter regime there is a well defined surjective map Ω : {Πj | 0 ≤ j ≤ 7} given by

and πj is a submodule of a cell module HΥ for some finite right cell Υ ⊆ Γ.

Proof. This follows immediately by comparing the Plancherel formula and the cell decomposition, using Proposition 5.3. For example, if 2 > r > 3/2 we have Ω(Π3) = Γe, Ω(Π4) = Ω(Π5) = Γ3, Ω(Π6) = Γ4, and Ω(Π7) = Γ6, and if r = 1 we have Ω(Π3) = Γe, and Ω(Π4) = Ω(Π5) = Ω(Π6) = Ω(Π7) = Γ3.

We will sometimes write Ω(π) in place of Ω(Π) if π ∈ Π.

Corollary 8.4. Each representation π appearing in the Plancherel Theorem for G2 admits a basis such that B1 and B2 hold, with bound aπ = a(w) for any w ∈ Ω(π).

Proof. This follows immediately from Proposition 8.3, Remark 5.4, and Theorem 5.5.

Henceforth we will assume that each representation appearing in the Plancherel Theorem is equipped with such a basis.

We note that the tempered irreducible representations of H are precisely the representations that appear in the Plancherel Theorem for G2. This can be seen directly by classifying, via central characters and weight spaces, all irreducible tempered representations of H in an analogous way to [START_REF] Ram | Representations of rank two affine Hecke algebras[END_REF] and comparing with the Plancherel Theorem from Theorem 8.1. Thus, using Proposition 8.3 we note the following.

Observation 8.5. Every tempered irreducible representation π in type G2 satisfies B1 with respect to Γ = Ω(π).

Remark 8.6. We note that any finite dimensional representation π satisfying B1 with respect to a finite cell Γ ∈ Λ is necessarily tempered. To see this, note that since π(Cw) = 0 whenever w / ∈ Γ ≥ LR , and thus π(Cw) is nonzero for only finitely many w ∈ W . Hence there is a bound on the matrix coefficients of π(Cw), and then Casselman's Criterion applies (see Opdam [START_REF] Opdam | On the spectral decomposition of affine Hecke algebras[END_REF]Lemma 2.20]). In fact one can check that π is square integrable (see Lusztig [24, §3] and [START_REF] Lusztig | Some examples of square integrable representations of semisimple p-adic groups[END_REF]).

The asymptotic Plancherel measure

Each rational function f (q) = a(q)/b(q) can be written as f (q) = q -N a ′ (q -1 )/b ′ (q -1 ) with N ∈ Z where a ′ (q -1 ) and b ′ (q -1 ) polynomials in q -1 nonvanishing at q -1 = 0. The integer N in this expression is uniquely determined, and is called the q-valuation of f , written νq(f ) = N . For example, νq((q 2 + 1)(q 3 + 1)/(q 7q + 1)) = 2. Definition 8.7. Let Π be a class of representations appearing in the Plancherel Theorem, and let C be the 'coefficient' of a generic character χπ with π ∈ Π. Consider this coefficient as a rational function C = C(q) in q by setting q = q. We define the q-valuation of Π to be νq(Π) = νq(C(q)). We also write νq(π) = νq(Π) for any π ∈ Π.

For example, consider the class Π2. The associated coefficient is

and thus

For another example consider the class Π7 = {π7}. We have νq(Π7) = νq q 2a-4b (q -2a+4b -1)(q 4a-6b -1) (q 2a + 1)(q -2b + 1)(q -4b + q -2b + 1)(q 4a-4b + q 2a-2b + 1) =

Note that the values of the a-function are arising in these examples. Indeed we have the following theorem, where a(Γ) denotes the constant value of Lusztig's a-function on the two-sided cell Γ, and Ω is as in Proposition 8.3. Note the similarity with the finite dimensional case described at the beginning of this section.

Theorem 8.8. For each classes Π appearing the the Plancherel formula in type G2 we have νq(Π) = 2a(Ω(π)).

Proof. This is by direct inspection using the formula in Theorem 8.1.

Definition 8.9. Using Theorem 8.8 we can define an asymptotic Plancherel measure on Irrep(H) by dµ ′ (π) = lim q→∞ q 2aπ dµ(π).

Proposition 8.10. The asymptotic Plancherel measure on the classes Π0 and Πi with i = 1, 2 is as follows:

For the classes of finite cells we have µ ′ (π3) = 1, µ ′ (π5) = 1 2 , and

Proof. This is a straightforward calculation.

Remark 8.11. Note that the measure dµ ′ on Π0 = Ω -1 (Γ0) is the Hall measure, and thus the Schur functions of type G2 are orthonormal with respect to this measure (see, for example, [START_REF] Ram | Kostka-Foulkes polynomials and Macdonald spherical functions[END_REF]Proposition 3.1]). Similarly, in the generic cases for Π1 and Π2 the measure dµ ′ is the Hall measure of type A1. In the non-generic cases dµ ′ is the Hall measure for the modified Schur functions s k (ζ 1/2 ).

The conjecture P1

We can now prove that P1 holds for G2.

Theorem 8.12. Lusztig's conjecture P1 holds for G2.

Proof. Recall that ∆(w) is defined by Pe,w = nwq -∆(w) + (strictly smaller powers of q), where nw = 0. We are required to prove that a(w) ≤ ∆(w). This is equivalent to showing that lim q→∞ q a(w) Pe,w(q) < ∞,

where we write Pe,w(q) for the specialisation of Pe,w at q = q. By the Plancherel Theorem we have q a(w) Pe,w(q) = q a(w) Tr(Cw) =

q a(w) χπ(Cw) dµ(π).

Suppose that w is in the two-sided cell Γ. In type G2 it follows from Corollary 8.4 that the integral above is over only those classes of representations associated to the cells Γ ′ with Γ ′ ≥LR Γ. For each such class of representations the Plancherel measure is, by Corollary 7.19 and Theorem 8.8, of the form

where dµ ′ is the asymptotic Plancherel measure. Thus the integrand (with respect to the asymptotic Plancherel measure) is q a(w)-aπ Γ ′ tr(cπ,w)(1 + O(q -1 )). Since Γ ′ ≥LR Γ we use B5 to give aπ Γ ′ ≥ aπ Γ = a(w) and thus the power of q in the integrand is at most 0. It is clear from the explicit G2 Plancherel Theorem that the limit may be passed under the integral sign, and the result follows.

The Duflo elements

In this section we extend the idea in the proof of P1 to compute the set D of Duflo elements. This calculation will be used in Section 9 when dealing with the conjectures involving D. We note that since we have proved P1 and computed Lusztig's a-function it is also possible to use a technique of Xie [START_REF] Xie | A decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2[END_REF] to compute D.

Theorem 8.13. For Γ ∈ Λ let DΓ = D ∩ Γ. For the infinite cells we have

and for the finite cells we have

Proof. Let w ∈ W , and let n ′ w denote the coefficient of q -a(w) in Pe,w. Thus w ∈ D if and only if n ′ w = 0. The formula

gives, as in the proof of Theorem 8.12,

(again, we are using Corollary 8.4 here). Thus in the case w ∈ Γ = Γ0 we have, by Theorem 6.6,

Thus n ′ w = 0 if and only if uw = vw (due to the trace) and τw = 0 (since the measure is the Hall measure, see Remark 8.11). Thus n ′ w = 0 if and only if w = u -1 w0u for some u ∈ B0 (moreover, in this case n ′ w = 1). The argument for w ∈ Γ1 or w ∈ Γ2 with r generic for the cell is similar, noting that the measure in this case is the Hall measure for Schur functions of type A1.

Consider the case w ∈ Γ1 with r = 2. Recall the notation from Section 4.4. Theorem 7.15 again forces uw = vw if n ′ w = 0 (where w is written in "+-form"). This forces w to be either of (+, +)-type of (-, -)-type. In the former case we have cπ 1 ,w = (sN (ζ) + sN-1(ζ))Eu w,uw with N = τw ≥ 0, and in the latter case we necessarily have w = w - 1 where by definition τw = -1 and hence cπ 1 ,w = s0(ζ) = 1. Recall from the proof of Proposition 7.17 that sN (ζ) + sN-1(ζ) = s2N (ζ 1/2 ). The measure from Proposition 8.10 is in this case is the Hall measure for these Schur functions, and thus we see that n ′ w = 0 if and only if either w is of (+, +)-type with τw = 0, or w = w - 1 . In the former case we have w = u -1 w + 1 u for some u ∈ B + 1 ∩ s1B - 1 , and hence the result. The case w ∈ Γ2 with r = 3/2 is analogous. The claims for finite cells follow by direct calculations. For example, consider the most complicated case Γ = Γ3 and r = 1. In this case (8.2) gives

The matrices for ρ + 3 , ρ - 3 , ρ ′ 3 , and ρ ′′ 3 are computed from the decomposition in Proposition 5.3, and the leading matrices are computed as

Thus a direct calculation using (8.3) gives n ′ s 0 = n ′ s 1 = n ′ s 2 = 1 and n ′ w = 0 for all w ∈ Γ3\{s0, s1, s2}, and hence the result.

The remaining cases are similar (in fact, easier). We note that some of the finite cells (for example Γ = Γ6) can be handled using cell factorisation, in an analogous way to the infinite cells.

An inner product on J and conjecture P7

In this section we extend the above ideas further to endow Lusztig's asymptotic algebra JΓ with a natural inner product inherited from the Plancherel Theorem (a kind of asymptotic Plancherel Theorem). As a consequence we obtain a proof of P7. Recall that we have proved in Corollary 2.6 that for each Γ ∈ Λ we have that JΓ is isomorphic to the Z-algebra spanned by the leading matrices {cπ Γ ,w | w ∈ Γ}. We thus identify JΓ with this concrete algebra, with Jw ↔ cπ Γ ,w . Define an involution * on JΓ by linearly extending J * w = J w -1 . tr(q -aπ π(Tx) • q -aπ π(T y -1 ))(1 + O(q -1 ))dµ ′ (π).

Taking limits as q → ∞, and using the explicit expression for the Plancherel Theorem for G2 to see that the limit may be passed inside the integral, we see that

The terms cπ,xc π,y -1 are zero if π / ∈ Ω -1 (Γ), and hence the integral is over Ω -1 (Γ). Thus the formula •, • Γ given in the statement of the theorem defines an inner product on JΓ, and {Jw | w ∈ Γ} is an orthonormal basis.

Corollary 8.15. Conjecture P7 holds.

We will give a more combinatorial proof of P7 in Section 9.

Proof of Lusztig's conjectures P2-P15

In this section we prove Lusztig's conjectures P2-P15 for G2. We will denote by Λ∞ the set of infinite two-sided cells and by Λ f the set of finite two-sided cells. Let (πΓ) Γ∈Λ be the system of balanced cell representations afforded by Theorem 7.18. When Γi ∈ Λ∞ we have πΓ i = πi and when Γ ∈ Λ f the representation πΓ is the Kazhdan-Lusztig representation associated to Γ with its natural basis. By Corollary 7.19 we have aπ Γ = a(w) for all w ∈ Γ, and by Corollary 2.6 we see that the coefficients γ x,y,z -1 are the structure constants of the ring JΓ generated by {cπ Γ ,w | w ∈ Γ}.

Remark 9.6. One can prove P15 for Γ3 when r = 1 (where there is no cell factorisation) using similar methods. Indeed let Υi be the right cell that contains si and let Xi = {x ∈ W | six ∈ Υi}. Then, the free R-module Mi = h ♭ u Cs i hv | u, v ∈ Xi R of residues modulo H< LR Γ 3 is a two-sided H-module and the right action (respectively the left action) of H only depends on v (respectively u). Further, one can check by computations that the change of basis matrix to the Kazhdan-Lusztig basis contains only integers and we can therefore define submodules Ei of E that are naturally (H1, H2)-bimodule. We obtain that E is a bimodule since E = E1 + E2 + E3.

Conjectures

We conclude this paper with some conjectures. Conjecture 9.7. For each affine Hecke algebra there exists a balanced system of cell representations for each choice of parameters.

As seen in Section 2, assuming the truth of this conjecture one can show that Lusztig's a-function satisfies a(w) ≤ aΓ whenever w ∈ Γ. Further, we have equality if the system of balanced cell representations satisfies the extra axiom B4 ′ . Conjecture 9.8. There exists a well defined surjective map Ω from the set of classes of representations appearing in the Plancherel formula to the set of two-sided cells generalising the map from Proposition 8.3. Conjecture 9.9. Let Π be class of representations appearing in the Plancherel Theorem, and let Γ = Ω(Π) where Ω is as in Conjecture 9.8. Then a(w) = νq(Π)/2 for all w ∈ Γ.

We note that P1 can be deduced from Conjecture 9.9 in a similar was as in Theorem 8.12. Continuing in this direction, we make the following final conjecture.

Conjecture 9.10. The construction of the inner product in Theorem 8.14 generalises to arbitrary affine type.

The analysis of this paper proves all four conjectures in type G2.