A proof of Lusztig's conjectures for affine type $G_2$ with arbitrary parameters

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A proof of Lusztig’s conjectures for aﬀine type G_2 with arbitrary parameters

Jeremie Guilhot, James Parkinson

To cite this version:

Jeremie Guilhot, James Parkinson. A proof of Lusztig’s conjectures for aﬀine typeG_2with arbitrary parameters. 2017. �hal-01638380v2�

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A proof of Lusztig’s conjectures for affine type G₂ with arbitrary parameters

Jérémie Guilhot, James Parkinson ^∗

Abstract

We prove Lusztig’s conjecturesP1–P15for the affine Weyl group of typeG˜2 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 typeG˜2 for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in typeG˜2, allowing us to prove P1and 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 typesG2 andA1, 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 [17] Kazhdan and Lusztig introduced the theory in the equal parameter case, and in [22] 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 [18], and the case of arbitrary Coxeter groups was settled only very recently by Elias and Williamson in [7]. 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 [25, 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 parametersP1–P15are only known to hold in the following situations:

• thequasisplitcase where a geometric interpretation is also available [25, Chapter 16];

• finite dihedral type [10] and infinite dihedral type [25, Chapter 17] for arbitrary parameters;

• typeBn in the “asymptotic” case [3, 10];

• F4 for arbitrary parameters [10].

Note that the only example of an infinite Weyl group whereP1–P15are 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 G˜2 for arbitrary parameters. This provides the very first example of an affine Weyl group of rank greater than 1in 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 caseB˜2.

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 [10]

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 Hdefined over Z[q,q⁻¹]. LetΛ be the set of two-sided cells ofW with respect toL, and letΓ∈Λ. We say that a representationπisΓ-balanced if it admits a basis such that (1) the maximal degree of the coefficients that appear in the matrixπ(Tw)is bounded by a constanta_π(hereTw denotes the standard basis ofH) and (2) this bound is attained if and only ifw∈Γ. A balanced systems of cell representations is a family (πΓ)Γ∈ΛofΓ-balanced representations that satisfy some extra axioms (see Section 2). We show that the existence of such a system is sufficient to compute Lusztig’sa-function, and as a byproduct we obtain an explicit construction of Lusztig’s asymptotic algebraJ [25, Chapter 18].

In the case of G˜2, 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 [15]. It turns out that

∗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.

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1 Kazhdan-Lusztig theory 2

the representations associated to finite cells naturally give rise to a balanced representation and so most of our work is concerned with the infinite cells. In typeG˜2 there are 3 such cells for each choice of parameters, the lowest two-sided cellΓ0 and two other cellsΓ1 andΓ2. To each of these cells we associate a natural finite dimensional representationπi

admitting an elegant combinatorial description in terms of alcove walks, which allows us to establish the balancedness of these representations.

Next we investigate connections between Kazhdan-Lusztig cells and the Plancherel Theorem, using the explicit formulation of the Plancherel Theorem in typeG˜2obtained by the second author in [31]. In particular, we show that in typeG˜2

there is a natural correspondence, in each parameter range, between two-sided cells appearing in the cell decomposition, and the representations appearing in the Plancherel Theorem (these are the tempered representations). Moreover we define aq-valuation on the Plancherel measure, and show that in typeG˜2 theq-valuation of the mass of a tempered representation is twice the value of Lusztig’sa-function on the associated cell. This observation allows us to introduce anasymptotic Plancherel measure, and we use this measure to proveP1and determine the setDof Duflo involutions.

Moreover we show that the Plancherel theorem “descends” to give an inner product on Lusztig’s asymptotic algebraJ. Once we have established the existence of a balanced system of cell representations forG˜2 for each choice of parameters, provedP1, and determined the setD, conjecturesP2–P14follow very naturally, essentially from combinatorics of Weyl characters of typesG2andA1and explicit computations for the finite cells. ConjectureP15is slighlty different in nature and follows from the generalised induction process [14].

We note that in [38], Xie uses a decomposition formula for the Kazhdan-Lusztig basis to reduce P1–P15 to proving P8 and determining Lusztig’s a-function. However we note that indeed the main work of this paper is precisely the determination of Lusztig’sa-function, which uses our balanced system of cell representations.

We conclude this introduction with an outline of the structure of this paper. In Section 1 we recall the basics of Kazhdan- Lusztig theory. In Section 2 we introduce the axioms of a balanced system of representations, and show in Theorem 2.5 that given these axioms Lusztig’s a-function can be computed. Section 3 provides background on affine Weyl groups and the affine Hecke algebra. In Section 4 we recall the partition of G˜2 into cells for all choices of parameters, and discuss cell factorisation properties for the infinite cells. In Section 5 we prove that each finite cell admits a balanced cell representation. Some of the result in this section requires explicit computations. These have been carried out using gap3[36] and the packageCHEVIE[11, 28].

Section 6 deals with the case of the lowest two-sided cell. We note that this case has already been investigated by Xie in [39], however we include our analysis here since it illustrates very clearly in this simpler case the combinatorial methodology that we will employ for the remaining more complicated infinite cells. Section 7 deals with these remaining cells. We introduce a model based on alcove walks to study the representations associated to these cells. This allows us to give combinatorial proofs of bounds for matrix coefficients and to compute leading matrices for these representations. The analysis of this section is involved due in part to interesting complications arising in the case of non-generic parameters.

In Section 8 we analyse the connections between the Plancherel Theorem and the cell decomposition. We use the Plancherel Theorem to proveP1and determine the set of Duflo involutionD. We also define the “asymptotic Plancherel measure”, and show that it induces an inner product on Lusztig’s asymptotic algebra in typeG˜2. Finally, in Section 9 we provide our proof of the remaining conjecturesP2–P15.

1 Kazhdan-Lusztig theory

In this section we recall the setup of Kazhdan-Lusztig theory, including the Kazhdan-Lusztig basis, Kazhdan-Lusztig cells, and the Lusztig’s conjecturesP1–P15. In this section(W, S)denotes an arbitrary Coxeter system (with|S|<∞) with length functionℓ:W →N. ForI⊆S letWI be the standard parabolic subgroup generated byI. LetL:W→N be apositive weight functiononW. ThusL:W −→NsatisfiesL(ww^′) =L(w) +L(w^′)wheneverℓ(ww^′) =ℓ(w) +ℓ(w^′).

Letqbe an indeterminate and letR=Z[q,q⁻¹]be the ring of Laurent polynomial inq.

1.1 The Kazhdan-Lusztig basis

TheHecke algebra Hassociated to(W, S, L)is the algebra overRwith basis{Tw|w∈W}and multiplication given by TwTs=

(Tws ifℓ(ws) =ℓ(w) + 1

Tws+ (q^L(s)−q^−L(s))Tw ifℓ(ws) =ℓ(w)−1.

The basis{Tw|w∈W}is called thestandard basis ofH. We setqs=q^L(s) fors∈S.

The involution¯onRwhich sendsqtoq⁻¹ can be extended to an involution onHby setting X

w∈W

awTw= X

w∈W

awT_w⁻¹−1.

In [17], Kazhdan and Lusztig proved that there exists a unique basis{Cw|w∈W}ofHsuch that, for allw∈W, Cw=Cw and Cw=Tw+X

y<w

Py,wTy wherePy,w∈q⁻¹Z[q⁻¹].

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1 Kazhdan-Lusztig theory 3

This basis is called theKazhdan-Lusztig basis ofH. The polynomialsPy,w are called theKazhdan-Lusztig polynomials, and to complete the definition we setPy,w= 0whenevery6< w (here≤denotes Bruhat order onW). We note that the Kazhdan-Lusztig polynomials, and hence the elementsCw, depend on the the weight function L. For example, in the dihedral groupI2(2m)withm≥2,L(s1) =a, andL(s2) =b, we have

Ps1,s1s2s1 =







q^−(b−a)+q^−a−b ifa < b

q^−2a ifa=b

q^−(a+b)−q^−(a−b) ifa > b.

In particular, this example shows that the positivity properties enjoyed byPy,z in the equal parameter case (that is, L=ℓ) do not transfer across to the general case.

Letx, y∈W. We denote byhx,y,z∈Rthe structure constants associated to the Kazhdan-Lusztig basis:

CxCy= X

z∈W

hx,y,zCz.

Definition 1.1([25, Chapter 13]). TheLusztiga-function is the functiona:W →Ndefined by a(z) := min{n∈N|q⁻ⁿhx,y,z∈Z[q⁻¹]for allx, y∈W}.

WhenW is infinite it is, in general, unknown whether thea-function is well-defined. However in the case of affine Weyl groups it is known thatais well-defined, and thata(z)≤L(w0)wherew0 is the longest element of the underlying finite Weyl groupW0 (see [25]). Thea-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. Forx, y, z∈W letγ_x,y,z−1denote the constant term ofq⁻^a^(z)hx,y,z.

The coefficientsγ_x,y,z⁻¹are the structure constants of theasymptotic algebraJ introduced by Lusztig in [25, Chapter 18].

1.2 Kazhdan-Lusztig cells and associated representations

Define preorders≤L,≤R,≤LR onW extending the following by transitivity:

x≤Ly ⇐⇒ there existsh∈ Hsuch thatCx appears in the decomposition in the KL basis ofhCy

x≤Ry ⇐⇒ there existsh∈ Hsuch thatCx appears in the decomposition in the KL basis ofCyh x≤LRy ⇐⇒ there existsh, h^′∈ Hsuch thatCxappears in the decomposition in the KL basis ofhCyh^′. We associate to these preorders equivalence relations∼L,∼R, and∼LR by setting (for∗ ∈ {L,R,LR})

x∼∗yif and only ifx≤∗yandy≤∗x

The equivalence classes of∼L,∼R, and∼LRare called left cells,right cells, andtwo-sided cells.

We denote byΛthe set of all two-sided cells (note thatΛdepends on the choice of parameters). Given any cellΓ(left, right, or two-sided) we set

Γ≤∗ :={w∈W |there existsx∈Γsuch thatw≤∗x}

and we defineΓ≥∗,Γ>∗ andΓ<∗ similarly.

Example 1.3. The following table records the decomposition of the dihedral groupW=I2(m) =hs1, s2iinto 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 rank3(dimension2) Weyl groups every two-sided cell intersects a finite parabolic subgroup (hence a dihedral group), and so assuming the Lusztig conjecturesP4and P12 the table below gives conjectural values of thea-function on all cells. These ‘conjectures’ become ‘theorems’ for typeG˜2

due to the results of this paper.

To each right cell Υ of W there is a natural rightH-module HΥ constructed as follows. Let H≤Υ and H<Υ be the R-modules

H≤Υ=hCx|x∈Υ≤Ri and H<Υ=hCx|x∈Υ<Ri.

ThenH≤ΥandH<Υ are naturally rightH-modules. ForH≤Υ this is immediate from the definition of≤R. ForH<Υwe note that ifx≤Rywithy∈Υand ifx /∈Υthen, forh∈ H,

Cxh= X

z≤Rx

azCz.

Ifz∈Γthen necessarilyaz= 0(for otherwisey∼Rz≤Rxand soy≤Rxandx≤Rygiving x∈Γ). ThusH<Γis a rightH-module. Hence the quotient

HΥ:=H≤Υ/H<Υ

is naturally a rightH-module with basis{Cw|w∈Υ}whereCw is the class ofCwinHΥ.

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2 Systems of balanced representations and Lusztiga-function 4

W two-sided cells values of thea-function

I²(2), a≥b {e},{s¹},{s²}, {w0} 0, a, b, a+b I²(m), a=b, m≥2 {e}, W\{e,w0},{w0} 0, a, ma

I²(2m), a > b, m≥2 {e}, W\{e, s²,w0s²,w0},{s²},{w0s²},{w0} 0, a, b, ma−(m−1)b, ma+mb

I²(∞), a=b {e}, W\{e} 0, a

I2(∞), a > b {e}, W\{e, s2},{s2} 0, a, b

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

1.3 Lusztig conjectures

Define∆ :W →Nandnz∈R\{0}by the relation

Pe,z=nzq^−∆(z)+ strictly smaller powers ofq.

This is well defined becausePx,y∈q⁻¹Z[q⁻¹]for allx, y∈W. Let D={w∈W|∆(w) =a(w)}.

The elements ofDare called Duflo elements(or, somewhat prematurely,Duflo involutions; seeP6below).

In [25, Chapter 13], Lusztig has formulated the following 15 conjectures, now known asP1–P15. P1. For anyz∈W we havea(z)≤∆(z).

P2. Ifd∈ Dandx, y∈W satisfyγx,y,d6= 0, thenx=y⁻¹.

P3. Ify∈W then there exists a uniqued∈ Dsuch thatγ_y−1,y,d6= 0.

P4. Ifz^′≤LRzthena(z^′)≥a(z). In particular thea-function is constant on two-sided cells.

P5. Ifd∈ D,y∈W, andγ_y−1,y,d6= 0, thenγ_y−1,y,d=nd=±1.

P6. Ifd∈ Dthend²= 1.

P7. For anyx, y, z∈W, we haveγx,y,z=γy,z,x.

P8. Letx, y, z∈W be such thatγx,y,z6= 0. Thenx∼Ly⁻¹,y∼Lz⁻¹, andz∼Lx⁻¹. P9. Ifz^′≤Lz anda(z^′) =a(z), thenz^′∼Lz.

P10. Ifz^′≤Rz anda(z^′) =a(z), thenz^′∼Rz.

P11. Ifz^′≤LRzanda(z^′) =a(z), thenz^′∼LRz.

P12. IfI⊆S then thea-function ofWI is the restriction of thea-function ofW.

P13. Each right cellΓofW contains a unique elementd∈ D. We haveγ_x−1,x,d6= 0for allx∈Γ.

P14. For eachz∈W we havez∼LRz⁻¹.

P15. Ifx, x^′, y, w∈W are such thata(w) =a(y)then X

y^′∈W

h_w,x^′_,y^′⊗h_x,y^′_,y= X

y^′∈W

h_y^′_,x^′_,y⊗h_x,w,y^′ inR⊗^ZR.

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 [25] and the updated version available on ArXiv), (2) in the

“quasisplit case” [25, Chapter 16] whereW is obtained by “twisting” a larger Coxeter groupW˜, andLis the restriction of the length function onW˜ toW, (3) whenW is a dihedral group (finite or infinite) for all choices of parameters (see [10, 25]), (4) whenW =Bnin the “asymptotic case” (see [3, 10]), (5) whenW =F4 for any choices of parameters (see [10]). 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 conjecturesP1–P15for G˜2 for all choices of parameters. Our approach extends naturally to all rank2affine Weyl groups, although the details in the three parameter caseB˜2 becomes rather involved due to the large number of distinct regimes of cell decompositions.

2 Systems of balanced representations and Lusztig a-function

In this section we define abalanced system of cell representations, inspired by the work of Geck [8, 10] 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 typeG˜2.

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2 Systems of balanced representations and Lusztiga-function 5

2.1 System of balanced representation

We now introduce the central notion of this paper, based on the work of Geck [8, 10] 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 anR-polynomial ringRΓ(where we could haveRΓ=R) and a basis(e_j)ofVΓsuch that the following holds, where[πΓ(Cw)]i,j denotes the matrix entries ofπΓ(Cw)with respect to the basis(e_j):

B1. Ifw /∈Γ≥LR thenπΓ(Cw) = 0.

B2. There exists boundsa_π

Γ∈Nsuch thatdeg[πΓ(Cw)]i,j ≤a_π

Γ for allw∈W and all1≤i, j≤dim(VΓ).

B3. We havemaxi,jdeg[πΓ(Cw)]i,j=a_π

Γ if and onlyw∈Γ. We define theleading matrices by c_π_Γ_,w =sp_|

q−1=0 q⁻^a^π^Γ[πΓ(Cw)]

. B4. The set{c_π_Γ_,w|w∈Γ}of leading matrices is free overZ.

B5. IfΓ^′≤LRΓthena_π

Γ′ ≥a_π

Γ. The natural numbersa_π

Γ are called thebounds 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 typeG˜2 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, letΥ be a right cell, and letHΥ be as in Section 1.2. If Cw acts nontrivially onHΥ then there existu, v∈Υsuch thatCu·Cw=P

zhu,w,zCz withhu,w,v 6= 0. Thusv≤LRw.

We say that a representation(π, V) is Γ-balanced for the two-sided cell Γif V admits a basis such that B2, B3 and B4 hold. We note that inB2 and B3 it is equivalent to replace Cw byTw, becauseCw = Tw+P

v<wpv,wTv with pv,w∈q⁻¹Z[q⁻¹]. However inB1one cannot replaceCw byTw.

Example 2.2. LetW be an affine Weyl group of typeG˜2 with diagram and weight function defined by

✐ ✐ ✐

a b b

s1 s2 s0

where a, b are positive integers. Let I ⊆ S be a union of conjugacy classes in S. We define the one dimensional representationρIofW by

ρI(Ts) =

(qs ifs∈I,

−q⁻¹s otherwise.

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 whereΓe is the two-sided cell that contains precisely the identity and (2) ρS cannot be balanced for any two-sided cellΓ.

Consider the representationρI whereI={s0, s2}. We will see in Section 4 thatΓ5 :={s0s2s0}is a two-sided cell inW when a/b >2. Forw∈W we haveρI(w) =q^bℓ²^(w)(−q)^−aℓ¹^(w) whereℓ2(w)is the number of s2 and s0 generators in any reduced expression ofwandℓ1(w)is the number ofs1 generators. Saying that the representationρI isΓ5-balanced fora/b >2means thatbℓ2(w)−aℓ1(w)≤3bfor allwand that there is equality if and only ifw=s0s2s0. This can be done by studying reduced expressions inW, and we will see another proof using Kazhdan-Lusztig theory in Section 5.

Proceeding as above, one can show that the representationρI where I={s1}is Γ7-balanced whenevera/b <1. Once again we will see in Section 4 thatΓ7:={s1}is a two-sided cell inW for this parameter range.

2.2 Computing the a-function

In this section we show that axiomsB1–B5, along with an additional axiomB4^′introduced below, are sufficient to show that Lusztig’sa-function is constant on two-sided cells, and moreover we are able to compute the value of thea-function in terms of the bounds(a_π

Γ)Γ∈ΛfromB2.

Let(πΓ)Γ∈Λbe a balanced system of cell representations forHwith boundsa_π

Γ for allΓ∈Λ. We have CxCy=X

Γ∈Λ

X

z∈Γ

hx,y,zCz. (2.1)

Proposition 2.3. Letx, y∈W andw∈ΓwhereΓ∈Λ. We havedeg(hx,y,w)≤a_π

Γ.

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3 Affine Weyl groups, affine Hecke algebras, and alcove walks 6

Proof. We proceed by downwards induction. Let Γ ∈ Λ and suppose that deg(hx,y,z) ≤ a_π

Γ′ for all z ∈ Γ^′ where Γ^′>LRΓ. Then applyingπΓto (2.1) using the fact thatπΓis a cell representation gives

πΓ(CxCy) =X

z∈Γ

hx,y,zπΓ(Cz) + X

Γ^′∈Λ, Γ^′>LRΓ

X

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 2a_π

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 bya_π

Γ′ +a_π

Γ ≤2a_π

Γ. Indeed the maximal degree that can appear in πΓ(Cz) is stricly less thana_π

Γ since z /∈Γand the bounds of the balanced system are decreasing with respect to≤LR. We now show that

deg(hx,y,z)≤a_π

Γ for allz∈Γ.

Let m = max{deg(hx,y,z) |z ∈ Γ} and let Z = {z^′ ∈ Γ|deg(h_x,y,z^′) =m} 6=∅. For z ∈ Z define ˜γ_x,y,z−1 ∈ Zby hx,y,z=q^m˜γ_x,y,z−1+lower terms. ByB3we haveπΓ(Cz) =q^a^π^Γc_π_Γ_,z+lower terms, withc_π_Γ_,z6= 0the leading matrix.

Then the right hand side of (2.2) is of the form q^m+^a^π^Γ X

z∈Z

˜

γ_x,y,z−1c_π_Γ_,z+lower terms,

and byB4 the expression in the sum (that is, the coefficient ofq^m+^a^π^Γ) is nonzero. By comparing with the lefthand side in (2.2) it follows thatm+a_π

Γ ≤2a_π

Γ that ism≤a_π

Γ as required.

Corollary 2.4. LetΓ∈Λ. The subsetJΓ ofMdim(πΓ)(RΓ) generated by{c_π_Γ_,w |w∈Γ}is aZ-subalgebra.

Proof. LetΓbe the two-sided cell an letx, y∈Γ. ApplyingπΓtoCxCy, usingB1, and multiplying byq⁻²â^π^Γ we get [q⁻â^π^ΓπΓ(Cx)][q⁻â^π^ΓπΓ(Cy)] =X

z∈Γ

[q⁻^a^π^Γhx,y,z][q⁻^a^π^ΓπΓ(Cz)] + X

Γ^′∈Λ, Γ^′>LRΓ

X

z∈Γ^′

[q⁻^a^π^Γhx,y,z][q⁻^a^π^ΓπΓ(Cz)].

Specialising atq⁻¹ = 0will annihilate all the terms in the double sum. Indeed for z ∈ Γ^′ with Γ^′ >LR Γ we have deg(hx,y,z)≤a_π

Γ′ ≤a_π

Γ and the maximal degree that can appear inπΓ(Cz)is strictly less thata_π

Γ. Thus we obtain c_π_Γ_,xc_π_Γ_,y=X

z∈Γ

˜

γ_x,y,z⁻¹c_π_Γ_,z

where˜γ_x,y,z−1 ∈Zis the coefficient of degreea_π

Γ ofhx,y,z.

We introduce the following additional axiom, where˜γ_x,y,z−1∈Zis the coefficient of degreea_π

Γ ofhx,y,z. B4^′. LetΓ∈Λ. For eachz∈Γ, there exists(x, y)∈Γ² such thatγ˜_x,y,z−16= 0.

We can now show that if all axiomsB1–B5andB4^′are satisfied, then we can compute Lusztig’sa-function in terms of the boundsa_π

Γ.

Theorem 2.5. Suppose thatB1–B5andB4^′ are satisfied. Thena(w) =a_π

Γ for allw∈Γ.

Proof. According to Proposition 2.3, we only need to show thata_π

Γ ≥a(w). To do so it is enough to findx, y∈W such thatdeg(hx,y,w) =a_π

Γ or equivalently, to findx, y∈W such thatγ˜x,y,w6= 0. Hence the result usingB4^′.

Corollary 2.6. AssumingB1–B5andB4^′, the ring JΓis isomorphic to Lusztig’s asymptotic algebra associated toΓ.

Proof. The elements ˜γ_x,y,z−1 are the coefficients of hx,y,z of degree a_π

Γ, and are the structure constants of JΓ with respect to the basis{c_π_Γ_,w|w∈Γ}. Indeed once we know thata_π

Γ=a(Γ)we know that the structure constants ofJΓ

areγ_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 thatHis affine cellular.

3 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 inG˜2 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 [34]. Finally we present a combinatorial formula for the Weyl character that will be used in Section 6.