Lecture notes in mathematics

Lecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
Subseries: Mathematisches Institut der Universit~.t und Max-Planck-lnstitut
fSr Mathematik, Bonn - vol. 7
Adviser:
F. Hirzebruch

1205
B.Z. Moroz

Analytic Arithmetic
in Algebraic Number Fields

Springer-Verlag
Berlin Heidelberg NewYork London Paris Tokyo


Author
B.Z. M o r o z
Max-Planck-lnstitut fLir Mathematik, Universit~.t Bonn
Gottfried-Claren-Str. 26, 5 3 0 0 Bonn 3, Federal Republic of G e r m a n y

Mathematics Subject Classification (1980): 11 D57, 11 R39, 11 R42, 11 R44,
11 R45, 2 2 C 0 5
ISBN 3 - 5 4 0 - 1 6 7 8 4 - 6 Springer-Verlag Berlin Heidelberg N e w York
ISBN 0 - 3 8 ? - 1 6 7 8 4 - 6 Springer-Verlag N e w York Berlin Heidelberg

Library of Congress Cataloging-in-Publication Data. Moroz, B.Z. Analytic arithmetic in algebraic
number fields. (Lecture notes in mathematics; 1205) "Subseries: Mathematisches lnstitut der
Universit&t und Max-Planck-lnstitut fur Mathematik, Bonn -vol. ? ." Bibliography: p. Includes index.

1. Algebraic number theory. I. Title. I1.Series: Lecture notes in mathematics (Springer-Verlag; 1205.
QA3.L28 no. 1205 [QA247] 510 [512'.74] 86-20335
ISBN 0-38?-16784-6 (U.S.)
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© Springer-Vertag Berlin Heidelberg 1986
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Introduction.
This book is an i m p r o v e d v e r s i o n of our memoir that a p p e a r e d in Bonner
M a t h e m a t i s c h e Schriften,

[64].

Its p u r p o s e is twofold:

first, we give

a complete r e l a t i v e l y s e l f - c o n t a i n e d proof of the t h e o r e m c o n c e r n i n g
a n a l y t i c c o n t i n u a t i o n and natural b o u n d a r y
in Chapter III of

of Euler products

(sketched

[64]) and d e s c r i b e a p p l i c a t i o n s of D i r i c h l e t series

r e p r e s e n t e d by Euler products under consideration;

secondly, we review

in detail c l a s s i c a l m e t h o d s of a n a l y t i c number theory in fields of alg e b r a i c numbers.

Our p r e s e n t a t i o n of these methods

b e e n most i n f l u e n c e d by the w o r k of E. Landau,
[24], and A. Weil,

[91]

(cf. also

[87]).

[40],

(see Chapter I) has
[42], E. Hecke,

In Chapter II we develop

f o r m a l i s m of Euler p r o d u c t s g e n e r a t e d by p o l y n o m i a l s w h o s e coefficients
lie in the ring of virtual c h a r a c t e r s of the

(absolute) Weil group of

a number field and apply it to study scalar products of A r t i n - W e i l Lfunctions.

This leads,

in particular,

to a s o l u t i o n of a l o n g - s t a n d i n g

p r o b l e m c o n c e r n i n g analytic b e h a v i o u r of the scalar products,

or con-

volutions,

[63] for

of L - f u n c t i o n s Hecke

the h i s t o r y of this problem;
C h a p t e r II, if you like).
scalar products

"mit G r ~ s s e n c h a r a k t e r e n "

(cf.

one may regard this note as a r~sum~ of

C h a p t e r III describes

a p p l i c a t i o n s of those

to the p r o b l e m of a s y m p t o t i c d i s t r i b u t i o n of integral

and prime ideals h a v i n g equal norms and to a c l a s s i c a l p r o b l e m about
d i s t r i b u t i o n of i n t e g r a l points on a v a r i e t y d e f i n e d by a s y s t e m of
norm-forms.

Chapter IV is d e s i g n e d to relate the contents of the b o o k

to the w o r k of other authors and to a c k n o w l e d g e our indebtedness

to

these authors.
should like to record here my sincere g r a t i t u d e to P r o f e s s o r P. Deligne
w h o s e remarks and e n c o u r a g e m e n t h e l p e d me to c o m p l e t e this work.
book,

as w e l l as

This

[64 ], has b e e n w r i t t e n in the quiet a t m o s p h e r e of the

M a x - P l a n c k - I n s t i t u t fur M a t h e m a t i k

(Bonn).

We are g r a t e f u l

to the

D i r e c t o r of the Institute P r o f e s s o r F. H i r z e b r u c h

for his h o s p i t a l i t y

and support of our work.

the h o s p i t a l i t y of

The author acknowledges


IV

the Mathematisches

Institut Universit~t

Z~rich, where parts of the

manuscript have been prepared.

Bonn-am-Rhein,

im M~rz 1986.

Table of contents

Chapter I.
§I.

C l a s s i c a l background.

On the m u l t i d i m e n s i o n a l a r i t h m e t i c in the sense of
E. Hecke.

p. I

§2.

G r o u p theoretic intermission,

p. 10

~3.

Weil's g r o u p and n o n - a b e l i a n L-functions.

p. 19

~4.

On c h a r a c t e r sums e x t e n d e d over integral ideals,

p. 32

§5.

On c h a r a c t e r sums e x t e n d e d over prime ideals,

p. 41

~6.

C o n s e q u e n c e s of the R i e m a n n Hypothesis.

p. 50

§7.

E q u i d i s t r i b u t i o n problems,

p. 60

A p p e n d i x I.

F r o b e n i u s classes in Well's groups,

p. 69

Appendix

Ideal classes and norm-forms,

p. 72

2.

C h a p t e r II.

Scalar p r o d u c t of L-functions.

§I.

D e f i n i t i o n and e l e m e n t a r y properties of scalar products,

p. 78

§2.

Digression:

p. 87

~3.

A n a l y t i c c o n t i n u a t i o n of Euler products,

p. 94

§4.

The natural b o u n d a r y of

p. 99

§5.

Explicit calculations

§6.

Proof of the theorem 4.2.

C h a p t e r III.

v i r t u a l characters of c o m p a c t groups,

L(s,H).

related to scalar products,

p. 107
p. 125

Ideals with equal norms and integral points
on n o r m - f o r m varieties.

§I.

O n c h a r a c t e r sums e x t e n d e d over ideals h a v i n g equal norms,

p. 141

§2.

E q u i d i s t r i b u t i o n of ideals w i t h equal norms,

p. 151

§3.

E q u i d i s t r i b u t i o n of integral points in the a l g e b r a i c
sets d e f i n e d by a s y s t e m of norm-forms.

C h a p t e r IV.

Remarks and comments.

p. 160
p. 168

L i t e r a t u r e cited.

p. 171

Index

p. 177


Notations

and conventions.

We shall use the f o l l o w i n g notations and abbreviations:
empty set
:=

"is d e f i n e d as"

A\B

the set t h e o r e t i c d i f f e r e n c e
the set of natural numbers. (including zero)
the ring of natural integers
the field of r a t i o n a l numbers
the field of real numbers

5+

the set of p o s i t i v e real numbers
the field of c o m p l e x numbers

A

the group of i n v e r t i b l e elements

in a ring

the set of all the simple

(continuous)

of a

G

(topological)

group

characters

a fixed a l g e b r a i c closure of the field
I

denotes

A

k

the unit element in any of the m u l t i p l i c a t i v e

groups to be c o n s i d e r e d
{xIP(x)}

is the set of objects

card S, or simply

IsI,

x

s a t i s f y i n g the p r o p e r t y

stands for the c a r d i n a l i t y of a finite set

is an e x t e n s i o n of number fields:
[E:F]

denotes the degree of

G(E[F)

denotes

(a)

is a p r i n c i p a l ideal g e n e r a t e d by

is the absolute

~

a finite e x t e n s i o n

norm,

that is

NE/~,

EIF

of a divisor

E

is the a b s o l u t e value of a c o m p l e x number
stand for finite sequences
characters,

Im

E ~ F

divides

in a number field

,×,k

S;

ElF

the Galois group o f

means divisor

Ixl

P(x)

fields, etc.

is the image of the map

x

(of a fixed length)

of divisors,


Vil

Ker

is the kernel of the h o m o m o r p h i s m

Re s

is the real part of

Im s

is the i m a g i n a r y part of

fog

denotes the c o m p o s i t i o n of two maps,

s

in
s

in
so that

(feg) (a) = f(g(a))

r(s)

is the Euler's g a m m a - f u n c t i o n

l.c.m.

is the least common m u l t i p l e

g.c.d.

is the g r e a t e s t c o m m o n divisor

ol H

denotes

Gc

denotes sometimes

the r e s t r i c t i o n of a map

of a (topological)
A@B
A~

B

(the closure of)
group

G

is the tensor p r o d u c t of

A

and

and

B

is the d i r e c t sum of

R e f e r e n c e s of the

P

A

form : theorem 1.2.3,

to the set

H

the c o m m u t a t o r s u b g r o u p

B

lemma 1.1, p r o p o s i t i o n 2,

c o r o l l a r y I.A2.1 m e a n theorem 3 in ~2 of Chapter I, lemma I in §I of
the s a m e chapter,
in A p p e n d i x
references

p r o p o s i t i o n 2 in the same p a r a g r a p h and c o r o l l a r y

2 of C h a p t e r I, respectively;
to n u m b e r e d formulae.

that R i e m a n n Hypothesis,

I

the same s y s t e m is used for

Relations p r o v e d under the a s s u m p t i o n

A r t i n - W e i l c o n j e c t u r e or L i n d e l ~ f H y p o t h e s i s

are valid shall be m a r k e d by the letters

R, AW, L,

respectively,

b e f o r e their number.

Every p a r a g r a p h is r e g a r d e d as a d i s t i n c t unit, a brief r e l a t i v e l y selfc o n t a i n e d article;

thus we try to be c o n s i s t e n t in our notations

out a p a r a g r a p h b u t not n e c e s s a r i l y over the w h o l e chapter.

through-

In the

first three chapters we avoid b i b l i o g r a p h i c a l and h i s t o r i c a l references
w h i c h are c o l l e c t e d in the Chapter IV.


C h a p t e r I.

C l a s s i c a l background.

§I. On the m u l t i d i m e n s i o n a l a r i t h m e t i c in the sense of E. Hecke.

Let

k

be an a l g e b r a i c number field of degree

Consider

n = [k:~].

the f o l l o w i n g objects:
v

is the ring of integers of

SI
S

and

S2

k;

are the sets of real and c o m p l e x places of

respectively,

k

= S I U $2;

SO

is the set of prime divisors of

k

i d e n t i f i e d w i t h the set of

n o n - a r c h i m e d e a n valuations;
S:= S O U S

is the set of all primes in

r j:= ISjl, j = 1,2,

so that

n = r I + 2r2;

kp

is the c o m p l e t i o n of

Up

is the g r o u p of units of

w

is the v a l u a t i o n f u n c t i o n on

P

k

k;

at
kp

p

for
for
k

P

p 6 S;
p 6 So;

n o r m a l i s e d by the c o n d i t i o n

w

(k s) = w (k ~) = ~
p E So;
P P
P
Io(k)
is the monoid of integral ideals of

k;

I(k)

k;

(~)

is the group of fractional ideals of
"'p~P(~)

=

is the p r i n c i p a l

ideal g e n e r a t e d by

in

k

,

P6S o
we extend the v a l u a t i o n f u n c t i o n

w

to

I

and w r i t e

P
for
Jk

RE

Jk/k~

K k
p~s
P

units

v~

pWp ( ~ )

PCS o

k;

is the id~le-class group of

d i a g o n a l l y in
X:=

~

I;

is the id~le group of

Ck:=

0~=

k,

where

k*

is e m b e d d e d

Jk;

is r e g a r d e d as a n - d i m e n s i o n a l

~-algebra.

The group of

acts freely as a discrete g r o u p of t r a n s f o r m a t i o n s on the

m u l t i p l i c a t i v e group

X ~,

the a c t i o n b e i n g g i v e n by

X ~+ EX r X

E X @, C C Vef


where

k

is e m b e d d e d

X

where
m

be

~

diagonally

(2Z/22Z)

rI

in

order

of

the

Obviously,

x T r2 × JR+

T = {exp(2zi~) I0 < ~ < I}
the

X.

maximal

r1+r 2

denotes

finite

,

(I)

the u n i t c i r c l e

subgroup

of

k

in

~.

Let

; by a theorem

of D i r i c h l e t ,

~
r1+r2-1
= ~

v

O n e c a n s h o w that,

x ~/m~

in a c c o r d a n c e

(2)

with

(I) and

(2) F

r

X*/v*

where

r ° _< m a x

The diagonal

~

(m/2m)

{0,ri-I}

embedding

o x JR+ x

and

~

k

into

of

~

,

is a real
X

gives

(3)

(n-1)-dimensional

torus.

r i s e to a m o n o m o r p h i s m

f : k~/v ~ ÷ X~/v ~
o

of the g r o u p of p r i n c i p a l
g: X ~ / v ~ + 7
~.

denote

The c o m p o s i t i o n

ideals

of

the n a t u r a l

k

into the g r o u p

projection

of t h e s e m a p s

gof O

m a p of

(3).

Let

X~/v ~

o n the torus

can be continued

to a h o m o m o r -

phism

f: I(k) + ~

where

Pk := k~/v~.

Let

4~

,

6 Io(k)

fip k

g'fo

a n d let

~

'

c S 1.

(4)

One defines

subgroup

I(444,) = {0Z I 0~ 6 I(k), Wp(0~)

= I

for

p[4~,

p 6 SO }

a

of

I(k)

and a s u b g r o u p

P(4~)

of

Pk'

p 6 S,

where

~ --- I (4~),

= {Ca)]0~ 6 k ~,

~

denotes

the

natural

~p(~)

embedding

of

> 0

k

for

in

P
and

~

:=

group

for

k
P

(4~,~).

H(~)

is a f i n i t e

p 6 ~

The

ray c l a s s

group

:= I (4~)/P(4~)

of o r d e r

[H(4~) I = h ~ ( 4 ~ } ,

where

h =

are

the c l a s s

number

]H(I,~)I
and

,

the class

• ( ~ ) := card

For

a smooth

defines

by

subset

T

H(I,~)

of

= I(k)/P k

group

of

k

respectively,

and

A

one

(I(44~)NPk)/P(~).

T

and a ray

class

in

two f u n c t i o n s

I(-;A,~) : JR+ + ~

,

~(.;A,Y):

,

2 + ÷ IN

letting

I(x;A,Y)

= card

{~I~6

A

N Io,

f(~)

6 T,[~

I < x}

H(~)

}


and

{x;A,T)

= card

W e are i n t e r e s t e d
(x;A,T)

studies

as

{PiP 6 A N So,

in o b t a i n i n g

x + ~.

L-functions

A 9rossencharacter

f{p)

asymptotic

E Y,

estimates

To this e n d o n e d e f i n e s

associated

modulo

with

4~ =

IpI < x}.

for

% (x;A,Y)

grossencharacters

and
and

these characters.

(4~,~)

is, b y d e f i n i t i o n ,

a character

^~

X

of

I(4#~)

for w h i c h

X{(e))

Let 4~ i =

(4#Pi,~i)

i = 1,2.

If

£ I ( ~ 2 ),

= I(R)

and

whenever

let

Xi

X

such t h a t

~ 6 k*,

be a g r o s s e n c h a r a c t e r

we write

A grossencharacter

X I < X 2.

X

if

X I <_ X

modulo

~

=

and

X I (~)

implies
(4M/,4~)

(5)

(~) E P ( ~ ) .

m o d u l o 4~i,

= X2(~)
X

X I = X.
w e call

for
is c a l l e d

a

Given a proper
~

the c o n d u c t o r

and write

One continues
X: Io(k)

a proper

u I(~(X))

To simplify

is a s u b g r o u p

g r o u p of

grossencharacter

÷ T U {O}

X ~.

> O

for

p 64~

(of f i n i t e index)
We embed

t I/n 6~R+.

~R+

X

by l e t t i n g

our n o t a t i o n s w e w r i t e

=- 1(4#~), ~p(~)

t 6JR+,

in

,

grossencharacter
X

l

4~I I4~2, 44~i _~ 4 ~ 2

proper grossencharacter

of

t h e r e is

Let
v

diagonally

The following

X(~)

e { I (~)
.

of

to a m u l t i p l i c a t i v e
=0
for

v~(~)

for

~6

~ 6 k~

function

Io(k)~I(~(X)) .
whenever

= {eI£ -= 1(~)};

it

regarded

as a t r a n s f o r m a t i o n

in

t~

X~:

(t I/n

t l/n)

t-..t

r e s u l t is a g e n e r a l i s a t i o n

of

(3).

•


Lemma

I.

The

character

X~* (4~)

group

E X* ; l(ex)

= {Ill

I (t) = I

= X(x)
for

for x e X * , e e v* ( ~ ) ;

t E ~+}

r
is i s o m o r p h i c

Proof.

to

For

(ZZ/2ZZ)

I 6 X*,

o x ZZ n-1

x 6 X*

with

tp E ~ + ,

denotes

=

I[
[
pES~ IXp

ap E ~ ;

the p - c o m p o n e n t

x,

x
P (~)

a

(6)

p

ap E {0,1}

moreover

of

r ° _< r I .

we have

it
I (x)

with

so t h a t

x

6 k
P

is e q u i v a l e n t

when
.

p E S I.
Condition

Here

xp

l(t)

= I

P

to the e q u a t i o n

Z
t = O.
p6S~ p

The

second

view
for

condi£ion

I (ex)

of the D i r i c h l e t
the e x p o n e n t s

of g e n e r a t o r s
(cf.

also

[23],

theorem

{tp,ap}.

for

X*(~).

x E X

on units,
Solving

to a s y s t e m

these

We refer

, 8 E v

for

of

equations

these

(4~)

leads,

linear

one

equations

finds

calculations

in

to

a system
[24]

X

satisfying

(5)

is s a i d

to b e n o r m a l i s e d

I E X*(4~).

Lemma
modulo

Proof.
The

for

59).

A grossencharacter
if

= I (x)

2.

For

~

every

such

See

[24]

following

Proposition

1.

1

that

in
X((~))

(cf.

also

assertion

The

X*(~)
= I(~)

[23],

c a n be

group

there

§9;

easily

is a g r o s s e n c h a r a c t e r

whenever

~ E k*

and

(~)

X
E P(4~).

[91]).
deduced

of n o r m a l i s e d

from Lemma

grossencharacters

I and Lemma

modulo

~

2.

is


isomorphic to
r
(~Z/2~)

For

x E k

P

/k
°

Let
IIxll =

x 6 Jk

~n-1

x

H(4~).

one writes

I

IlxI

x

--

and let

H IIXpll .
p6S
P

Ix]

when

p 6 SI

IXI~wp(X )

when

p 6 S2

!Pl

when

p E SO

Xp

be the p-component of

we set then

By the product formula,

I1~1f

therefore the map

x,

~+

= 1

for

0~ 6 k *

,

II~II is well defined on

C k1 =

{~1~

c k,

•

I1~11

=

C k.

Let

1}

be the subgroup of id~le-classes having unit volume.
is known to be compact.
group

{xlx 6 Jk' Xp = 1

diagonally in

X*

The group
for

X*

p E So }

The group

C kI

can be identified with the subof

Jk'

so that

may be regarded as a subgroup of

~+

C k.

embedded

It follows

then that

Ck = ~ + x Clk"

There is a natural homomorphism
by the equation

id: Jk ÷ I(k)

(7)

of

Jk

on

I(k)

given


id x =

Let

; 6 Ck'

let

~(~)

in [93], p. 133) and let
which

~

is ramified;

a character

Xp

on

p

from definitions
id-1(~)

for
2.

and

=

~(X~)

~(;)

(defined as, e.g.,

= { ~(;), £~(~)}.

for

~e

of

I(~(~)),

Jk

in

S1

at

One can define

x 6 id-1(0£),

(trivial on

is well defined

The function
~(~);

X~

~

by the equation

= ~(x)

Xp

of

since

p

k~).

It follows

is constant

on

I(~(~)).

to the restriction
particular,

x E Jk-

be set of those primes

as a character

~6

Proposition

~=(~)

write

that

for

be the conductor

I(~(~))

X~(~)

if one regards

H

pWp(Xp)
PES o

~ ~

it satisfies

of

~

to

X~

is normalised

proper grossencharacter,

X~(~)

is a proper

(5) with
(regarded

~=

~(~)

and

as a subgroup

if and only if

there

grossencharacter

of

~ + ~ Ker g.

is one and only one

~

l

equal

Jk ),
If

X

in
is a

in

Ck

such that

grossencharacters

by

gr(k)

X =X~.
Proof.

See

We denote

[91], p. 9 - 10

(or [23],

the group of proper

and remark

normalised

that

gr(k)

Proposition

I defines
Let

of the shape

(6); we call

ap = ap(X),

~ C^1k -

a fibration

conductors.

write

§9).

X 6 gr(k)

of

gr(k)

and suppose
ap,tp

tp = tp(X) •

over the set of

that

appearing

in

X

satisfies

(6) exponents

(generalised)
(5) with
of

X

and


Let now

ap(X)

+ itp(X)

,

p 6 SI

1(lap(X) I + itp(X)},
2

P 6 S2

Sp(X)=

and let

= l
" z - S / 2 F (s/2) ,
Gp(S)

For

s 6 ~,

L

P 6 SI

(27)I-SF (s) ,

X 6 gr(k)

one defines

P 6 S2

a D i r i c h l e t series

oo

L(s,x)

=

Z C (x)n -s
n= 1 n

(8)

,

where

Cn(X)

=

Z

i~I--n

X(~),

@~6 Io(k),

is a finite sum extended over the integral
equal

to

n.

The series

in this h a l f - p l a n e

it can be d e c o m p o s e d

L(s,x)

One extends

(8) converges

=

ideals of

absolutely

k

whose norm is
Re s > I

K
(I-x(p) {pl-S) -I
P6S o

= L(s,x)

and

in an Euler product:

(10) by adding the gamma-factors

A(s,x)

for

(9)

(10)

at infinite places:

K Gp(S+Sp).
peS

(11)


g

By a t h e o r e m of E. Hecke,

[24]

s~

(cf. a l s o L93],

the f u n c t i o n

$7)

A (s,x)

can be meromorphically

contihued

satisfies

equation:

a functional

VII

to the w h o l e

complex

plane

•

and

I
----S
2
A(s,x)

where

a(X)

IW(x) I = I.

=

IDI.I ~ ( x ) I,

residue
2

g(x)
of

r1+r 2 r 2
z R

D

denotes

(12)

the d i s c r i m i n a n t

of

k

and

The function

s~

where

A(1-s,~),

= W(x) a(x)

= 0

L(s, X) - e ( k ) ~ ( ~ )
s-1

for

L(s,1)

X ~ I
at

h(mTIDl)
- I , ~

and

g(1)

= I,

is h o l o m o r p h i c

in

~.

s = I

is g i v e n by the e q u a t i o n :

e(k)

=

where

R

and

m

the o r d e r of the g r o u p of roots

is the r e g u l a t o r
of u n i t y

contained

of

k

in

k ~.

The

denotes

We w r i t e ,

for b r e v i t y ,

~k(S)

= L(s,1),

~(s)

= ~(s),

and let

L

(s,x)

=

~ G (S+Sp(X)) ,
p6S
p

(I 3)

so that

A (s,x)

= L (s,x)L~ (s,x) .

(14)


§2.

Group t h e g r e t i c

Let

G

intermission.

be a compact group and let

m a l i z e d by the c o n d i t i o n
uish b e t w e e n

equivalent

loss of generality,
Let
U)

L2(G)

p(G)

p

= 1.

b~ the Haar measure on

representations

and consider,

only finite d i m e n s i o n a l

G;

for

f

and

nor-

In w h a t follows we do not disting-

g

in

as we may w i t h o u t

unitary representations.

be the Hilbert space of square integrable

functions on

G

L2(G)

(with respect to

we write

(flg) = f f(x)g(x)dp(x).

The m a t r i x elements
orthogonal

of

basis of

(unitary)

L2(G);

=I

irreducible

representations

form an

we have also

O ,

X#X'

I ,

X =X'

(×lx')

for any two irreducible
of finite index

d(H)

characters

= [G:H];

d(H)~(H)

Given a r e p r e s e n t a t i o n

X

and

X'.

H

be a subgroup

obviously,

= I.

B: H + GL(m,~),

(1)

we let

and define a r e p r e s e n t a t i o n

A: G + GL(nm,~),

by the relation

Let

n:= d(H),

B(x)

= 0

for

x 6 G%H


11

I

B(tlxt[1)

A(X) =

"'" B(tlxtnl)

1

. ......................

,

B(tnXt[1. ) ... B(tn xt-ln )

where

{tjll <_ j < n}

of

in

H

is a set of representatives for the right cosets
n
so that G = U Ht.
and Ht ~ Ht i for i ~ j. One
j=1
3
J

G,

writes

A = IndG(B);

we denote the character of

denotes

the character of

B.

G
× (x) =

Lemma I.

Proof.

by

X G,

n
Z X (tj xt-1-J
j=1

this representation

irreducible

(2)

B

representation

G

representations

in the basis of matrix elements of

can not coincide.

Since every representation

in a direct sum of irreducible

ones, we see that a character determines

its representation.

Lemma 2.

Then

Let

~

be a character of

~(H)~G(x)

H.

= f ~(yxy-1)dp(y),
G

x 6 G.

(3)

We have
n
f ~(yxy-1)dy = Z f ~(ut.xt?lu-1)du.
l l
G
i=I H

y 6 H

= ~(y)

of

(up to equivalence).

of a compact group can be decomposed

But

X

The characters of two different irreducible representations

having different decompositions

Proof.

where

Obviously,

The character of a finite dimensional

determines

A

if and only if

whenever

u 6 H.

uyu -I 6 H
Thus

for

(4) gives

u 6 H,

(4)

therefore

~(uyu -I)

12

f ,(yxy-1)dy
G
Let
to

~
H

be a c h a r a c t e r
(sometimes

Proposition
of

G.

I.

=

of

we w r i t e

Let

~

n
X
i=I

f ~(tixtil)du
H

= p(H)~G(x).

G.

We d e n o t e

~H

(~]H)

for

by

the r e s t r i c t i o n

of

~H ) .

be a c h a r a c t e r

of

H

and

let

~

be a c h a r a c t e r

Then

]l (H) <~G I q)> = <4[ ~°H>

Proof.

Write

I =

By lemma

2,

f ~ (yxy-1) ~ (x) d~ (x) dp (y) .
GxG

I = p(H)

I =

~(y-lxy)

Remark

I.

<$GI~>

= <~I~H>H,
3.

irreducible
Proof.

= ~(x)

Equation

If

Let

H

<~G I~>.

for

where

<'['>H

is an a b e l i a n
of

be a simple

= <$1~H>,

x 6 G, y 6 G.

(5), in v i e w of

representation
X

On the o t h e r hand,

f $(x)~(y-lxy)d~(x)d~(y)
G× G

since

Lemma

(5)

(I), may be r e w r i t t e n

denotes

subgroup
G

of

doesn't

character

the s c a l a r

of

G,

product

in

then d i m e n s i o n

exceed
G

as follows:

L2(H).
of an

[G:H].

and let

m

,.YH = j=l
Z ajxj,

be t h e

decomposition

of

XH

a.3 6 ~

into s i m p l e

'

characters

of

H.

Suppose

that


13

Xj ~ X i

for

j ~ i

and

that,

say,

a I > O.

By P r o p o s i t i o n

I,

<x~lx> = <xl[xH >

In v i e w

aI.

of o r t h o g o n a l i t y

Since

×

is a s i m p l e

G

Xl

On
one

the o t h e r

relations,

hand,

(H is a b e l i a n ! ) ,

character,

= alX

X~

+ ~

so t h a t

B

be

R At.B
3

= ~

Proposition

of f i n i t e

2.

Let

representation

of

p
B.

1.

(6)

because

XI

is of d e g r e e

X(1) <_ [G:H].

of f i n i t e

of

for

index

=

[G:H]

G

i ~ j.

in

index

in d o u b l e

in

G

cosets

Suppose,

C. = t_IAt.- n B,
1
1
1

is a s u b g r o u p

=

(6) g i v e s

two s u b g r o u p s

a decomposition

At.B
z

<XIt~>

is of d e g r e e

therefore

G
<XIlX>

so t h a t

we have

,

X(1)a 1 <_. [G:H],

Let
A
and
n
U At.B
be
i=I
z

< X I I X H > H = al,

and

let

modulo

moreover,

G =

(A,B),

that

1 < i < n,
---

G.

be a representation

of

A

and

let

%

be a

Then

n

IndAG(p) ~

where

~i = P i ~

ations

of

Proof.

Let

Ci,

IndG(8)

8i ' Pi:

=

Z @ Ind G (oi),
i=I
i

x + P(tixtil),

8i:

(7)

x ÷ 8(x)

are represent-

I ~ i ! n.

~ = tr

p, X = tr 8

and

~i = tr ui

be

the c h a r a c t e r s


14
the representations

O,8

and

Ci'

In view of lemma 1,

respectively.

it is enough to show that
n

G~G =

We have

~i(x)

= ~(tixtil)~(x)

~(Ci)~G(x)

with

v • B.

Taking into account

from

v,w

I,

(v,w) 6 A x B.

be the characteristic

over

(9)

identities:

v • A,

w • B

(9')

Let

x 6 A

I
and

O,

Integration

the obvious

~(wxw -I) = ~(x),

=

Ji(x) :=

By lemma 2,

= f d~ (u)$ (uxu -1) ~((wtiv)uxu -I (wtiv)-l)
G

such that

fA(x)

x • C i.

(9) an equation

(Ci)~G(x)

for any

for

(8)

= ~G~i(uxu-1)dp(u ) = [~i(vuxu-lv-1)dp(u)

G

~(vxv -I) = ~(x),

one obtains

Z ~iG "
i=I

of

A

x 6 B

O,

x ~ B

fB(x) =

x ~ A

functions

I,

and

B,

and consider

an integral

f fA (Y) fB (z) • ((YtiZ) x (YtiZ)-I) d~ (Y) dP (z) •
G~G
AXB

P (C i ) ~ ( A ) ~ ( B ) ~ ( x )

in the both sides of

(g) gives:

= f ~(uxu -1)Ji(uxu-1 )d~(u) .
G

(10)


15

Let

T.

=

l

At. B

l

and let

J. (x) =
1

gx

(u) = ~(uxu-1).

fA(y) f B ( Z ) g x ( Y t i Z ) d p (y)d~(z),
GxG

so that a t r a n s l a t i o n

y ~

YtiZ

gives

Ji (x) = S gx (u)d~(u)
T.
l
We remark

One obtains:

S fA(uz-lt?11)d~(z)B

now that

Bf fA(uz-ltil- )d~(z)

and that

~(B

N u - l A t i ) = ~ (C i)

= ~(B N u-IAt.)l

for

u E T i.

Thus

(11)

Ji (x) = ~(C i) S ~ ( u x u - 1 ) d ~ ( u ) .
T.
l
Equations

(10)

and

(11) give

~(A)~(B)~(x) = f ~(uxu-1)d~(u) S
G
since
over

~(C i) ~ O.
i

Equation

and takes

into

Proposition

If

is a one d i m e n s i o n a l

ation

(12) and

T. N T. = @

l
]

statement

Corollary

I.

Let
Suppose

It follows

(3) w h e n one sums

for

i ~ j.

be a s u b g r o u p

that

H (r)

H,

w e say

This I

that r e p r e s e n t 2, if its

from P r o p o s i t i o n

s u m of m o n o m i a l

is a p a r t i c u l a r
Hi

of

that tensor p r o d u c t

in a d i r e c t

following

I ~ j ~ r.

that

from

representation

is monomial.

are s a t i s f i e d ,

can be d e c o m p o s e d

(I 2)

2.

A = Ind,(B)

conditions

(8) follows

account

proves
B

~ (vuxu -I v -I ) d~ (v) ,

T.
1

of m o n o m i a l

representations

representations.

The

case of this o b s e r v a t i o n .
of

G

is of finite

and

let

i n d e x in

H (j) =
G

J
D Hi,
i=I

and that if


16

I <_ j < r-1,

then

G = Hj +I H(j) .

Pi = Ind G. (X i) , I < i < r,
Xi: Hi--p GL(I,~)

Proof.

Let

X

(J) =

"'" ~ P r

J

r

monomial

induced by one dimensional
r
(r))
X (r) = K (XiIH
. Then
i=I

and let

Pl ~

Consider

=

(XiIH(J)),

representations

representations

Ind G, , (x(r)) .
H~r)

I <_ j < r.

(13)

We prove that

i=I
Pl ~

for

I ~ j ~ r.

holds

For

for some

j

.--~

j = I

equation

in the interval

IndG(j) (X (j)) @

(14) is obvious.

I < j < r-1.

(14) implies

If

G

(I 4)

Since

H

2 (in view of the condition

H(J)Jj+ I = G), equation

that

Pl @

This proves

Suppose

IndGj+ I (Xj+ I) = IndG(j+1) (X (j+1))

H

by Proposition

(14)

Pj = IndG(.) (X (j))
H 3

"'" ~

(14) for any

j,

is a finite group,

E^ X(gl)x(g2 )

=

Pj+I = IndG('+1 (x(J+I))
H 3 )
in particular,

the following

I

O

we obtain

(13).

relation holds:

when

gl ~ {g2 }
•

XCG

{

IG--~

when

(15)

gl E {g2 }

l{gl}l
where
Theorem

{g} = {hgh-11h
I.

of monomial

E G}

Every character
characters.

denotes

the conjugacy

class of

g

in

G.

of a finite group is a linear combination


17

Proof.
Lemma

See, e.g.,
4.

Let

A

[83], §10.

e GL(n,~).

If

A

is s e m i s i m p l e ,

then

oo

d e t ( I - A t ) -I =

Z

tmtr(smA) ,

(16)

m=O

and

o0

det(I-At)

=

Z

(-I) m tr(AmA) t m =

m=O

n

(-I) TM tr(AmA) t m,

(17)

m=O
where
A

SmA

AmA

respectively.

mally

in

Proof.
£n }

and

denote

the m - t h

Identities

(16) and

symmetric

and exterior

(17) are u n d e r s t o o d

powers

to h o l d

of
for-

~[[t]].

Let

V

be an n - d i m e n s i o n a l

be a ~ - b a s i s

of

V.

Suppose

complex

that

d e t (1-At) -1 (Z1 ^ "'" ^ in)

=

vector

space

and let {£I.~.,

Ai.z = (~igi , I --< i _< n.

(l-At) -I£ I ^ ... ^ ( 1 - A t ) - I g n

Then

(18)

and

oo

(I-At)-I£.

Identities

(18) a n d

(16)

follows.

Z ~mtmgi,
m=O

I < i < n.

(19) give:

d e t (I -At) -I =

and

=

Identity

Z
m=O
(17)

tm

mI
mn
Z

'~I "''an
m I +... + m n = m

follows

'

f r o m the e q u a t i o n s :

(19)


18

d e t ( l + A t ) i I A ... A in =

(I+At) ZI ^ "'" ^ (1+At) in

and

(1+At)£j

=

(1+ajt)£j

,

I ! J ! n.