Affine cellularity of affine Hecke algebras of rank two

Let $\mathcal{A}=\mathbb{C}[v,v^{-1}]$ where $v$ is an indeterminate. Let $\mathcal{H}$ be the Iwahori-Hecke algebra associated to $W$, with $\mathcal{A}$-basis $\{T_{w}|w\in W\}$ and multiplication rule given by

for all $s\in S$ and $w\in W$. Let $\bar{\ }$ be the ring involution of $\mathcal{A}$ which takes $v$ to $v^{-1}$. It can be extended to a ring involution of $\mathcal{H}$ via

We set

For each $w\in W$ there exists a unique element $C_{w}\in\mathcal{H}$ (see [11, Theorem 5.2]) such that

1. (1)

   $\bar{C}_{w}=C_{w}$

2. (2)

   $C_{w}\equiv T_{w}\mod\mathcal{H}_{<0}$.

For any $w\in W$ we set

It is well known ([11, §5.3]) that $P_{y,w}=0$ whenever $y\nleq w$ (here $\leq$ denotes the Bruhat order). It follows that $\{C_{w}|w\in W\}$ forms an $\mathcal{A}$-basis of $\mathcal{H}$ (the “Kazhdan-Lusztig basis”). The coefficients $P_{y,w}$ are known as the Kazhdan-Lusztig polynomials.

We denote by $h_{x,y,z}$ the structure constant with respect to the Kazhdan-Lusztig basis. That is, we set

Note that $\bar{h}_{x,y,z}=h_{x,y,z}$ and, similarly to Remark 3.2, we have $h_{x,y,z}=h_{y^{-1},x^{-1},z^{-1}}$. We write $z\leftarrow_{\mathcal{L}}y$ if there exists some $s\in S$ such that $h_{s,y,z}\neq 0$, that is $C_{z}$ appears with a non-zero coefficient in the expression of $C_{s}C_{y}$ in the Kazhdan-Lusztig basis. The Kazhdan-Lusztig left pre-order $\leq_{\mathcal{L}}$ on $W$ is the transitive closure of this relation. The equivalence relation associated to $\leq_{\mathcal{L}}$ will be denoted by $\sim_{\mathcal{L}}$, that is

The corresponding equivalence classes are called the left cells of $W$. Similarly, we can define a pre-order $\leq_{\mathcal{R}}$ multiplying on the right in the defining relation. The associated equivalence relation will be denoted by $\sim_{\mathcal{R}}$ and the corresponding equivalence classes are called the right cells of $W$. Using the antiautomorphism $\flat$, we have (see [11, §8])

Finally we write $x\leq_{\mathcal{LR}}y$ if there exists a sequence $x=x_{0},x_{1},...,x_{n}=y$ of $W$ such that for each $0\leq i\leq n-1$ we have either $x_{i}\leftarrow_{\mathcal{L}}x_{i+1}$ or $x_{i}\leftarrow_{\mathcal{R}}x_{i+1}$. The equivalence relation associated to $\leq_{\mathcal{LR}}$ will be denoted by $\sim_{\mathcal{LR}}$ and the equivalence classes are called the two-sided cells of $W$.
The preorders $\leq_{\mathcal{L}},\leq_{\mathcal{R}},\leq_{\mathcal{LR}}$ induce partial orders on the left, right and two-sided cells, respectively.

We will follow the notation of [1]. Let $?\in\{\mathcal{L},\mathcal{R},\mathcal{LR}\}$. We define a $?$-ideal to be a left ideal if $?=\mathcal{L}$, a right ideal if $?=R$ and a two-sided ideal if $?=\mathcal{LR}$. Similarly, an $\mathcal{H}$-?-module is a left $\mathcal{H}$-module if $?=\mathcal{L}$, a right $\mathcal{H}$-module if $?=R$ and an $\mathcal{H}$-$\mathcal{H}$-bimodule if $?=\mathcal{LR}$.

Let $\Gamma$ be a $?$-cell of $W$. We set

Then by definition of $\leq_{?}$, we see that $\mathcal{H}_{\leq_{?}\Gamma}$ and $\mathcal{H}_{<_{?}\Gamma}$ are $?$-ideals of $\mathcal{H}$. Therefore

is naturally an $\mathcal{H}$-$?$-module. It is called the Kazhdan-Lusztig cell module associated to $\Gamma$. Note that it is a free $\mathcal{A}$-module with basis the images of the elements $C_{w}$ for $w\in\Gamma$.

Let $\Gamma$ be a two-sided cell of $W$ and let

be its decomposition into left cells. We denote by $[\ .\ ]$ the natural projection onto $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$. (Then $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ has an $\mathcal{A}$-basis $\{[C_{w}]\mid w\in\Gamma\}$.) As an $\mathcal{A}$-module we have

Further if one assumes that the Lusztig conjecture

holds, then the $\mathcal{A}$-submodules $\mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}$ are left $\mathcal{H}$-submodules of $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ for all $1\leq i\leq m$. Indeed, for all $h\in\mathcal{H}$ and $y\in\Gamma_{i}$, we have

which, using $(\ast)$, yields that $h[C_{y}]\in\mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}$.

Finally, since the $\mathcal{H}$-$\mathcal{H}$-bimodule $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ is a two-sided ideal in $\mathcal{H}/\mathcal{H}_{<_{\mathcal{LR}}\Gamma}$, it can be viewed as an algebra (possibly without identity element). The multiplication is given by

If one assumes that $(\ast)$ holds, then the $\mathcal{A}$-submodules $\mathcal{M}^{\mathcal{L}}_{\Gamma_{i}}$ are left ideals of $\mathcal{M}^{\mathcal{LR}}_{\Gamma}$ for all $1\leq i\leq m$.

Let $\bar{S}=\{\bar{{\bf s}}_{1},...,\bar{{\bf s}}_{r}\}$ be the set of conjugacy classes in $S$. Any weight function on $W$ is completely determined by its values on $\bar{S}$. Let $V=\mathbb{R}^{r}$ be the Euclidean space of dimension $r$ and let $\omega_{1},...,\omega_{r}$ be the standard basis of $V$. We identify the set of weight functions on $W$ with the set of points in $V$ with integer coordinates via

where $s_{i}\in\bar{{\bf s}}_{i}$ for all $i$. The element of the $r$-tuple $(L(s_{1}),...,L(s_{r}))$ are called the parameters. To any choice of parameters one can associate a partition of $W$ into left, right and two-sided cells.

According to Bonnafé’s semicontinuity conjecture [1], there exists a minimal finite set of hyperplanes $\mathfrak{H}$ in $V$ such that the partition of $W$ into cells is the same for all parameters $\mathcal{P},\mathcal{P}^{\prime}\in\mathbb{N}^{r}$ belonging to the same $\mathfrak{H}$-facet (we refer to [2] for the definition of facets). The elements of this minimal set are called essential hyperplanes. The conjecture also states that the partition into cells for non-generic parameters can be recovered from the partition with respect to generic parameters. We refer to [1] for details on this conjecture. In the following definition we assume that Bonnafé’s conjecture holds.

In this paper, we are only concerned with affine Weyl groups of rank 2 where the semicontinuity conjecture is known to hold and where the generic parameters have been determined in [8].

In this section we introduce the relative Kazhdan-Lusztig polynomials as in [3]. Let $S^{\prime}\subsetneq S$. We denote by $W^{\prime}$ the subgroup of $W$ generated by $S^{\prime}$ and by $X^{\prime}$ the set of distinguished left coset representative of $W^{\prime}$ in $W$. Every element of $w\in W$ can be written uniquely $w=xu$ where $x\in X^{\prime}$ and $u\in W^{\prime}$. Note that $\ell(w)=\ell(x)+\ell(u)$.
Let $\preceq^{\prime}$ be the Kazhdan-Lusztig left preorder relation defined with respect to the Coxeter group $(W^{\prime},S^{\prime})$ and the corresponding Hecke algebra. We define a relation $\sqsubseteq$ on $W$ as follows. Let $x,y\in X^{\prime}$ and $u,v\in W^{\prime}$. We write $xu\sqsubset yv$ if $x<y$ and $u\preceq v$. We write $xu\sqsubseteq yv$ if $xu\sqsubset yv$ or $x=y$ and $u=v$. Then we have

Using this theorem, Geck obtained the following result (see [3, Section 4]).

We now introduce the Generalised Induction Theorem. The idea is to generalise the construction above to some subsets of $W$ which may not be parabolic subgroups. We refer to [7, 8] for details.

We consider a subset $U\subseteq W$ and a collection $\{X_{u}\ |\ u\in U\}$ of subsets of $W$ satisfying the following conditions

1. I1.

   for all $u\in U$, we have $e\in X_{u}$,

2. I2.

   for all $u\in U$ and $x\in X_{u}$ we have $\ell(xu)=\ell(x)+\ell(u)$,

3. I3.

   for all $u,v\in U$ such that $u\neq v$ we have $X_{u}u\cap X_{v}v=\emptyset$,

4. I4.

   the submodule $\mathcal{M}:=\mathcal{h}T_{x}C_{u}|\ u\in U,\ x\in X_{u}\mathcal{i}_{\mathcal{A}}\subseteq\mathcal{H}$ is a left ideal.

One can easily see that the set $\mathcal{B}:=\{T_{x}C_{u}|u\in U,x\in X_{u}\}$ is an $\mathcal{A}$-basis of $\mathcal{M}$. Thus for all $y\in W$ and all $v\in U$, we can write

Let $\preceq$ be the relation on $U$ defined as follows. Let $u,v\in U$. We write $u\preceq v$ if there exist $y\in W$ and $x\in X_{u}$ such that $T_{x}C_{u}$ appears with a non-zero coefficient in the expression of $T_{y}C_{v}$ in the basis $\mathcal{B}$. We still denote by $\preceq$ the pre-order induced by this relation (i.e. the transitive closure). For $u,v\in U$, $x\in X_{u}$ and $y\in X_{v}$ we write $xu\sqsubset yv$ if $u\preceq v$ and $xu<yv$. We write $xu\sqsubseteq yv$ if $xu\sqsubset yv$ or $x=y$ and $u=v$.

Now if one assumes that

1. I5.

   for all $v\in U$, $y\in X_{v}$ we have

   $$T_{y}C_{v}\equiv T_{yv}\mod\mathcal{H}_{<0}$$

then we have $\tilde{C}_{yv}=C_{yv}$.

In this section we present a very nice description of $\Gamma$. It is rather surprising that most of the two-sided cells in $W$ can be described in such a uniform way. Note that this description is vital in the proof of affine cellularity. We refer to the next section for examples of this description.

By inspection of the different partitions into cells given in [6, 8], one can show that there exist two subsets $\mathcal{T}$ and $\mathcal{Z}=\{z_{1}=e,z_{2},\ldots,z_{m}\}$ of $W$ and an element $w_{\Gamma}\in W$ in a parabolic subgroup $W^{\prime}$ such that for all $1\leq i,j\leq m$ and all $\tau\in\mathcal{T}$ we have

and the following map is bijective

The left cells lying in $\Gamma$ are of the form

and we have

The set $\mathcal{T}$ can be expressed in one of the following forms:

1. (1)

   $\mathcal{T}=\{t_{1}^{n}t_{2}^{m}\mid n,m\in\mathbb{N}\}$

2. (2)

   $\mathcal{T}=\{t^{n}\mid n\in\mathbb{N}\}$

3. (3)

   $\mathcal{T}=\{e,t\}$

4. (4)

   $\mathcal{T}=\{e\}$

In case (1) we set $T=\{t_{1},t_{2}\}$, in case (2) and (3) we set $T=\{t\}$ and in case (4) we set $T=\{e\}$. In all cases it can be checked that for $\tau\in\mathcal{T}$, we have

It is clear that if $\Gamma$ is infinite, then we must be in case (1) or (2) and if $\Gamma$ is finite then we must be in case (3) or (4). We refer to Section 5 for examples of the different cases: when $\Gamma=\tilde{c}_{0}$ we are in case (1), when $\Gamma=\tilde{c}_{1}$ or $\tilde{c}_{2}$ we are in case (2), when $\Gamma=\tilde{c}_{3}$ we are in case (3) and when $\Gamma=\tilde{c}_{4}$ we are in case (4).

Let $X^{\prime}$ be the set of distinguished left coset representative of $W^{\prime}$ in $W$ and let $\leq^{{}^{\prime}}_{\mathcal{L}}$ be the Kazhdan-Lusztig left preorder relation defined with respect to $W^{\prime}$. Then explicit computations show that either

1. (1)

   $w_{\Gamma}$ is the longest element in $W^{\prime}$;

2. (2)

   $w_{\Gamma}=sw^{\prime}$ where $w^{\prime}$ is the longest element of $W^{\prime}$, $w_{\Gamma}^{2}=1$ and $\{w_{\Gamma},w^{\prime}\}$ is a left ideal of $W^{\prime}$ with respect to $\leq^{\prime}_{\mathcal{L}}$.

The element $\mathbf{P}$ in Case (1). Assume we are in Case (1). By well-known properties of the longest element in a Coxeter group, for all $u\in W^{\prime}$ we have $u\leq^{\prime}_{\mathcal{L}}w_{\Gamma}$ implies that $u=w_{\Gamma}$. Further, one can easily check from the definition of $\sqsubset$ that this implies that if $w\in W$ satisfies $w\sqsubseteq yw_{\Gamma}$ then $w=xw_{\Gamma}$ with $x<y$ and $x\in X^{\prime}$. Thus by Proposition 3.8 we get for all $y\in X^{\prime}$

We set for $y\in\mathcal{Z}^{-1}\cup T$

so that we have

The last equality holds since

The element $\mathbf{P}$ in Case (2). Assume that we are in Case (2). We set $U=\{w_{\Gamma}\}$ and $X=X^{\prime}\cup X^{\prime}s$.

Claim. The set $U$ together with $X$ satisfy condition I1–I5.

Assume for now that it is the case, then for all $y\in X$ we have (note that in the sum below the element $x$ is chosen in $X$ and not in $X^{\prime}$ as above!)

For $y\in\mathcal{Z}^{-1}\cup T$ we set

so that we have

The last equality holds since

(here we need the fact that $w_{\Gamma}^{2}=1$).

Proof of Claim. In order to verify that $U=\{w_{\Gamma}\}$ together with $X=X^{\prime}\cup X^{\prime}s$ satisfies condition I1–I5, note that $Xw_{\Gamma}=X^{\prime}w^{\prime}\cup X^{\prime}w_{\Gamma}$, from where conditions I1–I3 follow easily. Next we know that $\{w^{\prime},w_{\Gamma}\}$ is a left ideal of $W^{\prime}$, thus by Corollary 3.10 we get that

is a left ideal of $\mathcal{H}$. Since

it follows that $\mathfrak{B}=\mathcal{h}T_{x}C_{w_{\Gamma}}\mid x\in X\mathcal{i}$ and I4 follows. Let $x\in X$. If $x\in X^{\prime}$ Condition I5 is clearly satisfied since $w_{\Gamma}\in W^{\prime}$. So assume that $x\in X^{\prime}s$, that is $x=x^{\prime}s$ for some $x^{\prime}\in X^{\prime}$. Then we have