SOME
DEGENERATE UNIPOTENT
BLOCKS
NAGOYA$\mathrm{U}$
NIVERSITYHYOHEMIYACHI(名古屋大学宮地兵衛)
1.
PREIJMINARY
Definition 1. Let
$G$be
a
finite
group
of
Lie
type.
Let
$A$be
a
unipotent
block ideal
of
$G$with
$\Phi_{e}$-defect
torus
$T$and
canonical
$ch,amct.er$A in
$M=$Zg(T).Let
$W(M, \mathrm{X})$be the
inertial group
of
A.
We say that
$A$is
a
Rouquier block
if
there exists a
Levi
subgroup
$L$of
$G$such that
(i)
there
ex.ist,sparabolic subgroup
$P$of
$G$with Levi decomposition
$P=LUp,$
(ii)
$L$contains
$M$,
(iii)
$H=L\cdot W$(A#,
$\lambda$)
is a
proper
subgroup
of
$G$,
(iv)
There
exists
a
block
$B$of
$H$with canonical character A such that
$A$is
Morita equivalent to
$B$.
Remark 2. By L. Puig
[Pui90],
if
$\ell$dose
not divide the order
of
Weyl group
$W$of
$G$and
dose
$q-1,$
then the principal
block
of
$G$is
Morita
(Puig)
equivalent to the principal block
of
$T.W$.
In
the
case
of
type
$A$,
there is
a
generalization
of
this
theorem
which
was conjectured
by
R. Rouquier
[ROu98]
in,the
context
of
symmetric
groups.
This
generalization
is intensively stud
$\dot{a}ed$by J. Chuang, R. Kessar, K.
Ta.
$n$,
W. Turner,
A.
Hida and the author [CK02],
$[\mathrm{C}\mathrm{T}02\mathrm{a}]$,
$[\mathrm{T}\mathrm{u}\iota\{12],[\mathrm{H}\mathrm{M}00],[\mathrm{M}\mathrm{i}\mathrm{y}01]$In th.i
$s$case
there
$.i\cdot s$also
an
interesting inter pretation
on
these
Rouquier
blocks in terms
of
Fock
space over
quantum
ajfine algebra
of
type
$A[\mathrm{C}’l\mathrm{T}02\mathrm{b}],[\mathrm{L}\mathrm{M}02]$.
One aim
of
this note is to report that there
are
Rouquier blocks in
type $E_{6}$aiid
$E_{8}$ (seeTheorem 4
and
$\mathrm{R}\epsilon \mathrm{n}\mathrm{l}\mathrm{a}.\mathrm{r}\mathrm{k}$ $5$below
)which
are
not
included in [Pui90].
In
tllr
$()$ng.ll this note,
we
asume
that
$\mathrm{a}$.
prime
number
$p$and
a
prime
power
$q$satisfy
the
following
C011ditiO11:
(1)
(i)
$P\neq 2,3,$(ii)
$p$divides
$q^{2}+1,$and
(iii)
$q$is odd.
Remark 3.
(i)
and
(ii)
are
essential in
this
note,(iii)
isneeded to
use Kawanaka
’s
general.ized
$Gel. \int$and
Graev character
[GP92]. So, this might
be removed once
one
gets
the lower unitriangulaiity
of
decorn.pO-sition
matrix.
The main strategy
(whichI learned from T.
Okuyama)
is analogous
to
that
in [KMOO], Namely,
we
construct
$\mathrm{a}$.
stable equivalence between two blocks by Broue’s
theorem
[Br092,
6.3.
Theorem]
and
checking the assumption, and then
we
chase the images of simple
modules.
The
most
powerful tool in
this
approach
is Linckelmann’s theorem
[Lin96].
In this note
we
choose
a
numbering of simple
roots of
type
$E_{8}$as
follows:
$s_{2}$
91
$s_{3}$ $s_{4}$ $s_{5}$ $s_{6}$ $s_{7}$ $s_{8}$TABLE
$E_{5}$.
120
2. THE
STATMENT OFMAIN
THEOREM
Let
$\mathrm{G}$be
the Chevalley
group
of type
$E_{6}$with its defining field
$\mathrm{F}_{q}$.
Let
$F$be its standard Frobenius
map.
Let
$\mathrm{L}$be the Levi subgroup corresponding to
$\{52, s_{3}, s_{4}, s_{5}\}$
.
It
is
of type
$D_{4}$.
Let
$A$(resp.
$B$)
be the principal block ideal of
$\mathrm{k}\mathrm{G}^{F}$(resp.
$\mathrm{k}\mathrm{L}^{F}.\mathfrak{S}_{3}$).
Moreover,
$\mathrm{L}^{F}.6_{3}$contains
$N_{\mathrm{C}\tau}^{F}(D)$.
Hence,
$\mathrm{f}$
(2)
there
is
a
Green
correspondence
$\mathrm{G}^{F}$ $arrow \mathit{2}arrow$ $\mathrm{L}^{F}.6_{3}$$\mathrm{g}$
(see
[A1])86].)
Let
$\Delta(D)$be
$\{(x, x)|x\in D\}$.
Let
$\mathrm{X}$be the
Green
correspondent of
an
indecon.lpos-able
$(A,A)-$bimodule
$A$in
$G\mathrm{x}H$with
vertex
$\Delta(D)$.
(In
other words,
$\mathrm{X}$is the
Scott
$(A, B)$-bimodule
$\mathrm{S}_{\mathrm{G}^{F}\mathrm{x}(\mathrm{L}^{P}.6_{3})}(\Delta(D))$
with vertex
$\Delta(D)$.
)
Now,
we
can
state
the main
result of
this note
as
follows:
Theorem
4.
The
functor
$\mathrm{X}$$\otimes_{B}-$
induces
a Morita
equivalence be
rween
$A$and
$B$.
Remark
5.
By
[Miy03]
we
know
that the principal
block
$B_{0}(\mathrm{k}\mathrm{G}^{F})$is
Morita
equivalent
to
the
unipotent
block ideal
of
$E_{8}(q)$with
canonical character
$\phi_{23,01}$in
the notation
[BMM93]. Moreover,
thanks
to
Broui ’s
abelian
defect
conjecture
and
our
mainchart [BMM93], our
mainresult
is
expected
to be
useful
to
settle
Broue’s abelian
defect
.conjecture
for
the
following
unipotent
$\Phi_{4}- bloc\lambda.s$:
(i)
Group
E7
(q)
:
can
onical characters
$/” \mathrm{j}$,
$\phi_{11}^{3}$,
(ii)
Group
Es(q):
canonical characters
$\mathrm{E}_{3.1}$,
$\phi_{123,0}13$,
$\phi_{12,03}$.
These
will be
discussed
in
ehewhere.
3. THE
DECOMPOSITION MATRIXNow,
we
recall
what is
known for the
decomposition
matrix of
$A$without
Theorem 4.
Using [Gec93], [GH97],
$[\mathrm{G}\mathrm{P}92],\mathrm{e}\mathrm{t}\mathrm{c}$
,
we
can
approximate
the decomposition matrix
as
follows:
Lemma 6. The
$decom,pos.itv.on$matrix
of
the
unipotent
characters
lying in the principal block
of
$\mathrm{G}^{F}$has
the following shape:
Here,
$*$I
would like
to
remove
these unknown
parameters
$*$’s
in
Lennna 6
as
possible. Using Theorem
4,
Le
una
6
and
the
precise
correspondence
on
the simple modules
over
$A$and
$B$,
we
get
the
following
llewTheorem
7.
The decomposition
$m.a,trix$of
the
unipotent
characters
lying
in
the principal
block
of
$\mathrm{G}^{F}$ $is$$g\nu\cdot,ver\tau$,
as
$fol_{1}l_{l}ows$:
’ $\iota s$ $u$ $n$ $lf$ $n$ $\iota$
4.
REMARKS
ONHECKE
ALGEBRASBy Theorem 4,
we
can
know tlia.t
Theorem 8. Let
$F$be
a
field
$uri$th
an
invertible
element
$q$.
We
assume
$Mt$(i)
The
characteristic
of
$F$is
not 2, 3.
(ii)
$q^{4}=1$,
$q^{2}\neq 1$.
(iii)
$Fconta\prime i.n.s$ $q^{1}F$.
(iv)
If
the
characteristic
of
$F$is positive, then,
$q$
lie in
the
prime
field of
$F$.
Then,
$B_{0}(\mathcal{H}_{F,q}(E_{6}))$and
$B_{0}(H_{F,q}(D_{4})).6_{3}$are
Morita
equivalent
We
can
construct
$B_{0}(7\{_{F,q}(D_{4})).6_{3}$as a
block ideal of
a
well-know
$\mathrm{n}$Iwahori-Hecke
algebra
in
the
following
way.
$H_{F,q}(D_{4})$is
a
$q$-defoni tion
of
the
group
algebra
of Weyl
group
$\mathrm{W}(\mathrm{D}\mathrm{a})$of type
$D_{4}$.
And,
in
our
situation,
$W(D_{4}).6_{3}$is nothing but the Weyl
group
$W(7\mathrm{J})$of type
$F_{4}$.
Moreover,
$W(D_{4})$is
realized
as
$\mathrm{a}$. reflection
subgroup of
$W(F_{4})$generated
by
all the reflections of
$W(F_{4})$whose
roots
$\mathrm{a}\mathrm{n}\cdot \mathrm{e}$long. So, let
us
recall
tlle definition of Iwahori-Hecke algebra
of
type
$F_{4}$.
Definition 9. Let
$R$be
an
integral domain ith
invertible
elements
$u\xi$,
$v^{1}2$.
The Iwahori-Hecke algebra
$H_{R,u}$,$v(F_{4})$
over
$R$eoith.
$p$arameter
$u$,
$v$is
an
associative algebra
with
generators
$T_{1}$,
$T_{2}$,
$T_{3}$,
$T_{4}$and
relations
$(T_{i}-u)(T_{i}+1)=0$
for
$i=1,2$,
$(T_{j}-v)(T_{j}+1)=0$for
$j=3,4,$
$T_{i}T.\cdot T_{j}.$. $=T_{i+1}T_{i}T_{i+1}$
for
$i=1,3$,
$T_{2}T_{3}T_{2}T_{3}=$
T3T2T3,
$T_{i}T_{j}=T_{j}T_{i}$
.
for
$1\leq i<j-1\leq 3.$From
now
on,
we
consider
the
Iwahori-Hecke
algebra.
$H_{\mathrm{k},q,1}(F_{4})$of
type
$F_{4}$with
parameter
$q$
and
1.
$\mathrm{P}n\mathrm{t}$$a_{1}$
,
$a_{2}$,
$a_{3}$,
$\mathrm{a}_{4}$.
respectively to
be
$a_{1}=T_{2}$
,
$a_{2}=T_{1}$,
$a_{3}$ $=T_{3}T_{2}T_{3}$,
$a_{4}=T_{4}T_{3}T_{2}T_{3}T_{4}$.
$Ft_{k}$,$q(D_{4})$
is isomorphic to the subalgebra
$?t’$of
$Ptl,,q,1$$(F_{4})$generated by all
$a_{2}$,
a3,$a_{4}$.
One
can
easily check
that Oi’s satisfy the quodratic relations and
braid
relations. Moreover, clearly, k(
$T_{3}$,
$T_{4}\rangle$is
isomorphic
to
the
group
algebra k&s since
$T_{3}^{2}=1=T_{4}^{2}$.
By
definition, the
action of
$T_{3}$and
$T_{4}$on
$H’$is also
clear.
Since the principal block ideinpotent of
$H_{\mathrm{k},q}(D_{4})$is normalized by
$\mathrm{k}\langle T_{3}, T_{4}\rangle$,
it is lifted to the idempotent
$\mathcal{H}_{\mathrm{k},q}(D_{4})$
is isomorphic to the subalgebra
$?t’$of
$\mathcal{H}_{\mathrm{k},q,1}(F_{4})$generated by
$a_{1}$
,
$\mathrm{a}_{2}$,
$a_{3}$,
$a_{4}$.
One
can
easily check
that
$a_{u}i’ \mathrm{s}$satisfy
$\mathrm{t}1^{-}1\mathrm{e}$quodratic relations and
braid
relations. Moreover, clearly,
$\mathrm{k}\langle T_{3}, T_{4}\rangle$is
isomorphic
to
the
group
algebra
$\mathrm{k}6_{3}$since
$T_{3}^{2}=1=T_{4}^{2}$.
By
definition, the
action of
$T_{3}$and
$T_{4}$on
$H’$is also
cleaJ.
122
of
tl.le whole algebra
$\mathcal{H}_{\mathrm{k},q,1}(F_{4})$.
The decomposition matrix of
$Lt_{l:,q}$,1$(F_{4})$is
first
calculated
by
Bremke
[Bre94, p.342].
So,
it is
worth
saying
the
correspondences
among
characters, simple modules, PIM’s,
etc
over
Bo(Hk,q{E6)
$)$alld
$B_{0}(\mathcal{H}_{\mathrm{k},q,1}(F_{4}))$.
The correspondence is given
as
follows:
/15,4
1
1
1
1
11
$\phi_{1_{0}^{r},16}\phi_{81,10}\phi_{6,2^{r}}\phi_{10,9}\phi_{90,8}\phi_{80,\tau}\phi_{81,6}\phi_{1\mathrm{s},\mathrm{s}}\phi_{1},0_{0}\phi_{61}E_{6}’$.
$11.\cdot..--.\cdot.\cdot- 111$ $11.\cdot$.
$111.\cdot$.
$11.\cdot.\cdot$ $1.\cdot.\cdot$.
$111.-\cdot.\cdot-111^{\cdot}$.
$rightarrow,,,’.\cdot\ovalbox{\tt\small REJECT}_{\phi_{96}’\ldots 1..11}^{11.1}\phi_{1,24}\phi_{2,16}\ldots.1\phi_{12}\ldots\ldots.\cdot\cdot 11\phi_{9,10}111\phi_{16,5}1111\phi_{112}’..1\phi_{8.3}’1..1\ldots 1\phi_{8,9}’111\phi_{9_{1}6}’\phi_{9,2}111\phi_{2,4}’1\phi_{1},0\overline{1}F_{4}---\cdot$
.
111
$\phi_{15,17}$ $/)_{1,\mathrm{S}36}$ $E_{6}$ $\phi$ 1,0 $\phi$ 6,1 $\phi$15,51
1
1
.
.
.
. .
.
.
$\phi$ 81,61
1
1
1
.
1
.
1
.
.
.
$\phi$ 90,8 $\phi$80,7 $\phi$ 10’91
.
1
1
.
.
1
1 1
1
.
1
1
1
.
.
$\phi$ 81,10 $\phi$ 15’16.
.
.
1
1
.
1
$\overline{1}-$1
1
.
$\phi$ 1s,17 $\phi$ 6,25 $\phi$ 1,.$\cdot$ 36.
.
.
.
.
1
1
1
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