Regular irreducible characters of a hyperspecial compact group

0.31 MB·

by  Koichi Takase

Regular irreducible characters
of a hyperspecial compact group
Koichi Takase ∗
7
1
0
2
Abstract

b
A parametrization of irreducible unitary representations associated
e
F with the regular adjoint orbits of a hyperspecial compact subgroup of
a reductive group over a non-dyadic non-archimedean local filed is pre-
5
sented. The parametrization is given by means of (a subset of) the char-
2
acter group of certain finite abelian groups arising from the reductive

group. Our method is based upon Cliffod’s theory and Weil representa-
]
T tions over finite fields. It works under an assumption of the triviality of
R certainSchurmultipliersdefinedforanalgebraic groupoverafinitefield.
. Theassumptionofthetrivialityhasgood evidencesinthecaseofgeneral
h
linear groups and highly probable in general.
t
a
m
1 Introduction
[

2 Let F be a non-dyadic non-archimedean local field. The integer ring of F is
v denoted by O with the maximal ideal p generatedby ̟. The residue class field
7
F= O/p is a finite field of q elements. Fix a continuous unitary character τ of
2
theadditivegroupF suchthat x F τ(xO) =1 =O,anddefineanadditive
1
6 character τ of F by τ(x) = τ(̟{ −∈1x).|For an inte}ger l > 0 put Ol = O/pl so
0 that F=O .
1
1. If a conbnected redbuctive quasi-split linear group G over F is split over an
0 unramified extension of F, then there exists a smooth affine group scheme G
7 overO suchthatG F =GandG FisaconnectedreductivegroupoverF.
O O
1 ⊗ ⊗
In this case the locally compact group G(F) = G(F) of the F-rational points
:
v has an open compact subgroup G(O) wich is called a hyperspecial compact
i subgroup of G(F) [19, 3.8.1]. An important problem in the harmonic analysis
X
on G(F) is to determine the irreducible unitary representationsof the compact
r
a group G(O). Such a representation π of G(O) factors through the canonical
grouphomomorphismG(O) G(O ) for some r >0 since the canonicalgroup
r
→
homomorphismis surjective due to the smoothness of the groupscheme G over
O and Hensel’s lemma, andπ is trivialonthe kernelof the canonicalgroupho-
momorphismforsomer >0. Hencetheproblemisreducedtodeterminetheset
Irr(G(O )) of the equivalence classes of the irreducible complex representations
r
of the finite group G(O ).
r
This problem in the case r = 1, that is the representation theory of the
finite reductive group G(F), has been studied extensively, starting from Green
[8] concerned with GL (F) to the decisive paper of Deligne-Lusztig [3].
n
∗Theauthor ispartiallysupportedbyJSPSKAKENHIGrantNumberJP16K05053
1
Thispapertreatsthecaser >1wherethestudyoftherepresentationtheory
ofthe finite groupG(O )isless complete. Shalika[14]treatsthe caseSL (O ),
r 2 r
Silberger [16]the case PGL (O ). Shintani [15] andG´erardin[6] treatcuspidal
2 r
representationsof GL (O ) in order to constructsupercuspidalrepresentations
n r
of GL (F). The last two papers use Clifford theory and Weil representations
n
over finite fields. In the series of papers [9, 10, 11, 12], Hill treats the case
GL (O ) systematically by means of Clifford theory, but different methods are
n r
used for representations associated with different type of adjoint orbits.
Inthispaper,wewillestablishaparametrizationoftheirreduciblerepresen-
tations of G(O ) (r > 1) associated with the regular (more precisely smoothly
r
regular) adjoint orbits. Taking a representative β of the adjoint orbit, the
parametrizationis givenby means ofasubsetofthe charactergroupofG (O )
β r
whereG isthecentralizerofβinGwhichissmoothcommutativegroupscheme
β
over O. Our theory is based on Clifford theory and Weil representations over
finite fields, and it works well under an assumption of the triviality of certain
Schur multiplier of a finite commutative group G (F). We can verify the as-
β
sumptioninthe caseofGL withn 4,andthe discussionsinthis papershow
n
≤
that the assumption is highly probable for the reductive groups in general
The main result of this paper is Theorem 2.4.1. The situation is quite
simple when r is even, and almost all of this paper is devoted to treat the case
ofr =2l 1beingodd. InthiscaseweneedWeilrepresentationoverfinitefield
−
to construct irreducible representations of G (O ) K (O ), where K (O )
β r l−1 r l−1 r
·
is the kernel of the canonical group homomorphism G(O ) G(O ), and at
r l−1
→
this point appears the Schur multiplier as an obstruction to the construction.
Hereweshallnotethatoverafinitefield,Weilrepresentationisarepresentation
ofa symplectic group,notof the 2-foldcoveringgroupofit, andthatthe Schur
multiplier is coming not from Weil representation but from certain twist which
occurs en route of connecting K (O ) with the Heisenberg group over finite
l−1 r
field (see section 3 for the details).
Several fundamental properties of the Schur multiplier will be discussed in
section 4. These properties, combined with the results of [18] in the case of
G = GL , shows that it is highly probable that the Schur multiplier is trivial
n
for all reductive group schemes over O provided that β is regular and that the
residue characteristic is big enough.
Wewillgivesomeexamplesofclassicalgroupswherethecharacteristicpoly-
nomialofβ isirreduciblemodulop. Inthiscasetheparametrizationisgivenby
a subset of the character group of unit groups of unramified extensions of the
base field F. See propositions 5.1.4 for a generallinear group,5.2.4 for a group
of symplectic similitudes, 5.3.4 and 5.3.5 for a general orthogonal group with
respectto aquadraticformofevenandoddvariablesrespectively,and5.4.4for
an unitary group associated with Hermitian form of odd variables.
2 Main results
2.1 Let G GL be a closed smooth O-group subscheme, and g the Lie
n
⊂
algebraofGwhichisaclosedaffineO-subschemeofgl theLiealgebraofGL .
n n
WemayassumethatthefibersG K (K =F orK =F)arenon-commutative
O
⊗
algebraic K-group(that is smooth K-group scheme).
For any O-algebra K, the set of the K-valued points gl (K) is identified
n
2
with the K-Lie algebra of square matrices M (K) of size n with Lie bracket
n
[X,Y] = XY YX, and the group of K-valued points GL (K) is identified
n
−
with the matrix group
GL (K)= g M (K) detg K×
n n
{ ∈ | ∈ }
where K× is the multiplicative group of K. Hence g(K) is identified with a
matrixLiesubalgebraofgl (K)andG(K)isidentifiedwithamatrixsubgroup
n
of GL (K). Let
n
B :gl gl =Spec(O[t])
n×O n →O
be the trace form on gl , that is B(X,Y) = tr(XY) for all X,Y gl (K)
n ∈ n
with any O-algebra K. The smoothness of G implies that we have a canonical
isomorphism
g(O)/̟rg(O) ˜ g(O )=g(O) O
r O r
→ ⊗
([4,Chap.II, 4,Prop.4.8])andthatthecanonicalgrouphomomorphismG(O)
§ →
G(O )issurjectiveduetoHensel’slemma. Thenforany0<l<rthecanonical
r
group homomorphism G(O ) G(O ) is surjective whose kernel is denoted by
r l
→
K (O ).
l r
For any g G(O) (resp. X g(O)), the image under the canonical surjec-
∈ ∈
tion onto G(O ) (resp. onto g(O )) with l >0 is denoted by
l l
g =g (mod pl) G(O ) (resp.X =X (mod pl) g(O )).
l l l l
∈ ∈
Since the reduction modulo p plays a fundamental role in our theory, let us use
the notation g =g(mod p) G(F) (resp. X =X(mod p) g(F)) if l =1.
∈ ∈
We will pose the following three conditions;
I) B :g(F) g(F) F is non-degenerate,
× →
II) foranyintegersr =l+l′with0<l′ l<r,wehaveagroupisomorphism
≤
g(Ol′) ˜ Kl(Or)
→
defined by X(mod pl′) 1+̟lX(mod pr),
7→
III) if r=2l 1 3 is odd, then we have a mapping
− ≥
g(O) K (O )
l−1 r
→
defined by X (1+̟l−1X +2−1̟2l−2X2l−2)(mod pr).
7→
The condition I) implies that B : g(O ) g(O ) O is non-degenerate for all
l l l
× →
l > 0, and so B : g(O) g(O) O is also non-degenerate. The mappings
× →
of the conditions II) and III) from Lie algebras to groups can be regarded as
truncations of the exponential mapping.
See section 5 for the examples of classical groups which satisfy these three
fundamental conditions of our theory.
The character group of an finite abelian group is denoted by .
G G
b
3
2.2 From now on we will fix an integer r 2 and put r = l+l′ with the
≥
smallest integer l such that 0<l′ l. In other word
≤
l :r =2l,
l′ =
(l 1 :r =2l 1.
− −
Take a β g(O) and define a character ψ of the finite abelian group K (O )
β l r
∈
by
ψ ((1+̟lX) (mod pr))=τ(̟−l′B(X,β)) (X g(O)).
β
∈
Then β(mod pl′) ψβ gives an isomorphismof the additive group g(Ol′) onto
the character grou7→p K (O ) . For any g =g(mod pr) G(O ), we have
l r r r
∈
ψ (g−1hg )=ψ (h) (h K (O )).
β r rb Ad(g)β ∈ l r
So the stabilizer of ψ in G(O ) is
β R
G(O ,β)= g G(O ) Ad(g)β β (mod pl′)
r r r
∈ ≡
n (cid:12) o
which is a subgroup of G(Or) containin(cid:12)g Kl′(Or).
Now let us denote by Irr(G(O ) ψ ) (resp. Irr(G(O ,β) ψ )) the set of
r β r β
| |
the isomorphism classes of the irreducible complex representation π of G(O )
r
(resp. σ of G(O ,β)) such that
r
hψβ,πiKl(Or) =dimCHomKl(Or)(ψβ,π)>0
(resp. ψ ,σ >0). Then Clifford’s theory says that
h β iKl(Or)
1) Irr(G(O )) = Irr(G(O ) ψ ) where is the disjoint
r r β
|
β(mGodpl′) β(mGodpl′)
union over the representatives β(mod pl′) of the Ad(G(Ol′))-orbits in
g(Ol′),
2) a bijection of Irr(G(O ,β) ψ ) onto Irr(G(O ) ψ ) is given by
r β r β
| |
σ IndG(Or) σ.
7→ G(Or,β)
So our problem is to give a good parametrization of the set Irr(G(O ,β) ψ )
r β
|
for β g(O) which is regular enough.
∈
For any β g(O), let us denote by G = Z (β) the centralizer of β in G
β G
∈
which is a closed O-group subscheme of G. The Lie algebra g = Z (β) of G
β g β
is the centralizer of β in g which is a closed O-subscheme of g.
2.3 InthissubsectionwewilldefineaSchurmultiplierwhichisanobstruction
to our theory.
Take a β g(O) such that g (F) (cid:0) g(F). Then non-zero F-vector space
β
∈
V =g(F)/g (F) has a symplectic form
β β
X˙,Y˙ =B([X,Y],β)
β
h i
4
where X˙ = X (mod g (F)) V with X g (F). Then g G (F) gives an
β β β β
∈ ∈ ∈
element σ of the symplectic group Sp(V ) defined by
g β
X (mod g (F)) Ad(g)−1X (mod g (F)).
β β
7→
Note that the group Sp(V ) acts on V from right. Let v [v] be a F-linear
β β
7→
section on V of the exact sequence
β
0 g (F) g(F) V 0. (1)
β β
→ → → →
For any v V and g G (F), put
β β
∈ ∈
γ(v,g)=γ (v,g)=Ad(g)−1[v] [vσ ] g (F).
g g β
− ∈
Take a ρ g (F) . Then there exists uniquely a v V such that
β g β
∈ ∈
ρ(γ(v,g))=τ( v,v )
b h giβ
for all v V . Note that v V depends on ρ as well as the section v [v].
∈ β g ∈ β b 7→
Let
G (F)(c) = g G(F) Ad(g)Y =Y for Y g (F)
β β
{ ∈ | ∀ ∈ }
bethe centralizerofg (F)inG(F), whichisasubgroupofG (F). Thenforany
β β
g,h G (F)(c), we have
β
∈
v =v σ−1+v (2)
gh h g g
because γ(v,gh)=γ(v,g)+γ(vσ−1,h) for all v V . Put
g ∈ β
c (g,h)=τ(2−1 v ,v )
β,ρ g gh β
h i
for g,h G (F)(c). Then the relation (2) shows that c Z2(G (F)(c),C×)
∈ β b β,ρ ∈ β
is a 2-cocycle with trivial action of G (F)(c) on C×. Moreoverwe have
β
Proposition 2.3.1 The Schur multiplier [c ] H2(G (F)(c),C×)is indepen-
β,ρ β
∈
dent of the choice of the F-linear section v [v].
7→
[Proof] Take another F-linear section v [v]′ with respect to which we will
define γ′(v,g) g and v′ V as abov7→e. Then there exists a δ V such
∈ β g ∈ β ∈ β
that ρ([v] [v]′)=τ( v,δ ) for all v V . We have v′ =v +δ δσ for all
− h iβ ∈ β g g − g
g ∈Gβ(F)(c). So if we put α(g)=τ 2−1hvg′ −vg−1,δiβ for g ∈Gβ(F)(c), then
we have b
(cid:0) (cid:1)
b
τ 2−1 v′,v′ =τ 2−1 v ,v α(h)α(gh)−1α(g)
h g ghiβ h g ghiβ ·
for all g,h G(cid:0) (F)(c). (cid:4) (cid:1) (cid:0) (cid:1)
b β b
∈
2.4 Now our main result is
Theorem 2.4.1 Suppose that a β g(O) satisfies the conditions
∈
1) G is commutative smooth O-group scheme, and
β
2) the Schur multiplier [c ] H2(G (F),C×) is trivial for all characters
β,ρ β
∈
ρ g (F) .
β
∈
b
5
Then we have a bijection θ σ of the set
β,θ
7→
θ G (O ) s.t. θ =ψ on G (O ) K (O )
β r β β r l r
{ ∈ ∩ }
onto Irr(G(O ,β) ψ ).
r | β b
The proof is givenin subsection 2.5 for even r and subsection 2.6 for odd r.
Remark 2.4.2 1) A sufficient condition for the first condition of Theorem
2.4.1 is given by Theorem 2.7.1.
2) The smoothness of G over O implies that the canonical group homomor-
β
phism Gβ(Or) Gβ(Ol′) is surjective. So we have
→
l :r =2l,
G(Or,β)=Gβ(Or) Kl′(Or), l′ = (3)
· (l 1 :r =2l 1.
− −
3) As presented in the following two subsections, the second condition in the
theorem is required only in the case of r being odd.
4) Since G F and G F are F-algebraic group, and the former is not
O β O
⊗ ⊗
commutative while the latter is, so we have
dimFg(F)=dimG OF>dimGβ OF=dimFgβ(F).
⊗ ⊗
That is g (F)(cid:0)g(F).
β
5) Since G is assumed to be commutative, we have G (F)(c) =G (F) in the
β β β
definition of the Schur multiplier [c ].
β,ρ
6) Assume that G (F)(c) is commutative. Then the cohomology class [c ]
β β,ρ
∈
H2(G (F)(c),C×)istrivialifandonlyifc issymmetric,thatisc (g,h)=
β β,ρ η,ρ
c (h,g) for all g,h G (F)(c). In fact, only if part is trivial. Let
β,ρ β
∈
1 T i j G (F)(c) 1 (4)
β
→ −→G −→ →
bethegroup extension associated withthe 2-cocyclec Z2(B (F)(c),T)
β,ρ β
where T is the subgroup of z C× such that z = 1.∈Then the groups
∈ | |
are compact commutative group, and we have a group extension of the
Pontryagin dual groups
1 G (F)(c) bj bi T 1. (5)
β
→ −→G −→ →
Since T ≃Z is free the group extebnsionb(5) isbtrivial and so is the group
extension (4).
b
2.5 Assume that r = 2l is even so that l′ = l. In this case the proof of
Theorem 2.4.1 is quite easy. Let us suppose more generally that there exists a
commutative subgroup of G(O ,β) such that
r
C
G(O ,β)= K (O ).
r l r
C·
6
Letusdenoteby thesubsetofthecharactergroup consistingoftheθ
Cβ C ∈C
such that θ = ψ on K (O ). Then any θ gives an one-dimensional
β C ∩ l r ∈ Cβ
representation σβ,θbof G(Or,β) defined by b b
b
σ (gh)=θ(g) ψ (h) (g ,h K (O )).
β,θ β l r
· ∈C ∈
Then we have a proposition of which our Theorem 2.4.1 is a special case;
Proposition 2.5.1 θ σ gives a bijection of onto Irr(G(O ,β) ψ ).
7→ β,θ Cβ r | β
[Proof] Take any σ Irr(G(O ,β) ψ ) with representationspace V . Then
∈ r | β b σ
V (ψ )= v V σ(g)v =ψ (g)vfor g K (O )
σ β σ β l r
{ ∈ | ∀ ∈ }
isanon-trivialG(O ,β)-subspaceofV sothatV =V (ψ ). Then,foranyone-
r σ σ σ β
dimensionalrepresentationχ ofG(O ,β) suchthatχ=ψ onK (O ), we have
r β l r
K (O ) Ker(χ−1 σ). On the other hand G(O ,β)/K (O ) is commutative,
l r r l r
⊂ ⊗
we have dim(χ−1 σ)=1 and then dimσ =1. Put θ =σ and we have
σ =σ . (cid:4) ⊗ |C ∈Cβ
β,θ
b
2.6 Assume that r = 2l 1 3 is odd so that l′ = l 1 1. We have a
− ≥ − ≥
chain of canonical surjections
:K (O ) K (O ) ˜ g(O ) g(F) (6)
l−1 r l−1 r−1 l−1
♥ → → →
defined by
1+̟l−1X (mod pr) 1+̟l−1X (mod pr−1)
7→
X (mod pl−1) X =X (mod p)..
7→ 7→
Let us denote by Z(O ,β) the inverse image under the surjection of g (F).
r β
♥
Then Z(O ,β) is a normal subgroup of K (O ) containing K (O ) as the
r l−1 r l r
kernel of .
♥
Let us denote by Y the set of the group homomorphisms ψ of Z(O ,β) to
β r
C× suchthatψ =ψ onK (O ). Thenabijectionofg (F) ontoY isgivenby
β l r β β
ρ ψ =ψ (ρ ),
7→ β,ρ β · ◦♥ b
where a group homomorphism ψβ :Z(Oer,β) C× is defined by
→
1+̟l−1X (mod pr) τ ̟−lB(X,β) (2̟)−1B(X2,β)
e
7→ −
with X =X(mod p) g (F). (cid:0) (cid:1)
β
∈
Take a ψ Y . For two elements
β
∈
x=1+̟l−1X (mod pr), y =1+̟l−1Y (mod pr)
ofK (O ),wehavex−1 =1 ̟l−1X+2−1̟2l−2X2(mod pr)sothatwehave
l−1 r
−
xyx−1y−1 =1+̟r−1[X,Y] (mod pr) K (O ) K (O )
r−1 r l r
∈ ⊂
and so ψ (xyx−1y−1)=τ ̟−1B(X,ad(Y)β) . Hence we have
β
ψ(xy(cid:0)x−1y−1)=ψ (xyx−(cid:1)1y−1)=1
β
7
for all x K (O ) and y Z(O ,β) so that we can define
l−1 r r
∈ ∈
D :K (O )/Z(O ,β) K (O )/Z(O ,β) C×
ψ l−1 r r l−1 r r
× →
by
D (g˙,h˙)=ψ(ghg−1h−1)=ψ (ghg−1h−1)=τ ̟−1B([X,Y],β)
ψ β
for g =(1+̟l−1X)(mod pr),h=(1+̟l−1Y)(mod(cid:0)pr) K (O ), w(cid:1)hich is
l−1 r
∈
non-degenerate. Then Proposition 3.1.1 of [18] gives
Proposition 2.6.1 For anyψ =ψ Y with ρ g (F) , there exists unique
β,ρ β β
∈ ∈
irreducible representation π of K (O ) such that ψ,π >0. Fur-
β,ρ l−1 r h β,ρiZ(Or,β)
thermore
b
dimπβ,ρ
IndKl−1(Or)ψ = π
Z(Or,β) β,ρ
and π (x) is the homothety ψ(x) for all xMZ(O ,β).
β,ρ r
∈
Fix a ψ = ψ Y with ρ g (F) . Let G (O ,β)(c) be a subgroup of
β,ρ β β l r
∈ ∈
G(O ,β) defined by
r
b Ad(g)β β (mod pl),
Gl(Or,β)(c) =(cid:26)g (mod pr)∈G(Or)(cid:12) Ad(g)X≡=Xfor∀X ∈gβ(F) (cid:27).
(cid:12)
Then, for any g =g(mod pr) G (O ,β(cid:12))(c) and x=(1+̟l−1X)(mod pr)
r l r (cid:12)
∈ ∈
Z(O ,β), we have
r
g−1xg x−1 = 1+̟l−1g−1Xg 1 ̟l−1X +2−1̟2l−2X2 (mod pr)
r r −
=(cid:0)1+̟l−1 Ad(g)−(cid:1)1(cid:0)X X (mod pr) Kl(O(cid:1)r),
− ∈
and (cid:0) (cid:1)
ψ(g−1xg x−1)=ψ (g−1xg x−1)=τ ̟−lB(X,Ad(g)β β) =1,
r r β r r −
that is ψ(g−1xg ) = ψ(x) for all x Z((cid:0)O ,β). This means th(cid:1)at, for any
g G (O ,rβ)(c),r the g-conjugate of π∈ is irsomorphic to π , that is, there
l r β,ρ β,ρ
∈
exists a U(g) GLC(Vβ,ρ) (Vβ,ρ is the representation space of πβ,ρ) such that
∈
π (g−1xg)=U(g)−1 π (x) U(g)
β,ρ β,ρ
◦ ◦
for all x K (O ), and moreover, for any g,h G (O ,β)(c), there exists a
l−1 r l r
c (g,h) ∈C× such that ∈
U
∈
U(g) U(h)=c (g,h) U(gh).
U
◦ ·
Then c Z2(G (O ,β)(c),C×) is a C×-valued 2-cocycle on G (O ,β)(c) with
U l r l r
∈
trivial action on C×, and the cohomology class [c ] H2(G (O ,β)(c),C×) is
U l r
∈
independent of the choice of each U(g).
In the next section, we will construct π by means of Weil representations
β,ρ
over the finite field F (see Proposition 3.3.1), and will show that we can choose
U(g) so that we have
c (g,h)=c (g,h)
U β,ρ
for all g,h G(O ,β)(c), where g G (F)(c) is the image of g G (O ,β)(c)
l r β l r
∈ ∈ ∈
under the canonical surjection G(O ) G(F) (see subsection 3.4).
r
→
Let us assume
8
Assumption 2.6.2 Thereexistsacommutativesubgroup G(O ,β)(c) such
l r
C ⊂
that
1) G(O ,β)= K (O ),
r l−1 r
C·
2) the cohomology class [c ] H2( ,C×) is trivial for all ρ g (F) ,
β,ρ|C×C ∈ C ∈ β
where G (F)(c) is the image of under the canonical surjection
β
C ⊂ C
G(Or) G(F). b
→
Under this assumption we have
Proposition 2.6.3 For any ρ g (F) , there exists a group homomorphism
β
∈
Uβ,ρ : GLC(Vβ,ρ) such that
C →
1) π (g−1xg)=U (g)−1 π (x) bU (g)for allg andx K (O )
β,ρ β,ρ β,ρ β,ρ l−1 r
◦ ◦ ∈C ∈
and
2) U (h)=1 for all h K (O ).
β,ρ l−1 r
∈C∩
[Proof]Becauseof2)inAssumption2.6.2thereexistsa grouphomomorphism
U : GLC(Vβ,ρ)suchthatπβ,ρ(g−1xg)=U(g)−1 πβ,ρ(x) U(g)forallg
andCx→ K (O ). Then for any h K (O ◦) there e◦xists a c(h) ∈C×C
l−1 r l−1 r
∈ ∈ C ∩ ∈
such that U(h)=c(h) π (h). On the other hand we have
β,ρ
·
G (O ,β)(c) K (O ) Z(O ,β)
l r l−1 r r
∩ ⊂
since (1+̟l−1X) G (O ,β)(c) K (O ) means that
r l r l−1 r
∈ ∩
β (1+̟l−1X)β(1+̟l−1X)−1 (mod pl)
≡
(β+̟l−1Xβ)(1 ̟l−1X) (mod pl)
≡ −
β+̟l−1[X,β] (mod pl)
≡
and then [X,β] 0(mod p), that is X(mod p) g (F). Then π (h) is the
β β,ρ
≡ ∈
homothety ψ (h) for all h K (O ). Extend the grouphomomorphism
β,ρ l−1 r
∈C∩
h c(h)ψ (h) of K (O ) to a group homomorphismθ : C×. Then
β,ρ l−1 r
g 7→U (g)=θ(g)−1CU∩(g) is the required group homomorphism.C(cid:4)→
ψ
7→
Let us denote by g (F) the set of (θ,ρ) g (F) such
C ×Kl−1(Or) β ∈ C × β
that θ = ψ on K (O ). Then (θ,ρ) g (F) defines an
β,ρ C ∩ l−1 r ∈ C ×Kl−1(Or) β
irreducible representatiobn σθ,ρ of G(Or,bβ)= Kl−1(Or) by b b
C·
b b
σ (gh)=θ(g) U (g) π (h)
θ,ρ β,ρ β,ρ
· ◦
for g and h K (O ). Then we have
l−1 r
∈C ∈
Proposition 2.6.4 Under Assumption 2.6.2, a bijection of g (F)
C ×Kl−1(Or) β
onto Irr(G(O ,β) ψ ) is given by (θ,ρ) σ .
r β θ,ρ
| 7→
b b
[Proof] Clearly π Irr(G(O ,β) ψ ) for all (θ,ρ) g (F) .
θ,ρ ∈ r | β ∈ C ×Kl−1(Or) β
Take a σ Irr(G(O ,β) ψ ). Then
r β
∈ |
b b
σ ֒ IndG(Or,β)ψ =IndG(Or,β) IndZ(Or,β)ψ
→ Kl(Or) β Z(Or,β) Kl(Or) β
= Ind(cid:16)G(Or,β)ψ (cid:17)
Z(Or,β) β,ρ
ρ∈Mgβ(F)ˆ
9
so that there exists a ρ g (F) such that
β
∈
σ ֒ IndG(Or,β)ψ =IndG(Or,β) IndKl−1(Or)ψ
→ Z(Or,β) β,ρ b Kl−1(Or) Z(Or,β) β,ρ
dimπβ,ρ (cid:16) dim(cid:17)πβ,ρ
= IndG(Or,β) π = σ ,
Kl−1(Or) β,ρ θ,ψ
θ
M M M
where is the direct sum over θ such that θ = ψ on K (O ).
β,ρ l−1 r
∈ C C ∩
θ
Then wMe have σ =σ for some (θ,ρ) g (F) . (cid:4)
θ,ρ ∈bC ×Kl−1(Or) β
Under the conditions of Theorem 2.4.1, we can put = G (O ). We have
β r
b C b
the following proposition by which our Theorem 2.4.1 is given as a special case
of Proposition 2.6.4.
Proposition 2.6.5 If G is commutative smooth over O, then (θ,ρ) θ gives
β
7→
a bijection of G (F) g (F) onto the set
β ×Kl−1(Or) β
θ G (O ) s.t. θ =ψ on G (O ) K (O ) .
β r β β r l r
{ ∈ b b ∩ }
[Proof] Take a (θ,ρ) G (F) g (F) . The smoothness of G over
∈ βb ×Kl−1(Or) β β
O implies that the canonical mapping g (O) g (F) is surjective. So Take a
β β
→
X gβ(F) with X gβ(O). Thben we have b
∈ ∈
g =1+̟l−1X +2−1̟2l−2X2 (mod pr) K (O ) G (O )
l−1 r β r
∈ ∩
so that
θ(g)=ψ (g)=τ ̟−lB(X +2−1̟l−1X2,β) 2−1̟−1B(X,β) ρ(X)
β,ρ
− ·
=τ(cid:0)̟−lB(X,β) ρ(X). (cid:1)
·
Hence we have (cid:0) (cid:1)
ρ(X)=τ ̟−lB(X,β) θ 1+̟l−1X +2−1̟2l−2X2 (mod pr) .
− ·
This means tha(cid:0)t the mapping(cid:1)(θ,ρ(cid:0)) θ is injective. Take X,X′ g (O(cid:1)) such
β
that X X′ (mod p). Then we have7→X′ =X +̟T with T g (∈O) and
β
≡ ∈
1+̟l−1X′+2−1̟2l−2X′2 (mod pr)
=1+̟−1X +2−1̟2l−2X2+̟lT (mod pr)
=(1+̟l−1X +2−1̟2l−2X2)(1+̟lT) (mod pr),
where 1+̟lT (mod pr) K (O ) and hence
l r
∈
θ(1+̟lT (mod pr))=ψ (1+̟lT (mod pr))=τ ̟−(l−1)B(T,β) .
β
(cid:16) (cid:17)
This and the commutativity of G show that
β
ρ(X)=τ ̟−lB(X,β) θ 1+̟l−1X+2−1̟2l−2X2 (mod pr)
− ·
with X g (F(cid:0)) with X g (cid:1)(O)(cid:0)gives an well-defined group homomor(cid:1)phism
β β
of g (F)∈to C×. Then (θ∈,ρ) G (O ) g (F) and our mapping in
quesβtion is surjective. (cid:4) ∈ β r ×Kl−1(Or) β
b b
10