# The Based Rings of Two-sided cells in an Affine Weyl group of type
$\tilde{B}_{3}$, II

Yannan Qiu∗ and Nanhua XI† ∗ School of Mathematical Sciences
Zhejiang University
Zhejiang 310058, China
China<EMAIL_ADDRESS>† Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Beijing 100190, China
and
School of Mathematical Sciences
University of Chinese Academy of Sciences
Chinese Academy of Sciences
Beijing 100049, China<EMAIL_ADDRESS>Dedicated to George Lusztig with
greatest respect.

###### Abstract.

We compute the based rings of two-sided cells corresponding to the unipotent
classes in $Sp_{6}(\mathbb{C})$ with Jordan blocks (33), (411), (222),
respectively. The results for the first two two-sided cells also verify
Lusztig’s conjecture on the structure of the based rings of two-sided cells of
an affine Weyl group. The result for the last two-sided cell partially
suggests a modification of Lusztig’s conjecture on the structure of the based
rings of two-sided cells of an affine Weyl group.

Y. Qiu was partially supported by National Natural Science Foundation of
China, No. 12171030.

N. Xi was partially supported by National Key R&D Program of China, No.
2020YFA0712600, and by National Natural Science Foundation of China, No.
11688101.

We are concerned with the based rings of two-sided cells in an affine Weyl
group of type $\tilde{B}_{3}$. In a previous paper we discussed the based ring
of the two-sided cell corresponding to the nilpotent element in
$Sp_{6}(\mathbb{C})$ with 3 equal Jordan blocks and showed that Lusztig’s
conjecture on the structure of the based rings of the two-sided cells of an
affine Weyl group needs modification (see section 4 in [QX]). In this paper we
compute the based rings of two-sided cells corresponding to the unipotent
classes in $Sp_{6}(\mathbb{C})$ with Jordan blocks (411), (33), (222),
respectively. The results for the first two two-sided cells also verify
Lusztig’s conjecture on the structure of the based rings of two-sided cells of
an affine Weyl group. The result for the last two-sided cell partially
suggests a modification of Lusztig’s conjecture on the structure of the based
rings of two-sided cells of an affine Weyl group. For the first two two-sided
cells, the validity of Lusztig’s conjecture on the based rings is already
included in the main theorem in [BO]. Here we construct the bijection in
Lusztig’s conjecture explicitly so that the results in this paper can be used
for computing certain irreducible representations of affine Hecke algebras of
type $\tilde{B}_{3}$. In section 5 we give a description for the based ring of
the two-sided cell corresponding to the nilpotent element in
$Sp_{6}(\mathbb{C})$ with Jordan blocks (222), which can also be used to
compute certain irreducible representations of affine Hecke algebras of type
$\tilde{B}_{3}$.

The contents of the paper are as follows. Section 1 is devoted to
preliminaries, which include some basic facts on (extended) affine Weyl groups
and their Hecke algebras and formulation of Lusztig’s conjecture on the
structure of the based ring of a two-sided cell in an affine Weyl group. In
section 2 we recall some results on cells of the (extended) affine Weyl group
of type $\tilde{B}_{3}$, which are due to J. Du. Sections 3, 4, 5 are devoted
to discussing based rings of two-sided cells corresponding to the unipotent
classes in $Sp_{6}(\mathbb{C})$ with Jordan blocks (411), (33), (222),
respectively.

##  1\. Affine Weyl groups and their Hecke algebras

In this section we fix some notations and refer to [KL, L1, L2, L3, QX] for
more details.

1.1. Extended affine Weyl groups and their Hecke algebras Let $G$ be a
connected reductive algebraic group over the field $\mathbb{C}$ of complex
numbers. Let $W_{0}$ be the Weyl group of $G$ and $W$ the affine Weyl group
attached to $G$. The set of simple reflections of $W$ is denoted by $S$. We
shall denote the length function of $W$ by $l$ and use $\leq$ for the Bruhat
order on $W$. We also often write $y<w$ or $w>y$ if $y\leq w$ and $y\neq w$.

Let $H$ be the Hecke algebra of $(W,S)$ over
$\mathcal{A}=\mathbb{Z}[q^{\frac{1}{2}},q^{-\frac{1}{2}}]$ $(q$ an
indeterminate) with parameter $q$. Let $\\{T_{w}\\}_{w\in W}$ be its standard
basis. Then we have $(T_{s}-q)(T_{s}+1)=0$ and $T_{w}T_{u}=T_{wu}$ if
$l(wu)=l(w)+l(u)$. Let $C_{w}=q^{-\frac{l(w)}{2}}\sum_{y\leq w}P_{y,w}T_{y},\
w\in W$ be the Kazhdan-Lusztig basis of $H$, where $P_{y,w}$ are the Kazhdan-
Lusztig polynomials. The degree of $P_{y,w}$ is less than or equal to
$\frac{1}{2}(l(w)-l(y)-1)$ if $y<w$ and $P_{w,w}=1$. Convention: set
$P_{y,w}=0$ if $y\not\leq w$.

If $y<w$, we write $P_{y,w}=\mu(y,w)q^{\frac{1}{2}(l(w)-l(y)-1)}+\text{lower
degree terms.}$ We shall write $y\prec w$ if $\mu(y,w)\neq 0$. We have

(a) Let $y\leq w$. Assume that $sw\leq w$ for some $s\in S$. Then

$\displaystyle P_{y,w}$ $\displaystyle=P_{sy,w},\ \text{if}\ sy>y;$
$\displaystyle P_{y,w}$
$\displaystyle=q^{1-c}P_{sy,sw}+q^{c}P_{y,sw}-\sum_{\stackrel{{\scriptstyle\stackrel{{\scriptstyle
z\in W}}{{y\leq z\prec
sw}}}}{{sz<z}}}\mu(z,sw)q^{\frac{l(w)-l(z)}{2}}P_{y,z},$

where $c=1$ if $sy<y$ and $c=0$ if $sy>y$.

(b) Let $y\leq w$. Assume that $ws\leq w$ for some $s\in S$. Then

$\displaystyle P_{y,w}$ $\displaystyle=P_{ys,w},\ \text{if}\ ys>y;$
$\displaystyle\ P_{y,w}$
$\displaystyle=q^{1-c}P_{ys,ws}+q^{c}P_{y,ws}-\sum_{\stackrel{{\scriptstyle\stackrel{{\scriptstyle
z\in W}}{{y\leq z\prec
ws}}}}{{zs<z}}}\mu(z,ws)q^{\frac{l(w)-l(z)}{2}}P_{y,z},$

where $c=1$ if $ys<y$ and $c=0$ if $ys>y$.

From the two formulas above one gets (see [KL])

(c) Let $y,w\in W$ and $s\in S$ be such that $y<w,\ sw<w,$ and $sy>y$. Then
$y\prec w$ if and only if $w=sy$. Moreover this implies that $\mu(y,w)=1$.

(d) Let $y,w\in W$ and $s\in S$ be such that $y<w,\ ws<w$, and $ys>y$. Then
$y\prec w$ if and only if $w=ys$. Moreover this implies that $\mu(y,w)=1$.

The following formulas for computing $C_{w}$ (see [KL]) will be used in
sections 3, 4, 5.

(e) Let $w\in W$ and $s\in S$. Then

(1) $\displaystyle
C_{s}C_{w}=\begin{cases}\displaystyle(q^{\frac{1}{2}}+q^{-\frac{1}{2}})C_{w},\quad&\text{if\
}sw<w,\\\ \displaystyle C_{sw}+\sum_{\stackrel{{\scriptstyle z\prec
w}}{{sz<z}}}\mu(z,w)C_{z},\quad&\text{if\ }sw\geq w.\end{cases}$ (2)
$\displaystyle
C_{w}C_{s}=\begin{cases}\displaystyle(q^{\frac{1}{2}}+q^{-\frac{1}{2}})C_{w},\quad&\text{if\
}ws<w,\\\ \displaystyle C_{ws}+\sum_{\stackrel{{\scriptstyle z\prec
w}}{{zs<z}}}\mu(z,w)C_{z},\quad&\text{if\ }ws\geq w.\end{cases}$

1.2. Cells of affine Weyl groups We refer to [KL] for definition of left
cells, right cells and two-sided cells of $W$.

For $h,\,h^{\prime}\in H$ and $x\in W$, write

$hC_{x}=\sum_{y\in W}a_{y}C_{y},\quad C_{x}h=\sum_{y\in W}b_{y}C_{y},\quad
hC_{x}h^{\prime}=\sum_{y\in W}c_{y}C_{y},\quad
a_{y},b_{y},c_{y}\in\mathcal{A}.$

Define $y\underset{L}{\leq}x$ if $a_{y}\neq 0$ for some $h\in H$,
$y\underset{R}{\leq}x$ if $b_{y}\neq 0$ for some $h\in H$, and
$y\underset{LR}{\leq}x$ if $c_{y}\neq 0$ for some $h,h^{\prime}\in H$.

We write $x\underset{L}{\sim}y$ if $x\underset{L}{\leq}y\underset{L}{\leq}x$,
$x\underset{R}{\sim}y$ if $x\underset{R}{\leq}y\underset{R}{\leq}x$, and
$x\underset{LR}{\sim}y$ if $x\underset{LR}{\leq}y\underset{LR}{\leq}x$. Then
$\underset{L}{\sim},\ \underset{R}{\sim},\ \underset{LR}{\sim}$ are
equivalence relations on $W$. The equivalence classes are called left cells,
right cells, and two-sided cells of $W$ respectively. Note that if $\Gamma$ is
a left cell of $W$, then $\Gamma^{-1}=\\{w^{-1}\,|\,w\in\Gamma\\}$ is a right
cell.

For $w\in W$, set $R(w)=\\{s\in S\,|\,ws\leq w\\}$ and $L(w)=\\{s\in
S\,|\,sw\leq w\\}.$ Then we have (see [KL])

(a) $R(w)\subset R(u)$ if $u\underset{L}{\leq}w$ and $L(w)\subset L(u)$ if
$u\underset{R}{\leq}w.$ In particular, $R(w)=R(u)$ if $u\underset{L}{\sim}w$
and $L(w)=L(u)$ if $u\underset{R}{\sim}w.$

1.3. $*$-operations The $*$-operation introduced in [KL] and generalized in
[L1] is a useful tool in the theory of cells of Coxeter groups.

Let $s,t$ be simple reflections in $S$ and assume that $st$ has order $m\geq
3$. Let $w\in W$ be such that $sw\geq w,\ tw\geq w$. The $m-1$ elements $sw,\
tsw,\ stsw,\ ...,$ is called a left string (with respect to $\\{s,t\\})$, and
the $m-1$ elements $tw,\ stw,\ tstw,\ ...,$ is also called a left string (with
respect to $\\{s,t\\})$. Similarly we define right strings (with respect to
$\\{s,t\\}$). Then (see [L1])

(a) A left string in $W$ is contained in a left cell of $W$ and a right string
in $W$ is contained in a right cell of $W$.

Assume that $x$ is in a left (resp. right ) string (with respect to
$\\{s,t\\})$ of length $m-1$ and is the $i$th element of the left (resp.
right) string, define ${}^{*}x$ (resp. $x^{*}$) to be the $(m-i)$th element of
the string, where $*=\\{s,t\\}$. The following result is proved in [X2].

(b) Let $x$ be in $W$ such that $x$ is in a left string with respect to
$*=\\{s,t\\}$ and is also in a right string with respect to
$\star=\\{s^{\prime},t^{\prime}\\}$. Then ${}^{*}x$ is in a right string with
respect to $\\{s^{\prime},t^{\prime}\\}$ and $x^{\star}$ is in a left string
with respect to $\\{s,t\\}$. Moreover ${}^{*}(x^{\star})=({}^{*}x)^{\star}$.
We shall write ${}^{*}x^{\star}$ for ${}^{*}(x^{\star})=({}^{*}x)^{\star}$.

The following result is due to Lusztig [L1].

(c) Let $\Gamma$ be a left cell of $W$ and an element $x\in\Gamma$ is in a
right string $\sigma_{x}$ with respect to $*=\\{s,t\\}$. Then any element
$w\in\Gamma$ is in a right string $\sigma_{w}$ with respect to $*=\\{s,t\\}$.
Moreover $\Gamma^{*}=\\{w^{*}\,|\,w\in\Gamma\\}$ is a left cell of $W$ and
$\Omega=\displaystyle\left(\cup_{w\in\Gamma}\sigma_{w}\right)-\Gamma$ is a
union of at most $m-2$ left cells.

Following Lusztig [L1] we set $\tilde{\mu}(y,w)=\mu(y,w)$ if $y<w$ and
$\tilde{\mu}(y,w)=\mu(w,y)$ if $w<y$. For convenience we also set
$\tilde{\mu}(y,w)=0$ if $y\nless w$ and $w\nless y$. Assume that
$x_{1},x_{2},...,x_{m-1}$ and $y_{1},y_{2},...,y_{m-1}$ are two left strings
with respect to $*=\\{s,t\\}$. Define

$a_{ij}=\begin{cases}\tilde{\mu}(x_{i},y_{j}),\quad&\text{if\ }\\{s,t\\}\cap
L(x_{i})=\\{s,t\\}\cap L(y_{j}),\\\ 0,\quad&\text{otherwise}.\end{cases}$

Lusztig proved the following identities (see Subsection 10.4 in [L1]).

(d) If $m=3$, then $a_{11}=a_{22}$ and $a_{12}=a_{21}$.

(e) If $m=4$, then

(3) $\displaystyle a_{11}=a_{33},\ a_{13}=a_{31},\ a_{22}=a_{11}+a_{13},\
a_{12}=a_{21}=a_{23}=a_{32}.$

1.4. Lusztig’s $a$-function For $x,y\in W$, write

$C_{x}C_{y}=\sum_{z\in W}h_{x,y,z}C_{z},\qquad
h_{x,y,z}\in\mathcal{A}=\mathbb{Z}[q^{\frac{1}{2}},q^{-\frac{1}{2}}].$

Following Lusztig ([L1]), we define

$a(z)={\rm min}\\{i\in\mathbf{N}\ |\
q^{-\frac{i}{2}}h_{x,y,z}\in\mathbb{Z}[q^{-\frac{1}{2}}]{\rm\ for\ all\
}x,y\in W\\}.$

If for any $i$,
$q^{-\frac{i}{2}}h_{x,y,z}\not\in\mathbb{Z}[q^{-\frac{1}{2}}]{\rm\ for\ some\
}x,y\in W$, we set $a(z)=\infty.$ The following properties are proved in [L1].

(a) We have $a(w)\leq l(w_{0})$ for any $w\in W$, where $w_{0}$ is the longest
element in the Weyl group $W_{0}$.

(b) $a(x)\geq a(y)$ if $x\underset{LR}{\leq}y$. In particular, $a(x)=a(y)$ if
$x\underset{LR}{\sim}y$.

(c) $x\underset{L}{\sim}y$ (resp. $x\underset{R}{\sim}y,\
x\underset{LR}{\sim}y$) if $a(x)=a(y)$ and $x\underset{L}{\leq}y$ (resp.
$x\underset{R}{\leq}y,\ x\underset{LR}{\leq}y$).

(d) If $h_{x,y,z}\neq 0$, then $z\underset{R}{\leq}x$ and
$z\underset{L}{\leq}y$. In particular, $a(z)>a(x)$ if
$z\not\underset{R}{\sim}x$, and $a(z)>a(y)$ if $z\not\underset{L}{\sim}y$.

Following Lusztig, we define $\gamma_{x,y,z}$ by the following formula,

$h_{x,y,z}=\gamma_{x,y,z}q^{\frac{a(z)}{2}}+{\rm\ lower\ degree\ terms}.$

Springer showed that $l(z)\geq a(z)$ (see [L2]). Let $\delta(z)$ be the degree
of $P_{e,z}$, where $e$ is the neutral element of $W$. Then actually one has
$l(z)-a(z)-2\delta(z)\geq 0$ (see [L2]). Set

$\mathcal{D}=\\{z\in W\ |\ l(z)-a(z)-2\delta(z)=0\\}.$

The elements of $\mathcal{D}$ are involutions, called distinguished
involutions of $(W,S)$ (see [L2]). The following properties are due to Lusztig
[L2] except the (j) (which is trivial) and (k) (proved in [X2]).

(e) $\gamma_{x,y,z}\neq 0\Longrightarrow x\underset{L}{\sim}y^{-1},\
y\underset{L}{\sim}z,\ x\underset{R}{\sim}z.$

(f) $x\underset{L}{\sim}y^{-1}$ if and only if $\gamma_{x,y,z}\neq 0$ for some
$z\in W$.

(g) $\gamma_{x,y,z}=\gamma_{y,z^{-1},x^{-1}}=\gamma_{z^{-1},x,y^{-1}}$.

(h) $\gamma_{x,d,x}=\gamma_{d,x^{-1},x^{-1}}=\gamma_{x^{-1},x,d}=1$ if
$x\underset{L}{\sim}d$ and $d$ is a distinguished involution.

(i) $\gamma_{x,y,z}=\gamma_{y^{-1},x^{-1},z^{-1}}.$

(j) If $\omega,\tau\in W$ has length 0, then

$\gamma_{\omega x,y\tau,\omega z\tau}=\gamma_{x,y,z},\ \ \gamma_{x\omega,\tau
y,z}=\gamma_{x,\omega\tau y,z}.$

(k) Let $x,y,z\in W$ be such that (1) $x$ is in a left string with respect to
$*=\\{s,t\\}$ and also in a right string with respect to
$\\#=\\{s^{\prime},t^{\prime}\\}$, (2) $y$ is in a left string with respect to
$\\#=\\{s^{\prime},t^{\prime}\\}$ and also in a right string with respect to
$\star=\\{s^{\prime\prime},t^{\prime\prime}\\}$, (3) $z$ is in a left string
with respect to $*=\\{s,t\\}$ and also in a right string with respect to
$\star=\\{s^{\prime\prime},t^{\prime\prime}\\}$. Then

$\gamma_{x,y,z}=\gamma_{{}^{*}x^{\\#},{}^{\\#}y^{\star},{}^{*}z^{\star}}.$

For $w\in W$, set $\tilde{T}_{w}=q^{-l(w)/2}T_{w}$. For $x,y\in W$, write

$\tilde{T}_{x}\tilde{T}_{y}=\sum_{z\in W}f_{x,y,z}\tilde{T}_{z},\qquad
f_{x,y,z}\in\mathcal{A}=\mathbb{Z}[q^{\frac{1}{2}},q^{-\frac{1}{2}}].$

(l) If $x,y,w$ are in a two-sided cell of $W$, $f_{x,y,w}=\lambda
q^{\frac{a(w)}{2}}+$ lower degree terms and as Laurent polynomials in
$q^{\frac{1}{2}}$, deg$f_{x,y,z}\leq a(w)$ for all $z\in W$, then

$\gamma_{x,y,w}=\lambda.$

(m) Each left cell (resp. each right cell) of $W$ contains a unique
distinguished involution.

(n) Each two-sided cell of $W$ contains only finitely many left cells.

(o) Let $I$ be a subset of $S$ such that the subgroup $W_{I}$ of $W$ generated
by $I$ is finite. Then the longest element $w_{I}$ is a distinguished
involution.

Let $d$ be a distinguished involution in $W$.

(p) For any $\omega\in\Omega$, the element $\omega d\omega^{-1}$ is a
distinguished involution.

(q) Suppose $s,t\in S$ and $st$ has order 3. Then $d\in D_{L}(s,t)$ if and
only if $d\in D_{R}(s,t)$. If $d\in D_{L}(s,t)$, then ${}^{*}d^{*}$ is a
distinguished involution.

1.5. Assume $s,t\in S$ and $st$ has order 4. Let $w,u,v$ be in $W$ such that
$l(ststw)=l(w)+4$ and $l(ststv)=l(v)+4$. We have (see [X2, 1.6.3])

  * (a)

$\gamma_{tsw,u,tv}=\gamma_{sw,u,stv},$

  * (b)

$\gamma_{tsw,u,tsv}=\gamma_{sw,u,sv}+\gamma_{sw,u,stsv},$

  * (c)

$\gamma_{tsw,u,tstv}=\gamma_{sw,u,stv},$

  * (d)

$\gamma_{tstw,u,tv}+\gamma_{tw,u,tv}=\gamma_{stw,u,stv},$

  * (e)

$\gamma_{tstw,u,tsv}=\gamma_{stw,u,stsv},$

  * (f)

$\gamma_{tstw,u,tstv}+\gamma_{tw,u,tstv}=\gamma_{stw,u,stv}.$

Assume $s,t\in S$ and $st$ has order 4. Let $w,u,v$ be in $W$ such that
$l(ustst)=l(u)+4$ and $l(vstst)=l(v)+4$. We have (loc.cit)

  * (a’)

$\gamma_{w,ut,vst}=\gamma_{w,uts,vs},$

  * (b’)

$\gamma_{w,ust,vst}=\gamma_{w,us,vs}+\gamma_{w,usts,vs},$

  * (c’)

$\gamma_{w,utst,vst}=\gamma_{w,uts,vs},$

  * (d’)

$\gamma_{w,ut,vtst}+\gamma_{w,ut,vt}=\gamma_{w,uts,vts},$

  * (e’)

$\gamma_{w,ust,vtst}=\gamma_{w,usts,vts},$

  * (f’)

$\gamma_{w,utst,vtst}+\gamma_{w,utst,vt}=\gamma_{w,uts,vts}.$

1.6. The based ring of a two-sided cell For each two-sided cell $c$ of $W$,
let $J_{c}$ be the free $\mathbb{Z}$-module with a basis $t_{w},\ w\in c$.
Define

$t_{x}t_{y}=\sum_{z\in c}\gamma_{x,y,z}t_{z}.$

Then $J_{c}$ is an associative ring with unit $\sum_{d\in\mathcal{D}\cap
c}t_{d}.$

The ring $J=\bigoplus_{c}J_{c}$ is a ring with unit
$\sum_{d\in\mathcal{D}}t_{d}$. Sometimes $J$ is called asymptotic Hecke
algebra since Lusztig established an injective $\mathcal{A}$-algebra
homomorphism

$\phi:H\to J\otimes\mathcal{A},\quad
C_{x}\mapsto\sum_{\stackrel{{\scriptstyle\stackrel{{\scriptstyle
d\in\mathcal{D}}}{{w\in W}}}}{{w\underset{L}{\sim}d}}}h_{x,d,w}t_{w}.$

1.7. Lusztig’s conjecture on the structure of $J_{c}$ In [L3] Lusztig states a
conjecture on $J_{c}$ using equivariant $K$-groups on finite sets.

Let $G$ be a connected reductive group over $\mathbb{C}$. Lusztig establishes
a bijection between the two-sided cells of the extended affine Weyl group $W$
and the unipotent classes of $G$.

For each two-sided cell $c$ of $W$, let $u$ be a unipotent element in the
unipotent class corresponding to $c$ and let $F_{c}$ be a maximal reductive
subgroup of the centralizer of $u$ in $G$.

Conjecture (Lusztig [L3]): Assume that $G$ is a simply connected simple
algebraic group over $\mathbb{C}$. Then there exists a finite set $Y$ with an
algebraic action of $F_{c}$ and a bijection

$\pi:c\to\text{{the set of} isomorphism classes of irreducible $F_{c}$-vector
bundles on}\ Y\times Y.$

such that

(i) The bijection $\pi$ induces a ring homomorphism

$\pi:J_{c}\to K_{F_{c}}(Y\times Y),\ \ t_{x}\mapsto\pi(x).$

(ii) $\pi(x^{-1})_{(a,b)}=\pi(x)_{(b,a)}^{*}$ is the dual representation of
$\pi(x)_{(b,a)}.$

##  2\. Cells in an extended affine Weyl group of type $\tilde{B}_{3}$

In this section $G=Sp_{6}(\mathbb{C})$, so that the extended affine Weyl group
$W$ attached to $G$ is of type $\tilde{B}_{3}$. The left cells and two-sided
cells are described by J. Du (see [D]). We recall his results.

2.1. The Coxeter graph of $W$. As usual, we number the 4 simple reflections
$s_{0},\ s_{1},\ s_{2},\ s_{3}$ in $W$ so that

$\displaystyle s_{0}s_{1}=s_{1}s_{0},\quad s_{0}s_{3}=s_{3}s_{0},\quad
s_{1}s_{3}=s_{3}s_{1},$
$\displaystyle(s_{0}s_{2})^{3}=(s_{1}s_{2})^{3}=e,\quad(s_{2}s_{3})^{4}=e,$

where $e$ is the neutral element in $W$. The relations among the simple
reflections can be read through the following Coxeter graph:

$\tilde{B}_{3}:$$2$$0$$1$$3$

There is a unique nontrivial element $\tau$ in $W$ with length 0. We have
$\tau^{2}=e,\ \tau s_{0}\tau=s_{1},\ \tau s_{i}\tau=s_{i}$ for $i=2,3.$ Note
that $s_{1},s_{2},s_{3}$ generate the Weyl group $W_{0}$ of type $B_{3}$ and
$s_{0},s_{1},s_{2},s_{3}$ generate an affine Weyl group $W^{\prime}$ of type
$\tilde{B}_{3}$. And $W$ is generated by $\tau,\ s_{0},s_{1},s_{2},s_{3}$.

2.2. Cells in $W$ According to [D], the extended affine Weyl group $W$
attached to $Sp_{6}(\mathbb{C})$ has 8 two-sided cells:

$A,\quad B,\quad C,\quad D,\quad E,\quad F,\quad G,\quad H.$

The following table displays some useful information on these two-sided cells.

|  | Number | Size of Jordan blocks | Maximal reductive subgroup
---|---|---|---|---
|  | of left | of the corresponding | of the centralizer of a unipotent
$X$ | ${a(X)}$ | cells in $X$ | unipotent class in $Sp_{6}(\mathbb{C})$ | element in the corresp. unipotent class
$A$ | 9 | 48 | (111111) | $Sp_{6}(\mathbb{C})$
$B$ | 6 | 24 | (21111) | $Sp_{4}(\mathbb{C})\times\mathbb{Z}/2\mathbb{Z}$
$C$ | 4 | 18 | (2211) | $SL_{2}(\mathbb{C})\times O_{2}(\mathbb{C})$
$D$ | 3 | 12 | (222) | $O_{3}(\mathbb{C})$
$E$ | 2 | 8 | (411) | $SL_{2}(\mathbb{C})\times\mathbb{Z}/2\mathbb{Z}$
$F$ | 2 | 6 | (33) | $SL_{2}(\mathbb{C})$
$G$ | 1 | 4 | (42) | $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$
$H$ | 0 | 1 | (6) | $\mathbb{Z}/2\mathbb{Z}$

The notations for two-sided cells in the table are the same as those in [D],
which will be replaced by other notations in subsequent sections, otherwise
confusion would happen since notations $C,F,G$ are already used for other
objects.

In subsequent sections, for a reduced expression $s_{i_{1}}s_{i_{2}}\cdots
s_{ik}$ of an element in $W$, we often write $i_{1}i_{2}\cdots i_{k}$ instead
of the reduced expression.

In the rest of the paper, $W$ always stands for the affine Weyl group attached
to $Sp_{6}(\mathbb{C})$, $\tau,\ s_{i}$ are as in Subsection 2.1, and all
representations in this paper are rational representations of algebraic
groups.

##  3\. The based ring of the two-sided cell containing $s_{0}s_{1}$

3.1. In this section $c$ stands for the two-sided cell of $W$ containing
$s_{0}s_{1}$. According to [D, Figure I, Theorem 6.4], $c$ has six left cells.
We list the six left cells and representative elements in the left cells given
in [D, figure I]:

$\Gamma_{1},\ 01;\quad\Gamma_{2},\ 012;\quad\Gamma_{3},\
0123;\quad\Gamma_{4},\ 01232;\quad\Gamma_{5},\ 012321;\quad\Gamma_{6},\
012320.$

The values of $a$-function on $c$ is 2.

The corresponding unipotent class in $Sp_{6}(\mathbb{C})$ has Jordan block
sizes (411). Maximal reductive subgroup of the centralizer of an element in
the unipotent class is $F_{c}=\mathbb{Z}/2\mathbb{Z}\times
SL_{2}(\mathbb{C})$. Let $\epsilon$ be the nontrivial one dimensional
representation of $\mathbb{Z}/2\mathbb{Z}$ and $V(k)$ be an irreducible
representation of $SL_{2}(\mathbb{C})$ with highest weight $k$. They can be
regarded as irreducible representations of $F_{c}$ naturally. Up to
isomorphism, the irreducible representations of $F_{c}$ are $V(k),\
\epsilon\otimes V(k),\ k=0,1,2,3,...$. We will denote $\epsilon\otimes V(k)$
by $\epsilon V(k)$.

Let $x_{k}=(s_{0}s_{1}s_{2}s_{3}s_{2})^{k}s_{1}s_{0},\ u_{1}=e,\ u_{2}=s_{2},\
u_{3}=s_{3}s_{2},\ u_{4}=s_{2}s_{3}s_{2},\ u_{5}=s_{1}s_{2}s_{3}s_{2},\
u_{6}=s_{0}s_{2}s_{3}s_{3}.$

According to [D, Theorem 6.4], we have

(a) $c=\\{u_{i}x_{k}u_{j}^{-1},\ u_{i}\tau x_{k}u_{j}^{-1}\,|\,1\leq i,j\leq
6,\ k=0,1,2,3,...\\}.$

(b) $\Gamma_{j}=\\{u_{i}x_{k}u_{j}^{-1},\ u_{i}\tau x_{k}u_{j}^{-1}\,|\,1\leq
i\leq 6,\ k=0,1,2,3,...\\},\ j=1,2,3,4,5,6.$

Let $Y=\\{1,\ 2,\ ...,\ 6\\}$ and let $F_{c}$ act on $Y$ trivially. Then
$K_{F_{c}}(Y\times Y)$ is isomorphic to the $6\times 6$ matix ring
$M_{6}(\text{Rep\,}F_{c})$, where $\text{Rep\,}F_{c}$ is the representation
ring of $F_{c}$. Recall that $F_{c}=\mathbb{Z}/2\mathbb{Z}\times
SL_{2}(\mathbb{C})$ in this section.

The main result in this section is the following theorem.

Theorem 3.2. Let $c$ be the two-sided cell of $W$ (the affine Weyl group $W$
attached to $Sp_{6}(\mathbb{C})$) containing $s_{0}s_{1}$. Then the map

$\pi:c\to M_{6}(\text{Rep\,}F_{c}),\quad u_{i}x_{k}u_{j}^{-1}\mapsto
V(k)_{ij},\ u_{i}\tau x_{k}u_{j}^{-1}\mapsto\epsilon V(k)_{ij}\ $

induces a ring isomorphism

$\pi:J_{c}\to M_{6}(\text{Rep\,}F_{c}),\quad t_{u_{i}x_{k}u_{j}^{-1}}\mapsto
V(k)_{ij},\quad t_{u_{i}\tau x_{k}u_{j}^{-1}}\mapsto\epsilon V(k)_{ij},$

where $V(k)_{ij}$ (resp. $\epsilon V(k)_{ij})$ is the matrix in
$M_{6}(\text{Rep\,}F_{c})$ whose entry at $(p,q)$ is $V(k)$ (resp. $\epsilon
V(k)$) if $(p,q)=(i,j)$ and is 0 otherwise.

Remark: The Theorem 4 in [BO] implies that Lusztig’s conjecture on the
structure of $J_{c}$ is true. Since under the isomorphism $K_{F_{c}}(Y\times
Y)\simeq M_{6}(\text{Rep\,}F)$, irreducible ${F_{c}}$-vector bundles on
$Y\times Y$ correspond to those $V(k)_{ij},\ \epsilon V(k)_{ij}$, hence
Theorem 3.2 provides a computable verification for Lusztig conjecture on the
structure of $J_{c}$.

We prove Theorem 3.2 by establishing three lemmas.

Lemma 3.3. Let $1\leq i,j,m,n\leq 6$ and $k,l$ be nonnegative integers. For
$z_{k}=x_{k}$ or $\tau x_{k}$, $z_{l}=x_{l}$ or $\tau x_{l}$, and
$z_{p}=x_{p}$ or $\tau x_{p}$, we have

(a) $\gamma_{u_{i}z_{k}u_{j}^{-1},u_{m}z_{l}u_{n}^{-1},z}=0\quad\text{if}\
j\neq m\ \text{or}\ z\neq u_{i}\tau^{a}z_{p}u^{-1}_{n},\ a=0,1,\ \text{for
some }p;$

(b)
$\gamma_{u_{i}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{i}z_{p}u_{n}^{-1}}=\gamma_{z_{k},z_{l},z_{p}},\quad\text{for
any nonnegative integer }p.$

Proof. Note that $z_{l}^{-1}=z_{l}$. If
$\gamma_{u_{i}z_{k}u_{j}^{-1},u_{m}z_{l}u_{n}^{-1},z}\neq 0$, then by 1.4(e)
we get
$u_{i}z_{k}u_{j}^{-1}\underset{L}{\sim}(u_{m}z_{l}u_{n}^{-1})^{-1}=u_{n}z_{l}u_{m}^{-1}$,
$u_{i}z_{k}u_{j}^{-1}\underset{R}{\sim}z,\
u_{m}z_{l}u_{n}^{-1}\underset{L}{\sim}z$. By (b) in Subsection 3.1 we see that
the first assertion is true.

Now we prove the second assertion. Let $*=\\{s_{1},s_{2}\\},\
\\#=\\{s_{2},s_{3}\\}$ and $\star=\\{s_{0},s_{2}\\}$. Then

(c)
$\Gamma_{2}=\Gamma_{1}^{*},\quad\Gamma_{4}=\Gamma_{2}^{\\#},\quad\Gamma_{5}=\Gamma_{4}^{*},\quad\Gamma_{6}=\Gamma_{4}^{\star}.$

Applying 1.4 (k) we see that (b) is true if none of $i,j,n$ is 3.

Now assume that $i=3$. By 1.5 (b) we get

$\gamma_{u_{3}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{3}z_{p}u_{n}^{-1}}=\gamma_{u_{2}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{2}z_{p}u_{n}^{-1}}+\gamma_{u_{2}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{4}z_{p}u_{n}^{-1}}.$

By Part (a) of the lemma, we have
$\gamma_{u_{2}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{4}z_{p}u_{n}^{-1}}=0.$
Then using (c) above and 1.4 (k) we get

$\gamma_{u_{3}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{3}z_{p}u_{n}^{-1}}=\gamma_{u_{2}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{2}z_{p}u_{n}^{-1}}=\gamma_{z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},z_{p}u_{n}^{-1}}.$

Similarly, if $n=3$, we have

$\gamma_{u_{i}z_{k}u_{j}^{-1},u_{j}z_{l}u_{3}^{-1},u_{i}z_{p}u_{3}^{-1}}=\gamma_{u_{i}z_{k}u_{j}^{-1},u_{j}z_{l}u_{2}^{-1},u_{i}z_{p}u_{2}^{-1}}=\gamma_{u_{i}z_{k}u_{j}^{-1},u_{j}z_{l},u_{i}z_{p}}.$

We have showed for any $1\leq i,n\leq 6$ the following identity holds:

$\gamma_{u_{i}z_{k}u_{j}^{-1},u_{j}z_{l}u_{n}^{-1},u_{i}z_{p}u_{n}^{-1}}=\gamma_{z_{k}u_{j}^{-1},u_{j}z_{l},z_{p}}.$

Note that $z_{k}^{-1}=z_{k}$. By above identity and 1.4 (g), we get

$\gamma_{z_{k}u_{j}^{-1},u_{j}z_{l},z_{p}}=\gamma_{u_{j}z_{l},z_{p}^{-1},u_{j}z_{k}^{-1}}=\gamma_{z_{l},z_{p},z_{k}}=\gamma_{z_{k},z_{l},z_{p}}.$

Assertion (b) is proved and the lemma is proved.∎

Lemma 3.4. For nonnegative integers, and $a,b=0,1$, we have

$\gamma_{\tau^{a}x_{k},\tau^{b}x_{l},\tau^{a+b}x_{p}}=\gamma_{x_{k},x_{l},x_{p}},\quad\gamma_{\tau^{a}x_{k},\tau^{b}x_{l},\tau^{c}x_{p}}=0\
\text{if }\tau^{c}\neq\tau^{a+b}.$

Proof. The assertion follows from 1.4 (j).∎

Lemma 3.5. For nonnegative integers $k,l$ we have

$t_{x_{k}}t_{x_{l}}=\sum_{0\leq p\leq\min\\{k,l\\}}t_{{x_{k+l-2i}}}.$

Proof. If $k=0$ or $l=0$, the identity above is trivial since $x_{0}$ is a
distinguished involution.

Now assume that $k=1$ and $l\geq 1$. Let
$\zeta=q^{\frac{1}{2}}-q^{-\frac{1}{2}}$. By a simple computation we see

$\tilde{T}_{x_{1}}\tilde{T}_{x_{l}}=\zeta^{2}(\tilde{T}_{x_{l+1}}+\tilde{T}_{x_{l-1}}+\tilde{T}_{s_{0}s_{1}s_{3}(s_{2}s_{3}s_{2}s_{0}s_{1})^{l}}+\tilde{T}_{s_{0}s_{2}s_{3}s_{2}s_{1}(s_{2}s_{3}s_{2}s_{0}s_{1})^{l}})+\text{lower
degree terms,}$

Since $a(s_{0}s_{1}s_{3}(s_{2}s_{3}s_{2}s_{0}s_{1})^{l})\geq
a(s_{0}s_{1}s_{3})=3,\
a(s_{0}s_{2}s_{3}s_{2}s_{1}(s_{2}s_{3}s_{2}s_{0}s_{1})^{l})\geq
a(s_{2}s_{1}s_{2})=3$, we see that
$s_{0}s_{1}s_{3}(s_{2}s_{3}s_{2}s_{0}s_{1})^{l}$ and
$s_{0}s_{2}s_{3}s_{2}s_{1}(s_{2}s_{3}s_{2}s_{0}s_{1})^{l}$ are not in the two-
sided cell $c$. By 1.4 (l), we have

$t_{x_{1}}t_{x_{l}}=t_{x_{l+1}}+t_{x_{l-1}}.$

For $k\geq 2$, since $t_{x_{k}}=t_{x_{1}}t_{x_{k-1}}-t_{x_{k-2}}$, we can use
induction on $k$ to prove the lemma. The argument is completed.∎

Proof of Theorem 3.2. Combining Lemmas 3.3, 3.4 and 3.5 we see that Theorem
3.2 is true.

##  4\. The based ring of the two-sided cell containing $s_{1}s_{3}$

4.1. In this section $c$ stands for the two-sided cell of $W$ containing
$s_{1}s_{3}$. According to [D, Figure I, Theorem 6.4], $c$ has eight left
cells. We list the eight left cells and representative elements in the left
cells given in [D, Figure I]:

$\begin{array}[]{lllllllll}&\Gamma_{1},&13;&\Gamma_{2},&132;&\Gamma_{3},&1323;&\Gamma_{4},&1320;\\\
&\Gamma_{5},&03;&\Gamma_{6},&032;&\Gamma_{7},&0323;&\Gamma_{8},&0321.\end{array}$

The values of $a$-function on $c$ is 2.

The corresponding unipotent class in $Sp_{6}(\mathbb{C})$ has Jordan block
sizes (33). Maximal reductive subgroup of the centralizer of an element in the
unipotent class is ${F_{c}}=SL_{2}(\mathbb{C})$. Let $V(k)$ be an irreducible
representation of ${F_{c}}=SL_{2}(\mathbb{C})$ with highest weight $k$. Up to
isomorphism, the irreducible representations of ${F_{c}}$ are $V(k),\
k=0,1,2,3,...$.

Let $x_{k}=(\tau s_{0}s_{3}s_{2})^{k}s_{1}s_{3},\ u_{1}=e,\ u_{2}=s_{2},\
u_{3}=s_{3}s_{2},\ u_{4}=s_{0}s_{2},\ u_{5}=\tau,\ u_{6}=\tau s_{2},\
u_{7}=\tau s_{3}s_{2},\ u_{8}=\tau s_{0}s_{2}.$

According to [D, Theorem 6.4], we have

(a) $c=\\{u_{i}x_{k}u_{j}^{-1}\,|\,1\leq i,j\leq 8,\ k=0,1,2,3,...\\}.$

(b) $\Gamma_{j}=\\{u_{i}x_{k}u_{j}^{-1}\,|\,1\leq i\leq 8,\ k=0,1,2,3,...\\},\
j=1,2,3,4,5,6,7,8.$

Let $Y=\\{1,\ 2,\ ...,\ 7,\ 8\\}$ and let ${F_{c}}$ act on $Y$ trivially. Then
$K_{F_{c}}(Y\times Y)$ is isomorphic to the $8\times 8$ matrix ring
$M_{8}(\text{Rep\,}{F_{c}})$, where $\text{Rep\,}{F_{c}}$ is the
representation ring of ${F_{c}}=SL_{2}(\mathbb{C})$.

The main result in this section is the following.

Theorem 4.2. Let $c$ be the two-sided cell of $W$ (the extended affine Weyl
group attached to $Sp_{6}(\mathbb{C})$) containing $s_{1}s_{3}$. Then the map

$\pi:c\to M_{8}(\text{Rep\,}F_{c}),\quad u_{i}x_{k}u_{j}^{-1}\mapsto
V(k)_{ij}$

induces a ring isomorphism

$\pi:J_{c}\to M_{8}(\text{Rep\,}F_{c}),\quad t_{u_{i}x_{k}u_{j}^{-1}}\mapsto
V(k)_{ij},$

where $V(k)_{ij}$ is the matrix in $M_{8}(\text{Rep\,}F_{c})$ whose entry at
$(p,q)$ is $V(k)$ if $(p,q)=(i,j)$ and is 0 otherwise.

Remark: The Theorem 4 in [BO] implies that Lusztig’s conjecture on the
structure of $J_{c}$ is true. Since under the isomorphism $K_{F_{c}}(Y\times
Y)\simeq M_{8}(\text{Rep\,}F_{c})$, irreducible ${F_{c}}$-vector bundles on
$Y\times Y$ correspond to $V(k)_{ij}$’s, Theorem 4.2 provides a computable
verification for Lusztig’s conjecture on the structure of $J_{c}$.

We prove Theorem 4.2 by establishing two lemmas.

Lemma 4.3. Let $1\leq i,j,m,n\leq 8$ and $k,l$ be nonnegative integers. Then

(a) $\gamma_{u_{i}x_{k}u_{j}^{-1},u_{m}x_{l}u_{n}^{-1},z}=0\quad\text{if}\
j\neq m\ \text{or}\ x\neq u_{i}x_{p}u^{-1}_{n}\ \text{for some }p;$

(b)
$\gamma_{u_{i}x_{k}u_{j}^{-1},u_{j}x_{l}u_{n}^{-1},u_{i}x_{p}u_{n}^{-1}}=\gamma_{x_{k},x_{l},x_{p}},\quad\text{for
any nonnegative integer }p.$

Proof. Note that $x_{l}^{-1}=x_{l}$. If
$\gamma_{u_{i}x_{k}u_{j}^{-1},u_{m}x_{l}u_{n}^{-1},z}\neq 0$, then by 1.4(e)
we get
$u_{i}x_{k}u_{j}^{-1}\underset{L}{\sim}(u_{m}x_{l}u_{n}^{-1})^{-1}=u_{n}x_{l}u_{m}^{-1}$,
$u_{i}x_{k}u_{j}^{-1}\underset{R}{\sim}z,$ and
$u_{m}x_{l}u_{n}^{-1}\underset{L}{\sim}z$. By (b) in Subsection 4.1 we see
that the first assertion is true.

Now we prove the second assertion. Let $*=\\{s_{1},s_{2}\\},\
\\#=\\{s_{2},s_{3}\\}$ and $\star=\\{s_{0},s_{2}\\}$. Then

(c)
$\Gamma_{2}=\Gamma_{1}^{*},\quad\Gamma_{3}=\Gamma_{1}^{\\#},\quad\Gamma_{4}=\Gamma_{2}^{\star},\quad\Gamma_{5}=\tau\Gamma_{1}\tau,\
\Gamma_{6}=\tau\Gamma_{2}\tau,\ \Gamma_{7}=\tau\Gamma_{3}\tau,\
\Gamma_{8}=\tau\Gamma_{4}\tau.$

Applying 1.4 (j) and 1.4 (k) (repeatedly if necessary) we get the following
identity.

$\gamma_{u_{i}x_{k}u_{j}^{-1},u_{j}x_{l}u_{n}^{-1},u_{i}x_{p}u_{n}^{-1}}=\gamma_{x_{k}u_{j}^{-1},u_{j}x_{l},x_{p}}.$

Note that $\tau^{2}=e$. Again using 1.4 (j) and 1.4 (k) (repeatedly if
necessary) we get the following identity.

$\gamma_{x_{k}u_{j}^{-1},u_{j}x_{l},x_{p}}=\gamma_{x_{k},x_{l},x_{p}}.$

Part (b) is proved and the lemma is proved. ∎

Lemma 4.4. For nonnegative integers $k,l$, we have

$t_{x_{k}}t_{x_{l}}=\sum_{0\leq p\leq\min\\{k,l\\}}t_{x_{k+l-2i}}.$

Proof. If $k=0$ or $l=0$, the identity above is trivial since $x_{0}$ is a
distinguished involution.

Now assume that $k=1$ and $l\geq 1$. Put
$\xi=q^{\frac{1}{2}}+q^{-\frac{1}{2}}$. Let $H^{<13}$ be the
$\mathcal{A}$-submodule of $H$ spanned by all $C_{w}$ with $a(w)\geq 3$. By
Subsection 1.2 and 1.4 (b) we know that $H^{<13}$ is a two-sided ideal of $H$.
Before continuing, we make a convention: we shall use the symbol $\Box$ for
any element in the two-sided ideal $H^{<13}$ of $H$. Then $\Box+\Box=\Box$ and
$h\Box=\Box$ for any $h\in H$.

First we have

$C_{x_{1}}=C_{\tau s_{3}s_{0}s_{2}s_{1}s_{3}}=C_{\tau
s_{3}}C_{s_{0}}C_{s_{2}}C_{s_{1}}C_{s_{3}}-C_{\tau s_{0}s_{1}s_{3}},\ \
\text{and}\ C_{\tau s_{0}s_{1}s_{3}}\in H^{<13}.$

Note that ${L}(x_{k})=\\{s_{1},s_{3}\\}$. Hence

(4) $\displaystyle C_{x_{1}}C_{x_{l}}=C_{\tau
s_{3}}C_{s_{0}}C_{s_{2}}C_{s_{1}}C_{s_{3}}C_{x_{l}}+\Box=\xi^{2}C_{\tau
s_{3}}C_{s_{0}}C_{s_{2}}C_{x_{l}}+\Box\in\xi^{2}C_{\tau
s_{3}}C_{s_{0}}C_{s_{2}}C_{x_{l}}+H^{<13}.$

We compute $C_{\tau s_{3}}C_{s_{0}}C_{s_{2}}C_{x_{l}}$ step by step.

Step 1. Compute $C_{s_{2}}C_{x_{l}}.$ We have

$C_{s_{2}}C_{x_{l}}=C_{s_{2}x_{l}}+\sum\limits_{\begin{subarray}{c}z\prec
x_{l}\\\ s_{2}z<z\end{subarray}}\mu(z,x_{l})C_{z}.$

Note that $L(x_{l})=\\{s_{1},s_{3}\\}$. Assume that $z\prec x_{l}$ and
$s_{2}z\leq z$. If $s_{1}z\leq z$, then $\\{s_{1},s_{2}\\}\subset L(z)$ and
$a(z)\geq a(s_{1}s_{2}s_{1})=3$. In this case, we have $C_{z}\in H^{<13}$. If
$s_{1}z\geq z$, by 1.1(c) we must have $s_{1}z=x_{l}$. Then $z=\tau
s_{3}s_{2}x_{l-1}$. This contradicts $s_{2}z\leq z$. Therefore we have

(5) $\displaystyle C_{s_{2}}C_{x_{l}}=C_{s_{2}x_{l}}+\Box\in
C_{s_{2}x_{l}}+H^{<13}.$

Step 2. Compute $C_{s_{0}}C_{s_{2}x_{l}}.$ We have

$C_{s_{0}}C_{s_{2}x_{l}}=C_{s_{0}s_{2}x_{l}}+\sum\limits_{\begin{subarray}{c}z\prec
s_{2}x_{l}\\\ s_{0}z<z\end{subarray}}\mu(z,s_{2}x_{l})C_{z}.$

Note that $L(s_{2}x_{l})=\\{s_{2}\\}$. Assume that $z\prec s_{2}x_{l}$ and
$s_{0}z\leq z$. If $s_{2}z\leq z$, then $\\{s_{0},s_{2}\\}\subset{L}(z)$ and
$a(z)\geq a(s_{0}s_{2}s_{0})=3$. In this case, we have $C_{z}\in H^{<13}$. If
$s_{2}z\geq z$, by 1.1(c) we must have $z=x_{l}$. This contradicts $s_{0}z\leq
z$. Therefore we have

(6) $\displaystyle C_{s_{0}}C_{s_{2}x_{l}}=C_{s_{0}s_{2}x_{l}}+\Box\in
C_{s_{0}s_{2}x_{l}}+H^{<13}.$

Step 3. Compute $C_{\tau s_{3}}C_{s_{0}s_{2}x_{l}}.$ We have

$C_{\tau
s_{3}}C_{s_{0}s_{2}x_{l}}=C_{x_{l+1}}+\sum\limits_{\begin{subarray}{c}z\prec
s_{0}s_{2}x_{l}\\\ s_{3}z<z\end{subarray}}\mu(z,s_{0}s_{2}x_{l})C_{\tau z}.$

Assume that $z\prec s_{0}s_{2}x_{l}$ and $s_{3}z\leq z$. Using 1.4 (b), 1.4
(c) and 1.4 (d), we see that $a(\tau z)\geq 2$ , and if $a(\tau z)=2$ then
$x_{l}\underset{L}{\sim}\tau z\underset{R}{\sim}x_{1}$. We are only concerned
with those $C_{\tau z}$ in above summation with $a(\tau z)=2$. Then $\tau
z=x_{m}$ for some $m<l$ and $L(z)=\\{s_{0},s_{3}\\}$. Note that
$L(s_{0}s_{2}x_{l})=\\{s_{0}\\}$. We then have
$\mu(z,s_{0}s_{2}x_{l})={\tilde{\mu}({}^{\star}z,{}^{\star}(s_{0}s_{2}x_{l}))}=\tilde{\mu}(s_{2}z,s_{2}x_{l})=\tilde{\mu}(^{*}(s_{2}z),^{*}(s_{2}x_{l}))=\tilde{\mu}(s_{1}s_{2}z,x_{l})$,
where $\star=\\{s_{0},s_{2}\\},\ *=\\{s_{1},s_{2}\\}$. Since $m<l$, we have
$\tilde{\mu}(s_{1}s_{2}z,x_{l})=\mu(s_{1}s_{2}z,x_{l})$. Noting that
$s_{3}s_{1}s_{2}z=s_{3}s_{1}s_{2}\tau x_{m}\geq s_{1}s_{2}\tau x_{m}$ and
$s_{3}x_{l}\leq x_{l}$, by 1.1(c) we see $s_{3}s_{1}s_{2}z=x_{l}$, which
implies that $\tau z=x_{l-1}$.

In conclusion, if $z\prec s_{0}s_{2}x_{l}$ and $s_{3}z\leq z$, then either
$C_{z}\in H^{<13}$ or $z=\tau x_{l-1}$. Hence we have

(7) $\displaystyle C_{\tau
s_{3}}C_{s_{0}s_{2}x_{l}}=C_{x_{l+1}}+C_{x_{l-1}}+\Box.$

Combining formulas (4)-(7) we get

$C_{x_{1}}C_{x_{l}}=\xi^{2}(C_{x_{l+1}}+C_{x_{l-1}})+\Box\in\xi^{2}(C_{x_{l+1}}+C_{x_{l-1}})+H^{<13}.$

Therefore we have $t_{x_{1}}t_{x_{l}}=t_{x_{l+1}}+t_{x_{l-1}}.$

For $k\geq 2$, since $t_{x_{k}}=t_{x_{1}}t_{x_{k-1}}-t_{x_{k-2}}$, we can use
induction on $k$ to prove the lemma. The argument is completed. ∎

Proof of Theorem 4.2. Combining Lemmas 4.3 and 4.4 we see that Theorem 4.2 is
true.

##  5\. The based ring of the two-sided cell containing $s_{1}s_{2}s_{1}$

5.1. In this section we consider the two-sided cell in $W$ containing
$s_{1}s_{2}s_{1}$. In [QX] we showed that Lusztig’s conjecture on the
structure of the based ring of the two-sided cell needs modification. In this
section we give a description of the based ring. For consistence, we keep the
notations in [QY] for the two-sided cell of $W$ containing $s_{1}s_{2}s_{1}$.
In particular, we denote $D$ for the two-sided cell of $W$ containing
$s_{1}s_{2}s_{1}$.

According to [D,Figure I, Theorem 6.4], we have the following result.

(a) There are 12 left cells in the two-sided cell $D$ and a representative of
each left cell in $D$ are:

$\begin{array}[]{lllllllll}&D_{013},&013;&D_{2},&0132;&D_{02},&01320;&D_{12},&01321;\\\
&&&&&&&&\\\ &D_{3},&01323;&D_{03},&013203;&D_{01},&013201;&D_{13},&013213;\\\
&&&&&&&&\\\
&D^{\prime}_{2},&0132032;&\widehat{D^{\prime}_{2}},&0132132;&D_{1},&01320321;&D_{0},&01321320.\end{array}$

The value of $a$-function on $D$ is 3.

Let $\Gamma$ and $\Gamma^{\prime}$ be two left cells of $W$. If
$\Gamma^{\prime}=\Gamma^{*}$ for some $*=\\{s,t\\}$ (see Subsection 1.3 for
definition of $*$-operation), then we write $\Gamma\
\overset{\\{s,t\\}}{\text{------}}\ \Gamma^{\prime}$. The following result is
easy to verify.

Lemma 5.2. Keep the notations as above. Then we have

$D_{3}\ \overset{\\{s_{2},s_{3}\\}}{\text{------}}\ D_{013}\
\overset{\\{s_{1},s_{2}\\}}{\text{------}}\ D_{2}.$

$\displaystyle D_{0}\ \overset{\\{s_{0},s_{2}\\}}{\text{------}}$
$\displaystyle\widehat{D^{\prime}_{2}}\
\overset{\\{s_{2},s_{3}\\}}{\text{------}}\ D_{12}\
\overset{\\{s_{0},s_{2}\\}}{\text{------}}\ D_{01}\
\overset{\\{s_{1},s_{2}\\}}{\text{------}}\ D_{02}\
\overset{\\{s_{2},s_{3}\\}}{\text{------}}\ D^{\prime}_{2}\
\overset{\\{s_{1},s_{2}\\}}{\text{------}}\ D_{1}.$ $\displaystyle\
|{\scriptstyle{\\{s_{1},s_{2}\\}}}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\
|{\scriptstyle{\\{s_{0},s_{2}\\}}}$ $\displaystyle
D_{13}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ D_{03}$

5.3. Let

$\begin{array}[]{lllllll}u_{k}=(s_{0}s_{1}s_{3}s_{2})^{k}s_{0}s_{1}s_{3},&&\\\
x_{k}=(s_{1}s_{2}s_{3}s_{0})^{k}s_{1}s_{2}s_{1},&\ x^{\prime}_{0}=\tau
s_{2}s_{0}s_{1}s_{2}s_{1},&\ x^{\prime}_{k+1}=\tau
s_{0}s_{2}s_{3}s_{0}x_{k},\\\ p_{1}=e,&p_{2}=s_{2},&p_{3}=s_{3}s_{2},\\\
p_{4}=s_{1}s_{2},&p_{5}=s_{0}s_{2},&p_{6}=s_{0}s_{1}s_{2},\\\
p_{7}=s_{3}s_{1}s_{2},&p_{8}=s_{3}s_{0}s_{2},&p_{9}=s_{2}s_{3}s_{1}s_{2},\\\
p_{10}=s_{2}s_{3}s_{0}s_{2},&p_{11}=s_{0}s_{2}s_{3}s_{1}s_{2},&p_{12}=s_{1}s_{2}s_{3}s_{0}s_{2};\\\
q_{4}=p_{1}=e,&q_{5}=\tau,&q_{6}=s_{0},\\\
q_{7}=s_{3},&q_{8}=s_{3}\tau,&q_{9}=s_{2}s_{3},\\\
q_{10}=s_{2}s_{3}\tau,&q_{11}=s_{0}s_{2}s_{3},&q_{12}=s_{1}s_{2}s_{3}\tau.\end{array}$

According to [D, Theorem 6.4], we have

(a) The two-sided cell $D$ consists of the following elements:

$p_{i}u_{k}p_{j}^{-1},\ p_{i}\tau u_{k}p_{j}^{-1},\ q_{l}x_{0}q_{m}^{-1},\
q_{l}x_{0}q_{6}^{-1},\ q_{l}x_{0}q_{6}^{-1}\tau,\
q_{l}x^{\prime}_{0}q_{m}^{-1},$

where $1\leq i,j\leq 12,\ 4\leq l\leq 12,\ 4\leq m\neq 6\leq 12,\ k\geq 0.$

For convenience, we number the left cells in $D$ as follows:

$\begin{array}[]{lllllll}\Gamma_{1}=D_{013},&\ \Gamma_{2}=D_{2},&\
\Gamma_{3}=D_{3},&\ \Gamma_{4}=D_{12},&\ \Gamma_{5}=D_{02},&\
\Gamma_{6}=D_{01},\\\ \Gamma_{7}=D_{13},&\ \Gamma_{8}=D_{03},&\
\Gamma_{9}=\widehat{D^{\prime}_{2}},&\ \Gamma_{10}=D^{\prime}_{2},&\
\Gamma_{11}=D_{0},&\ \Gamma_{12}=D_{1}.\end{array}$

Then (loc. cit) we have

(b1) For $j=1,\ 2,\ 3$, the left cell $\Gamma_{j}$ consists of the following
elements:

$p_{i}u_{k}p_{j}^{-1},\quad p_{i}\tau u_{k}p_{j}^{-1},\qquad 1\leq i\leq 12,\
k\geq 0.$

(b2) For $j=4,\ 5,\ 7,\ 8,\ ...,\ 12$, the left cell $\Gamma_{j}$ consists of
the following elements:

$p_{i}u_{k}p_{j}^{-1},\quad p_{i}\tau u_{k}p_{j}^{-1},\quad
q_{l}x_{0}q_{j}^{-1},\quad q_{l}x^{\prime}_{0}q_{j}^{-1},\qquad{1\leq i\leq
12},\ 4\leq l\leq 12,\ k\geq 0.$

Note that $p_{4}u_{k}p_{4}^{-1}=x_{k+1},\ p_{4}\tau
u_{k}p_{4}^{-1}=x^{\prime}_{k+1}.$

(b3) The left cell $\Gamma_{6}$ consists of the following elements:

$p_{i}u_{k}p_{6}^{-1},\quad p_{i}\tau u_{k}p_{6}^{-1},\quad
q_{l}x_{0}q_{6}^{-1},\quad q_{l}x_{0}q_{6}^{-1}\tau,\qquad{1\leq i\leq 12},\
4\leq l\leq 12,\ k\geq 0.$

5.4. For the two-sided cell $D$, the corresponding unipotent class in
$Sp_{6}(\mathbb{C})$ has Jordan block sizes (222). Maximal reductive subgroup
of the centralizer of an element in the unipotent class is
${F_{c}}=O_{3}(\mathbb{C})=\mathbb{Z}/2\mathbb{Z}\times SO_{3}(\mathbb{C})$.
Let $\tilde{F}_{c}=\mathbb{Z}/2\mathbb{Z}\times SL_{2}(\mathbb{C})$ be the
simply connected covering of $F_{c}$.

Let $Y$ be a set of 12 elements and let $\tilde{F}_{c}$ act on $Y$ trivially.
Then $K_{\tilde{F}_{c}}(Y\times Y)$ is isomorphic to the $12\times 12$ matrix
ring $M_{12}(\text{Rep\,}\tilde{F}_{c})$, where $\text{Rep\,}\tilde{F}_{c}$ is
the representation ring of $\tilde{F}_{c}$.

For nonnegative integer $k$, let $V(k)$ be an irreducible representation of
$SL_{2}(\mathbb{C})$ with highest weight $k$. Let $\epsilon$ be the sign
representation of $\mathbb{Z}/2\mathbb{Z}$. Regarding $V(k)$ and $\epsilon$ as
representations of $\tilde{F}_{c}$ naturally, then, up to isomorphism, the
irreducible representations of $\tilde{F}_{c}$ are $V(k),\ \epsilon V(k),\
k=0,\ 1,\ 2,\ ....$ When $k$ is even, $V(k)$ and $\epsilon V(k)$ are also
irreducible representation of $F_{c}$.

Let $V(k)_{ij}\in M_{12}(\text{Rep\,}\tilde{F}_{c})$ be the matrix whose entry
at $(i,j)$ is $V(k)$ and is 0 elsewhere. Similarly we define $\epsilon
V(k)_{ij}$. The main result in this section is the following theorem.

Theorem 5.5. There is a natural injection

$\displaystyle\pi:c\hookrightarrow$ $\displaystyle M_{12}(\text{Rep\,}\
\tilde{F}_{c}),$ $\displaystyle p_{i}u_{k}p_{j}^{-1}\longmapsto$
$\displaystyle\begin{cases}{V(2k)_{ij},}&{1\leq i,j\leq 3,}\\\
{V(2k+2)_{ij},}&{4\leq i,j\leq 12,}\\\
{V(2k+1)_{ij},}&{\text{otherwise};}\end{cases}$ $\displaystyle p_{i}\tau
u_{k}p_{j}^{-1}{\longmapsto}$ $\displaystyle\begin{cases}{\epsilon
V(2k)_{ij},}&{1\leq i,j\leq 3,}\\\ {\epsilon V(2k+2)_{ij},}&{4\leq i,j\leq
12,}\\\ {\epsilon V(2k+1)_{ij},}&{\text{otherwise};}\end{cases}$
$\displaystyle y{\longmapsto}$ $\displaystyle V(0)_{lm},\quad\text{if $y$ can
be obtained from $x_{0}$ by a}$ sequence of left and/or right star operations,
$\displaystyle y{\longmapsto}$ $\displaystyle\epsilon V(0)_{lm},\quad\text{if
$y$ can be obtained from $x^{\prime}_{0}$ by a}$ sequence of left and/or right
star operations,

where $y=q_{l}x_{0}q_{m}^{-1}$ or $q_{l}x^{\prime}_{0}q_{m}^{-1}\ (m\neq 6)$
or $q_{l}x_{0}q_{6}^{-1}$ or $q_{l}x_{0}q_{6}^{-1}\tau$, $4\leq l,m\leq 12$.

The injection $\pi$ induces an injective ring homomorphism

$\Pi:J_{c}\rightarrow M_{12}(\text{Rep\,}\tilde{F}_{c})\simeq
K_{\tilde{F}_{c}}(Y\times Y),\quad t_{w}\mapsto\pi(w),$

where $Y$ is a set of 12 elements with trivial $\tilde{F}_{c}$ action.

Proof: We need to prove that

(8) $\Pi(t_{w}t_{u})=\pi(w)\cdot\pi(u),\quad\text{for all }w,u\in D.$

Since $D$ is the union of all $\Gamma_{i},\ 1\leq i\leq 12$ and $D=D^{-1}$, we
know that $D$ is the union of all $\Gamma_{i}^{-1}\cap\Gamma_{j},\ 1\leq
i,j\leq 12.$

Assume that $w\in\Gamma_{i}^{-1}\cap\Gamma_{j}$ and
$u\in\Gamma_{k}^{-1}\cap\Gamma_{l}$. Using 1.4(j), 1.4(k) and Lemma 5.2, we
know that it suffices to prove formula (8) for
$w\in\Gamma_{i}^{-1}\cap\Gamma_{j}$, $u\in\Gamma_{k}^{-1}\cap\Gamma_{l}$,
$i,j,k,l\in\\{1,4\\}$. When $j\neq k$, by 1.4 (e) we see that $t_{w}t_{u}=0$,
hence formula (8) holds in this case. When $i=j=k=l$, according to Theorem 3.1
in [QX], we know that formula (8) holds in this case. To complete the proof of
the theorem we need to prove formula (8) for the following cases:

(i) $i=1,\ j=k=1,\ l=4$;

(ii) $i=1,\ j=k=4,\ l=4;$

(iii) $i=1,\ j=k=4,\ l=1;$

(iv) $i=4,\ j=k=1,\ l=1;$

(v) $i=4,\ j=k=1,\ l=4;$

(vi) $i=4,\ j=k=4,\ l=1.$

Keep the notations in the above paragraph. Applying 1.4 (g) and 1.4 (i) we see
that to prove formula (8) we only need to prove it for the following two
cases: ($\clubsuit$) $w\in\Gamma_{4}^{-1}\cap\Gamma_{1}$ and
$u\in\Gamma_{1}^{-1}\cap\Gamma_{1}$; ($\spadesuit$)
$w\in\Gamma_{4}^{-1}\cap\Gamma_{1}$ and $u\in\Gamma_{1}^{-1}\cap\Gamma_{4}$.

Lemma $\clubsuit$: We have

(a) $\Gamma_{4}^{-1}\cap\Gamma_{1}=\\{s_{1}s_{2}u_{k},\ s_{1}s_{2}\tau
u_{k}\,|\,k\geq 0\\}$ and $\Gamma_{1}^{-1}\cap\Gamma_{1}=\\{u_{k},\ \tau
u_{k}\,|\,k\geq 0\\}$.

(b) $\displaystyle t_{s_{1}s_{2}u_{k}}t_{u_{l}}=t_{s_{1}s_{2}\tau
u_{k}}t_{\tau u_{l}}=\sum_{0\leq
i\leq\min\\{2k+1,2l\\}}t_{s_{1}s_{2}u_{k+l-i}}.$

(c) $\displaystyle t_{s_{1}s_{2}\tau
u_{k}}t_{u_{l}}=t_{s_{1}s_{2}u_{k}}t_{\tau u_{l}}=\sum_{0\leq
i\leq\min\\{2k+1,2l\\}}t_{s_{1}s_{2}\tau u_{k+l-i}}.$

###### Proof.

Part (a) is obtained from 5.3 (b1) and 5.3 (b2).

Now we prove (b). Since $u_{0}$ is a distinguished involution, (b) is true for
$l=0$.

Assume that $l>0$. First we will prove

(9) $\displaystyle
t_{s_{1}s_{2}u_{0}}t_{u_{l}}=t_{s_{1}s_{2}u_{l}}+t_{s_{1}s_{2}u_{l-1}},\quad{\text{for
any}}\ l>0.$

Let $\xi=q^{\frac{1}{2}}+q^{-\frac{1}{2}}$. By a simple computation we get

(10) $\displaystyle
C_{s_{1}s_{2}u_{0}}=(C_{s_{1}}C_{s_{2}}-1)C_{s_{0}s_{1}s_{3}}$ (11)
$\displaystyle C_{s_{0}s_{1}s_{3}}C_{u_{l}}=\xi^{3}C_{u_{l}}.$

Hence

(12) $\displaystyle
C_{s_{1}s_{2}u_{0}}C_{u_{l}}=\xi^{3}(C_{s_{1}}C_{s_{2}}-1)C_{u_{l}}.$

Before continuing, we make a convention: we shall use the symbol $\Box$ for
any element in the two-sided ideal $H^{<013}$ of $H$ spanned by all $C_{w}$
with $a(w)>3$. Then $\Box+\Box=\Box$ and $h\Box=\Box$ for any $h\in H$.

In [QX, subsection 3.3, Step 1], we have shown the following identity:

(13) $\displaystyle C_{s_{2}}C_{u_{l}}=C_{s_{2}u_{l}}+\Box\in
C_{s_{2}u_{l}}+H^{<013}.$

Now we compute $C_{s_{1}}C_{s_{2}u_{l}}$. By formula (1) in 1.1 (e), we have

$C_{s_{1}}C_{s_{2}u_{l}}=C_{s_{1}s_{2}u_{l}}+\sum\limits_{\begin{subarray}{c}y\prec
s_{2}u_{l}\\\ s_{1}y<y\end{subarray}}\mu(y,s_{2}u_{l})C_{y}.$

Note that ${L}(s_{2}u_{l})=\\{s_{2}\\}$. First we have
$\mu(u_{l},s_{2}u_{l})=1$ and $s_{1}u_{l}<u_{l}$. By 1.4 (c) and 1.4 (d), if
$C_{y}$ appears in the above summation with nonzero coefficient, $y\neq u_{l}$
and $C_{y}\not\in H^{<013}$, then $y\in\Gamma_{4}^{-1}\cap\Gamma_{1}$.

Assume $y\prec s_{2}u_{l}$, $s_{1}y<y$ and
$y\in\Gamma_{4}^{-1}\cap\Gamma_{1}$. By (a) we must have
$y{=s_{1}s_{2}u_{k}}=s_{1}s_{2}(s_{0}s_{1}s_{3}s_{2})^{k}s_{0}s_{1}s_{3}$ for
some nonnegative integer $k\leq l-1$. Since $s_{2}s_{0}y\geq s_{0}y\geq y$, by
1.3 (d) we get $\mu(y,s_{2}u_{l})=\mu(s_{0}y,u_{l})$. Now $s_{3}u_{l}\leq
u_{l}$ and $s_{3}s_{0}y\geq s_{0}y$, by 1.1(c) we get $s_{3}s_{0}y=u_{l}$.
Hence $y=s_{1}s_{2}u_{l-1}$.

We have shown

(14) $\displaystyle
C_{s_{1}}C_{s_{2}u_{l}}=C_{s_{1}s_{2}u_{l}}+C_{u_{l}}+C_{s_{1}s_{2}u_{l-1}}+\Box.$

Combining formulas (12) (13) and (14), we get formula (9).

Recall the following formula in [QX, 3.3]:

(15) $\displaystyle t_{u_{k}}t_{u_{l}}=\sum_{0\leq
i\leq\min\\{2k,2l\\}}t_{u_{k+l-i}}.$

Now we employ formulas (9) and (15) to prove the identity in (b). We use
induction on $k$. When $k=0$, it is just formula (9). Assume that the formula
in (b) is true for nonnegative integer less than $k$. We have

$\displaystyle t_{s_{1}s_{2}u_{k}}t_{u_{l}}=$
$\displaystyle(t_{s_{1}s_{2}u_{0}}t_{u_{k}}-t_{s_{1}s_{2}u_{k-1}})t_{u_{l}}$
$\displaystyle=$ $\displaystyle t_{s_{1}s_{2}u_{0}}\cdot\sum_{0\leq
i\leq\min\\{2k,2l\\}}t_{u_{k+l-i}}-t_{s_{1}s_{2}u_{k-1}}t_{u_{l}}$
$\displaystyle=$ $\displaystyle\sum_{0\leq
i\leq\min\\{2k,2l\\}}t_{s_{1}s_{2}u_{k+l-i}}+\sum_{0\leq
i\leq\min\\{2k,2l\\}}t_{s_{1}s_{2}u_{k+l-i-1}}-\sum_{0\leq
j\leq\min\\{2k-1,2l\\}}t_{s_{1}s_{2}u_{k+l-1-j}}$ $\displaystyle=$
$\displaystyle\sum_{0\leq
i\leq\min\\{2k,2l\\}}t_{s_{1}s_{2}u_{k+l-i}}+\sum_{1\leq
i\leq\min\\{2k+1,2l+1\\}}t_{s_{1}s_{2}u_{k+l-i}}-\sum_{1\leq
j\leq\min\\{2k,2l+1\\}}t_{s_{1}s_{2}u_{k+l-j}}$ $\displaystyle=$
$\displaystyle\sum_{0\leq i\leq\min\\{2k+1,2l\\}}t_{s_{1}s_{2}u_{k+l-i}}$

Since $\tau u_{k}=u_{k}\tau$, by 1.4 (j) we have $t_{s_{1}s_{2}\tau
u_{k}}t_{\tau u_{l}}=t_{s_{1}s_{2}u_{k}}t_{u_{l}}$. Part (b) is proved.

Since $\tau u_{k}=u_{k}\tau$ and $\tau u_{k+l-i}=u_{k+l-i}\tau$, using 1.4 (j)
we see that Part (c) follows from Part (b).

The proof is completed. ∎

Lemma $\spadesuit$. (a) For $k\geq 0$, we have
$s_{1}s_{2}u_{k}s_{2}s_{1}=x_{k+1}$ and $s_{1}s_{2}\tau
u_{k}s_{2}s_{1}=x^{\prime}_{k+1}$. Moreover,
$\Gamma_{4}^{-1}\cap\Gamma_{4}=\\{x_{k},\ x^{\prime}_{k}\,|\,k\geq 0\\}$.

For nonnegative integers $k,l$ we have

(b) $\displaystyle t_{s_{1}s_{2}u_{k}}t_{u_{l}s_{2}s_{1}}=t_{s_{1}s_{2}\tau
u_{k}}t_{u_{l}\tau s_{2}s_{1}}=\sum_{0\leq
i\leq\min\\{2k+1,2l+1\\}}t_{x_{k+l+1-i}}.$

(c) $\displaystyle t_{s_{1}s_{2}\tau
u_{k}}t_{u_{l}s_{2}s_{1}}=t_{s_{1}s_{2}u_{k}}t_{u_{l}\tau
s_{2}s_{1}}=\sum_{0\leq i\leq\min\\{2k+1,2l+1\\}}t_{x^{\prime}_{k+l+1-i}}.$

###### Proof.

Part (a) follows from the discussion in Subsection 5.3.

Now we prove Part (b). First we prove

(16) $\displaystyle
t_{s_{1}s_{2}u_{0}}t_{u_{l}s_{2}s_{1}}=t_{x_{l+1}}+t_{x_{l}}.$

In [QX, Subsection 4.2], it is shown
$t_{s_{1}s_{2}u_{0}}t_{u_{0}s_{2}s_{1}}=t_{x_{1}}+t_{x_{0}}$. Now assume that
$l\geq 1$. As before, $\xi=q^{\frac{1}{2}}+q^{-\frac{1}{2}}$. Since
$C_{s_{0}s_{1}s_{3}}C_{u_{l}s_{2}s_{1}}=\xi^{3}C_{u_{l}s_{2}s_{1}}$, using
formula (10) we get

(17) $\displaystyle
C_{s_{1}s_{2}u_{0}}C_{u_{l}s_{2}s_{1}}=\xi^{3}(C_{s_{1}}C_{s_{2}}-1)C_{u_{l}s_{2}s_{1}}.$

We compute the right hand side of equality (17) step by step. As in the proof
of Lemma $\clubsuit$, we shall use the symbol $\Box$ for any element in the
two-sided ideal $H^{<013}$ of $H$ spanned by all $C_{w}$ with $a(w)>3$.

Step 1: Compute $C_{s_{2}}C_{u_{l}s_{2}s_{1}}$.

By formula (1) in 1.1 (e), we have
$C_{s_{2}}C_{u_{l}s_{2}s_{1}}=C_{s_{2}u_{l}s_{2}s_{1}}+\sum\limits_{\begin{subarray}{c}y\prec
u_{l}s_{2}s_{1}\\\ s_{2}y<y\end{subarray}}\mu(y,u_{l}s_{2}s_{1})C_{y}.$ Note
that ${L}(u_{l}s_{2}s_{1})=\\{s_{0},s_{1},s_{3}\\}$.

Assume $y\prec u_{l}s_{2}s_{1}$ and $s_{2}y<y$.

  * •

If $s_{0}y>y$, then by 1.1(c) we get $s_{0}y=u_{l}s_{2}s_{1}$. This
contradicts the assumption. So $s_{0}y>y$ would not occur.

  * •

If $s_{1}y>y$, then by 1.1(c) we get $s_{1}y=u_{l}s_{2}s_{1}$. This
contradicts the assumption. So $s_{1}y>y$ would not occur.

  * •

If $s_{0}y<y,s_{1}y<y$ and $s_{2}y<y$, then $a(y)\geq a(w_{012})=6$. So
$C_{y}\in H^{<013}.$

Therefore,

(18) $\displaystyle
C_{s_{2}}C_{u_{l}s_{2}s_{1}}=C_{s_{2}u_{l}s_{2}s_{1}}+\Box.$

Step 2: Similar to the proof for formula (14), we have

(19) $\displaystyle
C_{s_{1}}C_{s_{2}u_{l}s_{2}s_{1}}=C_{s_{1}s_{2}u_{l}s_{2}s_{1}}+C_{u_{l}s_{2}s_{1}}+C_{s_{1}s_{2}u_{l-1}s_{2}s_{1}}+\Box.$

Note $s_{1}s_{2}u_{l}s_{2}s_{1}=x_{l+1}$. Combining formulas (17)-(19), we get
(16).

Now we can prove part (b) using induction on $k$. By 1.4 (j), we know
$t_{s_{1}s_{2}u_{k}}t_{u_{l}s_{2}s_{1}}=t_{s_{1}s_{2}\tau u_{k}}t_{u_{l}\tau
s_{2}s_{1}}$. Thus for $k=0$, Part (b) is equivalent to formula (16), which is
true. Now assume that $k\geq 1$ and Part (b) is true for $k-1$. Using Lemma
$\clubsuit$ and 1.4(i), induction hypothesis and formula (16), we get

$\displaystyle t_{s_{1}s_{2}u_{k}}t_{u_{l}s_{2}s_{1}}=$
$\displaystyle(t_{s_{1}s_{2}u_{0}}t_{u_{k}}-t_{s_{1}s_{2}u_{k-1}})t_{u_{l}s_{2}s_{1}}$
$\displaystyle=$ $\displaystyle{t_{s_{1}s_{2}u_{0}}\cdot}\sum_{0\leq
i\leq\min\\{2l+1,2k\\}}t_{u_{k+l-i}s_{2}s_{1}}-t_{s_{1}s_{2}u_{k-1}}t_{u_{l}s_{2}s_{1}}$
$\displaystyle=$ $\displaystyle\sum_{0\leq
i\leq\min\\{2l+1,2k\\}}(t_{x_{k+l+1-i}}+t_{x_{k+l-i}})-\sum_{0\leq
i\leq\min\\{2l+1,2k-1\\}}t_{x_{k+l-i}}$ $\displaystyle=$
$\displaystyle\sum_{0\leq i\leq\min\\{2l+1,2k+1\\}}t_{x_{k+l+1-i}}.$

This completes the proof for Part (b).

Proof for Part (c) is similar. First, it is easy to check that

$C_{s_{0}s_{2}u_{0}}C_{u_{0}s_{2}s_{1}}=\xi^{3}(C_{\tau
x^{\prime}_{1}}+C_{\tau x^{\prime}_{0}}),$

which implies

$t_{s_{1}s_{2}\tau
u_{0}}t_{u_{0}s_{2}s_{1}}=t_{x^{\prime}_{1}}+t_{x^{\prime}_{0}}.$

Further, we prove that

$t_{s_{1}s_{2}\tau
u_{0}}t_{u_{l}s_{2}s_{1}}=t_{x^{\prime}_{1+1}}+t_{x^{\prime}_{l}}.$

Then using induction on $k$, as the proof for Part (b), we prove Part (c). The
proof for Lemma $\spadesuit$ is completed. ∎

We have completed the proof for Theorem 5.5.

5.6. Motivated by Theorem 5.5 and the discussion of the cocenter of $J$ in
[BDD, Section 5] and some other evidences, we suggest a modification of
Lusztig’s conjecture on the structure of $J_{c}$ which is stated for any
connected reductive groups over $\mathbb{C}$.

Let $W$ be the extended affine Weyl group attached to a connected reductive
group over $\mathbb{C}$ (see Subsection 1.1) and let $c$ be a two-sided cell
of $W$. Let $F_{c}$ be a maximal reductive subgroup of the centralizer of an
element in the corresponding unipotent class of $G$. Then there should exist a
reductive group $\tilde{F}_{c}$ with the following properties:

(i) The reductive group $\tilde{F}_{c}$ is a simply connected covering of
$F_{c}$. That is, the identity component $\tilde{F}_{c}^{\circ}$ has simply
connected derived group, and there is a natural surjective homomorphism
$\tilde{F}_{c}\to F_{c}$ with finite kernel. In particular, if $F_{c}^{\circ}$
has simply connected derived group, then $\tilde{F}_{c}=F_{c}$.

(ii) There exists a finite set $Y$ with an algebraic action of $\tilde{F}_{c}$
and an injection

$\pi:c\hookrightarrow\text{isomorphism classes of irreducible
$\tilde{F}_{c}$-vector bundles on}\ Y\times Y.$

such that

(iii) The injection $\pi$ induces a ring injection

$\Pi:J_{c}\to K_{\tilde{F}_{c}}(Y\times Y),\ \ t_{x}{\mapsto}\pi(x).$

(iv) $\pi(x^{-1})_{(a,b)}=\pi(x)_{(b,a)}^{*}$ is the dual representation of
$\pi(x)_{(b,a)}.$

(v) $K_{\tilde{F}_{c}}(Y\times Y)$ is a finitely generated left (and right as
well) $\Pi(J_{c})$-module.

It seems natural that the $F_{c}$-set $\mathbf{B}_{e}$ defined in a recent
paper (see [L4]) would have an $\tilde{F}_{c}$-action compatible with the
$F_{c}$-action and then $\mathbf{B}_{e}$ should be a good candidate for the
set $Y$ above.

Acknowledgement: Part of the work was done during YQ’s visit to the Academy of
Mathematics and Systems Science, Chinese Academy of Sciences. YQ is very
grateful to the AMSS for hospitality and for financial supports.

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