Integral $u$-deformed involution modules

TL;DR

This paper extends Lusztig's $u$-deformed involution modules from the indeterminate $u$ to any nonzero complex number outside \\{0,1,-1\\}, for all Coxeter systems, broadening the applicability of the module construction.

Contribution

It proves Lusztig's speculation that the $u$-deformed involution module construction can be generalized to any nonzero scalar outside \\{0,1,-1\\} over arbitrary fields.

Findings

Confirmed the extension of the involution module to any scalar in the specified set.

Established the isomorphism of the module with a submodule of formal sums over the Coxeter group.

Generalized Lusztig's construction beyond the original indeterminate parameter.

Abstract

Let $(W,S)$ be a Coxeter system and $\ast$ an automorphism of $W$ with order $\leq 2$ and $S^{\ast}=S$. Lusztig and Vogan ([11], [14]) have introduced a $u$-deformed version $M_u$ of Kottwitz's involution module over the Iwahori-Hecke algebra $\mathscr{H}_{u}(W)$ with Hecke parameter $u^2$, where $u$ is an indeterminate. Lusztig has proved that $M_u$ is isomorphic to the left $\mathscr{H}_{u}(W)$-submodule of ${\hat{\mathscr{H}}}_u$ generated by $X_{\emptyset}:=\sum_{w^*=w\in W}{u^{-\ell(w)}T_w}$, where ${\hat{\mathscr{H}}}_u$ is the vector space consisting of all formal (possibly infinite) sums $\sum_{x\in W}{c_xT_x}$ ($c_x\in\mathbb{Q}(u)$ for each $x$). He speculated that one can extend this by replacing $u$ with any $\lambda\in \mathbb{C}\setminus\{0,1,-1\}$. In this paper, we give a positive answer to his speculation for any $\lambda\in K\setminus\{0,1,-1\}$ and any $W$, where $K$ is an arbitrary ground field.

Equations

$$\ell (w):=\min \{k\mid w=s_{i_1}\dots s_{i_k}\text{ for some }{s}_{i_1},\dots ,s_{i_k}\in S\}.$$

$$T_w{T}_{w^{\prime }}=T_{w{w}^{\prime }}\text{if ℓ(ww′)=ℓ(w)+ℓ(w′);}$$

$$(T_s+1)(T_s-u^2)=0\text{if s∈S.}$$

$$T_s{a}_w=u{a}_w+(u+1)a_{sw}\text{if sw=ws∗>w;}$$

$$T_s{a}_w=(u^2-u-1)a_w+(u^2-u)a_{sw}\text{if sw=ws∗<w;}$$

$$T_s{a}_w=a_{sw{s}^{\ast }}\text{if sw=ws∗>w;}$$

$$T_s{a}_w=(u^2-1)a_w+u^2{a}_{sw{s}^{\ast }}\text{if sw=ws∗<w.}$$

$$X_{\varnothing }:=\sum _{x\in W,x^{\ast }=x}{u}^{-\ell (x)}{T}_x\in {\hat{\mathcal{H}}}_{\mathcal{A},u}\subseteq {\hat{\mathcal{H}}}_u.$$

$${\mathcal{A}}_{\pm 1}:=\mathbb{Z}[u,u^{-1},(u+1{)}^{-1},(u-1{)}^{-1}].$$

$$M_{{\mathcal{A}}_{\pm 1},u}:={\mathcal{A}}_{\pm 1}{\otimes }_{\mathcal{A}}{M}_{\mathcal{A},u},{\mathcal{H}}_{{\mathcal{A}}_{\pm 1},u}:={\mathcal{A}}_{\pm 1}{\otimes }_{\mathcal{A}}{\mathcal{H}}_{\mathcal{A},u},{\hat{\mathcal{H}}}_{{\mathcal{A}}_{\pm 1},u}:={\mathcal{A}}_{\pm 1}{\otimes }_{\mathcal{A}}{\hat{\mathcal{H}}}_{\mathcal{A},u}.$$

$$M_{\lambda }:=$$

$${\mathcal{H}}_{\lambda }:=$$

$${\hat{\mathcal{H}}}_{\lambda }:=$$

$${\iota }_K:K{\otimes }_{{\mathcal{A}}_{\pm 1}}{\mathcal{H}}_{{\mathcal{A}}_{\pm 1},u}{X}_{\varnothing }$$

$$r{\otimes }_{{\mathcal{A}}_{\pm 1}}h{X}_{\varnothing }$$

$$s⋉w:=\{\begin{array}{ll}sw& \text{if sw=ws∗;}\\ sw{s}^{\ast }& \text{if sw=ws∗.}\end{array}$$

$$s_{i_{1}}\ltimes s_{i_{2}}\ltimes\cdots\ltimes s_{i_{k}}\ltimes w:=s_{i_{1}}\ltimes\bigl{(}s_{i_{2}}\ltimes\cdots\ltimes(s_{i_{k}}\ltimes w)\cdots\bigr{)}.$$

$$z_a:=s_{i_a}⋉s_{i_{a+1}}⋉\cdots ⋉s_{i_k}⋉1=s_{i_a}{s}_{i_{a+1}}\dots s_{i_k}{s}_{i_{k+1}}{s}_{i_{k+2}}\cdots s_{i_{t_a}},$$

$${\sigma }_z:=s_{i_1}\dots s_{i_k}\in W.$$

$$w_0:=w,w_t:=s_{i_t}⋉w_{t-1},\text{for 1≤t≤k.}$$

$${\ell }^{\ast }(w):=\#\{1\le t\le k\mid s_{i_t}{w}_t=w_t{s}_{i_t}^{\ast }\}.$$

$$T_x{a}_1=\sum _{z\in {\mathbf{I}}_{\ast }}{L}_z^x{a}_z,\forall x\in W.$$

$${\tilde{L}}_z^x:=(-1{)}^{\ell (x)}ϵ(z)\overline{L_z^x}.$$

$$\mu (a_z)=\sum _{x\in W}{\tilde{L}}_z^x{T}_x.$$

$$\mathbf{I}_{\ast}({\prec_{\sigma}z}):=\Bigl{\{}w\in\mathbf{I}_{\ast}\Bigm{|}\begin{matrix}\text{$\rho(w)<\rho(z)$ and there exists a reduced $\mathbf{I}_{*}$-expression $\sigma^{\prime}$}\\ \text{of $w$ such that $\sigma^{\prime}_{w}<\sigma_{z}$.}\end{matrix}\Bigr{\}}$$

$$T_{{\sigma }_z}{a}_1=(u+1{)}^{{\ell }^{\ast }(z)}{a}_z+\sum _{w\in {\mathbf{I}}_{\ast }({\prec }_{\sigma }z)}{L}_w^{{\sigma }_z}{a}_w,L_w^{{\sigma }_z}\in u\mathbb{Z}[u].$$

$$T_{{\sigma }_z}{a}_1=$$

$$=$$

$$T_{s_{i_1}}{a}_{z^{\prime }}=u{a}_{z^{\prime }}+(u+1)a_{s_{i_1}{z}^{\prime }}=u{a}_{z^{\prime }}+(u+1)a_z,$$

$$T_{s_{i_1}}{a}_{z^{\prime }}=a_{s_{i_1}{z}^{\prime }{s}_{i_1}^{\ast }}=a_z,$$

$$T_{{\sigma }_z}{a}_1=(u+1{)}^{{\ell }^{\ast }(z)}{a}_z+\sum _{w\in {\mathbf{I}}_{\ast }({\prec }_{\sigma }z)}{L}_w^{{\sigma }_z}{a}_w,$$

$$T_{{\sigma }_z}{a}_1=(u+1{)}^{{\ell }^{\ast }(z)}{a}_z+\sum _{\begin{array}{c}w\in {\mathbf{I}}_{\ast }\\ \rho (w)<\rho (z)\end{array}}{L}_w^{{\sigma }_z}{a}_w,L_w^{{\sigma }_z}\in u\mathbb{Z}[u],$$

$$T_{{\sigma }_z}{a}_1\equiv T_{{\hat{\sigma }}_z}{a}_1(\text{mod }\sum _{\begin{array}{c}w\in {\mathbf{I}}_{\ast }\\ \rho (w)<\rho (z)\end{array}}u\mathbb{Z}[u]a_w).$$

$${\mathcal{A}}_{-1}:=\mathbb{Z}[u,u^{-1},(u+1{)}^{-1}],{\mathcal{A}}_1:=\mathbb{Z}[u,u^{-1},(u-1{)}^{-1}].$$

Full text

1. Introduction

Let $(W,S)$ be a fixed Coxeter system and $\ast$ be a fixed automorphism of $W$ with order $\leq 2$ and such that $S^{\ast}=S$. That is, $s^{\ast}\in S$ for any $s\in S$. Let $\ell:W\rightarrow\mathbb{N}$ be the usual length function on $W$. If $w\in W$ then by definition

[TABLE]

Definition 1.1**.**

We define \mathbf{I}_{\ast}:=\bigl{\{}w\in W\bigm{|}w^{\ast}=w^{-1}\bigr{\}}. The elements of $I_{\ast}$ will be called twisted involutions relative to $\ast$.

Let $u$ be an indeterminate over $\mathbb{Q}$ (the field of rational numbers).

Definition 1.2** ([1, 8]).**

Let $\mathscr{H}_{u}:=\mathscr{H}_{u}(W)$ be the associative unital $\mathbb{Q}(u)$-algebra with a $\mathbb{Q}(u)$-basis $\{T_{w}\mid w\in W\}$and multiplication defined by

[TABLE]

We call $\mathscr{H}_{u}(W)$ the Iwahori–Hecke algebra over $\mathbb{Q}(u)$ associated to $(W,S)$ with Hecke parameter $u^{2}$.

Let $\mathcal{A}:=\mathbb{Z}[u,u^{-1}]$ be the ring of Laurent polynomials on $u$. Let $\mathscr{H}_{\mathcal{A},u}$ be the $\mathcal{A}$-subalgebra of $\mathscr{H}_{u}$ generated by $\{T_{w}\mid w\in W\}$. Then $\mathscr{H}_{\mathcal{A},u}$ is a natural $\mathcal{A}$-form of $\mathscr{H}_{u}$ and isomorphic to the abstract $\mathcal{A}$-algebra defined by the same generators and relations as in Definition 1.2. For any field $K$ and any $\lambda\in K^{\times}$, there is a unique ring homomorphism $\phi:\mathcal{A}\rightarrow K$ satisfying that $\phi(u)=\lambda$. We define $\mathscr{H}_{\lambda}:=K\otimes_{\mathcal{A}}\mathscr{H}_{u}$ and call $\mathscr{H}_{\lambda}$ the specialized Iwahori–Hecke algebra associated to $(W,S)$ with Hecke parameter $\lambda^{2}$.

Let $M_{u}$ be a $\mathbb{Q}(u)$-linear space with a $\mathbb{Q}(u)$-basis $\{a_{z}\mid z\in\mathbf{I}_{\ast}\}$.

Lemma 1.3** ([11, 14]).**

There is a unique $\mathscr{H}_{u}$-module structure on $M_{u}$ such that for any $s\in S$ and any $w\in\mathbf{I}_{\ast}$,

[TABLE]

When $u$ is specialized to $1$, the module $M_{u}$ degenerates to the involution module introduced more than fifteen years ago by Kottwitz [9]. Kottwitz found the module by analyzing Langlands theory of stable characters for real groups. He gave a conjectural description of it (later established by Casselman) in terms of the Kazhdan-Lusztig left cell representations of the Weyl group $W$. One interesting fact about the module $M_{u}$ is that if $W$ is of finite classical type then any irreducible representation $V$ appears as a component of $M_{u}$ if and only if $V$ is a special irreducible representation of $W$ in the sense of [10]. For this reason, we call $M_{u}$ the $u$-deformed involution modules.

In a series of papers [11], [12], [13], [14], Lusztig and Vogan have studied the $u$-deformed involution modules systematically. A bar invariant canonical basis for $M_{u}$ and certain coefficient polynomials $P_{y,w}^{\sigma}$ were introduced, which can be regarded as some twisted analogue of the classical well-known Kazhdan–Lusztig basis and Kazhdan–Lusztig polynomials ([8]).

Let $\hat{\mathscr{H}}_{\mathcal{A},u}$ (resp., $\hat{\mathscr{H}}_{u}$) be the free $\mathcal{A}$-module (resp., the $\mathbb{Q}(u)$-vector space) consisting of all formal (possibly infinite) sums $\sum_{x\in W}c_{x}T_{x}$, where $c_{x}\in\mathcal{A}$ (resp., $c_{x}\in\mathbb{Q}(u)$) for each $x\in W$.

Definition 1.4**.**

([12]) We define

[TABLE]

Theorem 1.5** ([13]).**

The map $\mu:M_{u}\rightarrow\hat{\mathscr{H}}_{u}$ which sends $a_{1}$ to $X_{\emptyset}$ can be extended uniquely to a left $\mathscr{H}_{u}$-module isomorphism $M_{u}\cong\mathscr{H}_{u}X_{\emptyset}$.

Note that the above theorem was proved in [7] by the first author and Jing Zhang in the special case when $W=\mathfrak{S}_{n}$ and $\ast=\operatorname{id}$.

Definition 1.6**.**

We define

[TABLE]

Let $M_{\mathcal{A},u}$ be the free $\mathcal{A}$-submodule of $M_{u}$ generated by $\{a_{z}\mid z\in\mathbf{I}_{\ast}\}$. By Lemma 1.3, it is clear that $M_{\mathcal{A},u}$ naturally becomes a left $\mathscr{H}_{\mathcal{A},u}$-module. We set

[TABLE]

For any ring homomorphism $\phi:\mathcal{A}_{\pm 1}\rightarrow K$ with $\lambda=\phi(u)\in K\setminus\{0,1,-1\}$, we define

[TABLE]

If $W$ is finite, then $\mathscr{H}_{\mathcal{A}_{\pm 1},u}=\hat{\mathscr{H}}_{\mathcal{A}_{\pm 1},u}$, $\mathscr{H}_{u}=\hat{\mathscr{H}}_{u}$ and $\mathscr{H}_{\lambda}=\hat{\mathscr{H}}_{\lambda}$. Note that $\mathscr{H}_{\mathcal{A},u}$ (resp., $\mathscr{H}_{\mathcal{A}_{\pm 1},u}$) is a free $\mathcal{A}$-module (resp., $\mathcal{A}_{\pm 1}$-module) with basis $\{T_{w}\mid w\in W\}$. For simplicity, we shall often abbreviate $1_{K}\otimes_{\mathcal{A}}T_{w}$ and $1_{K}\otimes_{\mathcal{A}_{\pm 1}}T_{w}$ as $T_{w}$.

By some calculations in small ranks, Lusztig has speculated in [13, §4.10] that Theorem 1.5 might be extended to the setting of specialized version $\hat{\mathscr{H}}_{\lambda}$ of $\hat{\mathscr{H}}_{u}$ for arbitrary $\lambda\in\mathbb{C}\setminus\{0,1,-1\}$ when $W$ is finite. Therefore, it is natural to make the following conjecture.

Conjecture 1.7**.**

Let $K$ be a field and $\lambda\in K\setminus\{0,1,-1\}$. Let $(W,S)$ be an arbitrary Coxeter system.

(1) The map $\mu$ restricts to a left $\mathscr{H}_{\mathcal{A}_{\pm 1},u}$-module isomorphism $M_{\mathcal{A}_{\pm 1},u}\cong\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$.

(2) For any ring homomorphism $\phi:\mathcal{A}_{\pm 1}\rightarrow K$ with $\lambda=\phi(u)$, the map which sends $1_{K}\otimes_{\mathcal{A}_{\pm 1}}a_{1}$ to $1_{K}\otimes_{\mathcal{A}_{\pm 1}}X_{\emptyset}$ can be extended uniquely to a well-defined left $\mathscr{H}_{\lambda}$-module isomorphism $M_{\lambda}\cong\mathscr{H}_{\lambda}(1_{K}\otimes_{\mathcal{A}_{\pm 1}}X_{\emptyset})$.

(3) For any ring homomorphism $\phi:\mathcal{A}_{\pm 1}\rightarrow K$ with $\lambda=\phi(u)$, the canonical map

[TABLE]

is a left $\mathscr{H}_{\lambda}$-module isomorphism.

(4) If $W$ is finite, then $\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$ is a pure and free $\mathcal{A}_{\pm 1}$-submodule of ${\mathscr{H}}_{\mathcal{A}_{\pm 1},u}$.

The purpose of this paper is to give a proof of the above conjecture and thus give a positive answer to Lusztig’s speculation. As an application of our main result, we obtain a new integral basis for the module $M_{u}$ and for the module $\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$, see Corollary 2.21 and Corollary 2.28.

2. Proof of Conjecture 1.7

The purpose of this section is to give a proof of Conjecture 1.7.

Definition 2.1** ([4]).**

For any $w\in\mathbf{I}_{\ast}$ and $s\in S$, we define

[TABLE]

For any $w\in\mathbf{I}_{\ast}$ and $s_{i_{1}},\cdots,s_{i_{k}}\in S$, we define

[TABLE]

It is clear that $s\ltimes w\in\mathbf{I}_{\ast}$ whenever $w\in\mathbf{I}_{\ast}$ and $s\in S$. Furthermore, $\ltimes$ is in general not associative.

Definition 2.2** ([2, 4]).**

Let $w\in\mathbf{I}_{\ast}$. If $w=s_{i_{1}}\ltimes s_{i_{2}}\ltimes\cdots\ltimes s_{i_{k}}\ltimes 1$, where $k\in\mathbb{N}$, $s_{i_{j}}\in S$ for each $j$, then $(s_{i_{1}},\cdots,s_{i_{k}})$ is called an $\mathbf{I}_{\ast}$-expression for $w$. Such an $\mathbf{I}_{\ast}$-expression for $w$ is reduced if its length $k$ is minimal.

We regard the empty sequence $()$ as a reduced $\mathbf{I}_{\ast}$-expression for $w=1$. Let “$\leq$” be the Bruhat partial ordering on $W$ defined with respect to $S$ (cf. [6]). We write $u<w$ if $u\leq w$ and $u\neq w$. It follows by induction on $\ell(w)$ that every element $w\in\mathbf{I}_{\ast}$ has a reduced $\mathbf{I}_{\ast}$-expression.

Lemma 2.3** ([4, 5]).**

Let $w\in\mathbf{I}_{\ast}$. Any reduced $\mathbf{I}_{\ast}$-expression for $w$ has a common length. Let $\rho:\mathbf{I}_{\ast}\rightarrow\mathbb{N}$ be the map which assigns $w\in\mathbf{I}_{\ast}$ to this common length. Then $(\mathbf{I}_{\ast},\leq)$ is a graded poset with rank function $\rho$. Moreover, if $s\in S$ then $\rho(s\ltimes w)=\rho(w)\pm 1$, and $\rho(s\ltimes w)=\rho(w)-1$ if and only if $\ell(sw)=\ell(w)-1$.

Corollary 2.4** ([7, Corollary 2.6]).**

Let $w\in\mathbf{I}_{\ast}$ and $s\in S$. Suppose that $sw\neq ws^{\ast}$. Then $\ell(sw)=\ell(w)+1$ if and only if $\ell(ws^{\ast})=\ell(w)+1$, and if and only if $\ell(s\ltimes w)=\ell(w)+2$. The same is true if we replace “$+$” by “$-$”.

Lemma 2.5**.**

Let $z\in\mathbf{I}_{*}$. Let $(s_{i_{1}},\cdots,s_{i_{k}})$ (where $k\in\mathbb{N}$) be an arbitrary reduced $\mathbf{I}_{*}$-expression of $z$. Then there exist $s_{i_{k+1}},s_{i_{k+2}},\dots,s_{i_{r}}\in S$ and integers ${k}\leq t_{k}\leq t_{k-1}\leq\dots\leq t_{1}=r$ such that $r=\ell(z)$, and for each $1\leq a\leq k$,

[TABLE]

$\rho(z_{a})=k-a+1$, $\ell(z_{a})=t_{a}-a+1$. In particular, $s_{i_{a}}s_{i_{a+1}}\dots s_{i_{t_{a}}}$ is a reduced expression of $z_{a}$, $s_{i_{a}}\cdots s_{i_{k}}$ is a reduced expression and $(i_{k+1},i_{k+2},\dots,i_{r})$ is uniquely determined by the reduced $\mathbf{I}_{*}$-expression $(s_{i_{1}},\cdots,s_{i_{k}})$.

Proof.

This follows from Lemma 2.3 and Corollary 2.4 and an induction on $k$. ∎

Definition 2.6**.**

For each $z\in\mathbf{I}_{\ast}$ and each reduced $\mathbf{I}_{\ast}$-expression $\sigma=(s_{i_{1}},\cdots,s_{i_{k}})$ of $z$, we define

[TABLE]

In particular, we have $\sigma_{1}=1$ and $\rho(z)=\ell(\sigma_{z})$ for any $z\in\mathbf{I}_{\ast}$. In general, $\sigma_{z}$ depends on the choice of the reduced $\mathbf{I}_{\ast}$-expression $(s_{i_{1}},\cdots,s_{i_{k}})$ of $z$.

Definition 2.8** ([5, Proposition 2.5], [15, Proposition 2.2]).**

Let $w\in\mathbf{I}_{\ast}$ and $(s_{i_{1}},\cdots,s_{i_{k}})$ be a reduced $\mathbf{I}_{*}$-expression of $w$. We define

[TABLE]

Define $\ell^{*}:\mathbf{I}_{\ast}\rightarrow\mathbb{N}$ by

[TABLE]

The notation $\ell^{*}(w)$ we used here was denoted by $\ell^{\theta}(w)$ in [4] and [5], and was denoted by $\phi$ in [13, §1.5]. By [5], $\ell^{*}(w)$ depends only on $w$ but not on the choice of the reduced $\mathbf{I}_{\ast}$-expression $(s_{i_{1}},\cdots,s_{i_{k}})$ of $w$.

Lemma 2.9** ([3, Theorem 4.8], [5, §2.2]).**

Let $w\in I_{\ast}$. Then $\rho(w)=(\ell(w)+\ell^{*}(w))/2$.

Let $\bar{}:\mathcal{A}\rightarrow\mathcal{A}$ be the ring involution such that $\overline{u^{n}}=(-u)^{-n}$ for any $n\in\mathbb{Z}$. Let $\epsilon:\mathbf{I}_{\ast}\rightarrow\{1,-1\}$, $z\mapsto(-1)^{\rho(z)}$, $\forall\,z\in\mathbf{I}_{\ast}$. By Lemma 2.9, our $\epsilon$ coincides with the function $\epsilon$ defined in [13, §1.5].

Definition 2.10** ([13, §1.1]).**

Let $\{L_{z}^{x}\mid z\in\mathbf{I}_{\ast},x\in W\}$ be a set of uniquely determined polynomials in $\mathbb{Z}[u]$ such that

[TABLE]

Definition 2.11** ([13, §1.6]).**

For $x\in W,z\in\mathbf{I}_{\ast}$, we set

[TABLE]

Lemma 2.12** ([13, §1.7, the 5th line above §1.8]).**

For $z\in\mathbf{I}_{\ast}$, we have

[TABLE]

Note that there is a typo in the identity on $\mu(a_{z})$ in [13, §1.7, the 5th line above §1.8]). The element $T_{z}$ in the right hand should be replaced by $T_{x}$.

Proposition 2.13**.**

Let $z\in\mathbf{I}_{\ast}$ and $\sigma=(s_{i_{1}},\cdots,s_{i_{k}})$ be a reduced $\mathbf{I}_{*}$-expression of $z$. Let $(i_{k+1},\cdots,i_{r})$ be the unique $(r-k)$-tuple determined by this reduced $\mathbf{I}_{*}$-expression as described in Lemma 2.5. Then $z=s_{i_{1}}\ltimes\cdots\ltimes s_{i_{k}}\ltimes 1=s_{i_{1}}\cdots s_{i_{k}}s_{i_{k+1}}\cdots s_{i_{r}}$ with $k=\rho(z)$, $r=\ell(z)$, where $s_{i_{1}},\dots,s_{i_{r}}\in S$, and we have $L_{z}^{\sigma_{z}}=(u+1)^{\ell^{*}(z)}$, and $L_{w}^{\sigma_{z}}\neq 0$ only if $\rho(w)<\rho(z)$ and there exists a reduced $\mathbf{I}_{*}$-expression $\sigma^{\prime}$ of $w$ such that $\sigma^{\prime}_{w}<\sigma_{z}$. Moreover, $L_{w}^{\sigma_{z}}\in u\mathbb{Z}[u]$ if $w\neq z$.

Definition 2.14**.**

Let $z\in\mathbf{I}_{\ast}$ and $\sigma=(s_{i_{1}},\cdots,s_{i_{k}})$ be a reduced $\mathbf{I}_{*}$-expression of $z$. We define

[TABLE]

Then Proposition 2.13 is equivalent to the following identity:

[TABLE]

Proof of Proposition 2.13: Let $z\in\mathbf{I}_{*}$. We prove the proposition by induction on $\rho(z)$. If $\rho(z)=0$, then $z=1$, $\sigma_{z}=1$ and $T_{1}a_{1}=a_{1}$.

Let $k\in\mathbb{N}^{*}$. Suppose that the statement holds when $\rho(z)<k$. Let $z\in\mathbf{I}_{*}$ with $\rho(z)=k$. We follow the notation and hypothesis in Lemma 2.5 and Definition 2.6. Then $z=s_{i_{1}}\ltimes\cdots\ltimes s_{i_{k}}\ltimes 1=s_{i_{1}}\cdots s_{i_{k}}s_{i_{k+1}}\cdots s_{i_{r}}$ with $k=\rho(z)$, $r=\ell(z)$ for some $s_{i_{1}},\dots,s_{i_{r}}\in S$. By definition, $\sigma_{z}=s_{i_{1}}\cdots s_{i_{k}}$. Let $x^{\prime}=s_{i_{1}}\sigma_{z}=s_{i_{2}}\cdots s_{i_{k}}$. Note that $(s_{i_{1}},\cdots,s_{i_{k}})$ is a reduced $\mathbf{I}_{*}$-expression implies that $\sigma^{\prime}:=(s_{i_{2}},\cdots,s_{i_{k}})$ is a reduced $\mathbf{I}_{\ast}$-expression of $z^{\prime}:=s_{i_{2}}\ltimes\cdots\ltimes s_{i_{k}}\ltimes 1$, then $\rho(z^{\prime})=k-1$ and $x^{\prime}=\sigma_{z^{\prime}}$ in the notation of Definition 2.6. Now $x^{\prime}<\sigma_{z}$ and

[TABLE]

We consider the first term in the above identity. There are two possibilities:

Case 1. If $s_{i_{1}}{z^{\prime}}={z^{\prime}}s_{i_{1}}^{*}$, then $z=s_{i_{1}}\ltimes{z^{\prime}}=s_{i_{1}}{z^{\prime}}$. Thus

[TABLE]

where $\ell^{*}(z)=\ell^{*}(z^{\prime})+1$ and $z^{\prime}\in\mathbf{I}_{\ast}(\prec_{\sigma}z)$, as required.

Case 2. If $s_{i_{1}}{z^{\prime}}\neq{z^{\prime}}s_{i_{1}}^{*}$, then $z=s_{i_{1}}\ltimes{z^{\prime}}=s_{i_{1}}{z^{\prime}}s_{i_{1}}^{*}$. Thus

[TABLE]

where $\ell^{*}(z)=\ell^{*}(z^{\prime})$, as required.

Therefore, it remains to consider the term $T_{s_{i_{1}}}a_{z^{\prime\prime}}$ for each $z^{\prime\prime}\in\mathbf{I}_{\ast}(\prec_{\sigma^{\prime}}z^{\prime})$. We know that $\sigma_{z}=s_{i_{1}}x^{\prime}=s_{i_{1}}\cdots s_{i_{k}}$ and $\sigma_{z^{\prime}}=x^{\prime}=s_{i_{2}}\cdots s_{i_{k}}$ are reduced expressions. Combining our assumption $z^{\prime\prime}\in\mathbf{I}_{\ast}(\prec_{\sigma^{\prime}}z^{\prime})$ and Lemma 1.3 together we can deduce that $T_{s_{i_{1}}}a_{z^{\prime\prime}}$ is a $\mathbb{Z}[u]$-linear combination of some $a_{w}$ with $w\in\mathbf{I}_{\ast}(\prec_{\sigma}z)$. Therefore, we get that

[TABLE]

where $L_{w}^{\sigma_{z}}\in u\mathbb{Z}[u]$ for each $w\in\mathbf{I}_{\ast}(\prec_{\sigma}z)$. This completes the proof of the proposition.

Note that $w\in\mathbf{I}_{\ast}(\prec_{\sigma}z)$ implies that $\rho(w)<\rho(z)$.

Corollary 2.16**.**

Let $z\in\mathbf{I}_{\ast}$ and $\sigma,\hat{\sigma}$ be two reduced $\mathbf{I}_{*}$-expressions of $z$. Then

[TABLE]

Corollary 2.17**.**

For each $z\in\mathbf{I}_{\ast}$ we fix a reduced $\mathbf{I}_{\ast}$-expression $\sigma$ of $z$ and define $\sigma_{z}$ as in Definition 2.6. Then the map $\sigma_{*}\colon z\mapsto\sigma_{z}$ defines an injection from $\mathbf{I}_{\ast}$ into $W$. In other words, $\sigma_{z_{1}}=\sigma_{z_{2}}$ if and only if $z_{1}=z_{2}$.

Proof.

This follows from Proposition 2.13. ∎

Definition 2.18**.**

We define

[TABLE]

Corollary 2.19**.**

For each $z\in\mathbf{I}_{\ast}$ we fix a reduced $\mathbf{I}_{\ast}$-expression $\sigma$ of $z$ and define $\sigma_{z}$ as in Definition 2.6. Then

[TABLE]

where for each $w\in\mathbf{I}_{\ast}(\prec_{\sigma}z)$, $\xi_{z}^{w}\in\mathcal{A}_{-1}$. In particular,

[TABLE]

Proof.

This follows from Proposition 2.13, (2.15) and Corollary 2.16. ∎

Let $M_{\mathcal{A},u}$ be the free $\mathcal{A}$-submodule of $M_{u}$ generated by $\{a_{z}\mid z\in\mathbf{I}_{\ast}\}$. It is clear that $M_{\mathcal{A},u}$ naturally becomes a left $\mathscr{H}_{\mathcal{A},u}$-module. We set

[TABLE]

For each $z\in\mathbf{I}_{\ast}$, we identify $1_{\mathcal{A}_{-1}}\otimes_{\mathcal{A}}a_{z}$, $1_{\mathcal{A}_{1}}\otimes_{\mathcal{A}}a_{z}$, $1_{\mathcal{A}_{\pm 1}}\otimes_{\mathcal{A}}a_{z}$ and $1_{\mathbb{Q}(u)}\otimes_{\mathcal{A}}a_{z}$ with $a_{z}$.

Corollary 2.21**.**

For each $z\in\mathbf{I}_{\ast}$ we fix a reduced $\mathbf{I}_{\ast}$-expression $\sigma$ of $z$ and define $\sigma_{z}$ as in Definition 2.6. Then the elements in the following set

[TABLE]

form an $\mathcal{A}_{-1}$-basis of $M_{\mathcal{A}_{-1},u}$, an $\mathcal{A}_{\pm 1}$-basis of $M_{\mathcal{A}_{\pm 1},u}$ and a $\mathbb{Q}(u)$-basis of $M_{u}$. The same is true if one replaces $\mathcal{A}_{-1}$ with any field $K$ and $u$ with any $\lambda\in K^{\times}$ whenever there is a ring homomorphism $\phi:\mathcal{A}_{-1}\rightarrow K$ with $\lambda=\phi(u)$.

Proof.

This follows from Proposition 2.13 and (2.15). ∎

Corollary 2.23**.**

For each $z\in\mathbf{I}_{\ast}$ we fix a reduced $\mathbf{I}_{\ast}$-expression $\sigma$ of $z$ and define $\sigma_{z}$ as in Definition 2.6. We have that

[TABLE]

In particular,

[TABLE]

and the map $\mu\downarrow_{M_{\mathcal{A}_{\pm 1},u}}:M_{\mathcal{A}_{\pm 1},u}\rightarrow\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$ is surjective.

Proof.

This follows from Corollary 2.21 and the surjectivity of $\mu\downarrow_{M_{\mathcal{A}_{-1},u}}:M_{\mathcal{A}_{-1},u}\twoheadrightarrow\mathscr{H}_{\mathcal{A}_{-1},u}X_{\emptyset}$. ∎

For any field $K$ and any ring homomorphism $\phi:\mathcal{A}_{\pm 1}\rightarrow K$ with $\lambda=\phi(u)$, we define

[TABLE]

to be the composition of the following surjection

[TABLE]

with the canonical surjective homomorphism

[TABLE]

introduced in Conjecture 1.7. By definition, we know that $\mu_{K}$ is surjective.

Proposition 2.24**.**

Let $K$ be a field. For any ring homomorphism $\phi:\mathcal{A}_{\pm 1}\rightarrow K$ with $\lambda=\phi(u)$, the elements in the following set

[TABLE]

form an $K$-basis of $\mathscr{H}_{\lambda}(1_{K}\otimes_{\mathcal{A}_{\pm 1}}X_{\emptyset})$. In particular, $\mu_{K}$ is a left $\mathscr{H}_{\lambda}$-module isomorphism. Furthermore, the elements in the following set

[TABLE]

form an $\mathcal{A}_{\pm 1}$-basis of $\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$.

Proof.

We consider the first part of the proposition. Recall that $\mu_{K}$ is surjective. Since $\{1_{K}\otimes_{\mathcal{A}_{\pm 1}}a_{z}\mid z\in\mathbf{I}_{\ast}\}$ is a $K$-basis of $M_{\lambda}$, it suffices to show that the elements in the subset (2.25) are $K$-linearly independent. Note that the assumption $\lambda\neq 0,-1$ is used here to ensure that $\mu_{K}$ is surjective (by Corollary 2.21 and Corollary 2.23).

Suppose that the elements in the subset (2.25) are $K$-linearly dependent. That says, we can find an positive integer $m$ and

[TABLE]

such that $Y_{K,z_{1}},\cdots,Y_{K,z_{m}}$ are $K$-linearly dependent. For each $z\in\{z_{1},\cdots,z_{m}\}$, we fix a reduced $\mathbf{I}_{\ast}$-expression of $z$ and define $\sigma_{z}$ as in Definition 2.6. Without loss of generality, we can assume that $\rho(z_{1})\leq\rho(z_{2})\leq\cdots\leq\rho(z_{m})$. Equivalently, $\ell(\sigma_{z_{1}})\leq\ell(\sigma_{z_{2}})\leq\cdots\leq\ell(\sigma_{z_{m}})$. Furthermore, we can find an integer $n\geq m$ and a finite subset $\{w_{m+1},\cdots,w_{n}\}$ of $W\setminus\{\sigma_{z_{1}},\cdots,\sigma_{z_{m}}\}$ such that for each $1\leq j\leq m$,

[TABLE]

By Definition 2.10, Definition 2.11, Lemma 2.12 and Proposition 2.13,

[TABLE]

where $D_{1}$ is the following $n\times n$ diagonal matrix:

[TABLE]

$D_{2}$ is the following $n\times n$ diagonal matrix:

[TABLE]

and $A_{u}$ is the following $n\times m$ matrix in $M_{n\times m}(\mathbb{Z}[u^{-1}])$:

[TABLE]

such that the top $m\times m$ submatrix is a lower triangular matrix with diagonal elements given by

[TABLE]

By assumption $\lambda=\phi(u)\not\in\{0,1,-1\}$. We define $A_{\lambda}:=A_{u}\downarrow_{u:=\lambda}$. Then

[TABLE]

By the above discussion and the assumption that $\lambda\neq 1$ we can see that $\operatorname{rank}A_{\lambda}=m$. Note that $\{1_{K}\otimes_{\mathcal{A}_{\pm 1}}T_{w}\mid w\in W\}$ is a subset of $K$-linearly independent elements in $\hat{\mathscr{H}}_{\lambda}$. Since $D_{1},D_{2}$ are invertible, it follows that

[TABLE]

is a set of $K$-linearly independent elements in $\mathscr{H}_{\lambda}(1_{K}\otimes_{\mathcal{A}_{\pm 1}}X_{\emptyset})\subseteq\hat{\mathscr{H}}_{\lambda}$. We get a contradiction. In particular, this implies that $\mu_{K}$ is a left $\mathscr{H}_{\lambda}$-module isomorphism. This proves the first part of the proposition.

Finally, taking $K=\mathbb{Q}(u)$ we see that $\mu_{\mathbb{Q}(u)}$ is an isomorphism by the first part of the proposition which we have just proved. This further implies that $\operatorname{id}_{\mathbb{Q}(u)}\otimes_{\mathcal{A}_{\pm 1}}\mu\downarrow_{M_{\mathcal{A}_{\pm 1},u}}$ is an isomorphism. Since

[TABLE]

it follows that $\operatorname{Ker}\mu\downarrow_{M_{\mathcal{A}_{\pm 1},u}}=0$. Hence $\mu\downarrow_{M_{\mathcal{A}_{\pm 1},u}}$ is an isomorphism and the elements in (2.26) form an $\mathcal{A}_{\pm 1}$-basis of $\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$. This proves the second part of the proposition and hence we complete the proof of the proposition. ∎

Proof of Conjecture 1.7: (1) and (2) follows from Proposition 2.24. Now (3) follows from (1) and (2). It remains to consider (4). For this purpose, we assume that $W$ is finite. Then $X_{\emptyset}\in\mathscr{H}_{\mathcal{A}_{\pm 1},u}$.

By (2.27) and Proposition 2.13, we easily see that the elements in the following set

[TABLE]

form an $\mathcal{A}_{\pm 1}$-basis of $\hat{\mathscr{H}}_{\mathcal{A}_{\pm 1},u}={\mathscr{H}}_{\mathcal{A}_{\pm 1},u}$. This implies that $\mathscr{H}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$ is a pure and free $\mathcal{A}_{\pm 1}$-submodule of ${\mathscr{H}}_{\mathcal{A}_{\pm 1},u}$. This completes the proof of Conjecture 1.7.

Corollary 2.28**.**

The elements in the following set

[TABLE]

form an $\mathcal{A}_{\pm 1}$-basis of ${\mathscr{H}}_{\mathcal{A}_{\pm 1},u}X_{\emptyset}$. The same is true if one replaces $\mathcal{A}_{\pm 1}$ with any field $K$ and $u$ with any $\lambda\in K^{\times}$ whenever there is a ring homomorphism $\phi:\mathcal{A}_{\pm 1}\rightarrow K$ with $\lambda=\phi(u)$.

Proof.

This follows from Corollary 2.21, Corollary 2.23 and Conjecture 1.7 (which we have just proved). ∎

By Lemma 2.12,

[TABLE]

where $\widetilde{L}_{z}^{x}\in\mathbb{Z}[u^{-1}]$. Following [13, Theorem 0.2(b)], we define $n_{z}^{x}:=\widetilde{L}_{z}^{x}\downarrow_{u^{-1}=0}\in\mathbb{Z}$. Then Lusztig has proved in [13, Theorem 0.2(c)] that there is a unique surjective function $\pi:W\twoheadrightarrow\mathbf{I}_{\ast}$ such that for any $x\in W,z\in\mathbf{I}_{\ast}$, we have $n_{z}^{x}=1$ if $z=\pi(x)$; and $n_{z}^{x}=0$ if $z\neq\pi(x)$.

Our next result shows that the map $\sigma_{*}$ which we introduced in Corollary 2.17 is a right inverse of $\pi$.

Corollary 2.30**.**

Let $\sigma_{*}\colon\mathbf{I}_{\ast}\hookrightarrow W$ be the injection defined in Corollary 2.17. Then $\pi\circ\sigma_{*}=\operatorname{id}_{\mathbf{I}_{*}}$.

Proof.

Let $z\in\mathbf{I}_{\ast}$. Following [13, §1.8], we use $\{\underline{T}_{w}\mid w\in W\}$ to denote the standard basis of the specialization $\mathscr{H}_{0}$ of $\mathscr{H}_{u}$ at $u:=0$, and use $M_{0}$ to denote the specialization of $M$ at $u:=0$. Then $M_{0}$ is a $\mathbb{Q}$-space with basis $\{\underline{a}_{w}\mid w\in\mathbf{I}_{\ast}\}$ and with $\mathscr{H}_{0}$-module structure given by

[TABLE]

where $s\in S,w\in\mathbf{I}_{\ast}$.

Setting $u:=0$ on both sides of (2.15), we get that

[TABLE]

On the other hand, since $\widetilde{L}^{\sigma_{z}}_{x}:=(-1)^{\ell(\sigma_{z})}\epsilon(x)\overline{L^{\sigma_{z}}_{x}}$ for any $x\in\mathbf{I}_{\ast}$, setting $u:=0$ in $L^{\sigma_{z}}_{x}$ is equivalent to setting $u^{-1}:=0$ in $\widetilde{L}^{\sigma_{z}}_{x}$. We can deduce from [13, Theorem 0.2(c)] that

[TABLE]

Note that $(-1)^{\ell(\sigma_{z})}\epsilon(z)=(-1)^{\rho(z)}(-1)^{\rho(z)}=1$. This completes the proof of the corollary. ∎

Acknowledgements

The research was carried out under the support from the National Natural Science Foundation of China (No. 11525102, 11471315).