数理解析研究所講究録
1190 巻 2001 年 35-49

35のプロヅクにおける圏同値の組合せ論的記述、 との独立性と分解定数について$GL_{n}(F_{q})$

$q$

宮地

兵衛 (Hyohe Miyachi)$(\neq \mathrm{F}\hslash_{\sim}\cdot(\neg]^{\lambda}\backslash \backslash \backslash \cdot\backslash \cdot\backslash ’\Theta^{\mathrm{t}}\overline{k})$Notation$1$

The result of this paper is a joint work with Akihiko Hida 1 Let$G$
be a finite group, andbe a splitting -modular system for $G$ .
, let $B_{0}(RG)$ be theHere char $(K)=0$ , char(k) $=\ell>0$ . For
principal block of $RG$ .
denotes the symmetric group on letters.
denotes a field withelements with \dag er . Let natural numbers $e(q)$ and $r(q)$ be as follows:$\ell$$(K, \mathcal{O}, \mathrm{k})$$R\in\{\mathcal{O}, \mathrm{k}\}$$n$$\mathfrak{S}_{n}$$\ell$$e(q):={\rm Min}$$q$$\mathrm{F}_{q}$$q${$i\in \mathrm{N}|q^{i}\equiv 1$(mod$r(q):={\rm Max}\{r\in \mathrm{N}|\ell^{\mathrm{r}}|q^{e(q)}-1\}$$\ell)$},: the -part of $q^{e(q)}-1$ .
$\ell$

Letandbe blocks ideals. “$A\sim_{M}B$ ” means thatis Morita“”$A\sim_{d}B$(Puig) equivalent to B.
means that is derived (splendid
Rickard) equivalent to (see $[34],[35]$ ).
We use results on representation theory of finite general linear groups
in non-defining characteristic due to Fong-Srinivasan and Dipper-James(see [14], [15], $[9],[10],[11],[12],$ $[13],[19]$ ).
$A$

$A$

$B$

$A$

$B$

2

Motivations
We wish to prove the following conjectures:

Conjecture 2.1 (Brou\’e). [2], [3], [4] Let be an -block ideal of withabelian defect group D. Then and its Brauer correspondent in $N_{c(}D$ )are derived equivalent?
$B$$\ell$$G$$B$

detailed version of this paper will be submitted for “Doctor thesis at Chiba
Univ., Japan”.
1Hida’s address is “Department of Mathematics, Faculty of Education,
University, Urawa city 338, Japan E–mail: ahida@post.saitama-u.ac.jp”$\mathrm{T}\mathrm{h}\mathrm{e}$$\mathrm{S}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{t}\wedge \mathrm{a}\mathrm{m}\mathrm{a}$36Conjecture 2.2 (James). [20] Suppose that char$(\mathrm{k})=\ell>n$ and $e(q)=$. Let be a primiiive e-th root of unity in C. Then, the decomposition
over is equal to that of
matrix of Dipper-James Schur algebra
over ?
Dipper-James Schur algebra$e$$\zeta$$\mathbb{C}$$S_{\zeta}(n,r)\mathrm{c}$$\mathrm{k}$$S_{\overline{q}}(n, r)_{\mathrm{k}}$connec..twith a Robeed reductive algebraic group overLet be ais connected. Let be anius map $F$ . We assume that the centre ofprime number with \dag er .
1.., :$\mathrm{G}$$\mathrm{F}_{q}$$\ell$$\mathrm{G}$$\ell$.

$ii_{\sim}$$\mathrm{r}$$q$,$-\mathit{1}.$Lusztig series
The following is so-called Lusztig series:.

$\mathcal{E}(\mathrm{G}^{F}, \{s\}):=(\mathrm{T},\theta\bigcup_{)}\{\chi\in\hat{\mathrm{G}}^{p}|\langle\chi,.R_{\mathrm{T}}^{\mathrm{G}}(\theta)\rangle\neq\backslash 0\}$

runs

and
is a generalized Deligne-Lusztig character.
Its modular version is given as follows:
.
For a
’ let

Here, the above pair$(\mathrm{T}, \theta)$.

$\theta\in\hat{\mathrm{T}}^{F}rightarrow s_{1}\in \mathrm{T}^{*F}$$s_{1}\in\{s\}$, and$R_{\mathrm{T}}^{\mathrm{G}}(\theta)$$t-$$\cdot\cdot\cdot$

$=$

.

.:.

$\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\backslash .\mathrm{n}\ell’-\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{t}Sr\in.\mathrm{G}^{*F^{*}}’$.:.

$\cdot$.

.

,$\cdot’.‘::^{r}..\cdot..:.*t\in(c’\sim \mathrm{G}*.(S)F^{*})_{\ell}-$$.\backslash .\cdot.-..\cdot;$$\mathcal{E}_{\ell}^{-}.(\mathrm{G}F.\{_{S}...\}\backslash ,’..)=.\cdot.4=\cup \mathcal{E}(\mathrm{G}^{F}, \{St\}t)$.

$\sim\wedge\cdot$

Theorem 2.3 (Brou\’e--Michel). [5] Each set.
-block of$\mathcal{E}_{\ell}.(.\cdot \mathrm{G}^{F}.’.’(\{s:.-\})$

is a union of$\mathrm{G}^{F}$$\ell$

Definition 2.1. An -blocksuch that$\ell$$\chi\in \mathcal{E}_{\ell}(\mathrm{G}^{p}, \{1\})$$B$

as an algebra is unipotent, if there exists
isbelongs to B. In$parti_{C}u\iota a_{t}rpB_{0}(\mathcal{O}\mathrm{G}^{F})$$\chi$

unipotent.
1

:.:’.

$Supposetha\}t$
the $cen.tre:jof\mathrm{G}$Theorem 2.4 (Bonnaf\’e--Rouquier). [1]is a Levi subgroup of G. Thenected andis$con\dot{n}$$C_{\mathrm{G}}*(S)^{*}F$

.:

${ }$

$l$

.

$i\cdot-$$(\mathrm{x}$$\mathcal{E}_{\mathit{1}}(\mathrm{G}^{F}, \{s\})\sim M\mathcal{E}_{\ell}(C_{\mathrm{c}*}(s)^{*F}, \{1\})$

as -block ideals. ($i.e$ . If a block
exists a unipotentof$\ell$$blo.c.k.\cdot B_{1}’,$$B_{s}$

belongs to$\mathcal{E}_{\ell}(\mathrm{G}^{F}, \{s‘\})$$C_{\mathrm{G}^{*}}..(s)^{*F},- S..uch..(_{:}^{1}th\ldots.a.tB_{s}..’$,.

$equiv\dot{a}l\dot{e}nt.)$$\ovalbox{\tt\small REJECT} \mathrm{t}.:_{}.\cdot.:$.

$’...\cdot:.\cdot$$..\backslash ,$$\Gamma..’$

,

,. ..

$r\mathrm{Y}$

.

$t$

and

$B_{1}’$, then there
are Morita

37

Remark 1. The Morita equivalence in the above theorem is not a Puig
equivalence in general.
In particular, for finite general linear groups we may concentrate unipotent blocks by Bonnaf\’e-Rouquier theorem.
We want to classify the block ideals of
up to Morita equivalence,and recover its structure as algebras from some small subgroups. So, wewish to prove the following conjecture:$\mathrm{k}\mathrm{G}(\mathrm{F}_{q})$Conjecture 2.5. If $e(q)–e(q’),$ $r(q)=r(q’)$ then for any unipotent
block ideal
there exists a unipotent block ideal $B’$ ofofsuch that $B\sim_{M}B’$ by an exact l-pe rmutation $(B, B’)-’\dot{b}im$odule. Thisequivalence preserves the natural indices of modules.
$B$$\mathrm{G}(\mathrm{F}_{q})$$\mathrm{G}(\mathrm{F}_{q’})$

In this article we deal the special case concerning these three conjectures for finite general linear groups.

3

Abacus and $[w:k]$ -pairs

Definition 3.1. For a -core and a non-negative integer
be the set of paritions of $kw+|\tau|$ whose -core is .
, define
Given partition$k$$\tau$$k$$\lambda=$$\beta$$\Lambda_{k,w,\mathcal{T}}$:$\beta=(\beta_{1},\beta_{2}, \ldots)$$(\lambda_{1}, \lambda_{2}, \ldots , \lambda_{r})$, let$\tau$$\beta_{i}:=r-i+\lambda_{i}(1\leq i\leq r)$

We call this$w$

as follows:.

an -element -set for .
$r$$\lambda$$\beta$Definition 3.2 (Scopes). For a non-negative integer $m$ and an m-core, let be the -element -set for , and suppose that whenis diplayed on an abacus with $m$ -runners there are more than beads inthe i-th column than in the $(i-1)$ -th column. Let m-core be displayedby an -element -set satisfying$\tau=(\tau_{1}, \ldots, \tau_{r})$$\Gamma$$r$$\beta$$\tau$$k$$\Gamma$$\overline{\tau}$$\overline{\Gamma}$$r$$\beta.$for$\overline{\Gamma}_{j}=\Gamma_{j}$$j\neq i,$$i-1$$\overline{\Gamma}_{i}=\Gamma_{i-1}$$\mathrm{V}$$\overline{\mathrm{r}}_{i-1}=\mathrm{r}_{i}$,is the number of beads on the j-th runner inwherelation for F. In these situation, we shall say ihata Scopes $[w:k]$ -pair.
..
.. .

$t\dot{h}e..$$\Gamma_{j}$$\Lambda m,w,\tau$$i$$\mathrm{t}$J’ab’acusand$co‘ n$figu-$\Lambda_{m,w,\overline{\tau}}f_{\mathit{0}7}m$:38$\overline{\tau}$$\tau$

$i-1$

$i$

$i-1$

$i$

$\mathrm{O}$$\mathrm{O}$$\mathrm{O}$$\mathrm{O}$$\mathrm{O}$$\mathrm{O}$$\Rightarrow$..
.

Scopes proved the$\mathrm{O}$$\mathrm{O}$$\mathrm{O}$$\mathrm{O}$:$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$Theorem 3.1 (Scopes). [with $k\geq w$ , then -blocks
equivalent.

$\mathit{3}\eta$$B^{w,\tau}$$p$

By Jost we ako know theand$\Lambda_{p,w,\tau}$

and$B^{w,\nu}$of$\Lambda_{p,w,\nu}$form a $[w : k]$ -pair$symmet_{\dot{\mathcal{H}}}C$

groups are Morita:$\mathrm{f}\mathrm{o}\mathrm{U}_{0}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$Theorem 3.2 (Jost). [23] If$k\geq w$ , then unipotent -blocks$l$

If$\Lambda_{e,w,\tau}$$B_{w,\tau}$

and
and

$\Lambda_{e,w,\nu}$form a $[w : k]$ -pair withare

$B_{w,\nu}$equivalent.

$Mo7\dot{\tau}ta$

Example 1. If is a unipotent block of $GL_{n}(q)$ with -weight 2, thenone of the following holds:
$B$

1.
2.

$e$

$B\cong B_{0}(\mathrm{k}cL_{2e}(q))$.

.
equivalent to its Brauer correponActually, these blocks are de) (The method wedent of the -local subgroup. (Hida-Miyachi)
used is different from J. Chuang’s forwith $m<n$ .
3. $B\sim_{M}B’$ for some unipotent block $B’$ of$(B,\overline{B})$forms [2: 1]-pair for some unipotent block$\overline{B}$of$\mathrm{k}GL_{n-1}(q)$$($$7^{\cdot}ived$$\ell$$(\mathit{1}g\mathit{9}g)$$\mathfrak{S}_{n}$$\mathrm{k}GL_{m}(q)$4A core and results of J. Chuang and
R. Kessar$\rho$. Let be the e-core
Definition 4.1
which satisfies the following property: has an abacus configulation inwhich each runner other than the leflmost one (the O-th runner) has atleast $w-1$ more beads than the runners to its immediate lefl.
$(\mathrm{C}\mathrm{h}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{g}-\mathrm{K}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{a}\mathrm{r}-\mathrm{R}_{0}\mathrm{u}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{e}\mathrm{r})$$\rho$$[\mathit{8}]$$\rho$

39

Chuang and Kessar considered the following setting up:$e=p>w$ .
$r:=|\rho|$.

$G:=\mathfrak{S}pw+r$$B^{w,\rho}$.

: the p–block of$D:=\mathrm{a}$$\mathrm{k}G$

defect group of$N:=\mathfrak{S}_{p}l\mathfrak{S}_{w}\supset D$with p–weight$B^{w,\rho}$and p–core .
$\rho$.

.

$L:=\mathfrak{S}\cross\cdots\cross \mathrm{p}p\mathrm{e}\cross \mathfrak{S}r$.

$H:=(\mathfrak{S}\iota \mathrm{P}\mathfrak{S}w)\mathrm{X}\mathfrak{S}r\supset Nc(D)$$\mathcal{O}Hf:=\mathrm{t}\mathrm{h}\mathrm{e}$$w$.

Brauer correspondent of$B^{w,\rho}$

in $H$ .

in $G\cross H$ with respect to
Let $X$ be the Green correspondent of$(G\cross G, \Delta(D),$ $G\cross H)$ . Chuang and Kessar proved the following:$B^{w,\rho}$

Theorem 4.1 (Chuang-Kessar). [8] Suppose that $p>w$ . Then, $we$
get an isomorphism$\mathcal{O}Hf\cong \mathrm{E}\mathrm{n}\mathrm{d}_{G(X)}Q$

by checking$Mo$ rita equivalent to. In particular,$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{\mathcal{O}}(\mathrm{E}\mathrm{n}\mathrm{d}c(\mathrm{x}))\leq w!\cdot \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{\mathcal{O}}(\mathcal{O}Lf)$$B^{w,\rho}$.

Remark 2. 1. $X$ is exact.
is a split
2.
.
3.
.
4. By Marcus [27]is known, but
5.
$\mathcal{O}Hfarrow \mathrm{E}\mathrm{n}\mathrm{d}_{G}(X)$$(\mathcal{O}Hf, \mathcal{O}Hf)$-monomo $7phism$ .

$w!\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathcal{O}(\mathcal{O}Lf)=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{\mathcal{O}}(\mathcal{O}Hf)$$\mathcal{O}Hf\sim_{d}B\mathrm{o}(\mathcal{O}N)$$(D^{\lambda}\otimes_{B^{w,\rho}}x)\downarrow_{L}$$D^{\lambda}\otimes_{B^{w},\rho}X$

is not known.

$\mathcal{O}Hf$

is

40

5A theorem of Chuang-Kessar type
We assume that char $(\mathrm{k})=\ell>w$ . Choose a prime power with$e(q)=e$ . Just mimicking Chuang and Kessar’s setting up, we consider
the following:$q$$r:=|\rho|$.

$G(q):=GL+r(ewq)$ .
$\mathrm{t}$$B^{w,\rho}(q)$$\rho$: the unipotent -block of$l$$\mathrm{k}G(q)\mathrm{W}\mathrm{i}\acute{\mathrm{t}}\mathrm{h}e$-weight$w$

and e-core.

$D(q):=\mathrm{a}$defect group of$B^{w,\rho}$.

.

$N(q):=cL_{e}(q)\mathit{1}\mathrm{e}w\supset D(q)$$L(q):=GLe(q)\cross\cdots\cross GL_{e}(q)\cross GL_{r}(q)$.

$H_{w}(q):=(GL_{e}(q)\iota \mathfrak{S}_{w})\cross GL_{r}(q)\supset N_{c(D(q}))$$\mathcal{O}H_{w}(q)f_{q}:=\mathrm{t}\mathrm{h}\mathrm{e}$Brauer correspondent of.

$B_{w,\rho}(q)$in $H_{w}(q)$ .

Once we believe that an anology of Chuang-Kessar theorem holdjs forfinite general linear groups, we can easily prove the following:
Proposition 5.1. (An anology of Chuang-Kessar theorem) Let$X(q)$ be the Green correspondent ofin $G(.q.)\cross H_{w}(q)$ with respectisomorphism
to $(G(q)\cross G(q), \Delta(D(q)),$ $c(q)\cross H_{w}(q))$ . Then, we et$B^{w,\rho}(q)$$\dot{g}$an.

$\mathcal{O}H_{w}(q)f_{q}\cong \mathrm{E}\mathrm{n}\mathrm{d}G(q)(x_{\mathcal{O}}(q))$

by checkingis. In particular,$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{\mathcal{O}}(\mathrm{E}\mathrm{n}\mathrm{d}_{G(q)}(X_{Q}(q)))\leq w!\cdot \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\mathcal{O}(\mathcal{O}L(q)f_{q})$$\mathcal{O}H_{w}(q)f_{q}$$Mo7\dot{\mathrm{v}}ta$equivalent to$B_{w,\rho}(q)$.

Remark 3. One must consider not only unipotent characters but alsocharacters indexed by semisimple -elements. We can know these characters by [9]. We also need some results by [15] in order to mimic Chuang
and Kessar’s argument.
$\ell$

41

6Indices of$B_{0}(GL(eq)\iota \mathfrak{S}_{w})$-modulesIn this section we reformulate indices of the simple $B_{0}(GL_{e}(q)l\mathfrak{S}w)-$modules to fit that ofvia the equivalence in Proposition 5.1. For$\nu_{i}=(i,1^{e-i})\vdash
e$ . The principal block$i=1,2,$, lethas
non-isomorphic irreducible modules$B_{w,\rho}(q)$$B_{0}(\mathrm{k}GL_{e}(q))$$e$$\ldots$

$e$

..: ;
$\cdot$.

$\{D_{\mathrm{k},q}(\nu_{i})|i--1,2, \ldots, w\}$

Fix $R\in\{K, \mathrm{k}\}$ . Let$\sum_{i=1}^{e}n_{i}=w$.

$n$be an -tuple non-negative integer of . i.e.
is an $R[GL_{e}(q)\cross w]$ -module.
$w$

$e$

$S_{R,q}(n):=\otimes(S_{R,q}(\nu_{i})\otimes \mathfrak{n}_{i})$In particular,ple
is a
subgroup
act on $S_{R,q}(n)$ . So, $S_{R,q}(n)$ is an
is decomposed into

. The parabolic$\mathrm{s}\mathrm{i}\mathrm{m}|$$S_{K,q}(\dot{n})$$K[GL_{e}(q)^{\mathrm{X}w}]-\mathrm{m}\mathrm{o}\dot{\mathrm{d}}\mathrm{u}\mathrm{l}\mathrm{e}$$\mathfrak{S}_{\mathfrak{n}}$$R[L_{(e^{w})}\rangle\triangleleft \mathfrak{S}_{n}]$-module.

$\mathrm{I}\mathrm{n}\mathrm{d}_{L_{(e}w}^{L_{()}\aleph\circ^{\vee}}\mathrm{e}^{w})\mathrm{n}S_{R,q}(n)$$\oplus(S_{R,q}(n)\lambda\vdash \mathfrak{n}\otimes_{R}(\dim_{R}S_{R}^{\lambda})\cdot S_{R}^{\lambda})$where$S_{R}^{\lambda_{i}}$$\bigotimes_{1}$.

means the Specht module ofcorresponding toandis the inner tensor product of$S_{R}^{\mu}$$R[\mathfrak{S}_{|\mu|}]$$S_{R,q}(n)\otimes_{R}S_{R}^{\lambda}$

modules $S_{R,q}(n)$ and$S_{R}^{\lambda}$

. Let

. .
$\backslash$$S_{R}^{\lambda_{i}}$$S_{R}^{\lambda_{i}’}$$\backslash ‘$

$,$

.

.

if $i+e$ is even,
if $i+e$ is odd.
,

Here,is the conjugate partition of . Let
For
let
be
be the $R[H_{w}(q)]$ -module.
Moreover, one can construct modules by using$\lambda_{i}$$\cdot$$\backslash$.

$\lambda_{i}’$$U_{R,q}(\lambda)$$S_{R}^{\lambda}=$$R[L_{(e^{w})}\rangle\triangleleft \mathfrak{S}_{n}]-$$\mathcal{I}_{R}^{\lambda_{i}}=\{$$\lambda\vdash n$$\mu,$.

.
, and let

$T_{R}^{\lambda}=\otimes_{i}\mathcal{I}_{R}^{\lambda_{i}}$$\mathrm{I}\mathrm{n}\mathrm{d}_{L_{()}\triangleleft}^{GL()\mathrm{c}^{\sim}}\mathrm{e}q\rangle l\mathrm{q}_{*}^{w}’(S_{R,q}(n)\otimes T_{R}^{x})$$U_{\mathrm{k},q}(\lambda)^{\rho}$$U_{R,q}(\lambda)\otimes_{R}S_{R,q}(\rho)$$\{D_{\mathrm{k},q}(\nu_{i})|i=1,2, \ldots, e\}$instead of.
by$\mathrm{k}[GL_{e}(q)]$-modules $\{S_{\mathrm{k},q}(\nu_{i})|i=1,2, \ldots, e\}$ . We denote it$V_{\mathrm{k},q}(\lambda)^{\rho}$

$.1$

:

. .
$\cdot$

$-.$

.

42

7

Results
Now we can state our main results of this article as follows:-moduleTheorem 7.1 (Hida-Miyachi). [18] For any simple
is inde, the Green correspondentofpendent of in the following sense:be the canonical
Assume that $e(q)=e(q’)$ and $r(q)=r(q’)$ . Let,-bimodule which induces
due to A. Marcus. Then
$B_{w,\rho}(q)$$D_{\mathrm{k},q}(\lambda)\otimes_{\mathrm{k}G}X(q)$$D_{\mathrm{k},q}(\lambda)$$D_{\mathrm{k},q}(\lambda)$$q$$\mathcal{M}_{q,q’}$$\mathrm{k}H_{w}(q)f_{q}\sim_{M}\mathrm{k}H(q’)f_{q}’$$(\mathrm{k}H_{w}(q)f_{q}, \mathrm{k}H(q’)f_{q}’)$.

$D_{\mathrm{k},q}(\lambda)\otimes_{B}(q)X(w,\rho q)\otimes \mathrm{k}Hw(q)fg\mathcal{M}_{q},’\otimes_{\mathrm{k}}qH(q’)fq;x(q)^{\mathrm{v}}’\cong D_{\mathrm{k}},q’(\lambda)$. Here,is the -quotient of
Actually,. Moreover, we know the decomposition numbers corresponding to the-core :$\overline{\lambda}$$e$$D_{\mathrm{k},q}(\lambda)\otimes_{B_{w,\rho()}}qx(q)\cong V_{\mathrm{k},q}(\overline{\lambda})^{\rho}$$\lambda$$e$$\rho$$d_{\lambda,\mu}=d_{\overline{\lambda}},=[\overline{\mu}U_{\mathrm{k},q}(\overline{\lambda}) :(The other parts of

V_{\mathrm{k},q}(\overline{\mu})]$.

can be calculated by Dipper-James theory.)$B_{w,\rho}(q)$Remark 4. First we can determine the Green correspondents of simple,, finding two trivial source modules of-modules inusing the decomposition numbers for Hecke algebras of type A by [33]and [22], chasing the image of Mullineux-Kleshchev map [29, p.120], the.
property of Specht modules [19] and induction on
Next we can determine the Green correspondents of simple
modules in $H_{w}(q)$ using induction on and some commutative diagrams
among $B_{w,\rho}(q),B0(cL_{e}(q))\otimes B_{w-1,\rho}(q)$ and their Brauer correspondents.
In order to prove $B_{w,\rho}(q)\sim_{M}B_{w,\rho}(q’)$ with the property in the above, and [36].
theorem we use$B_{2,\rho}(q)$$H_{2}(q)$$B_{2,\rho}(q)$$\Lambda_{e,2,\rho}$$B_{w,\rho}(q)-$$w$$[\mathit{1}\mathit{4}],l\mathit{2}7\mathit{1}$Corollary 7.2. [18] If there exist a sequence of e-cores$\beta^{=\tau^{0}},T^{1}$: andsuch that
conjecture is true for$\Lambda_{e,w,\tau}$$\Lambda_{e,w,\tau}\dot{.}+1$

form a$B_{w,\tau^{S}}(q)$

.

,-..

,

$\mathcal{T}^{S}$$[w : k_{i}]$-pair with $k_{i}\geq w-1$ , Brou\’e’s43

Theorem 7.3 (Hida-Miyachi). [18] Assume that $e=e(q)=e(q’)$ and$r(q)=r(q’)$ . If there exist a sequence of e-cores$\rho=\tau^{01},$$\tau,$ $\ldots,\tau^{S}$

such that$\Lambda_{e,w,\tau^{i}}$and$\Lambda_{e,w,\tau^{:+1}}$

form a$[w:k_{i}]$-pair with $k_{i}\geq w-1$ , then$B_{w,\tau^{\iota}}(q)\sim_{M}Bw,\mathcal{T}\theta(q’)$.

Here, each $[w : w-1]$ -pair is a derived (spelendid) equivalence between two unipotent blocks. Moreover, the above $Mo$ rita equivalence preserves natural indices (partitions) of modules. ( $i.e$ . The simple mod” $lik$, the Young module(resp. the $‘\prime s_{pe}eCht$
ule
module$PIMP_{q}(\mu))$ indexed by a partition
corresponds to), $P_{q’}(\mu))$ . )(resp.
$D_{\mathrm{k},q}(\mu)$$S_{\mathrm{k},q}(\mu)$$X_{q}(\mu),$$D_{\mathrm{k},q’}(\mu)$$\mu$$S_{\mathrm{k},q’}(\mu),$ $x_{q^{\prime(}}\mu$

Remark 5. Just mimicking an argument in [7], constructing a generalization of [38] and using Theorem 7.1, we deduce the above results. (seealso [32] .
$)$8

A conjecture

8.1

The theory of$\mathrm{L}\mathrm{a}\mathrm{s}\mathrm{C}\mathrm{o}\mathrm{u}\mathrm{x}-\mathrm{L}\mathrm{e}\mathrm{c}\mathrm{l}\mathrm{e}\Gamma \mathrm{c}$-Thibonbe the quantized enLet be an indeterminate over Q. Let.
corresponding to the Dynkin diagram
veloping algebra over
The so-called “Fock space”$U_{v}(\overline{\epsilon\iota_{e}})$$v$$A_{e-1}^{(1)}$$\mathbb{Q}(v)$$\mathcal{F}_{v}=\bigoplus_{\lambda}\mathbb{Q}(v)|\lambda\rangle$

indexed by the set of all partitions.
-vector space with basis
is theIn [24] Lascoux, Leclerc and Thibon introduced an algorithm to computeand conjectured
the canonical basis of the basis representationthat it also compute the decomposition matrix of the Iwahori-Hecke althe LLTgebras of type A at a root of unity over C. This is soconjeture.
The LLT conjecture is now a theorem (see, for example, [29, Chap. 6]
and the references of Chapter 6).
$\mathbb{Q}(v)$$|\lambda\rangle$$M_{v}(\Lambda_{0})$$\mathrm{c}\mathrm{a}\mathrm{U}\mathrm{e}\mathrm{d}$44

In [25], Leclerc and Thibon define a canonical basis of the v-deformed
Fock space representationof the affine Lie algebra . They conjectured that the entries of the transition matrix between these basis andare also crystalized decomposition numbers of the Dipper-James’
Schur algebra for specialized at a primitive e-th root of unity.
This LT conjecture is now a theorem overwhere is a primitivee-th root of umity over , due to Varagnolo and Vasserot [39].
Leclerc and Thibon showd that1.

$\mathfrak{g}1_{e}\wedge$$F_{v}$$\{|\lambda\rangle\}_{\lambda}$$\zeta$$\mathbb{Q}(\zeta)$$\zeta$$\mathbb{C}$

Theorem 8.1 (Leclerc-Thibon). There exist basesof characterized by:$\{G(\lambda)\}$and$\{G^{-}(\lambda)\}$$\mathcal{F}_{v}$$\sim$.

$\mathit{1}^{\cdot}.G^{\cdot}\overline{(}-\lambda)=G(\lambda),\cdot\overline{G-(\lambda)}=c^{-(\lambda}..)$

2.

$G(\lambda)\equiv|\lambda\rangle$

Here,$\{|\lambda\rangle\}$$L$(resp.

mod $vL$ ,$L^{-}$$G^{-}(\lambda)\underline{=}|\lambda\rangle$) denotes the$\mathbb{Z}[v]$mod(resp.

$v^{-1}L^{-}$$\mathbb{Z}[v^{-1}]$.

) -lattice in$F_{v}$

with basis$\Gamma$Let$G( \mu)=\sum_{\lambda}d_{\lambda,\mu}(v)|\lambda\rangle$,$G^{-}( \lambda)=\sum_{\lambda}e\lambda,\mu(v)|\mu\rangle$.

$\mathrm{v}$and$\mathrm{D}_{m,e,0}(v)=[d_{\lambda,\mu}(v)]_{\lambda,\mu}\vdash m$.

Theorem 8.2 (Leclerc-Thibon, Varagnolo-Vasserot). The mat rix.
is equal to the decomposition$\mathrm{D}_{m,e,0}(1)$$mat\dot{m}\mathrm{D}_{m,\zeta,0}$RemarkIf both andare primitive e-th roots of unity in.
. So, we may write
instead of$\zeta$$\zeta’$$\mathrm{D}_{m,e,0}$$\mathrm{D}_{m,\zeta^{\prime,0}}$8.2Let$\mathbb{C}$, then$\mathrm{D}_{m,\zeta,0}=$$\mathrm{D}_{m,\zeta,0}$

Our hope$l_{\mathrm{k},q}(\lambda)$

be the Loewy length ofas follows:$S_{\mathrm{k},q}(\lambda)$.

For $\ell>w$ , we define$rad_{\lambda,\mu}(v).\in \mathrm{N}[v]$$rad_{\lambda,\mu}(v)$$- \sum_{=k0}^{\iota_{\mathrm{k},q(\lambda}}[\mathrm{R}\mathrm{a}\mathrm{d}^{k}(U_{\mathrm{k}},(\overline{\lambda}))/\mathrm{R}\mathrm{a}\mathrm{d}k+.1(U_{\mathrm{k}.’q}.(\overline{\lambda})))-1q.\cdot :V_{\mathrm{k},q}(\overline{\mu})]v^{k}$45

is given explicitly by some products ofRemark 6. Note that
Littlewood-Richardson coefficients and . Moreover, an explicit formula
will be written in [30].
for, we
By the construction of the Loewy series offoralso know$rad_{\lambda,\mu}(v)$$v^{i}$$rad_{\lambda,\mu}(v)$$\lambda\in\Lambda_{e,w,\rho}$$S_{\mathrm{k},q}(\lambda)$

Rad.

$(S\mathrm{k},q(\lambda))/\mathrm{R}\mathrm{a}\mathrm{d}^{i+1}(S_{\mathrm{k}},(q\lambda))\cong \mathrm{S}\mathrm{o}\mathrm{C}^{\mathrm{t}_{\mathrm{k}}(}-(\lambda)is\mathrm{k},q(\lambda))/\mathrm{S}\mathrm{o}\mathrm{c}^{l_{\mathrm{k}}(}-i-1(\lambda)S_{\mathrm{k}},q(\lambda))$Theorem 8.3 (Geck). [16] There exists a square lower
matrix A such that each entry of A is non-negative andA.

$unit_{\dot{\mathcal{H}}}angular$$\mathrm{D}_{m,\overline{q},\ell}=\mathrm{D}m,e,0^{\cdot}$

By the above theorem and Theorem 7.1, we deduce
Corollary 8.4. If $rad_{\lambda,\mu}(v)=0$ , thenfor any$d_{\lambda,\mu}(v)--\mathrm{o}$$\lambda,$$\mu\in\Lambda_{e,w,\rho}.$.

Not only do we want to show that James’ conjecture is true, but wewhich is now known to be awant to know an explicit formula forcertain parabolic Kazhdan-Lusztig polynomial.
According to James conjecture and Rouquier-Leclerc-Thibon conjec, we hope thefiltrations
also 6..27)] on
ture [29, $6.3.3-.$ .(following:,.
$d_{\lambda,\mu}(v)$$\mathrm{J}\mathrm{a}\mathrm{n}.\mathrm{t}\mathrm{z}\mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}..\cdot.\mathbb{C}$$.\mathrm{s}\mathrm{e}\mathrm{e}\backslash$$\iota$Conjecture 8.5.

for any$d_{\lambda,\mu}(v)=rad_{\lambda},(\mu v)$$\lambda,$$\mu\in\Lambda_{e,w,\rho}$.

The first announcement of this was stated in the author’s lecture “On”the unipotent blocks of finite general linear groups” at a conference
Alg\’ebres de Hecke affines et groups r\’eductifs (CIRM,Lulniny,16-20
bre 2000)“ organized by M. Geck and R. Rouquier.

$\mathrm{O}\mathrm{C}\mathrm{t}_{\mathrm{C}\succ}$

Acknowledgments
The author would like to thank M. Geck, B. Leclerc, and R. Rouquierfor stimulating discussions on James’ conjecture and canonical bases.
$\iota$.

$\star\prime\prime\doteqdot\vee \mathrm{x}\ovalbox{\tt\small REJECT}^{\backslash }$[1] C. BONNAF\’E AND R. ROUQUIER, Cat\’egode Deligne-Lusztig, preprint, (2000).

:$7\dot{\mathrm{u}}es.d\acute{e}r\dot{\mathrm{v}}v\acute{e}eS$$\prime i_{-}$

.,

et va

$r\dot{\tau}\acute{e}t\acute{e}s$46[2] M. BROU\’E, Isom\’etries parfaites, types de blocs, cat\’egories
Ast\’erisque, 181-182 (1990), 61-92.

,$d\acute{e}\dot{\mathcal{H}}v\acute{e}eS$[3] –, Equivalences of blocks of group algebras, in In Proceedings ofthe International Conference on Representations of Algebras, V.Dlab.Scott,, Finite Dimensional Algebra and Related Topics,
andOttawa, Aug 1992, Kluwer Academic Publishers, pp. 1-26.
$\mathrm{L}.\mathrm{L}$$\mathrm{e}\mathrm{d}\mathrm{s}.$[4] M. BROU\’E AND G. MALLE,
Ast\’erisque, 212 (1993), 119-189.

Zyklotomische Heckealgebren,[5] M. BROU\’E AND J. MICHEL, Blocs et s\’eris de Lusztig dans un grouper\’eductif fini, J. Reine Angew. Math., 395 (1989), 56-67.

[6] R. W. CARTER, Finite Groups of Lie Type: Conjugacy Classes andComplex Characters, John Wiley, New York, 1985.
[7] J. CHUANG, The derived categories of some blocks of symmetricgroups and a conjecture of Brou\’e, J. Algebra, 217 (1999), 114-155.
[8] J. CHUANG AND R. KESSAR, Symmetric groups, wreath products,
Morita equivalences, and Brou\’e’s abelian defect group conjecture,
preprint, (2000).
[9] R. DIPPER AND G. D. JAMES, Identification of the irreducible modular representations of $GL_{n}(q)$ , J. Algebra, 104 (1986), 266-288.
[10] –, Representations of Hecke algebras of general linear groups,
Proc. London Math. Soc., 52 (1986), 20-52.
[11] –, Blocks and idemportents of Hecke algebras of general linear
groups, Proc. London Math. Soc., 54 (1987), 57-82.

[12] –, The -Schur algebra, Proc. London Math. Soc., 59 (1989), 23$q$

50.
[13] –, The -Tensor and -Weyl modules, rbans. Amer. Math. Society, 327 (1991), 23-50.
$q$

$q$

[14] P. FONG AND B. SRINIVASAN, Blocks with cyclic defect groups in$GL(n, q)$ , Bull. Amer. Math. Soc., 3 (1980), 1041-1044.

47[15] –, The blocks of finite general linear and unitary groups, Invent.
math., 69 (1982), 109-153.

[16] M. GECK, Brauer trees of Hecke algebras, Comm. Algebra., 20(1992), 2937-2973.
[17] –, Kazhdan-Lusztig cells, -Schur algebras, and James’ conjeture,
preprint, (1999).
$q$[18] A. HIDA AND H. MIYACHI, Some blocks of finite general linear
groups in non-defining characteristic, preprint, (2000).
[19] G. D. JAMES, The irreducible representations of the finite generallinear groups, Proc. London Math. Soc., 52 (1986), 236-268.

[20] –, The decomposition matrices of $GL_{n}(q)$ for $n\leq 10$ , Proc. London Math. Soc., 10 (1989), 225-265.
[21] G. D. JAMES AND A. KERBER, The representation theory of thesymmetric group, Encyclopedia of Mathematics and its Applications
16 (Cambrige University Press), 1981.
[22] G. D. JAMES AND A. MATHAS, A -analogue of the JantzenSchaper theorem, Proc. London. Math. Soc., 74 (1997), 241-274.
$q$[23] T. JOST, Morita equivalence for blocks of finite general linear groups,
manuscripta math., 91 (1996), 121-144.
[24] A. LASCOUX, B. LECLERC, AND J.-Y. THIBON, Hecke algebras atroots of unity and crystal bases of quantum affine algebras, Commun.
Math. Physics, 181 (1996), 205-263.

[25] B. LECLERC AND J.-Y. THIBON, Canonical base $ofq$ -deformed Fockspace, Int. Math. Res. Notices, (1996), 447-498.
[26] M. LINCKELMANN, Stable equivalences of Morita type for selfinjective algebras and -groups, Math.Z., 223 (1996), 87-100.
$p$[27] A. MARCUS, On equivalences between blocks of group algebras: reduction to the simple components, J. Algebra, 184 (1996), 372-389.

48[28] A. MATHAS, Decomposition matrices of Hecke algebras of type A ,in Gap : groups, algorithms and programming, M. Sch\"onert et al.,, vol. 3.4.4., RWTH Aachen, 1997.
$\mathrm{e}\mathrm{d}\mathrm{s}.$[29] –,-Hecke algebras and Schur algebras of the $symmet’\dot{\eta}C$groups, vol. 15 of University Lecture Series, AMS, 1999.
$Iwah_{\mathit{0}\dot{\mathcal{H}}}$[30] H. MIYACHI, Some remarks on James conjecture, preprint, (2000).
$f$[31] H. NAGAO IAND Y. TSUSHIMA, Representations of Finite Groups,
Academic Press, New York, 1990.
[32] T. OKUYAMA, Some examples of derived equivalent blocks of finitegroups, preprint, (1998).
[33] M. J. RICHARDS, Some decomposition numbers for Hecke algebras
of general linear groups, Math. Proc. Camb. Phil. Soc., 119 (1996),

383-402.

[34] J. RICKARD, $Mo$ rita theory for derived categories, J. London Math.
Soc, 39 (1989), 436-456.
.
$i$[35] –, Splendid equivalences: Derived categories and $pe\gamma mutaion$ modules, Proc. London Math. Soc, 72 (1996), 331-358.
[36] R. ROUQUIER, From stable equivalences to Rickard equivalences forblocks with cyclic defect: ”Groups ’93, Galway-St Andrews” conference, vol. 2 of London Math. Soc. series 212, Cambridge University, 1995, 512-523.
$\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{S}.\mathrm{S}$equivalece for blocks of theand[37] J. SCOPES, Cartan
symmetric groups, J. Algebra, 142 (1991), 441-455.
$mat_{7\dot{\mathrm{B}}ce}S$$M_{\mathit{0}\dot{\mathcal{H}}}ta$[38] –, Symmetric group blocks of defect two, Quart. J. Math. Oxford. $(199.5)|"\sim’ 20_{:}1.-2.3.4$$\mathrm{S},.\mathrm{e}_{2}\mathrm{r},\cdot,.4=6‘$.

[39] M. VARAGNOLO AND E. VASSEROT, On the decomposition$..\cdot$

.

Duke.

.

$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}$$mat\gamma\dot{\eta}ces$$\mathrm{J}.,.100\urcorner \mathrm{j}:.\ldots..\cdot \mathrm{c}.\backslash \cdot\cdot’-.t(199.9).’269-297\vee.\cdot$$o.ft..h.equantizeds_{C}hura\iota.geb.ra.’$49

DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE ANDTECHNOLOGY, CHIBA UNIVERSITY, YAYOI-CHO, CHIBA CITY 263-8522, JAPANE–mail: mmiyachi@g.math.s.chiba-u.ac.jp