# The Leading Coefficients for an Affine Weyl Group of TypeG2˜: The Lowest Two-Sided Cell Case

Authors: Pengfei Guo; Zhu-Jun Zheng
Journal: Journal of Mathematics
 (2021)

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## Abstract

This study focusses on the leading coefficientsμu,w of the Kazhdan–Lusztig polynomials Pu,w for the lowest cell c0 of an affine Weyl group of type G2˜ and gives an estimation μu,w≤3 for u,w∈c0.


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## Body

## 1. Introduction

The Hecke algebraℋ of the Coxeter group W over ℤq1/2,q−1/2 is a free A-module and has a basis Tww∈W˜. Its multiplication is defined by the relations Ts−qTs+1=0 if s∈S and TwTu=Twu if lwu=lw+lu. David Kazhdan and George Lusztig proved that there exist an unique basis Cw in ℋ, such that Cw=q−lw/2∑u≤wPu,wTu and Cw¯=Cw [1] for each w∈W, where Pu,w∈ℤq of degree less than or equal to 1/2lw−lu−1 for u≤w and Pw,w=1 in [1]. It is called Kazhdan–Lusztig basis. The corresponding polynomial Pu,w is called Kazhdan–Lusztig polynomial.The leading coefficientμu,w of Pu,w is the coefficient of the term q1/2lw−lu−1 in Pu,w. It plays an important role in Kazhdan–Lusztig theory, Lie theory, and representation theory. It can be used to estimate the dimensions of the first extension groups for irreducible modules of algebraic groups and finite groups of Lie type [2]. By the recursive formula of Kazhdan–Lusztig polynomials in [1], we see that some μu,w play crucial roles in understanding these polynomials.However, there are few results for the leading coefficients in general. In [3], in order to understand the local finite problem of B2˜ type, George Lusztig computed some special leading coefficients for the affine Weyl group of this type. In [4], for an affine Weyl group of type A5, some nontrivial leading coefficients are found. Timothy McLarnan and Gregory Warrington have shown that μu,w can be greater than 1 for a symmetric group which is a counter example for 0,1conjecture [5]. In [6], Nanhua Xi showed that when u≤w and au≤aw, μu,w≤1 if W is a symmetric group or an affine Weyl group of type An. In [2], Leonard Scott and Nanhua Xi showed that the leading coefficient μu,w of Kazhdan–Lusztig polynomial Pu,w for an affine Weyl group of type An is n+2 if n≥4. Recently, Brant Jones and Richard Green [7, 8] showed that the coefficients are ≤1 for some special pairs of elements in the symmetric group and other Weyl groups. In [9, 10], Liping Wang computed most of the leading coefficients for an affine Weyl group of type B2˜ and A2˜.In order to use the conclusions in [5] to calculate μu,w for u,w∈c0, we need to compute CxCy for some x,y∈c0. It is difficult to compute these multiplications CxCy, especially for lx,ly large enough, since we need to compute the amount of μu,v for lu,lv large enough. The calculation is so complicated that we cannot do it manually. Luckily, it can be carried out by a powerful computer. Although there are some programs to compute Pu,v, such as Mark Goresky’s Pascal and Fokko du Cloux’s Coxeter 3.0, these algorithms are not efficient for our problem. So we work out a new algorithm to compute Pu,v and CsCy for the affine Weyl group of type G2˜. This algorithm is realized by visual C++ program, and this program should run on the main types of workstation under Windows. By this algorithm, we take some time to get all CsCy which we need. Moreover, we obtain all δx,y,z defined in [11] to prove our main result. We have a bound of μu,v for u,w∈c0 by some conclusions of representation of algebraic groups [7].We now outline the structure of this study. In Section2, we recall the definitions and properties Kazhdan–Lusztig basis, Kazhdan–Lusztig polynomial, Kazhdan–Lusztig cell, and Kazhdan–Lusztig coefficient. We compute all elements of c0 in Section 2.3 by the method mentioned in [2]. In Section 3, we give an explicit description of the algorithm which is used to compute Kazhdan–Lusztig polynomials and multiplications of Kazhdan–Lusztig basis. In Section 4, we compute all multiplications of Kazhdan–Lusztig basis which we need. From these multiplications, we get the main theorem of this study that μu,v≤3 for u,w∈c0.


## 2. Preliminaries

In this section, we review some basic facts for Kazhdan–Lusztig theory, properties of Kazhdan–Lusztig polynomial and Kazhdan–Lusztig coefficient, lowest two-sided cell, and the formula of Kazhdan–Lusztig coefficient for the lowest two-sided cell.


### 2.1. Basic Definitions and Conventions

LetR be a root system, and W0 be its Weyl group. The semidirect product W=W0∝Λr is the affine Weyl group, and W˜=W0∝Λ is the extended affine Weyl group, where Λ is the weight lattice of R, and Λr=ℤR is its corresponding root lattice.There is an abelian subgroupΩ of W˜, such that ωS=Sω for any ω∈Ω, where S is the set of simple reflections of W, and W˜=W∝Ω. The length function l of W and the Bruhat order ≤ on W can be extended to W˜ by setting lωw=lw and ωw≤ω′u if and only if ω=ω′ and w≤u, where ω,ω′ are in Ω and w,u are in W. The strict Bruhat order is similar defined as above.Letℋ˜ be the Hecke algebra of W˜,S over Laurent polynomial ring A=ℤq1/2,q−1/2 (q is an indeterminate). ℋ˜ is a free A-module and has the basis Tww∈W˜. Its multiplication is defined by the relations Ts−qTs+1=0 if s∈S and TwTu=Twu if lwu=lw+lu. The combination Cw=q−lw/2∑u≤wPu,wTu,w∈W˜ be its Kazhdan–Lusztig basis, where Pu,w∈ℤqu≤w are the Kazhdan–Lusztig polynomials (here Cw is just Cw′ in [1]). The degree of Pu,w is less than or equal to 1/2lw−lu−1 if u<w and Pw,w=1. The subalgebra ℋ of ℋ˜ generated by all Tss∈S is the Hecke algebra of the Coxeter system W,S. For ω in Ω and w in W, we have Cωw=TωCw. Thus, we have Pωu,ωw=Pu,w and Pω′u,ωw=0 for any different ω′,ω in Ω and u,w in W. It is known that the coefficients of these polynomials are all nonnegative [12].The Kazhdan–Lusztig polynomials can be written asPu,w=μu,wq1/2lw−lu−1 +lower degree terms. The coefficient μu,w is very interesting, which can be seen even from the recursive formula for Kazhdan–Lusztig polynomials [1]. We call μu,w the leading coefficient of Pu,w. Write u≺w if u≤w and μu,w≠0. Define μ˜u,w=μu,w if u≤w, −1lw−lu<0 and μ˜u,w=μw,u if w≤u.Forμu,w, there are the following conclusions [1].(1)
Letu,w∈W,s∈S, such that u<w,su>u,sw<w. Then, u≺w if and only if w=su. Moreover, μu,w=1 in this case.(2)
Letu,w∈W,s∈S, such that u<w,us>u,ws<w. Then, u≺w if and only if w=us. Moreover, μu,w=1 in this case.The elementsCw have the following properties [1]. For s∈S, we have(1)CsCw=q1/2+q−1/2Cw,sw≤w,Csw+∑y≺w,sy≤yCy,sw≥w,CwCs=q1/2+q−1/2Cw,ws≤w,Cws+∑y≺w,ys≤yCy,ws≥w.Givenx,y∈W, CxCy=∑z∈Whx,y,zCz, hx,y,z∈A. For any z∈W, define az  = min{i∈ℕ|q−1/2hx,y,z∈ℤq−1/2, for any x,y∈W}. Denote hx,y,z=γx,y,zqaz/2+δx,y,zqaz−1/2 +lower degree terms, where γx,y,z and δx,y,z are all nonnegative.


### 2.2. The Lowest Two-Sided Cell

It is known thatc0=w∈W˜|aw=lw0 is a two-sided cell, which is the lowest one with respect to the partial order ≤LR, where w0 is the longest element of W0 [13–15]. The set c0 is called the lowest two-sided cell of W˜.In [2], Leonard Scott and Nanhua Xi gave a description of c0. Let R+ (respectively, R+ and Δ) be the set of positive (respectively, negative and simple) roots in the root system R of W0. The set of dominant weights Λ+ is the set z∈Λ|lzw0=lz+lw0. For each simple root α, sα is the corresponding simple reflection in W0 and xα is the corresponding fundamental weight. For each w∈W0, set dw=w∏α∈Δ,wα∈R−xα. Then,(2)c0=dwzw0du−1|w,u∈W0,z∈Λ+.Moreover, the setc0,w′=dwzw0du−1|u∈W0,z∈Λ+ is a right cell of W˜ and c0,u=dwzw0du−1|w∈W0,z∈Λ+ is a left cell of W˜. The distinguished involutions of c0 are dww0dw−1, w∈W0.Forx∈Λ+, let Vx be a rational irreducible G-module of the highest weight x and Sx be the corresponding element defined in [16]. For any x∈Λ, Bernstein Joseph gave an element θx∈ℋ, defined as(3)θx=q−lx′Tx′q−lx″Tx″−1,where x′,x″∈Λ+ satisfies x=x′−x″. For x∈Λ+, xW0 is the W0-orbit in Λ which contains x. Set zx=∑x′∈xW0θx′. For x∈Λ+, define(4)Sx=∑x′∈Λ+,x′≤xdx′xzx′,where dx′x is the multiplicity of x′ in Vx. Then, Sx|x∈Λ+ is the A-base of the center of ℋ.For everyw∈W0, define two cell-related combinations:(5)Edw=q−ldw/2∑y≤dw,lyw0=ly+lw0Pyw0,dww0Ty,Fdw=q−ldw/2∑y≤dw,lyw0=ly+lw0Pyw0,dww0Ty−1,Then, the following facts were concluded in [16]:(a)
EdwSxCw0Fdu=Cdwxw0du−1 for any w,u∈W and x∈Λ+(b)
SxSy=∑z∈Λ+mx,y,zSz for any x,y∈Λ+. Here, mx,y,z is defined to be the multiplicity of Vz in the tensor product Vx⊗Vy;(c)
For anyy,w∈c0, if μy,w≠0, then we have y∼wL and y≁wR or y∼wR and y≁wL;(d)
Letw,w′,u∈W0, y,z∈Λ+ and let dwyw0du−1,dw′zw0du−1 be elements of the left cell c0,u of the extended affine Weyl group W˜. Then, μdwyw0du−1,dw′zw0du−1=μdwyw0,dw′zw0;(e)
Letx=duxw0,y=du′x′w0, where u,u′,v∈W0 and x,x′∈Λ+. Then, the set z∈W|hw0du−1,du′w0,z≠0 is finite, and we know that z=z1w0 for some z1∈Λ+ by hw0du−1,du′w0,z≠0. Moreover, μy,w=∑z∈Λ+mx∗,x,z∗δw0du−1,du′w0,zw0, where x∗=w0x−1w0.


### 2.3. The Lowest Two-Sided Cell ofG˜2

For the affine Weyl group of typeG˜2, W0=e,s,t,st,ts,sts,tst,stst,tsts,ststs,tstst,ststst, and the simple reflection set Δ=α,β.The fundamental weights ofG˜2 are xα=2α+β=ststsr and xβ=3α+2β=tstsr2 in W.As above subsection mentioned, for eachw∈W0, dw=w∏α∈Δ,wα∈R−xα. Then, we get the following results.(6)de=e,ds=sxα=sststsr=tstsr,dt=txβ=ststsr2=ststrstsr,dst=stxβ=sttstsr2=tstrstsr,dts=nntsxα=tsststsr=stsr,dsts=nnstsxα=stsststsr=tsr,dtst=ntstxβ=tsttstsr2=strstsr,dstst=nststxβ=ststtstsr2=trstsr,dtsts=nntstsxα=tstsststsr=sr,dststs=nststsxα=ststsststsr=r,dtstst=tststxβ=tststtstsr2=rstsr,dststst=stststxαxβ=rstsrtstsr.From these results andc0=dwzw0du−1|w,u∈W0,z∈Λ+, the two-sided cell c0 consists of the following elements:(7)vi,j,a,b=vistststxaybvj−1,where 1≤i,j≤12, a,b∈ℕ, and v1=e,v2=r,v3=sr,v4=tsr,v5=stsr, v6=tstsr,v7=rstsr,8=trstsr,v9=strstsr,v10=tstrstsr, v11=ststrstsr,v12=rststrstsr, x=rststs,y=rstst2.


## 2.1. Basic Definitions and Conventions

LetR be a root system, and W0 be its Weyl group. The semidirect product W=W0∝Λr is the affine Weyl group, and W˜=W0∝Λ is the extended affine Weyl group, where Λ is the weight lattice of R, and Λr=ℤR is its corresponding root lattice.There is an abelian subgroupΩ of W˜, such that ωS=Sω for any ω∈Ω, where S is the set of simple reflections of W, and W˜=W∝Ω. The length function l of W and the Bruhat order ≤ on W can be extended to W˜ by setting lωw=lw and ωw≤ω′u if and only if ω=ω′ and w≤u, where ω,ω′ are in Ω and w,u are in W. The strict Bruhat order is similar defined as above.Letℋ˜ be the Hecke algebra of W˜,S over Laurent polynomial ring A=ℤq1/2,q−1/2 (q is an indeterminate). ℋ˜ is a free A-module and has the basis Tww∈W˜. Its multiplication is defined by the relations Ts−qTs+1=0 if s∈S and TwTu=Twu if lwu=lw+lu. The combination Cw=q−lw/2∑u≤wPu,wTu,w∈W˜ be its Kazhdan–Lusztig basis, where Pu,w∈ℤqu≤w are the Kazhdan–Lusztig polynomials (here Cw is just Cw′ in [1]). The degree of Pu,w is less than or equal to 1/2lw−lu−1 if u<w and Pw,w=1. The subalgebra ℋ of ℋ˜ generated by all Tss∈S is the Hecke algebra of the Coxeter system W,S. For ω in Ω and w in W, we have Cωw=TωCw. Thus, we have Pωu,ωw=Pu,w and Pω′u,ωw=0 for any different ω′,ω in Ω and u,w in W. It is known that the coefficients of these polynomials are all nonnegative [12].The Kazhdan–Lusztig polynomials can be written asPu,w=μu,wq1/2lw−lu−1 +lower degree terms. The coefficient μu,w is very interesting, which can be seen even from the recursive formula for Kazhdan–Lusztig polynomials [1]. We call μu,w the leading coefficient of Pu,w. Write u≺w if u≤w and μu,w≠0. Define μ˜u,w=μu,w if u≤w, −1lw−lu<0 and μ˜u,w=μw,u if w≤u.Forμu,w, there are the following conclusions [1].(1)
Letu,w∈W,s∈S, such that u<w,su>u,sw<w. Then, u≺w if and only if w=su. Moreover, μu,w=1 in this case.(2)
Letu,w∈W,s∈S, such that u<w,us>u,ws<w. Then, u≺w if and only if w=us. Moreover, μu,w=1 in this case.The elementsCw have the following properties [1]. For s∈S, we have(1)CsCw=q1/2+q−1/2Cw,sw≤w,Csw+∑y≺w,sy≤yCy,sw≥w,CwCs=q1/2+q−1/2Cw,ws≤w,Cws+∑y≺w,ys≤yCy,ws≥w.Givenx,y∈W, CxCy=∑z∈Whx,y,zCz, hx,y,z∈A. For any z∈W, define az  = min{i∈ℕ|q−1/2hx,y,z∈ℤq−1/2, for any x,y∈W}. Denote hx,y,z=γx,y,zqaz/2+δx,y,zqaz−1/2 +lower degree terms, where γx,y,z and δx,y,z are all nonnegative.


## 2.2. The Lowest Two-Sided Cell

It is known thatc0=w∈W˜|aw=lw0 is a two-sided cell, which is the lowest one with respect to the partial order ≤LR, where w0 is the longest element of W0 [13–15]. The set c0 is called the lowest two-sided cell of W˜.In [2], Leonard Scott and Nanhua Xi gave a description of c0. Let R+ (respectively, R+ and Δ) be the set of positive (respectively, negative and simple) roots in the root system R of W0. The set of dominant weights Λ+ is the set z∈Λ|lzw0=lz+lw0. For each simple root α, sα is the corresponding simple reflection in W0 and xα is the corresponding fundamental weight. For each w∈W0, set dw=w∏α∈Δ,wα∈R−xα. Then,(2)c0=dwzw0du−1|w,u∈W0,z∈Λ+.Moreover, the setc0,w′=dwzw0du−1|u∈W0,z∈Λ+ is a right cell of W˜ and c0,u=dwzw0du−1|w∈W0,z∈Λ+ is a left cell of W˜. The distinguished involutions of c0 are dww0dw−1, w∈W0.Forx∈Λ+, let Vx be a rational irreducible G-module of the highest weight x and Sx be the corresponding element defined in [16]. For any x∈Λ, Bernstein Joseph gave an element θx∈ℋ, defined as(3)θx=q−lx′Tx′q−lx″Tx″−1,where x′,x″∈Λ+ satisfies x=x′−x″. For x∈Λ+, xW0 is the W0-orbit in Λ which contains x. Set zx=∑x′∈xW0θx′. For x∈Λ+, define(4)Sx=∑x′∈Λ+,x′≤xdx′xzx′,where dx′x is the multiplicity of x′ in Vx. Then, Sx|x∈Λ+ is the A-base of the center of ℋ.For everyw∈W0, define two cell-related combinations:(5)Edw=q−ldw/2∑y≤dw,lyw0=ly+lw0Pyw0,dww0Ty,Fdw=q−ldw/2∑y≤dw,lyw0=ly+lw0Pyw0,dww0Ty−1,Then, the following facts were concluded in [16]:(a)
EdwSxCw0Fdu=Cdwxw0du−1 for any w,u∈W and x∈Λ+(b)
SxSy=∑z∈Λ+mx,y,zSz for any x,y∈Λ+. Here, mx,y,z is defined to be the multiplicity of Vz in the tensor product Vx⊗Vy;(c)
For anyy,w∈c0, if μy,w≠0, then we have y∼wL and y≁wR or y∼wR and y≁wL;(d)
Letw,w′,u∈W0, y,z∈Λ+ and let dwyw0du−1,dw′zw0du−1 be elements of the left cell c0,u of the extended affine Weyl group W˜. Then, μdwyw0du−1,dw′zw0du−1=μdwyw0,dw′zw0;(e)
Letx=duxw0,y=du′x′w0, where u,u′,v∈W0 and x,x′∈Λ+. Then, the set z∈W|hw0du−1,du′w0,z≠0 is finite, and we know that z=z1w0 for some z1∈Λ+ by hw0du−1,du′w0,z≠0. Moreover, μy,w=∑z∈Λ+mx∗,x,z∗δw0du−1,du′w0,zw0, where x∗=w0x−1w0.


## 2.3. The Lowest Two-Sided Cell ofG˜2

For the affine Weyl group of typeG˜2, W0=e,s,t,st,ts,sts,tst,stst,tsts,ststs,tstst,ststst, and the simple reflection set Δ=α,β.The fundamental weights ofG˜2 are xα=2α+β=ststsr and xβ=3α+2β=tstsr2 in W.As above subsection mentioned, for eachw∈W0, dw=w∏α∈Δ,wα∈R−xα. Then, we get the following results.(6)de=e,ds=sxα=sststsr=tstsr,dt=txβ=ststsr2=ststrstsr,dst=stxβ=sttstsr2=tstrstsr,dts=nntsxα=tsststsr=stsr,dsts=nnstsxα=stsststsr=tsr,dtst=ntstxβ=tsttstsr2=strstsr,dstst=nststxβ=ststtstsr2=trstsr,dtsts=nntstsxα=tstsststsr=sr,dststs=nststsxα=ststsststsr=r,dtstst=tststxβ=tststtstsr2=rstsr,dststst=stststxαxβ=rstsrtstsr.From these results andc0=dwzw0du−1|w,u∈W0,z∈Λ+, the two-sided cell c0 consists of the following elements:(7)vi,j,a,b=vistststxaybvj−1,where 1≤i,j≤12, a,b∈ℕ, and v1=e,v2=r,v3=sr,v4=tsr,v5=stsr, v6=tstsr,v7=rstsr,8=trstsr,v9=strstsr,v10=tstrstsr, v11=ststrstsr,v12=rststrstsr, x=rststs,y=rstst2.


## 3. Algorithm for ComputingCsCw

We describe our algorithm in this section in order to compute Kazhdan–Lusztig polynomials for the affine Weyl group and some special multiplications of Kazhdan–Lusztig basis. Before doing this, we give some lemmas for preparation.The following lemma makes sure whether two subwordsw1 and w2 is equal in W˜.Lemma 1.
Given a pointv0∈A0+, where A0+=x∈ℌ∗|x,αˇ>0,α∈Π,x,α0ˇ<1, then w1v0=w2v0 if and only if w1=w2.Proof.
Ifw1=w2, it is easy to see that w1v0=w2v0. If w1v0=w2v0, then w1v0=w2v0∈w1A+∩w2A+≠∅ and w1A+=w2A+. Since Wa acts on all alcoves simply transitively, we get w1=w2.Lemma 2 (see [17]).
LetW,S be a Coxeter group and w∈W. Fix any reduced expression w=s1,…,sn. Then, v≤w if and only if there exists 1≤j1<j2<⋯<jp≤n, such that v=s1,…,s^j1,…,s^jp,…,sn. In fact, for v≤w, j=j1<j2<⋯<jp can be choosed, such that the above expression is a reduced expression of v.
The following lemma makes sure whetherw is a reduced form.Lemma 3 (see [18]).
Assume thatw=s1,s2,…,sq with si∈S. We can find a subsequence i1<i2<⋯<ir of 1,2,…,r, such that w=si1,si2,…,sir is a reduced expression.
The following lemma increases the speed of the algorithm.Lemma 4.
Letx,y∈W and lx−ly≥3. If there is a s∈S, such that sx<x and sy>y (respectively, xs<x and ys>y), then μx,y=0.Proof.
If there iss∈S, such that sx<x and sy>y, then Py,x=Psy,x. Since lx−ly≥3, the degree of Py,x is equal to the degree of Psy,x≤lx−lsy−1/2=lx−ly−2/2<lx−ly−1/2, then μx,y=0.
We describe our algorithm as follows.First step: all subwords ofw are obtained by following two cases(1)
If the length ofw is small (lw≤10), all subwords are obtained by the algorithm of binary tree. For example, in G2˜, if w=s0s1s2s1s0, then(8)A=e,s0,s1,s2,s0s1,s0s2,s0s0,s1s0,s1s2,s1s1,s2s1,s2s0,s0s1s2,s0s1s1,s0s1s0,s0s2s1,s0s2s0,s1s2s1,s1s2s0,s1s1s0,s2s1s0,s0s1s2s1,s0s1s2s0,s0s1s1s0,s0s2s1s0,s1s2s1s0,s0s1s2s1s0is the set of all subwords ofw.(2)
If the length ofw is large enough (lw>10), w is split into two smaller subwords w1,w2. We obtain the set A (respectively, B) of all subwords of w1 (respectively, w2) by the algorithm of binary tree. The generated relations of the Coxeter group can be used to do some preliminary reduces for A and B. The results are denoted by A′ and B′. All subwords of w are given by gluing the elements of A′ and B′. For example, in G2˜, w=s0s1s2s1s0s1s2s1s2s1. We split w into w1=s0s1s2s1s0 and w2=s1s2. Then,(9)A′=e,s0,s1,s2,s0s1,s0s2,s1s0,s1s2,s2s1,s0s1s0,s0s2s1,s1s2s1,s1s2s0,s2s1s0,s0s1s2s1,s0s1s2s0,s0s2s1s0,s1s2s1s0,s0s1s2s1s0,B′=e,s1,s2,s1s2.We get all subwords ofw,(10)e,s0,s1,s2,s0s1,s0s2,s1s0,s1s2,s2s1,s0s1s0,s0s2s1,s1s2s1,s1s2s0,s2s1s0,s0s1s2s1,s0s1s2s0,s0s2s1s0,s1s2s1s0,s0s1s2s1s0,s0s0,s1s0,s2s0,s1s2s0,s2s1s0,s0s1s0s0,s0s2s1s0,s1s2s1s0,s1s2s0s0,s2s1s0s0,s0s1s2s1s0,s0s1s2s0s0,s0s2s1s0s0,s1s2s1s0s0,s0s1s2s1s0s0,s0s2,s1s2,s2s2,s0s1s2,s0s2s2,s1s0s2,s1s2s2,s2s1s2,s0s1s0s2,s0s2s1s2,s1s2s1s2,s1s2s0s2,s2s1s0s2,s0s1s2s1s2,s0s1s2s0s2,s0s2s1s0s2,s1s2s1s0s2,s0s1s2s1s0s2,s1s2,s0s1s2,s1s1s2,s2s1s2,s0s1s1s2,s0s2s1s2,s1s0s1s2,s1s2s1s2,s2s1s1s2,s0s1s0s1s2,s0s2s1s1s2,s1s2s1s1s2,s1s2s0s1s2,s2s1s0s1s2,s0s1s2s1s1s2,s0s1s2s0s1s2,s0s2s1s0s1s2,s1s2s1s0s1s2,s0s1s2s1s0s1s2,s0s1s0,s0s2s0,s1s0s0.Second step: get all unique and reduced subwords ofwAll subwords ofw are denoted by M. For any m∈M, m is the length of m. Then, an increasing order of all elements of M is given with respect to ⋅. The elements of M can be reordered as w11,w12,…,w1r, where w11≤w12≤⋯≤w1r. There is a correspondence sequence w11v0,w12v0,…,w1rv0, where v0 is a rational point of A0+. By the lemma above, we delete w1i if there exists j<i, such that w1iv0=w1jv0. Then, all unique and reduced subwords of w are obtained by this way.Third step: compute Kazhdan–Lusztig polynomialPx,y for x,y∈W by following cases(i)
Ifly<lx, Px,y=0(ii)
Forly=lx, if yv0=xv0, then y=x, Px,y=1. Otherwise, Px,y=0.(iii)
Forlx<ly, y=sw and ly=lw+1. We use the following recursive formula to compute the corresponding Kazhdan–Lusztig polynomial:(ii)
Forly=lx, if yv0=xv0, then y=x, Px,y=1. Otherwise, Px,y=0.(iii)
Forlx<ly, y=sw and ly=lw+1. We use the following recursive formula to compute the corresponding Kazhdan–Lusztig polynomial:Wherea=1 if sx<x, a=0 if sx>x.By Lemma2, if xv0=xi1v0, where xi1 is a subword of sx, then sx>x and a=0; otherwise, sx<x, and sx is not a reduced form; then, a=1.In our algorithm, we computePx,y for reduced words x,y. So we get the reduced form of sx by following proceeding. By the second step, we get the reordered subwords set M and the corresponding sequence x11v0,x21v0,…,xn1v0 of sx. The by Lemma 3, the reduced form of sx is xi1, where xi1 is the first subword of sx that satisfies the condition xi1v0=sxv0.Thez in the above recursive formula is obtained by following proceedings.(1)
All unique reduced subwords ofw are obtained by the second step. The results are denoted by the set A;(2)
We search allu∈A satisfying the condition lw−lu is odd. The results are denoted by the set B;(3)
We search allu1∈B satisfying the condition su1v0=u1rv0, where u1r is a reduced subword of u1. The results are denoted by the set C;(4)
We search allu2∈C satisfying the condition xv0=u2rv0, where u2r is a reduced subword of u2. The results are denoted by the set D;(5)
We search allu3∈D which do not satisfy the condition Lemma 4. The results are denoted by the set E;(6)
We search allu4∈E satisfying the condition that the term μwi,wqlw−lwi+1/2 of Pwi,w is nonzero. The results are denoted by the set Z. The elements of Z are all z in the above recursive formula.IfPsx,w, Px,w, Pz,w, and Px,z are known, Px,y are computed by the above induction formula. Otherwise, we compute Px,y through the recursive algorithm.Fourth step: computeCsCx for s∈S,x∈W by following induction formula:(12)Csw=CsCw−∑y≺w,sy≤yμy,wCy,sw≥w.


## 4. Computationμy,w for y,w∈c0

In this section, we use the algorithm mentioned inSection 3 to calculate CsCx for some s∈S,x∈W. These multiplications are used to manually compute Cw0du−1Cdu′w0 for u,u′∈W0, in order to get δw0du−1,du′w0,z1w0, which is the coefficient of the term q1/2az1w0−1 of hw0du−1,du′w0,z1w0. By (e) in Section 2.2, μy,w=∑z∈Λ+mx∗,x′,z∗δw0du−1,du′w0,zw0, where x∗=w0x−1w0. Our main result Theorem 1 is obtained by Lemmas 5 and 6.We use an algebra antiautomorphism ofℋ which is defined by George Lusztig in [11] to prove the following lemma.Lemma 5.
Letx=duxw0,y=du′x′w0, where u,u′,v∈W0 and x,x′ belong to Λ+. Then, CxCy=Cy−1Cx−1.Proof.
There is a unique algebra antiautomorphism ofH which is denoted by φ, such that φTx=Tx−1, for all x∈W; it maps Kazhdan–Lusztig basis element Cx⟶Cx−1. For any x,y∈W, φCxCy=φCyφCx=Cy−1Cx−1, that is, φCxCy=φ∑z∈Whx,y,zCz=∑z∈Whx,y,zCz−1=∑z∈Why−1,x−1,zCz, and we get hx,y,z=hy−1,x−1,z−1. By (d) in Section 2.2, we know that hx,y,z≠0, z=z1w0, for some z1∈Λ+. Then, if hx,y,z≠0, the relation hx,y,z=hx,y,z1w0=hy−1,x−1,z−1=hy−1,x−1,w0z−1 holds. In the case of G2˜, we know that z1w0=w0z−1, that is, hx,y,z=hy−1,x−1,z, for all z∈W.
In the following, for anyu,u′∈W0, all Cw0du−1Cdu′w0 for u,u′∈W0 are computed to obtain δw0du−1,du′w0,z1w0, which is the coefficient of the term q1/2az1w0−1 of the multiplication polynomial hw0du−1,du′w0,z1w0.Proposition 1.
For anyu,u′∈W0, if δw0du−1,du′w0,z1w0 is not equal to zero, we must have δw0du−1,du′w0,z1w0=1 and z1∈0,xα,xβ, where xα,xβ∈Λ+ are fundamental weights.Proof.
We computeCxCy through the following steps.(a)
We use the formulaCwCs=Cx+∑u≺w,us<uμu,wCu for x=ws and lx=lw+1 [1] to get a decomposition of Cx(b)
For the reflections in step (a), we compute CsCy by the formula CsCy=Csy+∑v≺y,sv<yμv,yCv for sy>y and CsCy=q1/2+q−1/2Cy for sy<y [1](c)
We getCxCy=∑u,v∈WCuCv, where lu<lx by step (a) and step (b). Repeat this proceeding until CxCy=∑u′,v′∈WCu′Cv′, where lu′=0; then, we obtain CxCy.
It is known thatPw,v=1 for any w,v∈W0. By this conclusion, the complete decomposition Cw0=C1C2C1C2C1C2−4C1C2C1C2+3C1C2 is obtained. This complete decomposition plays a crucial role in the proof of the following proposition. Cw0du−1Cdu′w0 for u,u′∈W0 will be calculated with the increase of lw0du−1 by following cases.(i)
IfCw0du−1=Cw0, we get Cw0du−1Cdu′w0=Cw0Cdu′w0=C1C2C1C2C1C2−4C1C2C1C2+3C1C2Cdu′w0. The proposed algorithm is applied step by step to compute C1C2C1C2C1C2Cdu′w0, C1C2C1C2Cdu′w0, and C1C2Cdu′w0. Then, the decomposition of Cw0Cdu′w0 is obtained;(ii)
Iflw0du−1>lw0, there are two types of calculation of Cw0du−1Cdu′w0(1)
Step (a) and step (b) mentioned above will be repeatedly used to get linear combinationCw0du−1Cdu′w0=∑u,v∈WCuCv, where CuCv can be computed by recursion from case (i)(2)
IfCw0du−1Cdu′w0 cannot be computed by (1), step (a) and step (b) mentioned above will be repeatedly used to get the linear combination Cw0du−1Cdu′w0=∑u,v∈WCuCv, where Cu=Cw0. All Cw0Cv can be computed by (i).
We set2≜q1/2+q−1/2, 1≜s, 2≜t, and 0≜r. Compute C121212C121201210121212 as follows:(13)C121212=C2C1C2C1C2C1−4C2C1C2C1+3C2C1,C1C121201210121212=2C121201210121212,C2C121201210121212=C2121201210121212+C21201210121212+Cw0,C2C1C121201210121212=2Cxβw0+C21201210121212+Cw0,C1C21201210121212=C121201210121212+C1201210121212,C1C2C1C121201210121212=22Cxβw0+C121201210121212+C1201210121212+2Cw0,C2C1201210121212=C21201210121212+C201210121212+Cxαw0,C2C1C2C1C121201210121212=222+1Cxβw0+2C21201210121212+C201210121212+Cxαw0+22+1Cw0,C1C201210121212=C1201210121212+C01210121212,C1C2C1C2C1C121201210121212=223+2Cxβw0+2C121201210121212+3C1201210121212+C01210121212+2Cxαw0+23+2Cw0,C2C01210121212=C201210121212,C2C1C2C1C2C1C121201210121212=224+22+2Cxβw0+5C21201210121212+4C201210121212+22Cxαw0+24+22+2Cw0,C121212C121201210121212=224+22+2Cxβw0+5C21201210121212+4C201210121212+22Cxαw0+24+22+2Cw0−4222+1Cxβw0+2C21201210121212+C201210121212+Cxαw0+22+1Cw0+32Cxβw0+C21201210121212+Cw0=25−323+2Cxβw0+23−2Cxαw0+25−323+2Cw0.
We obtain the following Kazhdan–Lusztig basis multiplication results through the proposed the Kazhdan–Lusztig polynomials and the Kazhdan–Lusztig basis multiplication program. All Kazhdan–Lusztig basis multiplication of the lowest cell of the affine Weyl group of typeG˜2 are computed, and we gave the step by step computation details in reference [19].(14)Cw0Cw0=26−424+322Cw0,C121212C0121212=C1212120C121212=2Cxαw0+25−323+2Cw0,C121212C10121212=C12121201C121212=22Cxαw0+24−222Cw0,C121212C210121212=C121212012C121212=23−2Cxαw0+23−2Cw0,C121212C1210121212=C1212120121C121212=24−222Cxαw0+22Cw0,C121212C21210121212=C12121201212C121212=25−323+2Cxαw0+2Cw0,C121212C01210121212=C12121201210C121212=2Cxβw0+23−2Cxαw0+2Cw0,C121212C201210121212=C121212012102C121212=22Cxβw0+24−22Cxα121212+22Cw0,C121212C1201210121212=C1212120121021C121212=23−2Cxβw0+23−2Cw0+25−223+2Cxαw0,C121212C21201210121212=C12121201210212C121212=24−222Cxβw0+24−22Cxαw0+24−222Cw0,C121212C121201210121212=C121212012102121C121212=25−323+2Cxβw0+23−2Cxαw0+25−323+2Cw0,C121212C0121201210121212=C1212120121021210C121212=Cxαxβw0+22Cxα2w0+222Cxαw0+24−222Cxβw0+24−222Cw0,C1212120C121201210121212=C121212012102121C0121212=Cxαxβw0+22Cxα2w0+322Cxαw0+24−222Cxβw0+224−422Cw0,C1212120C0121201210121212=C1212120121021210C0121212=2C121212C0121201210121212,C12121201C10121212=24−22Cxαw0+26−324+222Cw0,C12121201C210121212=C121212012C10121212=25−223+2Cw0+2Cxβw0+223−2Cxαw0,C12121201C1210121212=C1212120121C10121212=24Cxαw0+22Cxβw0+24−22Cw0,C12121201C21210121212=C12121201212C10121212=2Cxα2w0+223−2Cxαw0+22Cxβw0+23Cw0,C12121201C01210121212=C12121201210C10121212=23−2Cw0+23−2Cxβw0+25−223+2Cxαw0,C12121201C201210121212=C121212012102C10121212=22Cxα2w0+224−22Cxαw0+222Cxβw0+24Cw0,C12121201C1201210121212=C1212120121021C10121212=23Cxα2w0+25+23−2Cxαw0+23+2Cxβw0+25−23+2Cw0,C12121201C21201210121212=C12121201210212C10121212=Cxαxβw0+222Cxα2w0+222Cxβw0+224−22Cw0+24+322Cxαw0,C12121201C121201210121212=C121212012102121C10121212=2Cxαxβ121212+23Cxα2121212+25−23−2Cw0+223+2Cxαw0+23−2Cxβw0,C12121201C0121201210121212=C1212120121021210C10121212=22Cxαxβw0+224−322Cw0+24+222Cxαw0+24−22Cxβw0+24Cxα2w0,C121212012C210121212=22Cxβw0+24Cxαw0+26−324+222Cw0,C121212012C1210121212=C1212120121C210121212=2Cxα2w0+22Cxβw0+223Cxαw0+25−223+2Cw0,C121212012C21210121212=C12121201212C210121212=C121212012C01210121212=C12121201210C210121212=22Cxα2w0+24Cxαw0+22Cxβw0+24−22Cw0,C121212012C201210121212=C121212012102C210121212=23Cxα2w0+25Cxαw0+23Cxβw0+25−23Cw0,C121212012C1201210121212=C1212120121021C210121212=Cxαxβw0+322Cxα2w0+322Cxβw0+224+22Cxαw0+224−22Cw0,C121212012C21201210121212=C12121201210212C210121212=2Cxαxβw0+23+2Cxα2w0+25+22Cxαw0+23+2Cxβw0+25−23+2Cw0,C121212012C121201210121212=C121212012102121C210121212=22Cxαxβw0+222Cxβw0+24Cw0+24+222Cxαw0+222Cxα2w0,C121212012C0121201210121212=C1212120121021210C210121212=23Cxαxβw0+223−2Cxβw0+25+2Cxαw0+223Cxα2w0+223−2Cw0,C1212120121C1210121212=22Cxα2w0+24Cxαw0+24Cxβw0+26−324+222Cw0,C1212120121C21210121212=C12121201212C1210121212=23−2Cxα2w0+223−2Cxαw0+223−2Cxβw0+25−223Cw0,C1212120121C01210121212=C12121201210C1210121212=23Cxα2w0+223−2Cxαw0+2Cxβw0+25−223+2Cw0,C1212120C0121212=22Cxαw0+26−324+22Cw0,C1212120C10121212=C12121201C0121212=23Cxαw0+25−223Cw0,C1212120C210121212=C121212012C0121212=24−22Cxαw0+24−22Cw0,C1212120C1210121212=C1212120C1210121212=2Cxβw0+223−2Cxαw0+23Cw0,C1212120C21210121212=C12121201212C0121212=C1212120C01210121212=C12121201210C0121212=22Cxβw0+24−22Cxαw0+22Cw0,C1212120C201210121212=C121212012102C0121212=23Cxβw0+25−23Cxαw0+23Cw0,C1212120C1201210121212=C1212120121021C0121212=22Cxα2121212+224−22Cxα121212+222Cxβ121212+24Cw0,C1212120C21201210121212=C12121201210212C0121212=23−2Cxα2w0+323−22Cxαw0+23−2Cxβw0+25−23−2Cw0,C12121201212C01210121212=C12121201210C21210121212=Cxαxβw0+222Cxα2w0+322Cxαw0+22Cxβw0+24−22Cw0,C12121201212C201210121212=C121212012102C21210121212=2Cxαxβw0+223Cxα2w0+323Cxαw0+23Cxβw0+25−23Cw0,C12121201212C1201210121212=C1212120121021C21210121212=22Cxαxβw0+24+22Cxα2w0+224−222Cw0+224+22Cxαw0+24Cxβw0,C12121201212C21201210121212=C12121201210212C21210121212=23−2Cxαxβw0+223−2Cxα2w0+223−2Cw0+25+23−22Cxαw0+223−2Cxβw0,C12121201212C121201210121212=C121212012102121C21210121212=24−222Cxαxβw0+222Cxβw0+324−522Cxαw0+24−22Cxα2w0+222Cw0,C12121201212C0121201210121212=C1212120121021210C21210121212=23+2Cxαxβw0+22Cxα3w0+223+2Cxα2w0+423−2Cxαw0+23+22Cw0+23+42Cxβw0,C12121201210C01210121212=24−22Cxα2w0+24−22Cxαw0+26−324+222Cw0,C12121201210C201210121212=C121212012102C01210121212=2Cxαxβw0+223Cxα2w0+23Cxβw0+25−23Cw0+323Cxαw0,C12121201210C1201210121212=C1212120121021C01210121212=22Cxαxβw0+224−22Cw0+224+22Cxαw0+24Cxβw0+24+22Cxα2w0,C12121201210C21201210121212=C12121201210212C01210121212=2Cxα3w0+423Cxαw0+223Cw0+23+32Cxβw0+22Cxαxβw0+23+32Cxα2w0,C12121201210C121201210121212=C121212012102121C01210121212=22Cxα3w0+224−22Cxαw0+322Cxβw0+22Cxαxβw0+222Cxα2w0+222Cw0,C12121201210C0121201210121212=C1212120121021210C01210121212=23Cxα3w0+23Cxαxβw0+223+2Cxβw0+25−2Cxαw0+223Cxα2w0+23+2Cw0,C121212012102C1201210121212=C1212120121021C201210121212=2Cxα3w0+823+2Cxαw0+323+22Cxβw0+42Cxαxβw0+423+32Cxα2w0+25+223−2Cw0,C121212012102C21201210121212=C12121201210212C201210121212=22Cxα3w0+424Cxαw0+24+322Cxβw0+222Cxαxβw0+24+322Cxα2w0+224Cw0,C121212012102C121201210121212=C121212012102121C201210121212=2Cxβ2w0+23+2Cxαxβw0+223+32Cxα2w0+22Cxα3w0+623−22Cxαw0+23+62Cxβw0+223+22Cw0,C121212012102C0121201210121212=C1212120121021210C201210121212=22Cxβ2w0+24+22Cxαxβw0+224+222Cxα2w0+424−22Cxαw0+24+422Cxβw0+24+22Cw0+222Cxα3w0,C1212120121021C1201210121212=22Cxα3w0+624Cxαw0+224+222Cxβw0+322Cxαxβw0+324+222Cxα2w0+26+22Cw0,C1212120121021C21201210121212=C12121201210212C1201210121212=2Cxβ2w0+23+32Cxαxβw0+523+32Cxα2w0+1023−2Cxαw0+323+52Cxβw0+25+223+2Cw0+22Cxα3w0,C1212120121021C121201210121212=C121212012102121C1201210121212=22Cxβ2w0+24Cxαxβw0+224Cw0+224+22Cxα2w0+424Cxαw0+24+322Cxβw0+22Cxα3w0,C1212120121021C0121201210121212=C1212120121021210C1201210121212=2Cxα2xβw0+22Cxβ2w0+423+2Cw0+523+52Cxβw0+923Cxαw0+523+52Cxα2w0+323+22Cxαxβw0+23+32Cxα3w0,C12121201210212C21201210121212=22Cxβ2w0+26−324+222Cw0+224+322Cxα2w0+324+422Cxαw0+24+322Cxβw0+322Cxαxβw0,C1212120121C201210121212=C121212012102C1210121212=Cxαxβw0+322Cxα2w0+222Cxβw0+224−22Cw0+24+322Cxαw0,C1212120121C1201210121212=C1212120121021C1210121212=2Cxαxβw0+223Cxα2w0+25−2Cw0+25+23+2Cxαw0+223−2Cxβw0,C1212120121C21201210121212=C12121201210212C1210121212=22Cxαxβw0+322Cxα2w0+322Cxβw0+24+22Cw0+224+22Cxαw0,C1212120121C121201210121212=C121212012102121C1210121212=23−2Cxαxβw0+23+2Cxβw0+23+2Cw0+25−2Cxαw0+23Cxα2w0,C1212120121C0121201210121212=C1212120121021210C1210121212=22Cxα3w0+224Cxαw0+422Cxβw0+222Cxαxβw0+422Cxα2w0+322Cw0,C12121201212C21210121212=24−222Cxα2w0+24−22Cxαw0+224−322Cxβw0+26−324+22Cw0,C12121201210212C121201210121212=C121212012102121C21201210121212=23+2Cxαxβw0+423Cxα2w0+423+22Cxαw0+423−32Cxβw0+23−2Cxβ2w0+25+23−22Cw0+23−2Cxα3w0,C12121201210212C0121201210121212=C1212120121021210C21201210121212=22Cxα2xβw0+224+322Cxβw0+622Cxα2w0+322Cxα3w0+22Cxβ2w0+422Cxαxβw0+224−22Cw0+324+622Cxαw0,C121212012102121C121201210121212=C121212012102121C121201210121212=24Cxαxβw0+224−22Cxα2w0+222Cxαw0+26−24−222Cw0+24−322Cxβ2w0+324−622Cxβw0+24−322Cxα3w0,C121212012102121C0121201210121212=C1212120121021210C121201210121212=2Cxα4w0+42Cxα2xβw0+723+72Cxα2w0+23+82Cxαxβw0+23Cxβ2w0+25+423+2Cxβw0+23+42Cxα3w0+25+223+112Cxαw0+25+23+22Cw0,C1212120121021210C0121201210121212=22Cxα4w0+422Cxα2xβw0+624+722Cxα2w0+24Cxβ2w0+24+722Cxαxβw0+26+324+222Cxβw0+26+24+922Cxαw0+26−24+522Cw0+24+422Cxα3w0.
We know thatδw0du−1,du′w0,z1w0 is the coefficient of q5/2 of hw0du−1,du′w0,z1w0. By the above process of calculation, we get the conclusion.
Define the following corresponding sets of nontrivial coefficients of the multiplications polynomials:(15)U1=du,du′,z|u,u′∈W0,∃!z∈0,xα,xβ,δw0du−1,du′w0,zw0=1,U2=du,du′|du−1,du′,z1,du−1,du′,z2,u,u′∈W0,z1,z2=0,xα,δw0du−1,du′w0,z1w0=δw0du−1,du′w0,z2w0=1,U3=du,du′|du−1,du′,z1,du−1,du′,z2,u,u′∈W0,z1,z2=0,xβ,δw0du−1,du′w0,z1w0=δw0du−1,du′w0,z2w0=1,U4=du,du′|du−1,du′,z1,du−1,du′,z2,du−1,du′,z3,u,u′∈W0,z1,z2,z3=0,xα,xβ,δw0du−1,du′w0,z1w0=1=δw0du−1,du′w0,z2w0=δw0du−1,du′w0,z3w0.
Using the conclusion of Proposition1, we can list the corresponding elements of nontrivial coefficients of the multiplications polynomials as follows:(16)U1=e,r,0,r,e,0,e,tstsr,xα,tstsr,e,xα,e,strstsr,xα,strstsr,e,xα,sr,r,0,r,trstsr,xα,trstsr,r,xα,r,tstrstsr,0,tstrstsr,r,0,r,trstsr,xα,trstsr,r,xα,sr,tsr,0,sr,rstsr,xα,rstsr,sr,xα,sr,ststrstsr,0,ststrstsr,sr,0,stsr,tsr,0,tsr,rststrstsr,xα,r,sr,0,tsr,sr,0,rststrstsr,tsr,xα,stsr,tstsr,0,tstsr,stsr,0,stsr,rstsr,0,rstsr,stsr,0,stsr,ststrstsr,xα,ststrstsr,stsr,xα,tstsr,trstsr,0,tsr,stsr,0,tstsr,tstrstsr,xα,trstsr,tstsr,0,tstrstsr,tstsr,xα,rstsr,trstsr,0,trstsr,rstsr,0,rstsr,rststrstsr,xα,rststrstsr,rstsr,xα,trstsr,strstsr,0,strstsr,trstsr,0,rstsrts,tstrstsr,0,tstrstsr,strstsr,0,strstsr,tstrstsr,0,tstrstsr,strstsr,0,tstrstsr,ststrstsr,0,ststrstsr,tstrstsr,0,U2=sr,strstsr,strstsr,sr,tsr,trstsr,trstsr,tsr,tsr,tstrstsr,tstrstsr,tsr,stsr,strstsr,strstsr,stsr,U3=e,ststrstsr,ststrstsr,e.U4=ststrstsr,rststrstsr,rststrstsr,ststrstsr.
The following lemma is a well-known result in representation theory [18].Lemma 6.
Assume thatL is a semisimple complex Lie algebra, and λ and λ′ are the dominant weights. Let Kλ,γ be the weight multiplicity of weight γ in the irreducible L-module Vλ, and mλ,λ′,ν be the tensor product multiplicity of irreducible module Vν in Vλ⊗Vλ′. Then, we have that 0≤mλ,λ′,ν≤Kλ,ν−λ′.Theorem 1.
For anyy,w∈c0, if y∼wL, that is, y=duxw0dv,w=du′x′w0dv, where u,u′,v∈W0, and x,x′∈Λ+. Then, we must have the following:(i)
If there isz∈0,xα,xβ, such that du,du′,z∈U1, then we have μy,w=mx,x′,z(ii)
If there isdu,du′∈U2, then we have μy,w=mx,x′,0+mx,x′,xα(iii)
If there isdu,du′∈U3, then we have μy,w=mx,x′,0+mx,x′,xβ(iv)
If there isdu,du′∈U4, then we have μy,w=mx,x′,0+mx,x′,xα+mx,x′,xβ
Otherwise,μy,w=0. Moreover, for any y,w∈c0, we get μy,w≤3.Proof.
For anyy,w∈c0, if μy,w≠0; then, we know that y∼wL or y∼wR. If y∼wR, then y−1∼wL−1. Since we have μy,w=μy−1,w−1, we can assume y∼wL.
Ify=duxw0dv,w=du′x′w0dv, for some u,u′,v∈W0, and x,x′∈Λ+. By (c) in Section 2.2, we also can assume y=duxw0,w=du′x′w0. By (d) in Section 2.2, we know that(17)μy,w=∑z∈Λ+mx∗,x′,z∗δw0du−1,du′w0,zw0,where x∗=w0x−1w0,z∗=w0z−1w0∈Λ+.
From this, we get the following:(1)
If there isz∈0,xα,xβ, such that du,du′,z∈U1, then μy,w=mx,x′,z(2)
If there isdu,du′,z∈U2, then μy,w=mx,x′,0+mx,x′,xα(3)
If there isdu,du′,z∈U3, then μy,w=mx,x′,0+mx,x′,xβ(4)
If there isdu,du′,z∈U4, then μy,w=mx,x′,0+mx,x′,xα+mx,x′,xβ
Since, we have(18)mx∗,x′,z∗=dimHomGVx∗⊗Vx′,Vz∗=dimHomGVz∗⊗Vx′,Vx∗=mz,x′,x.
By the Freudenthal formula in Lie algebra and Lemma6, we can get if z∈0,xα,xβ, then we have dimVzλ for any weight of Vz, then by the fact (e) in Section 2.2, we get mx∗,x′,z∗=mz,x′,x≤1 for x,x′∈Λ+ and z∈0,xα,xβ. Then, we get the theorem.



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*Source: 5905276-2021-07-27.xml*