数理解析研究所講究録 1438 巻 2005 年 19-36

19

JMMO Fock space and Geck-Rouquier classification of
simple modules for Hecke algebras
Nicolas Jacon
Using Lusztig a-function, M. Geck and R.Rouquier have recently
proved the existence of a canonical set in natural bijection with the set ofsimple modules for Hecke algebras. In this paper, we recall the definition ofthis set and we report recent results which show that the definition of canonical
basic set can be extended to the case of Ariki-Koike algebras. Moreover, wegive an explicit description of this set for all Hecke algebras and for all ArikiKoike algebras.
ABSTRACT,$B$

1. Introduction
Let $W$ be a finite Weyl group with set of simple reflections $S\subset W$ , let be
an indeterminate, $u=v^{2}$ and let $A=\mathbb{C}[v, v^{-1}]$ . The Hecke algebra $H$ of $W$ over$A$ is the associative A-algebra with basis $\{T_{w}|w\in W\}$ and the multiplication$s\in S$
between two elements of the basis is determined by the following rule. Let
and $w\in W$ , then:$v$$T_{s}T_{w}= \int_{[}T_{\mathrm{s}w}uT_{sw}+(u-1)T_{w}$if $l(sw)$ $>l(w)_{9}$if $l(sw)<l(w)$ ,where is the usual length function of $W$ . Such algebras play an important role,for example in the representation theory of finite groups of Lie type (see [DG] and[Gb] or in the theory of knots and links (see [GP, Chapter 4]).
be a homomorphism
and let 0.
Let $K$ be the field of fractions of.
into the field of complex numbers. Let $H_{K}:=K\otimes_{A}H$ and let
itrelatively
well-understood:isOn the one hand, the representation theory ofis known that this is a split semisimple algebra which is isomorphic to the groupand its simple modules are in natural bijection with the simple
algebraare much more complicatedmodules. On the other hand, the simple modules ofis not semisimple in general. In fact, this problem is linked
to describe because
between the Grothendieck group $R_{0}(HK)$to the problem of determining a map-modules and the Grothendieck group $R_{0}(Hc)$ of finitelyof finitely generated-modules. This map is called the “decomposition map” and it relates
generated-modules via a process of modular-modules with the sim ple
the simple
$l$

$)$

$Aarrow \mathbb{C}$$A$$H_{\mathbb{C}}.--\mathbb{C}\otimes_{A}H$$H_{K}$$\mathbb{C}[W]-$$\mathbb{C}[W]$$H_{\mathbb{C}}$$H_{\mathbb{C}}$$d_{\theta}$$H_{K}$$H_{\mathbb{C}}$$H_{\mathbb{C}}$$H_{K}$

reduction:$d_{\theta}$:$R_{0}(H_{K})arrow R_{0}(H_{\mathbb{C}})$.

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NICOLAS JACON

The decomposition maps associated to Hecke algebras of exceptional types arealmost all explicitely known, see [Mu], [Gel] , [Ge2] and [GL] (in fact, for $W=E_{8}$ ,we only have an “approximation” of this maP, see [Mu] . In this PaPer, we are
and type
mainly interesting about the classical types that is type $A_{n-1}$ , tyPe.
For these types, works of Ariki [Ab], Dipper-James [DJ], Dipper-James-Murphy-modules. In fact, Hecke[DJM], and Hu [H] provide a description of the simple
are particular cases of Ariki-Koike algebras (or cycloalgebras of type $A_{n-1}$ andtom ic Hecke algebras of type $G(d, 1, n)$ , see [AK] . In [A2], Ariki has shown thatthe computation of the decomposition map for these algebras can be easily deduced
from the computation of the Kashiwara-Lusztig canonical basis of irreducible high. In particular, these results lead to a parametrization
est weightof the simple modules for Ariki-Koike algebras (and so, for Hecke algebras of type
$A_{n-1}$ and) using a class of multipartitions which appears in the crystal graph, Hu-modules, namely the Kleshchev multipartitions. For typetheory ofhas obtained a classification of the simple modules by using the fact that the Hecke(withalgebra of typecan be seen as a subalgebra of a Hecke algebra of type, theseandunequal parameters). One of the main problem is that, for type
results lead to a recursive parametrization of the simple modules.
In [Gk] and in [GR], Geek and Rouquier have given another approch for thedescription of the simple-modules. They have shown the existence of a canonicalby using Lusztig a-fonction and Kazhdan-Lusztig theory. This set
set
is called the “canonical basic set” and it is in natural bijection with Irr(HC). Hence,
it gives a way to classify the simple-modules. Moreover, the existence of thecanonical basic set implies that the matrix associated tohas a lower triangularshape with 1 along the diagonal.
The description of the canonical basic set is now complete for all finite Weylgroups $W$ and for all specializations 0. In this paper, we report these recent results
which show that this set can be indexed by another class of multipartitions which
also appears in the crystal graph theory of the-modules. The proof also
requires Ariki’s theorem but what we obtain here is a non recursive parametrizationof the simple-modules. Moreover, we will see that all these results can beextended to the case of Ariki-Koike algebras even if we don’t have Kazhdan-Lusztig
type basis for Ariki-Koike algebras.
$)$$B_{n}$$D_{n}$$H_{\mathbb{C}}$$B_{n}$$)$$\mathcal{U}_{q}(\hat{sl}_{e})- \mathrm{m}\mathrm{o}\mathrm{d}111\mathrm{e}\mathrm{s}$$B_{n}$$D_{n}$$\mathcal{U}_{q}(sl_{e})$$B_{n}$$D_{n}$$B_{n}$$D_{n}$$H_{\mathbb{C}}$$B$$\subseteq \mathrm{I}\mathrm{r}\mathrm{r}(H_{K})$$H\mathbb{C}$$d_{\theta}$$\mathcal{U}_{q}(\hat{sl}_{e})$$H_{\mathbb{C}}$

2. Representations of semisimple Hecke algebras
Let $H$ be an Iwahori-H ecke algebra of a finite Weyl group $W$ over $A:=\mathbb{C}[v, v^{-1}]$
as it is defined in the introduction. Let
be the corresponding
and letHecke algebra. Thenis integrally closed in $K$ andis a split semisimple
algebra. By Tits deformation theorem (see, Theorem 8. L7]),
is isomorphic
to the group algebra. Hence, the simple -modules are in natural bijection
with the simple-modules. In fact, for type $A_{n-1}$ and type, the simple-modules can be explicitly described by using the theory of cellular algebras (see, the simple
while for type-modules are obtained by using Clifford
theory. We obtain the following parametrizations for the classical types of Weyl
$K=\mathrm{C}(\mathrm{v})$$A$$H_{K}$$[\mathrm{G}\mathrm{P}$$\mathbb{C}[W]$$B_{n}$$H_{K}$

group$D_{n}$$\mathrm{s}$$H_{K}$$H_{K}$$\mathbb{C}[W]$$[\mathrm{G}\mathrm{r}\mathrm{L}])$$H_{K}$$H_{K}$

2]
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS

2.1. Type $A_{n-1}$ . Assume that W is a Weyl group of type$\mapsto s_{1}s_{2}$$A_{n-1}$.

$\underline{s}_{n-1}$

Let A be a partition of rank , then, we can construct a $H$ module , free overwhich is called a Specht module (see the construction of “dual Specht modules” in[Ab, Chapter 13] in a more general setting). Moreover, we have:$A$$S^{\lambda}$$n$$\mathrm{I}\mathrm{r}\mathrm{r}(H_{K})=$

where we denote by

2.2. Type$B_{n}$$\Pi_{n}^{1}${$S_{K}^{\lambda}:=K\otimes_{A}S^{\lambda}|$the set of partitions of rankA$n$$\in\Pi_{n}^{1}$},.

. Assume that W is a Weyl group of type.

$B_{n}$$\underline{s}_{n}$

Let

be a
, free over$(\lambda^{(0)}, \lambda^{(1)})$$S^{(\lambda^{(\mathrm{O})},\lambda^{(1)})}$-partition of rank , then, we can construct a if-module
which is called a Specht module. Moreover, we have:$n$$\mathrm{b}\mathrm{i}$$A$,$\mathrm{I}\mathrm{r}\mathrm{r}(H_{K})=\{S_{K}^{(\lambda^{(0\rangle},\lambda^{(1)})}.--K\otimes_{A}S^{(\lambda^{(0)},\lambda^{(1)})}|(\lambda^{(0)}, \lambda^{(1)})\in\Pi_{n}^{2}\}$where we denote by

2.3. Type$D_{n}$$\Pi_{n}^{2}$the set of$\mathrm{b}\mathrm{i}$partitions of rank$n$.

. Assume that W is a Weyl group of type

$D_{n}$.

$-\underline{s}_{n}$with unequalof typeThen, $H$ can be seen as a subalgebra of a Hecke algebraparameters (see [Gex] for more details).
, we can construct
Similary to the equal parameter case, for all, free over $A$ which is called a Specht module. We have:amodule$B_{n}$$H_{1}$$(\lambda^{(0)}, \lambda^{(1)})$$\in\Pi_{n}^{2}$$S^{(\lambda^{\langle 0\rangle},\lambda^{(1\rangle})}$$H_{1}$.

$\mathrm{I}\mathrm{r}\mathrm{r}(H_{1,K})=\{S_{K}^{(\lambda^{(\mathrm{o})},\lambda^{(1)})}:=K\otimes_{A}S^{(\lambda^{(0)},\lambda^{(1)})}|(\lambda^{(0)}, \lambda^{(1)})\in\Pi_{n}^{2}\}$

We have an operation of restriction-modules, for
set of$(\lambda^{(0)}, \lambda^{(1)})\in\Pi_{n}^{2}$$H_{K}$$\bullet$if$\lambda^{(0)}\neq\lambda^{(1)}$module$\bullet$if, we have$H_{1,K}$-modules and the:${\rm Res}(S_{K}^{(\lambda^{(\mathrm{O}\rangle},\lambda^{(1)}\rangle})\simeq{\rm Res}(S_{K}^{(\lambda^{(1)},\lambda^{(0)})})$$V^{[\lambda^{\langle 0)},\lambda^{(1)}]}:={\rm Res}(S_{K}^{(\lambda^{\langle 0)},\lambda^{(1)})})$

is a simple, we haveare non isomorphic simple
and$\lambda^{(0)}=\lambda^{(1)}$$V^{[\lambda^{(0)},+]}$

between the set of${\rm Res}$and themodule.

$H_{K}$${\rm Res}(S_{K}^{(\lambda^{(0)},\lambda^{(1)})})=V^{[\lambda^{(\mathrm{O})},+]}\oplus V^{[\lambda^{\langle 0)},-]}$$V^{[\lambda^{(0)},-]}$$H_{K}$-modules.

Moreover we have:,$\mathrm{I}\mathrm{r}\mathrm{r}(H_{K})=\{V^{[\lambda,\mu]}|\lambda\neq\mu$$H_{K^{-}}$$(\lambda, \mu)\in\Pi_{n}^{2}\}\cup\{V^{[\lambda,\pm]}|$A$\in\Pi\frac{1n}{2}\}$

.

where

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NICOLAS JACON

Now, we turn to the problem of determining a classification of the set of simple
modules for Hecke algebras in the case where is no longer an indeterminate but
a complex number.
$v$

3,

Modular representations and canonical basic sets for Hecke algebras
Let 2:$Aarrow \mathbb{C}$

be a ring homomorphism. We put:$\mathcal{O}:=\{\frac{f}{g}|f$,$g\in \mathbb{C}[v_{\rfloor}^{\rceil},$

0 is a discrete valuation ring and we haveobtain a well-defined decomposition map$d_{\theta}$:.

$g(\theta(v))\neq 0\}$

$A\subseteq O$. By [$R_{0}(H_{K})arrow R_{0}(H_{\mathbb{C}})$$\mathrm{G}\mathrm{P}$, Theorem 7.4.3], we.

) is the Grothendieck group of finitely generated
where $R_{0}(H_{K})$ (resp.
modules (resp.
-modules). This is defined as follows: letbe a simple $H_{K}-$, there exists a $Ho$ -module such that $K\otimes 0\hat{V}=V$ .
module. Then, by
By reducing
modulo the maximal ideal of 0, $\mathfrak{m}:=(v-\theta(v))O$ , we obtain a-module (&o . Then, we put:
$H_{K^{-}}$$R_{0}(H_{\mathbb{C}})$$V$$H\mathbb{C}$$\hat{V}$$[\mathrm{G}\mathrm{P}, \S 7.4]$$\hat{V}$$\mathbb{C}$$H_{\mathbb{C}}$$\hat{V}$$d_{\theta}([V])=[\mathbb{C}\otimes_{\mathit{0}}\hat{V}]$is well-defined and forthat:$d_{\theta}$$V\in \mathrm{I}\mathrm{r}\mathrm{r}(\mathrm{H}\mathrm{k})$.

, there exist numbers$d_{\theta}([V])= \sum_{M\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathrm{C}})}d_{V,M}[M]$The matrix$(d_{V,M})_{V\in 1\mathrm{r}\mathrm{r}(H_{K})}$$(d_{V,M})_{M\in \mathrm{J}\mathrm{r}\mathrm{r}(H_{\mathrm{C}})}$

such

.

is called the decomposition matrix. For more details$\mathrm{A}.\mathrm{f}\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathrm{C}})$

about the construction of decomposition maps, even in a more general setting, see[Gb].
Now, we will recall results of Geek and Rouquier which show that the decomposition map has always a unitriangular shape with one along the diagonal.
Let
be the Kazhdan-Lusztig basis of $H$ . For , $y\in W$ , the multiplication between two elements of this basis is given by:$\{C_{w}\}_{w\in W}$$x$$C_{x}C_{y}= \sum_{z\in W}h_{x,y},{}_{z}C_{z}$where $h_{x,y,z}\in A$ for all $z\in W$ . For any $z\in W$ , there is a well-defined integer$a(z)\geq 0$ such that$v^{a(z)}h_{x,y,z}\in \mathbb{Z}[v]$for all , $y\in W$,$v^{a\langle z)-1}h_{x,y,z}\not\in \mathbb{Z}[v]$$x$for some , $y\in W$.
$x$

We obtain a function which is called the Lusztig a-function:
$a$

:

$W$

$arrow$$z$$\mapsto$%$a(z)$

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GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS

Now, following [ , Lem ma 1.9], to any$a(M)$ by the requirement that:$\mathrm{L}\mathrm{I}$$M\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathbb{C}})$, we can attach an a-value

CW.M $=0$ for all $w\in W$ with $a(w)>a(M)$ ,$CW.M\neq 0$ for some $w\in W$ with $a(w)=a(M)$ .

, in an analogous way. Note
We can also attach an a-value $a(V)$ to any-module usingthat there is an equivalent definition of the a-value of a simple
the fact that Hecke algebras are symmetric algebras, this will be important in thhecontext of Ariki-Koike algebras where we don’t have Kazhdan-Lusztig theory. Let
of $H$ (see: $H_{K}arrow A$ be the symmetrizing\S 7.1]) which is defined by, there exists a$\tau(T_{w})=0$ if $w\neq 1$ and $\tau(T_{1})=1$ . Then, for each$s_{V}\in
A$
that:
such
polynomial
Laurent$V\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{K})$$H_{K}$$[\mathrm{G}\mathrm{P},$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\tau$$V\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{K})$$\tau=\sum_{V\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{K})}\frac{1}{s_{V}}\chi_{V}$wheresimple$\chi_{V}$$H_{K}$,.
is called the Schur element of theis the character of$V$ . Then, Lusztig ([L2]) has shown that we have:-module$V\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{K})$$s_{V}$$a(V)= \frac{1}{2}\min\{l\in \mathbb{Z}|v^{l}s_{V}\in \mathbb{Z}[v]\}$.

We can now give the theorem of existence of the canonical basic set. The main toolof the proof is the Lusztig asymptotic algebra.
THEOREM 3.1 (Geek [Gk], Geck-R.Rouquier [GR]). We define the following:
subset of$\mathrm{I}\mathrm{r}\mathrm{r}(H_{K})$$B$ $:=${$V\in$

Irr(H$K$) $|d_{V,M}\neq 0$ and $a(V)=a(M)$ for some$M\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathbb{C}})$}.

Then there exists a unique bijection$\mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathbb{C}})M$$\mapstoarrow$$V_{M}B$

such that the following two conditions holds:
we have $d_{V_{M},M}=1$ and $a(V_{M})=a(M)$ .
(1) For all
are such that $d_{V,M}\neq 0$ , then we have(2) If $V\in$ Irr(H ) and$\leq a(V)$ , with equality only for $V=V_{M}$ .
The set 8 is called the canonical basic set with respect to the specialization 0.
$V_{M}\in B_{2}$$M\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathbb{C}})$$K$$\mathrm{a}(\mathrm{M})$

Note that a description of the set would lead to a natural parametrization ofis semisimple, we know by Tits deformation-modules. If
the set of simple
is just the identity. Hence, we obtain thematrixdecomposition
theorem that thefollow ing result.
$B$$H\mathbb{C}$$H_{\mathbb{C}}$PROPOSITION 3.2. Assume that 0 is such that
Then, we have:.
$B$$H_{\mathbb{C}}$is a split semisimple algebra.

$=\mathrm{I}\mathrm{r}\mathrm{r}(H_{K})$Now, We want to give an explicit description of in the non semisimple case.
is a root of unity. Thus, we
is semisimple unless, Theorem 7.4.7],
By [can restrict ourselves to the case where 71 (u) is a root of unity.
The idea is to use results of Ariki which give an interpretation of the decomposition map using the theory of canonical basis of Fock spaces. This theory can$B$$\mathrm{G}\mathrm{P}$$H_{\mathbb{C}}$$\theta(u)$24
NICOLAS JACON

not be applied to all Heeke algebras. In fact, this is concerned with the class of
Ariki-Koike algebras which contains Hecke algebras of type An-i andas special
cases:$B_{n}$

4. Ariki-Koike algebrasa) First, we recall the definition of Ariki-Koike algebras (see [Ma] for a complete
survey of the representation theory of these algebras). Let $R$ be a commutative ring,,, $u_{d-1}$ be $d+1$ parameters in $R$ . The Ariki,
letand letKoike algebra $7\{_{R,n}:=7\{_{R,n}$ $(v;u_{0}, \ldots, u_{d-1})$ (or cyclotomic Hecke algebra of type$G(d, 1, n))$ over $R$ is the unital associative $R$ -algebra with presentation by:, $T_{n-1}$ ,generators: To,
braid relations symbolised by the following diagram:$n\in \mathrm{N}$$d\in \mathrm{N}_{>0}$$u_{1},\ldots$$v_{?}u_{0}$..

$T_{1},\ldots$$\ovalbox{\tt\small REJECT} T_{0}T_{1}T_{2}$$\underline{T}_{n-1}$

and the following ones:$(T_{0}-u_{0})(T_{0}-u_{1})\ldots(T_{0}-u_{d-1})=0$$(T_{i}-v)(T_{i}+1)=0(\mathrm{i}\geq 1)$,.

These relations are obtained by deforming the relations of the wreath product. We have the following special cases:?
if $d=1$ , $H_{R,n}$ is the Hecke algebra of type $A_{n-1}$ over $R$ ,if $d=2$ ,is the Hecke algebra of typeover $R$ .
It is known that the simple modules ofare indexed by the -tuples of?
partitions. The same is true for the semisimple Ariki-Koike algebras defined overa field. We say that A is a -partition of rank if:
where, forA, , $d-1$ ,is apartition of ranksuch that,$(\mathbb{Z}/d\mathbb{Z})$$\mathfrak{S}_{n}$$\bullet$$B_{n}$$\mathcal{H}_{R,n}$$\bullet$$(\mathbb{Z}/d\mathbb{Z})$

.

$d$

$=$

.

$d$

$\mathfrak{S}_{n}$$n$$(\lambda^{(0)}, .., \lambda^{(d-1)})$$\mathrm{i}=0$$\lambda^{\langle i)}=(\lambda_{1}^{(?)}, \ldots, \lambda_{r_{7}}^{(i)})$$\ldots$$\lambda_{1}^{(i)}\geq..,$$|\lambda^{(i)}|$$\sum_{k=0}^{d-1}|\lambda^{(k)}|=n$$\geq\lambda_{r_{i}}^{(i)}>0$.

We denote by
the set of -partitions of rank .
For each -partition A of rank , we can associate a-modulewhich isfree over . This is called a Specht modulel. These modules generalize the Specht
modules previously defined for Hecke algebras of type An-i and.
Assume that is a field. Then, for each -partition of rank , there is a naturalbilinear form which is defined over each . We denote by rad the radical associatedto this bilinear form. Then, the non zeroform a complete setof non-isomorphic simple-modules (see for example [Ab, chapter 13]). In
particular, if $H_{R,n}$ is semisim, we havefor all A
and theset of simple modules are given by the. We have the following criterion ofsemisimplicity:
THEOREM 4.1 (Ariki [A1]).
is split semisimple if and only if we have:$d$$\Pi_{n}^{d}$

$n$

$d$

$n$

$S \frac{\lambda}{R}$$\prime H_{R,n}$$R$$B_{n}$

$R$

$d$

$n$

$S \frac{\lambda}{R}$$D \frac{\lambda}{R}:=S\frac{\lambda}{R}/\mathrm{r}\mathrm{a}\mathrm{d}(S\frac{\lambda}{R})$$\prime H_{R,n}$$\mathrm{r}\mathrm{a}\mathrm{d}(S\frac{\lambda}{R})=0$$\mathrm{p}\mathrm{l}\mathrm{e}$$\in\Pi_{n}^{d}$$S \frac{\lambda}{R}$$’\kappa_{R,n}$

we use the definition of the classical Specht modules. The Passage from classical Specht
modules to their duals is provided by the map (A (o),, ,)(A $(d-1)’$ ,, ..., $d-1$ ,where, fordenotes the conjugate Partition$1\mathrm{H}\mathrm{e}\mathrm{r}\mathrm{e}_{9}$$\lambda^{(d-1)}$$\lambda^{(1)}$$\ldots$$\mathrm{i}=0$$\lambda^{\langle i)’}$$\mapsto$$\lambda^{(d-2)’},$$\lambda(0\}’)$$\ldots\}$

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GECK-ROUQUIER CLASSIFICATION OP SIMPLE MODULES FOR HECKE ALGEBRAS.

for alland for all$\mathrm{i}\neq j$$2\in \mathbb{Z}$such that $|l|<n$ , we have:,$v^{l}u_{i}\neq u_{j}$$\bullet\prod_{i=1}^{n}$$(1+v+\ldots+v^{x-1})$$\neq 0$.

is a field of characteristic 0, using results of Dipper-Mathas
Assume that
[DM], Ariki-Mathas [AM] and Mathas [Ms] , the case where $H_{R,n}$ is not semisimple
are powers of . In
and where all the
can be reduced to the case where
this paper, we will mostly concentrate upon the case where is a primitive root ofunity of order .
,be integers such that $0\leq v_{0}\leq$
and let ,b) Letover with the following$...\leq v_{d-1}<e$ . We consider the Ariki-Koike algebra
choice of parameters:
for $j=0$ , , $d-1$ ,$R$$R=\mathbb{C}$$v$$u_{i}$$v$$e$$\eta_{e}:=\exp(\frac{2\mathrm{i}\pi}{e})$$v_{1},\ldots$$v_{0}$$v_{d-1}$$\mathbb{C}$$H_{\mathbb{C},n}$$u_{j}=\eta_{e}^{v_{j}}$$v=\eta_{e}$$\ldots$.

This algebra is not semisimple in general. Hence the simple. We denote:by the non zero$tt_{\mathbb{C},n}$-modules are given$D \frac{\lambda}{\mathbb{C}}$

Y3

$:=$

{A

$\in\Pi_{n}^{d}|D\frac{\lambda}{\mathbb{C}}\neq 0$}.

Now we wish to describe the notion of decomposition map in the context of ArikiKoike algebras.
$A=\mathbb{C}[y, y^{-1}]$ . We assume that
Let be an indeterminate and let a $:=y^{d}$ . Letandin such that we have for all
we have invertible elements , ,$|l|\leq
n$
:
with
$y$

$d$

$\mathrm{u}_{0}$$\mathrm{u}_{d-1}$$\ldots$$A$$\mathrm{i}\neq j$$\mathit{1}\in \mathbb{Z}$$0^{l}\mathrm{u}_{i}\neq \mathrm{u}_{j}$We consider the Ariki-Koike algebra $H_{A,n}$ with the following choice of parameters:
for $j=0$ , , $d-1$ ,$u_{j}=\mathrm{u}_{j}$$v=\mathfrak{v}$$\ldots$.

Let $K$ be the field of fractions of $A$ and let $’\mu_{K,n}:=K\otimes_{A}H_{A,n}’$ . By theis semisimple and its simple modules areabove criterion of semisimplicity,such thatdefined over $K$ . Now, let 0 :given by the Specht modules$d-1$. Then,
,
,
$j=0$
for
. Assume that we haveand the decomposition map is defined as follows:
we have$\prime H_{K,n}$$Aarrow \mathbb{C}$$S \frac{\lambda}{K}$$\theta(\mathrm{u}j)=\eta_{e^{j}}^{v}$$\theta(\mathrm{u})=\eta_{e}:=\exp(\frac{2i\pi}{e})$$\ldots$$\mathcal{H}_{\mathbb{C},n}=\mathbb{C}\otimes_{A}?\mathrm{f}_{A,n}$$d_{\theta}$:$R_{0}(?l_{K,n})[S \frac{\lambda}{K}]$$\mapstoarrow$$[S \frac{\lambda}{\mathbb{C}}]=\sum_{\underline{\mu}\in\Gamma_{0}^{n}}d_{S^{\frac{\lambda}{K}},D\frac{\mu}{\mathrm{c}}}[D^{\frac{\mu}{\mathbb{C}}]}R_{0}(\mathcal{H}_{\mathbb{C},n})$We will now explain the connections between this decomposition map and thetheory of Fock spaces.

5. Quantum groups and Fock spaces
The aim of this part is to introduce Ariki’s theorem which provides a way tocompute the decomposition maps for Ariki-Koike algebras. For details, we refer to[Ab]. We keep the notations of the previous section

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NICOLAS JACON

5.1. Quantum group of type$\{h_{i}, 0 |0\leq \mathrm{i}<e\}$and let$A_{e-1}^{(1)}$. Let

be a free -module with basis
be the dual basis with respect to the$\{\Lambda_{\mathrm{i}}, \delta|0\leq i<e\}$$\mathbb{Z}$$\mathfrak{h}$pairing:$\langle$,$\rangle$:$\mathfrak{h}^{*}\mathrm{x}$$=1$ and,such that, we define the simple roots of$0\leq \mathrm{i}<e$

for $0\leq \mathrm{i},j<e$ . For$\langle\Lambda_{i}, \not\supset\rangle=\langle\delta, h_{j}\rangle=0$$\langle\delta, 0\rangle$$\langle\Lambda_{x}, h_{j}\rangle=\delta_{ig}$$\mathfrak{h}arrow \mathbb{Z}$$\mathfrak{h}^{*}$

by:if $i=0$ ,if $i>0$ ,$2\Lambda_{0}-\Lambda_{e-1}-\Lambda_{1}+\delta$$\alpha_{i}=\{$$2\Lambda_{i}-\Lambda_{i-1}-\Lambda_{i+1}$. The are called the fundamental weights.
wherebe the quantum group of typeLet be an indeterminate and let
which is generated by elpments
is a unital associative algebra over$\{0, \ldots, e-1\}\}$ and(see for example [Ab] for the relations).
, we define:For
and$\Lambda_{e}:=\Lambda_{0}$$\Lambda_{i}$$A_{e-1}^{(1)}$$\mathcal{U}_{q}$$q$$\mathbb{C}(q)$$\{e_{i}$,. This$f_{i}|\mathrm{i}\in$$\{k_{h}|h\in \mathfrak{h}\}$..

$l\in \mathrm{N}$$j\in \mathrm{N}$.

$[j]_{q}.-- \frac{q^{j}-q^{-j}}{q-q^{-1}}$,$[j]_{q}^{!}:=[1]_{q}[2]_{q}\ldots[j]_{q}$$\{\begin{array}{l}lJ-\end{array}\}$,$q= \frac{[l]_{q}^{!}}{[i]_{q}^{!}[l-j]_{q}^{!}}$.

which is
Let $A=\mathbb{Z}[q, q^{-1}]$ . We consider the Kostant-Lusztig $A$ form of$A$-subalgebra ofpowersby
thedividedgenerated
:
is
a
by
this
denoted$\mathcal{U}_{q}$$u_{A}$$e_{l}^{(l)}:= \frac{e_{i}^{l}}{[l]_{q}^{!}}$,$\mathcal{U}_{q}$$f_{j}^{(l\rangle}:= \frac{f_{j}^{l}}{[l]_{q}^{!}}$

for$<e$ and$0\underline{<_{\backslash }}\mathrm{i},j$$l\in \mathrm{N}$and by$k_{h_{i}}$,$k_{\delta}$,$k_{h_{i}}^{-1}$

,

, for

$k_{7}^{-1}$. Now, if is a ring and an invertible element in , we can form theby specializing the indeterminate to $u\in S$ .
specialized algebra$n\geq0$, let
For anybe thevector space with basis consisting of all the-paxtitions of rank . The Fock space is the direct sum:$S$

$0\leq \mathrm{i}<e$

$S$

$u$

$\mathcal{U}s,u:=S\otimes Ay_{A}$

$d$

$q$

$\mathbb{C}(q)$$\mathcal{F}_{n}$$n$$\mathcal{J}^{}:=\oplus_{n\in \mathrm{N}}F_{n}$.

We will now see that this space can be endowed with two different structures of-module. To describe these actions, we need some combinatorial definitions.
, ...,
Let
be a -partition of rank . The diagram of A is thefollowing set :$l\mathit{4}_{q}$$\underline{\lambda}=$$d$$\lambda^{(d-1)})$$(\lambda^{(0)}$$n$[A] $=\{(a, b, c)|0\leq c\leq d-1$ , $1\leq b$$\leq\lambda_{a}^{(\mathrm{c})}\}$.

The elements of this diagram are called the nodes of A. Let $\gamma=(a, b, c)$ be anode of . The residue of associated to the set $\{e;v_{0_{7}}\cdots, vd-1\}$ is the element ofdefined by:$\underline{\lambda}$$\gamma$$\mathbb{Z}/e\mathbb{Z}$$\mathrm{r}\mathrm{e}\mathrm{s}(\gamma)=(b-a+v_{c})$(mod ).
$e$

If is a node with residue , we say that is an -node. Let A and be two-partitions of rank and $n+1$ such that [A]. There exists a node such
, we say that is anthat. Then, we denote. If
addable -node for A and a removable node for .
$\mathrm{i}$$\mathrm{i}$$\gamma$$\gamma$$d$$n$$[\mu]=[\underline{\lambda}]\cup\{\gamma\}$$i$$\underline{\mu}$$\subset[\underline{\mu}]$$\mathrm{r}\mathrm{e}\mathrm{s}(7)=\mathrm{i}$$[\underline{\mu}]/[\underline{\lambda}]=\gamma$$\mathrm{i}$$\underline{\mu}$$\gamma$$\gamma$

27
GECK-ROUQUIER CLASSIFICATION OP SIMPLE MODULES FOR HECKE ALGEBRAS

5.2. Hayachi realization of Fock spaces. In this part, we consider thefollowing order on the set of removable and addable nodes of a -partition we saythat $\gamma=(a,$b, c) is below $\gamma’=(a’,$b,$c’)$ if c $<c’$ or if c $=c’$ and a $<a’$ .
This order will be called the AM-order and the notion of normal nodes andgood nodes below are linked with this order (in the next paragraph, we will give
another order on the set of nodes which is distinct from this one).
Let A and be two -partitions of rank n and $n+1$ such that there exists an. We define:-node such that
$d$

$d$

$\underline{\mu}$$[\underline{\mu}]=[\underline{\lambda}]\cup\{\gamma\}$$\mathrm{i}$$\gamma${addable i- nodes of A above }$-\#$ {removable i- nodes ofabove 7},
{addable i- nodes of A below }{removable i- nodes of 7 below },{addable i- nodes of }-{removable i- nodes of },{0 nodes of }.
$\gamma$$N_{i}^{a}(\underline{\lambda},\underline{\mu})=\beta$$\underline{\mu}$$N_{i}^{b}(\underline{\lambda},\underline{\mu})=\#$$\gamma$$-\#$$\gamma$$\underline{\lambda}$$N_{i}(\underline{\lambda})=\#$$\underline{\lambda}$-$N_{\Phi}(\underline{\lambda})=\beta$$\underline{\lambda}$

THEOREM 5.1 (Hayashi [Ha]).
following action:$F$

becomes$a\mathcal{U}_{q}$-module with respect to the,,$f_{i} \underline{\lambda}=\sum_{res([\underline{\mu}]/\square \lambda)=i}q^{N_{\dot{f}}^{\mathrm{b}}\{\underline{\lambda},\underline{\mu})}\underline{\mu}$$e_{i} \underline{\lambda}=\sum_{res(\llcorner\lambda]/[\underline{\mu}]\rangle=i}q^{-N_{i}^{a}(\underline{\mu},\underline{\lambda})}\underline{\mu}$$k_{h_{i}}\underline{\lambda}=q^{N_{i}(\underline{\lambda})}\underline{\lambda}$

where$\mathrm{i}=0$,$\ldots$,,$k_{I\prime}\underline{\lambda}=q^{-N_{\mathrm{D}}(\underline{\lambda})}\underline{\lambda}$, $e-1$ .

generated by the empty -partition It isbe the -submodule ofLet
isomorphic to an integrable highest weight module. In [K] and [L3], Kashiwara andLusztig have independantly shown the existence of a remarkable basis for this classof modules: the canonical basis. We will see the links between the canonical basis. First, it is known that the elements ofand the decomposition map forofgraph called the crystal graph.
acertainthis basis are labeled by the vertices of
Based on Misra and Miwa’s result, Ariki and Mathas observed that the verticesof this graph are given by the set of Kleshchev -partitions which we will now define.
Let be a -partition and let be an -node, we say that is a normal i-nodeof A if, whenever ny is an -node of below , there are more removable i-nodes
between arid than addable -nodes between and . If is the highest normal-node of , we say that is a good i-node.
We can now define the notion of Kleshchev -partitions associated to the set:
DEFINITION 5.2. The Kleshchev -partitions are defined recursively as follows.
$d$$\mathcal{F}$$\Lambda 4$$\mathcal{U}_{q}$$tt_{\mathbb{C},n}$$\mathcal{M}$

$d$

$d$

$\underline{\lambda}$$\mathrm{i}$$\gamma$$\gamma$$i$$\underline{\lambda}$$\gamma$$\mathrm{i}$$\eta$$\mathrm{i}$$\underline{\lambda}$$\gamma$$\eta$$\gamma$$\gamma$$\gamma$$d$$\{e;v_{0}, \ldots, v_{d-1}\}$.

$\bullet$$d$

is Kleshchev.
The empty partition
and a good -node
If A is Kleshchev, there exist
that if we remove from , the resulting -partition is Kleshchev.
$\underline{\emptyset}:=$$(\emptyset, \emptyset, \ldots, \emptyset)$$\mathrm{i}\in\{0, \ldots, e-1\}$$\mathrm{i}$$\gamma$

such

$d$

$\underline{\lambda}$$\gamma$the set of Kleshchev -partitions associated to theset $\{e;v_{0}, \ldots, v_{d-1}\}$ . If there is no ambiguity concerning $\{e;v_{0}, \ldots, v_{d-1}\}$ , we denote
is given by:. Now, the crystal graph of
it by

We denote by$\Lambda^{0_{\rangle}n}$$d$$\Lambda_{\{e_{j}v_{0},\ldots,v_{t-1}\}}^{0,n}$$\lambda \mathit{4}$

28

..

NICOLAS JACON

vertices: the Kleshchev d-partitions,[AJ is a good i-node.
edges:
if and only if
is labeled by the Kleshchev d-partitions:Thus, the canonical basis of$\underline{\lambda}arrow\underline{\mu}i$$[\underline{\mu}]/$$\lambda 4$$\mathfrak{B}${$\mathfrak{B}=$$G(\underline{\lambda})|$A$\in\Lambda_{\{e;v_{0},\ldots,v_{d-1}\}}^{0}$,$n\in \mathrm{N}$}.

generated by the empty d-partition
moduleThis set is a basis of theand for any specialization of into an invertible element of a field , we obtainby specializing the set S.
a basis of the specialized module, there exist
basis, for each A
canonicalofthe
By the characterizationbasis suchcanonical
theof
and a unique element
polynomials
that:(mod ).
and$\mathcal{M}_{A}$$\mathcal{U}_{A}$

$R$

$u$

$q$

$\mathcal{A}4_{R,u}$$\in\Lambda^{0,n}$$G(\underline{\lambda})$$d_{\underline{\mu},\underline{\lambda}}(q)\in \mathbb{Z}[q]$$\mathrm{G}(\mathrm{X})=$$G(\underline{\lambda})=\underline{\lambda}$$d_{\underline{\mu},\underline{\lambda}}(q)\underline{\mu}$$q$$\sum_{\underline{\mu}\in\Omega_{\eta}^{d}},$Now we have the following theorem of Ariki which shows that the problem of
can be translated to that of comcomputing the decomposition numbers ofputing the canonical basis of U. This theorem was first conjectured by Lascoux,
Leclerc and Thibon ([LLT]) in the case of Hecke algebras of type $A_{n-1}$ .
$\mathcal{H}_{R,n}$

THEOREM 5.3 (Ariki [A2]). There exist a bijection

for all$\underline{\mu}\in\Pi_{n}^{d}$and$\underline{\lambda}\in\Lambda^{0,n}$$j_{0}$:$\Lambda^{0,n}arrow\Gamma_{0}^{n}$such that, we have:$d_{\underline{\mu},\underline{\lambda}}(1)=d_{S\frac{\mu}{K},D_{\mathrm{c}}^{j_{0}(\underline{\lambda})}}$,

where we recall that:$\Gamma_{0}^{n}:=${A$\in\Pi_{n}^{d}|D\frac{\lambda}{\mathbb{C}}\neq 0$}.

Moreover we have:
THEOREM 5.4 (Ariki [A3], Ariki-Mathas [AM] ). We have$j_{0}=Id$$\Gamma_{0}^{n}=\Lambda^{0,n}$.

andHence the above theorem gives a first classification of the simple modules by
the set of Kleshchev -partitions. As noted in the introduction, the problem of thisparametrization of the simple-modules is that we only know a recursive description of the Kleshchev -partitions. We now deal with another parametrizationof this set found by Foda et al. which uses almost the same objects as Ariki andMathas.
$d$$\mathcal{H}_{\mathbb{C},n}$$d$5.3. JMM O realization of Fock space. This action has been defined in[JM MO] and has been used and studied in [FL OTW]. We need to define anotherorder on the set of nodes of a d-partitions.
Here, we say that $\gamma=(a,$b, c) is aboveb,$c’)$ if:$\gamma’=(\mathrm{a},$$b-a+v_{c}<b’-a’+v_{c’}$ or if $b-a+v_{c}=b’-a’+v_{\mathrm{c}’}$ and $c>c’$ .

This order will be called the FLOTW order and it allows us to define functions
andetfor the AM
gives by the same way asorder.
Now, we have the following result:$\overline{N}_{i}^{a}(\underline{\lambda},\underline{\mu})$$\overline{N}_{i}^{b}(\underline{\lambda},\underline{\mu})$$\mathrm{n}$$N_{i}^{a}(\underline{\lambda},\underline{\mu})$$N_{i}^{b}(\underline{\lambda},\underline{\mu})$

28
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRASTHEOREM 5.5 (Jimbo, Misra, Miwa, Okado [JMMOj). $F$ isrespect to the action:$a\mathcal{U}_{q}$-module with,,

$f_{i} \underline{\lambda}=\sum_{\mathrm{r}\mathrm{e}\mathrm{s}([\underline{\mu}]/\square \lambda)=i}q^{\overline{N}_{i}^{b}(\underline{\lambda},\underline{\mu})}\underline{\mu}$$e_{i} \underline{\lambda}=\sum_{\mathrm{r}\mathrm{e}\mathrm{s}(\mathrm{L}\lambda]/[\underline{\mu}])=i}q^{-\overline{N}_{\dot{\mathrm{t}}}^{\alpha}(\underline{\mu}_{)}\underline{\lambda})}\underline{\mu}$$k_{h_{i}}\lambda=v^{N_{\dot{x}}(\underline{\lambda})}\underline{\lambda}$

where$0\leq \mathrm{i}\leq n-1$,,$k_{\mathfrak{D}}\underline{\lambda}=q^{-N_{\Phi}(\underline{\lambda}\rangle}\underline{\lambda}$. This action will be called the JMMO action.

the -module generated by the empty #partition with reWe denote byspect to the above action. This is a highest weight module which is isom orphic to. However, the elements of the canonical basis are differents in general. Here,the -partitions of the crystal graph are obtained recursively by adding good nodes
to -partitions of the crystal graph with respect to the FLOTW order.
Foda et al. showed that the analogue of the notion of Kleshchev d-partitions
for this action is as follow :$\overline{\mathcal{M}}$$\mathcal{U}_{q}$$\mathcal{M}$

$d$

$d$

$\mathrm{s}$DEFINITION 5.6 (Foda, Leclerc, Okado, Thibon, Welsh [FLOTW]). We say
is a FLOTW -partition associated to the set $\{e;v0, \ldots, vd-1\}$, ...,
that A
if and only if:(1) for all $0\leq j\leq d-2$ and $i=1$ , 2, , we have:
$=$

$d$

$\lambda^{(d-\mathrm{I})})$$(\lambda^{(0)}$$\ldots$$\lambda_{\iota}^{(j)}\geq\lambda_{i+v_{j+1}-v_{j}}^{(j+1)}$,$\lambda_{\mathrm{i}}^{(d-1)}\geq\lambda_{i+e+v\mathrm{o}-v_{d-1}}^{(0)}$;(2) for all $k>0$ , among the residues appearing at the right ends of the lengthrows of , at least one element of 0, 1, ..., $e-1\}$ does not occur.
$k$

$\{$

$\underline{\lambda}$

We denote by
to the set
denote it by.

the set of FLOTW -partitions of rank associated. If there is no ambiguity concerning $\{e;v0, \ldots, vd-1\}$ , we$d$$\Lambda_{\{e,v_{0},\ldots,v_{d-1}\}}^{1,n}$$\{e;v_{0}, \ldots, v_{d-1}\}$$\mathrm{A}^{1,n}$$n$.

is given by:Hence, the crystal graph ofvertices: the FLOTW #-partitions,is good -node with respect to theif and only ifedges: A
FLOTW order.
are labeled by the FLOTW J-partitions:So, the canonical basis elements of$\overline{\mathrm{A}4}$$\bullet$$\mathrm{i}$$\underline{i}\underline{\mu}$$[\underline{\mu}]/[\underline{\lambda}]$$\bullet$$\overline{\mathcal{M}}$$\overline{\mathfrak{B}}=${$\overline{G}(\underline{\lambda})|$A$\in\Lambda_{\{e_{}v\mathrm{o},\ldots,v_{d-1}\}}^{1,n}$,$n\in \mathrm{N}$}.

If we specialize these elements to $q=1$ , we obtain the same elements as in Theorem
5.3 (note that the action of the quantum group on the Fock space specialized at$q=1$ leads to the same module stucture for the Hayashi action and for the JMMO
action).
, there exist
By the characterization of the canonical basis, for each Aof the canonical basis suchand a unique element
polynomials$\in\Lambda^{1,n}$$\overline{G}’(\underline{\lambda})$$\overline{d}_{\underline{\mu},\underline{\lambda}}(q)\in \mathbb{Z}[q]$

that:
and
$\overline{G}(\underline{\lambda})=\sum_{\underline{\mu}\in\Pi_{\eta}^{d}}\overline{d}_{\underline{\mu},\underline{\lambda}}(q)\underline{\mu}$By Ariki’s theorem, we have:$\overline{G}(\underline{\lambda})=$A (mod ).
$q$

30
NICOLAS JACON

THEOREM 5.7 (Ariki [A2]). There exist a bijection, toe haveand A 6that for all$\underline{\mu}\in$n9$j_{1}$:$\Lambda^{1,n}arrow$$\mathrm{I}_{0}^{n}=\Lambda^{0,n}$such$\Lambda^{0,n}$$\overline{d}_{\underline{\mu},\underline{\lambda}}(1)=d_{S\frac{\mu}{K},D_{\mathrm{C}}^{j_{1}(\underline{\lambda})}}$.

Hence, we can alternatively use the JMMO action instead of the Hayashi action
to compute the decomposition matrix for Ariki-Koike algebras. A natural question
is now to ask if there is an interpretation of the FLOTW multipartitions in the
representation theory of Ariki-Koike algebras. An answer will be given by extending
the results of Geek and Rouquier to the case of Ariki-Koike algebras.

6. Canonical basic sets for Ariki-Koike algebras, vd-i be integers
and let ,Let be a positive integer,. We consider the Ariki-Koike algebra, oversuch that $0\leq v_{0}\leq..,$parameters:
following
choiceofthewith
far $j=0$ , , $d-1$ ,$\eta_{e}:=\exp(\frac{2i\pi}{e})$$e$$\mathrm{v}\mathrm{o}$$v_{1},\ldots$$H_{\mathbb{C}}$$\leq \mathrm{v}\mathrm{d}-\mathrm{i}<e$$n$$\mathbb{C}$$u_{j}=\eta_{e}^{v_{i}}$$\ldots$,$v=\eta_{e}$In this part, we show that there exists a “canonical basic set” of Specht modules-modules. To do this, we consider
which is in bijection with the set of simple
and we study the Kashhiwara-Lusztig canonical basis of
the Ariki-Koike algebra-module. First, we have to define a semisimple
the associated highest weight
as in the end of \S 4 ). Let
Ariki-Koike algebra which can be specialized to$A=\mathbb{C}[y,y^{-1}]$$0=y^{d}$.
. Letbe an indeterminate and let
We consider the Ariki-Koike algebra $H_{A,n}$ with the following choice of param$?t_{\mathbb{C},n}$$\mathcal{H}_{\mathbb{C},n}$$\mathcal{U}(\hat{sl}_{e})$$\mathrm{b}$$7\mathit{4}_{\mathbb{C},n}$$y$eters:$u_{j}=\eta_{d}^{j}y^{dv_{j}-je+de}$$v=\mathfrak{v}$for $j=0$ ,$\ldots$, $d-1$ ,,. It is easy
. Let $K:=\mathbb{C}(y)$ and let
where
to see that this algebra is semisimple and that, under the specialization $y\in A\mapsto$. Hence, we have a decomposition map, we obtain the algebra
as in \S 4 )::$\eta d:=\exp(\frac{2\mathrm{i}\pi}{d})$$\exp(\frac{2i\pi}{de})$$\mathcal{H}_{K,n}:=K\otimes_{A}\mathcal{H}_{A,n}$$\mathrm{C}-\mathbb{C}$$H_{\mathbb{C},n}$$\mathrm{b}$$d_{\theta}$$R_{0}( \mathcal{H}_{K,n})[S\frac{\lambda}{K}]$$\mapstoarrow$$[S \frac{\lambda}{\mathbb{C}}]=\sum_{\underline{\mu}\in\Gamma_{\mathrm{O}}^{n}}d_{S^{\frac{\lambda}{K}},D\frac{\mu}{\mathbb{C}}}[D\frac{\mu}{\mathbb{C}}]R_{0}(H_{\mathbb{C},n})$Now, we have to define an a-value on the simple modules (which are the Specht
modules defined over $K$ ). The main problem here is that we don’ have KazhdanLusztig type bases for Ariki-Koike algebras in general but we do have Schur elements: they have been computed by Geek, Iancu and Malle in [GIM], This leads
to the foliow ing definition of a-values (see [J2, \S 3.3]):$\mathrm{t}$where for $j=0$ , , $d-1$ we. We assume that the rank of A is . For $j=0$ , , $d-1$
we define the following rational numbers:

DEFINJTION 6.1. Let

have
and $p=1$ , ...,$\lambda^{(j)}:=$$\underline{\lambda}:=$$(\lambda^{(0)}, \lambda^{(1\rangle}, \ldots, \lambda^{(d)})\in\Lambda^{1,n}$$\ldots$$(\lambda_{1}^{\langle j)}, \ldots, \lambda_{h^{\langle j)}}^{\langle j)})$$n_{1}$$n$$m^{(j)}:=v_{j}- \frac{\mathrm{i}e}{d}+e$,$B_{p}^{\prime(j)}:=\lambda_{p}^{(j)}-p+n+m^{(j\rangle}$,$\ldots$

31
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS

where we use the convention that
. Then, we define:$\lambda_{p}^{(j)}:=0$$B^{\prime(j)}=$if $p>h^{(j)}$ . For $j=0$ ,$\ldots$, $d-1$ , let$(B_{1}^{\prime(j)}, \ldots, B_{n}^{\prime(j)})$.
$a_{1}( \underline{\lambda}):=(a_{j}b)\in B’(\mathrm{z})_{\cross B}r(j\rangle a>bi\mathrm{f}i=j\sum_{0\leq i\leq \mathrm{j}<d}\min\{a, b\}-\mathrm{o}_{1\underline{<}k\leq a}a\in B\leq\tau,j,<d\sum_{\langle i)}\min\{k, m^{(j)}\}$

whereis the rational number
The a-value associated to$\{e;v0,\ldots,v_{d-1}\}$$f(n)$ is a rational number which only depends on the parameters
and on (the expression of is given in [J2]).
a sequence of residues which will have
Next, we associate to each A“nice” properties with respect to the a-value:$a(\underline{\lambda}):=a_{1}(\underline{\lambda})+f(n)$$S \frac{\lambda}{K}$

$f$

$n$

$\in\Lambda^{1,n}$PROPOSITION 6,2 ([J2]). Let A$\in\Lambda^{1,n}$

and let:$l_{\max}:= \max\{\lambda_{1}^{(0)}, \ldots, \lambda_{1}^{(l-1)}\}$.

with lengthwith residue on a part
Then, there exists a removable nodewith length
part
a
end
$k-1$
right
the
at
-node
exist
a
, such that there doesn’tof(the existence of such a node is proved in [J2, Lemma 4.2]/
be the $k-1$ -nodes at the right ends of parts
Let ,be the removable -nodes of A on parts. Let ,
such that:$k$$\xi_{1}$$\lambda_{i_{1}}^{(i_{1})}$$l_{\max}$$l_{\max}$$\lambda_{p_{1}}^{(t_{1})}\geq\lambda_{p2}^{(l_{2})}\geq\ldots\geq$$\gamma_{1}$$\lambda_{p_{\tau}}^{(l_{\tau})}$$\xi_{1}$$\gamma_{2,\ldots;}\gamma_{r}$$\lambda_{j_{1}}^{\langle \mathrm{i}_{1}\}}\geq\lambda_{j_{2}}^{(i_{2})}\geq\ldots\geq\lambda_{j_{s}}^{\langle i_{s})}$$k$$\xi_{2,,..;}\xi_{s}$$\lambda_{j_{s}}^{(i_{\mathrm{s}})}>\lambda_{p_{1}}^{(l_{1})}$.

,
We remove the nodes ,from . Let be the resulting d-partition.
and we define recursively the -sequence of residues of A by:
Then,$\underline{\lambda}’$$\xi_{1}$$\xi_{2},\ldots$$\underline{\lambda}$$\xi_{\mathrm{S}}$$\lambda’\in\Lambda^{1,n-s}$

$a$

$a$

-sequence(A)$=a- sequence(\underline{\lambda}’),\underline{k,\ldots,k}$.

$s$

Example:Let $e=4$ , $d=3$ , $v_{0}=0$ , $v_{1}=2$ and $v_{2}=3$ . We consider the 3-partition
A $=(1,3.1,2.1.1)$ with the following diagram:$(\overline{\prod 0}$,

A is a FLOTW 3-partition.
We want to determine the sequence of : we have to find $k\in\{0, 1, 2, 3\}$ ,and a 2-partition such that:, .
a-sequence(A)$a$$\underline{\lambda}$$\underline{\lambda}’$$s\in \mathrm{N}$$k$$=a- \mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}(\underline{\lambda}’),k,m$$S\ldots$

The part with maximal length is the part with length 3 and the residue of theassociated removable node is 0. We remark that there are two others rem ovable$0-1\equiv$
0-nodes on parts with length 1 and 2, Since there is no node with residue(mod ) at the right ends of the parts of , we must remove these three O-nodes.
, hence:Thus, we have to take $k=0$ , $s=3$ and-sequence(A) $=a$-sequence(\emptyset , 2.1, 1.1.1), 0, 0, 0.
$3$$\underline{\lambda}$$e$$\underline{\lambda}’=$$(\emptyset, 2.1, 1.1.1)$$a$Observe that the 3-partition$(\emptyset, 2.1, 1.1.1)$

is a FLOTW 3-partition.

32
NICOLAS JACON

Now, the residue of the removable node on the part with maximal length is 3.
Thus, we obtain:
a-sequence(A) $=a$-sequence(\emptyset , 1.1, 1.1.1), 3, 0, 0, 0.

Repeating the same procedure, we finally obtain:$a$-sequence(A) $=3,2,2$ , 1, 1, 3, 0, 0, 0.

, let
and let a-sequence(_\lambda )PROPOSITION 6.3 ([J2]). Let, , , , , , , , , be its -sequence of residues where we assume that for$n\in \mathrm{N}$$\infty-$$\mathrm{i}_{1}$$\mathrm{i}_{1}$$\mathrm{i}_{2}$all$a_{1}j=1$,$\ldots$–$\ldots$’– 1, ate have

$=$

$a$

$\mathrm{i}_{S}$$\mathrm{i}_{s}$$\mathrm{i}_{2}$$\ldots$$\ldots$$\underline{\lambda}\in\Lambda^{1,n}$$\ldots a_{2,s},$$\mathrm{i}_{j}\neq \mathrm{i}_{j+1}$. Then, we have:$A( \underline{\lambda}):=f_{i_{s}}^{(a_{\mathrm{s}})}f_{i_{s-1}}^{(a_{\epsilon-1})}\ldots f_{i_{1}}^{(a_{1})}\underline{\emptyset}=\underline{\lambda}+\sum_{a\langle\underline{\mu})>a(\underline{\lambda})}c_{\underline{\lambda},\underline{\mu}}(q)\underline{\mu}$where$c_{\underline{\lambda},\underline{\mu}}(q)\in \mathbb{Z}[q, q^{-1}]$,.

. Using,A
It is obvious that the set {} is a basis of
the characterization of the canonical basis, we obtain the following theorem:$A(\underline{\lambda})|$PROPOSITION 6.4 ([J2]). Let$n\in \mathrm{N}$$\in\Lambda^{1,n}$$n\in \mathrm{N}$and let A$\mathcal{M}_{A}$$\in\Lambda^{1,n}$, then we have:$\overline{G}(\underline{\lambda}):=\underline{\lambda}+\sum_{a_{4}’\underline{\mu})>a(\underline{\lambda})}\overline{d}_{\underline{\mu},\underline{\lambda}}(q)\underline{\mu}$Now, assume thatis a Kleshchev multipartition and let(mod ). Letelement of the canonical basis ofsuch thatbe the element of the canonical basis ofsuch thatwithat $v=1$ . This defines a bijection:$\underline{\mu}\in\Lambda^{0,n}$$\Lambda \mathrm{t}$$v$$G(\underline{\mu})=\underline{\mu}$$\overline{\mathcal{M}}$$c(\underline{\mu})\in\Lambda^{1,n}$be thewithcoincides$G(\underline{\mu})$$\overline{G}(c(\underline{\mu}))$$\overline{G}(c(\underline{\mu}))$$G(\underline{\mu})$$c$:$\Lambda^{0,n}$$\Lambda^{1,n}$$arrow$

This bijection can be described by reading the crystal graphs of(seeand[FLOTW] ).
We can now define an -value on the set of simple-modules by setting for:$\overline{\mathcal{M}}$$\mathcal{M}$$a$$\mathcal{H}_{\mathbb{C},n}$$\underline{\mu}\in\Lambda^{0,n}=\Gamma_{0}^{n}$.

$a(D \frac{\mu}{\mathbb{C}}):=a(S_{K}^{\mathrm{c}(\underline{\mu})})$

Combining the above proposition with Theorem 5.7, we obtain the following
result which shows the existence of the canonical basic set for Ariki-Koike algebras
and gives an explicit description of this set.
THEOREM 6.5 ([J2]). We define the following subset of
$B$

Then:
(1) For all(2) If$:=${$S \frac{\lambda}{K}\in \mathrm{I}\mathrm{r}\mathrm{r}(7\mathrm{i}\mathrm{c},\mathrm{n})$$|$A$\mathrm{A}^{1,n}$$\in$$\mathrm{I}\mathrm{r}\mathrm{r}(H_{K,n})$:}.

, we have.
andare such that $d_{V,M}\neq 0$ , then we have$a(M)$ $\leq a(V)$ , with equality only
with
and
if
(3) We have
whereis as in Theorem 5. 7,$\underline{\mu}\in\Lambda^{0,n}$$d_{S_{K}^{\mathrm{c}(\underline{\mu})},D_{\mathrm{C}}^{\mathrm{A}}}=1$$V\in \mathrm{I}\mathrm{r}\mathrm{r}(H_{K,n})$$M\in \mathrm{I}\mathrm{r}\mathrm{r}(\mathcal{H}_{\mathbb{C},n})$$M=D \frac{\mu}{\mathbb{C}}$$c=j_{1}^{-1}$$i_{1}$$\underline{\mu}\in\Lambda^{0,n}$$V=S_{K}^{c(\underline{\mu})}$33
GECK-ROUQUIER CLASSIFICATION OP SIMPLE MODULES FOR HECKE ALGEBRASThe set is called the canonical basic set with respect to the specialization.
is in natural bijection with
$B$$\theta$and it$\mathrm{I}\mathrm{r}\mathrm{r}(H_{\mathbb{C},n})$This theorem shows that the decomposition matrix for Ariki-Koike algebras
has a lower triangular shape if we order the rows and columns with respect to thea-value.
Note also that the results of this section induce a purely combinatorial algorithm for the computation of the canonical basis and of the decomposition matrices
of Ariki-Koike algebras (see [J4]) which generalizes the LLT algorithm.

7. Consequences
We obtain an explicit description of the canonical basic set for all Hecke algebras
of classical type in characteristic 0.

7.1. Type $A_{n-1}$ . Assume that W is a Weyl group of type $A_{n-1}$ and that-root of unity. H is a special case of Ariki-Koike algebras. Hence,is a primitive
we can use Theorem 6.5 to find the canonical basic sets. We note that we can alsofind this set using results of Dipper and James as it is expained in [Gk, Example$\theta(u)$$e^{\mathrm{t}\mathrm{h}}$3.5].

PROPOSITION 7.1. Assume that W is a Weyl group of type $A_{n-1}$ and that-root of unity. Then, we have:is a primitive$\theta(u)$$e^{th}$

B

$=$

{

$S_{K}^{\lambda}$|A$\in\Lambda_{\{e,0\}}^{1,n}$}.

Note that:and$|\lambda|=n$ andfor all i$\lambda=$$|\lambda|=n$$\Leftrightarrow$$\lambda\in\Lambda_{\{e_{j}0\}}^{1,n}$$\Leftrightarrow$.

is e-regular.
, we can’t have$(\lambda_{1},$\ldots , $\lambda_{r})$$\in \mathrm{N}$$\lambda_{\dot{f}}=\ldots=\lambda_{\mathrm{z}+e-1}\neq 0$.

isand that
Assume that W is a Weyl group of type7.2. Type
Hence,
algebras.
Ariki-Koikeof-root of unity. H is a special casea primitive
we can use Theorem 6.5 to find the canonical basic sets.
$B_{n}$$\theta(u)$$B_{n}$$e^{\mathrm{t}\mathrm{h}}$PROPOSITION 7.2. Assume that $W$ is a Weyl group of tyPe
-root of unity. Then, we have:is a primitive$B_{n}$and that$\theta(u)$$e^{th}$$\bullet$

if

$e$

$B$

$\bullet$is odd:$=\{S_{K}^{(\lambda^{(\mathrm{o})},\lambda^{(1)})}|\lambda^{(0)}\in\Lambda_{\{\acute{e}j0\}}^{1n_{0}}, \lambda^{(1)}\in\Lambda_{\{e0\}}^{1,n_{1}}, n_{0}+n_{1}=n\}$.

if is even:$e$$B=\{S^{(\lambda^{(0)},\lambda^{(1)})}|(\lambda^{(0)}, \lambda^{(1)})\in\Lambda_{\{ej}^{1,n_{1,\frac{\epsilon}{2}\}}}\}$

Recall that(1) for all$(\lambda^{(0)}, \lambda^{(1)})\in\Lambda_{\{e_{j}1,\frac{\mathrm{e}}{2}\}}^{1,n}$$\mathrm{i}=1,2$,$\ldots$

if and only if.

$|\lambda^{(0)}|+|\lambda^{(1)}|=n$and:: toe have:$\lambda_{i}^{(0)}\geq\lambda_{i+\frac{e}{2}-1}^{(1)}$$\lambda_{i}^{(1)}\geq\lambda_{i+_{\tilde{2}}^{e}+1}^{(0)}$,;(2) for all $k>0$ , among the residues appearing at the right ends of theat least one element of $\{0, 1, \ldots, e-1\}$
length rows ofdoes not occur$k$$(\lambda^{(0)}, \lambda^{(1)})_{\lambda}$

34
NICOLAS JACON

is. Assume that W is a Weyl group of type and that
7.3. Type-root of unity. Then, H can be seen as a subalgebra of an Heckea primitiveof typeas in \S 2.3. The specialization 0 induces a decomposition
algebra:map for.
:$\theta(u)$$D_{n}$$D_{n}$$e^{\mathrm{t}\mathrm{h}}$$H_{1}$$B_{n}$$H_{1}$$d_{\theta}^{1}$$R_{0}(H_{1,K})arrow R_{0}(H_{1,\mathbb{C}})$

with unequal parameters are special cases of Ariki-Koike
Hecke algebras of type
algebras. Hence, we can also define a canonical basic set for these algebras (theexistence has been previously proved in [Gex]). Furthermore, in [Gex], Geek has
are those-modules in the canonical basic set for typeshown that the simple
basic
of
the
canonical
simple-modulestheofappearin the restriction,
which. We obtain the following description of B.
set for type
$B_{n}$$D_{n}$$H_{K}$$H_{1}$$K$$D_{n}$PROPOSITION 7.3. Assume that $W$ is a Weyl group of type-root of unity. Then:
is a primitive$D_{n}$and that$\theta(u)$$e^{th}$$\bullet$if$e$

is odd, we have:$B$ $=\{V^{[\lambda^{(0)},\lambda^{(1)}]}|\lambda^{(0)}\neq\lambda^{(1)}$.

,$\lambda^{(0)}\in\Lambda_{\{e,0\}}^{1,n\mathrm{o}}.$,$\lambda^{(1\}}\in\Lambda_{\{e,0\}}^{1,n_{1}}.$$\cup\{V^{[\lambda^{(0\rangle},\pm]}|\lambda^{(0)}\in\Lambda_{\{e,0\}}^{1,\frac{n}{2}}.\}$

if

$e$

, $n_{0}+n_{1}=n\}$.

is even, ate have:$B=\{V^{[\lambda^{\langle 0)},\lambda^{\langle 1\rangle}]}|\lambda^{(0)}\neq\lambda^{(1)}$,$(\lambda^{(0\rangle}, \lambda^{(1)})\in\Lambda_{\{ej}^{1,n_{0,\frac{e}{2}\}}}\}$$\cup\{V^{[\lambda^{(0)},\pm]}|(\lambda^{(0)}, \lambda^{(0)})\in\Lambda_{\{e,0,\frac{e}{2}\}}^{1,n}.\}$

Recall that(1) for all$(\lambda^{(0)}, \lambda^{(1)})\in\Lambda_{\{e;0,\frac{e}{2}\}}^{1}$$\mathrm{i}=1,2$,$\ldots$, we have:

if and only if$\lambda_{i}^{(0)}\geq\lambda_{l+\frac{\mathrm{e}}{2}}^{(1)}$.

$|\lambda^{(0)}|+|\lambda^{/1\rangle}\backslash |=n$

and:

,

$\lambda_{i}^{(1)}\geq\lambda_{\iota+\frac{\mathrm{e}}{2};}^{(0)}$(2) for all $k>0$ , among the residues appearing at the right ends of the, at least one element of 0, 1, ..., $e-$
length rows ofdoes not occur.
$(\lambda^{(0)}, \lambda^{(1)})$

$k$

$1\}$

$\{$

-modulesNote that Hu [H] has given another parametrization of the simple
by using the Kleshchev -partitions. The connection with the above result is
explained in [J3, \S 4.3]
$H_{\mathbb{C}}$$\mathrm{b}\mathrm{i}$7.4. Exceptional types. An explicit description of the canonical basic set
for all exceptional types and for all specializations can be found in [J3, Chapter 3].

7.5. Positive characteristic. The existence of the canonical basic set for
Hecke algebras of finite Weyl group W can also be proved when the Hecke algebra
is defined over a field of characteristic p where p is a “good” prime number for $W$(see [GR]). In fact, it is easy to see that the parametrization of the canonical basic
set in characteristic 0 holds in “good” characteristic (see [J5])35
GECK-ROUQUIER CLASSIFICATION OF SIMPLE MODULES FOR HECKE ALGEBRAS

7.6. Cyclotomic Hecke algebras of type $G(d,p,$n). Both Ariki-Koike alare particular cases of cyclotomic Hecke
gebras and Hecke algebras of type$p|d$kind of algebras have been defined in).
This$G(d,p,$(withn)
algebras of type[Ac] and can be seen as subalgebras of Ariki-Koike algebras. By using Clifford
theory and results on graded algebras proved by Genet in [Gg], we can obtained aparametrization of the simple modules for all Hecke algebras of type $G(d,p,$ n) in themodular case (see [GJ]). Note also that results of Hu provide another parametrization of the simple modules for these algebras in the case where p $=d$ (see [H] and[H2], see also [J3] for the connections between these two classifications).
$D_{n}$

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$\mathcal{U}_{q}(\hat{sl}(n))$$q$$\vee \mathrm{Q}$$q$

LABORATOIRE DE MATH\’EMATIQUES NICOLAS ORESME, UNIVERSITE’ DE CAEN, BP 5186,
14032 CAEN CEDEX, FRANCE.
-mail address: iaC0nmath. unicaen.fr$E$$\mathrm{F}$