Publicacions
Ma emá iques,
Vol
37
(1993),
75-82
.
NORMAL
FORMS
OF
INVARIANT
VECTOR
FIELDS
UNDER
A
FINITE
GROUP
ACTION
Abs ac
FEDERICO
SÁNCHEZ-BRINGAS
Le
I'
be
a
ini e
subg oup
o
GL(n,
(C)
.
This
subg oup
ac s
on
he
space
o
ge ms
o
holomo phic
ec o
ields
anishing
a
he
o igin
in
Cn
and on
he
g oup
o
ge ms
o
holomo phic
di eomo phisms
o ((Cn, 0)
.
We
p o e
a
heo em
o
in a ian
conjugacy
o
a
no -
mal
o m
and
linea iza ion
o
he
subspace o
in a ian
ge ms
o
holomo phic
ec o
ields
and we
gi e
a
desc ip ion
o his
ype
o
no mal
o ms
in
dimension
n
=
2
.
In oduc ion
The
goal
o his
pape
is
o
show
ha
he
classic
heo ems
o
Poinca é-
Dulac
[DU]
and
Siegel
[SI]
o
conjugacy
o
a no mal o m and
linea iza-
ion
o
ge ms
o
holomo phic
ec o
ields
a 0
E
Cn
hold
o
he
quo ien
space
Cn/I',
whe e
I' is
a
ini e
subg oup
o
GL(n,
C)
.
In
his
si ua ion
we
conside
he
ge ms
o
holomo phic
ec o
ields
and
he
ge ms
o
con-
juga ing
di eomo phism
o
Cl
in a ian
by
he ac ion
o
he
subg oup
.
I
is
well
known
ha
Cn/I'
has he
s uc u e
o
an
algeb aic
a ie y
and
u he mo e
any
a ie y
which
is
he
quo ien
o
a
ini e
g oup
o
local
di eomo phisms
o
Cn
is
o his
o m
(in
a
speci ic
sys em
o
coo di-
na es)
[CA],
hen
we
ob ain he e
esul s
o
conjugacies
o
no mal
o ms
and
linea iza ions
o
ge ms
o
holomo phic
ec o
ields
in his
kind
o
algeb aic
a ie ies
.
In
a
di e en
con ex ,
like
bi u ca ion
heo y,
some imes
conjugacy
o
ano mal
o m
o
ge ms
o
holomo phic
ec o
ields
which
p ese es
symme ies
a e
needed,
his
esul s
can
also
be
applied
.
In
he
i s
sec ion
we
p o e
he
main
heo em
using
he
algeb aic
app oach
de eloped
in
[CH]
.
In
he
second
sec ion
we
analyse
ca e ully
he
case
C'/I`
and
we
gi e
a
desc ip ion
o
no mal
o ms
.
Finally
we
wish
o
hank
Xa ie
Gomez-Mon
o
líis
help ul
commen s
and
ema ks
conce ning
his
wo k
.
7
6

F
.
SÁNCHEZ-BRINCAS
1
.
In a ian
conjugacy
o
a
no mal
o m
and
linea iza ion
in
Cn
Le
X(C
n
,
0)
be
he
space
o
ge ms
o
holomo phic
ec o
ields
a
0
E
Cn
anishing
a
he
o igin
.
Le
F
be a
ini e
subg oup
o
GL(n,
C)
which
ac s
na u ally
on
X(Cn,
0),
we
say
ha
X
is
in a ian
i i is
in a ian
by
his ac ion,
namely
i
o
all ,y
E
F,
d
"
y-1(X(1'(z))
=
X(z)
.
Le
X(C
n
/F,
0)
be
he
subspace
o
in a ian
elemen s
o
X(Cn,
0)
.
Gi en
X
E
X(Cn,0)
deno e
by
X
1
i s
linea pa ,
dX(0)
and
suppose
i
belongs
o
GL(n,
C)
.
Le
S
be
he
semisimple
pa
o
X1,
we
say
X
is
a
no mal
o m
i
LSX
=
0,
whe e
L
s
is
he Lie
de i a i e
o
S
.
X1
is
said
o be
esonan
i
i s
eigen alues
sa is y a
ela ion
( esonan e)
like his
:
A
=

M
i
ñ2,
u
=
1,
. . . ,
n,
(m1,
.
.
.
,
m,,)
E
Nn,

m
i
i
2
.
1

1
The
ec o
ield
zi
1
. .
.
zm
~
aáu
,
u
=
1,
.
. . ,
n
is
he
monomial
ec o
ield
associa ed
o his
esonan e
.
Suppose
ha
X
E
X(C',
0)
is
a
no mal
o m
and
he
coo dina es
o
Cn
a e
gi en
by
a
basis
o
eigen ec o s
o
he
semisimple
pa
S
o
X1
.
Then
condi ion
LSX
=
0
implies ha
X
-
X
1is
a
sum
o
esonan
monomial
ec o
ields
.
Le
Di
(Cn,
0)
be
he
g oup
o
ge ms
o
holomo phic
di eo mo phisms
which
ix
he
o igin
o
Cn,
F
ac s
by
conjuga ion
he e
.
We
say
ha
0
E
Di (Cn,0)
is
in a ian
i i is
in a ian
by
his ac ion,
namely
i
o
all
,y
E
F,
,y-10
y
=
0
.
Deno e by
Di (Cn/F,
0)
he
g oup
o
in a ian
elemen s
o
Di (Cn,
0)
.
Fo
any
X,
Y
E
X(C
n
,
0)
we
say
X
is
conjuga e
o
Y
i
he e
is
a
E
Di (Cn,
0)
such ha
0*
X
=
Y, whe e
0*
X
=
do-
1
XO
.
When
Y
is
he
linea
pa
o
X
we
say
X
is
linea izable
.
I
X
and
Y
a e
in a ian ,
he conjugacy
(linea iza ion)
is
called
in a ian
.
Theo em

1
.
Le

F

be

a
ini e

subg oup

o
GL(n,
C)

and
X
E
X(C
n
/I',
0)
.
Suppose
he linea
pa
X
1
o
X
is
in e ible
.
Then
:
1
.1
.
X
is
in a ian ly
conjuga e,
possibly
o mally
o a
no mal
o m
.
I
X
1
is
non- esonan
hen
his
conjugacy
is
an
in a ian
linea iza-
ion
.
1
.2
.
I
X
is
holomo phically
conjuga e
o a
no mal
o m,
hen
i
can
be
conjuga e
in
an
in a ian
holomo phic
way
.
The
p oo
o
his
heo em
is
a
consequence
o
he ollowing
lemma
.
Le
OC
.,o
be
he
algeb a
o
ge ms
o
holomo phic
unc ions
a
0
E
C"
,
and
NORMAL
FORMS
OF
INVARIANT
VECTOR
FIELDS

77
m
=
{
E
0c_,o
;
(0)
=
0}
i s
maximal
ideal
.
Fo
each
non-nega i e
in ege
k
deno e
,7c-
o
=
0cn,o/m
k
he
algeb a
o
ini e
dimension
o
k-je s
o
holomo phic
unc ions
.
The
elemen
X
E
X(Cn,
0)
de ines
a
de i a ion
X*
o
Oc~,o,
X*
=
LX
and
in a
na u al
way
he
k-je
o
X
de e mines
a
de i a ion
X
*
o
,7c_
,o,
hen
X
*
has
a
canonical
decomposi ion
:
Xk
=
Sk
+
Nk
,
whe e
Sk
is
he
semisimple
pa
and
Nk
is
he
nilpo en
pa
.
A
ema kable
ac
p o ed
in
[CH]
is
ha
S*
is
a
de i a ion
.
In a
simila
way,
we
deno e
by
Di k(Cn,
0)
he
g oup
o
k-je s o
ge ms
o
holomo phic
di eomo phisms
o
Cn
.
The
de ini ion
o
conjugacy
o
a
no mal
o m
(linea iza ion)
is
ex ended
in a
na u al
way
o
he
space
o
k-je s
o
ge ms
o
ec o
ields,
Xk(Cn,o)
.
Lemma
2
.
Le
F
be a
ini e
subg oup
o
GL(n,
C),
k
a
non-nega i e
in ege
and
Xk
E
X
k
(C
n
/I',
0)
hen
2
.1
.

The
semisimple
pa
S*
o
X
*
is
in a ian
.
2
.2
.
S*
is
in a ian
linea izable
i
X
1
is
in e ible
.
P oo
.
.
2
.1
.
On
one
handwe
ha e
he ollowing
ac
[Hu]
:
Le
V
be
a
C- ec o space
o
ini e
dimension
and
T
an
endomo phism
o
V
.
Then
he
semisimple
pa o
T
has
a
polynomial
exp ession
in
T,
p(T) wi h
coe icien s
in
C
.
On
he
o he
hand,
as
X
is
in a ian
and
y
E
F
is
linea
we
ha e
d-y
-
'Xy=
y
-1
X-y
=
X
hen
o
any
non-
nega i e
in ege
k,
y
-
1
Xky
=X*
and
-y-1X%
o
.
.
.
oXky
=
X*
o
. . .
oX*
hen
any
polynomial
exp ession
in
X*
wi h
coe iicien s
in
C
is
in a ian
.
2
.2
.

Le
Ok
E
Di k(Cn,
0)
be
he
Poinca é-Dulac di eomo phism
which
exis s
because
X
1
is
in e ible
.
Ok
is
angen
((k
-
1)-o de )
o
he
iden i y
di eomo phism,
and
linea izes
he
semisimple
pa o
Xk
.
De ine
he
a e age
~k
=
1
FI
-1
1
:,,
E
,
y
-1
Oky
.
Ok
is
in a ian
and
angen
((k
-
1)-o de )
o
he
iden i y
di eomo -
phism
.
Besides
y-1
0ky
I
(S*)
=
¡FI-1
57
(y
-1
0k
1
y)*S*(y
-1
0ky)
7EF

yE
=
IFI
-1

ySiy_
1
-yEF
whe e Si
is
he
linea
pa
o
Sk which
is
in a ian
because
o
Sk
.
7 8

F
.
SÁNCHEZ-BRINGAS
P oo
o
he
heo em
:
1
.1
.
Ls,Xk
=
0
i
and
only
i
0
=
(Ls,Xk)*
=
Si
X¡*
-Xk
Si
,
hen
he
canonical
decomposi ion
o
X¡
*
implies
ha
conjuga ing o
a no mal o m
in
xk(C
,
,0)
is
equi alen
o
linea izing
he
semisimple
pa
Sk*
.
Now
le
4
be
like
in
lemma
2
.
Rema k
ha
i
we
w i e
010
. . .
0
w2
=
Id
+02
+
. . .
+
01
+
. . .
hen
01+1 0
01
0
. . .
o
02
=
(Id+02+
. . .
+~~+~
1
+
1
+
.. .
)-}-~ +1(Id+
. .
.
)+
. . .
=Id+02+
. .
.+01+
.. .
so
his
wo
composi ions
lla e
he
same
1-je
and
Ok
o
. . .
o
02 conjuga es
in a ian ly
Xk
o
a no mal o m
because
as
we
showed,
his
di eomo -
phism
linea izes
S¡
.
Finally
he
limi
limk
-w
(&
o
.
. .
o
w2)
de ines
a
di eomo phism
0,
e en ually
o mal
which
conjuga es
in a ian ly
X
o
a no mal o m
.
1
.2
.
Le
0
be
he
holomo phic
conjugacy
(in
any
sys em
o
coo di-
na es)
hen a
simila
a gumen
as in
1
implies ha
¡F¡
-1
E~,
EI
,
"
y
-
'Oy
E
Di (C',
0)
conjuga es
X
holomo phically
and
in a ian ly
o
he
espec-
i e
no mal o m
.
2
.
Desc ip ion
in
C
2
:
In a ian
no mal
o ms
Le
Xl
be
a linea
ec o
ield
in
C
2
wi h
eigen alues
>11,
1 2
.
Choose
a
base
o
C
2
,
{el,
e2}
o
eigen ec o s
o S,
he
semisimple
pa
o
Xl
.
We
say
ha
X
l
belongs
o
he
Poinca é
domain
i
0
is
no
in
he
seg-
men
[Al,
A2]
.
O he wise
we
say
Xl
belongs
o
he
Siegel
domain
.
The
eigen alues
A1,
A2
a e
o
ype
(C,
),
C,
>_
0
i
:
¡w -
mlal
-
MA21>
Qml1
+
Im2j)
-
"
o
all
(ml,
m2)
E
(N')*, u
=
1,
2
.
Be o e
applying
heo em
1
in
his
con ex
we
poin
ou
ha
condi ions
o
X
be
conjuga e
holomo phically
o a
no mal o m
lla e
been
es ab-
lished
in
[DU]
i
X
l
is
in
he
Poinca é
domainand
in
[SI]
i
X
l
belongs
o
he
Siegel
domain
.
Theo em
3
.
Le
Xl
be he linea
pa
o
X
E
x(Cn/F,
0)
.
Le
A1,
A2
be he
eigen alues
o
X
l
.
3
.1
.
I
X
l
is
no
esonan ,
hen
X
is
linea izable in
a
holomo phic
in a ian
way
in
he
ollowing
cases
:
i)
X
l
belongs
o
he
Poinca é
domain
.
ii)
Xl
belongs
o
he Siegel
domain
and
Al,
A2
a e
o
ype
(C,
)
o
some
C,
>
0
.
NORMAL
FORMS
OF
INVARIANT
VECTOR
FIELDS

79
3
.2
.
I
X
l
is
esonan ,
hen
X
is
conjuga e
in
a
holomo phic
in a ian
way
o
a
no mal
o m
in
he
ollowing
cases
:
i)
Xl
belongs
o
he
Poánca é
domain
.
ii)
X
l
belongs
o
he Siegel
domain
and
he
no mal
o m
is col-
inea
o
Xl
.
Rema k
.
The e
exis
cases
whe e
he
in a ian
conjugacy
o
a
no mal
o m
is
only o mal
.
Fo
example
i
F
is
a
diagonal
g oup
(Le
.
each
o
i s
elemen s
a e
diagonal)
we
a e
going
o
show
he e a e
no mal
oms
wi h
linea
pa
in
he
Siegel
domain
which
do
no
e i y
condi ion
2,ii)
.
In
his
case
he
conjuga ing
di eomo phism
0
may
be
di e gen
because
one
o
i s
coo dina e
unc ions
can
ha e
coe icien s
which
g ow
like
he
Eule
unc ion,
[B ]
:
00
We
exp ess
he
in a iance
condi ion
in
X(C
2
,
0)
wi h
an
a e age
mo -
phism
o
he
g oup
ac ion
.
Le
II
:
X(C
2
,
0)
-->
X(C
2
/F,
0)
be
he
mo -
phism
o
C- ec o ial
spaces
de ined
by
II(X)
_
¡I7¡-1
~
y
E
y*X
.
Then
X
is
in a ian
i
and
only
i
i(X)
=
X
.
The e
a e
wo
di e en
cases
o
he amily
o
ini e
subg oups
o
GL(2,
C)
:
i)
I
F
is
diagonal,
he
monomial
ec o
ields
a e
eigen ec o s
o
II
and
II(X)
=
0
i
X
is
no
in a ian
.
ii)
I
F
is
no
diagonalizable
he
eigen ec o s
o
II
a e
no
monomials
and
II
does
no
anish
monomials
.
Le
us
ega d
i s
he case o
diagonalizable
g oups
.
P oposi ion
4
.
I
F
is
no
diagonalizable
and
Xl
is
a linea
ec o
ield,
hen
II
(X
1)
=
Xi
i
and
only
i
Xl
=
1
(Z1,
z2)
.
P oo£
Suppose
X
l
is
gi en
in
i s
Jo dan
canonical
o m
.
I
Xl
has
di e en
eigen alues
he
condi ion
X
1
-y
=
yXl
implies
y
is
diagonal,
he e o e
F
mus
be
diagonal
.
I
X
l
y
=
-yX,
implies
bu
F
is
ini e
hen
yn
=
Id
and
b
mus
be
0
.

8
0
F
.
SÁNCHEZ-BRINGAS
Rema k
.
This
p oposi ion
implies
ha
o
non
diagonalizable
g oups
ou
heo em
is
a
linea izing
heo em
illus a ed
by
he
ollowing
example
:
Le
F
be
he
bina y
dihed al
g oup
gene a ed
by
_
a0l
and
_
(0

1l
10
-i/

11 Ól
The
ec o
ield
X(zl,
z2)
=
(zl,
z2)
+
(zi
;
z2)
is
no
a
mul iple
o
he
adial
ec o
ield'
which
belongs
o
x(C
2
/F,
0)
.
Suppose
now
he
g oup
is
diagónal
.
In
o de
o
simpli y
he
desc ip ion
o
in a ian
no mal
o ms
we
will
Suppose
F
is
cyclic
and
gene a ed
by
e
(27 li
'-1)/ni
0
e(27 l2 ~---1-)/n2
n

,
E
N,
1,,
E
Z,
(l
i
,,
n
i ,)
=
1,
u
=
1,
2
.
I
X(zl,
z2)
=
Eu=1,2(E¡~_o
CI'j
.ziz2)eu,
he
equi a iance condi ion
is
imposed
independen ly
o
monomial
ec o
ields
.
The
nex
p oposi ion
desc ibes in a ian
ec o
ields
.
P oposi ion
5
.
Le
n
be
he
leas
common
mul iple
o
n1,
n2
and
lu
=
nlu
/n
u
.
The
monomial
ec o
ield
zu(zúz )eu
,
u
=
1,
2,
u
=,A
,
i >_
-1,
j
>
0',
i
+
j
>_
0
is
in a ian
'i
and
only
i
lui
=
-?7
j
(mod
n)
.
P oo
. .
Suppose
belongs
o
F
hen
II(z1
2e
u
)
=
( 1 /
n
Z
:7E
wl,yi72)zizjeu,
he e o e
zizieu
is
in a ian
i
and
only
i
n-1
7EP
7u
1
i7z
=
1
.

I
u
=
1,
-^YE
^Yi
-1
Y2
-
k=1
e2~ ~k((li/ni)(a-1)+(12/n2)
.Í)

This
sum
is
n
i
(n/n1)lli
+
(n/n2)l2j
-
(n/n1)l1(modn)
and
anishes
o he wise
.
Simi-
la ly
o
u
=
2
.
When
X
1
belongs
o
he
Poinca é
domain
and
i s
eigen alues
a e
eso-
nan ,
hen
he
o igin,
A1 and A2
a e
colinea ,
besides
0
1
[A1, a2],
he e-
o e
he e
is
only
one
possible
ype
o
esonance
:
Au
=
m w,
u
=,/=
.
When
X
1
belongs
o
he
Siegel
domain
he
esonance
Au
=
muAu
+
m
A,
gene a es
an
in ini e
amily
o
esonan es
o
ype
:
Au
=
k((mu
-
1)Au
+
m w)
+
ñuñ
=
k((mu
-
1)Au
+
m
ñ )
+
w,
keN,u7~
.
Finally
le
us
make
he
-
ollowing
classi ica ion
:
NORMAL
FORMS
OF
INVARIANT
VECTOR
FIELDS

8 1
P oposi ion
6
.
Le
I'
be a
ini e
diagonal
cyclic
subg oup
o
GL(2,
C)
.
The
in a ian
no mal o ms
a e
:
1
.
I
X1
belongs
o
he
Poinca é
domain
.
X
(z1,
z2)
=
X1(
2
1,
z2)
+
a¡z eu
whe e
17
i
-
7
u
(modn),
u
qÉ
and
u,
=
1,
2
.
2
.
I
X1
belongs
o
he
Siegel
domain
.
X(z1,
z2)
=
X1(z1,
z2)
+
Cz1
(É
akz1iz2j/
,
z2
(E
bkz1
i
z2
j
k=1

k=1
whe e
71i
-
-772
j
(mod
n)
o
k
-
0(mod
n)
and
i,
j a e
like
in
p oposi ion
5
.
Fo
all
cases
i
l1/n1
-
l2/n2
0
Z,
hen
X1
is
diagonal
.
This
sec ion
can
be
applied
o
ob ain
conjugacies
o
no mal
o ms
in
su aces
o
ype
C
2
/P whe e
I'
is
a
ini e
subg oup
o
SU(2)
.
These
su aces
a e
embedded
in
C
3
wi h
an
isola ed
singula i y
a
he
o igin
[KLE]
.
I
I' is
diagonal,
hen
i is
cyclic
and
gene a ed
by
e
27
V
I'---
1
/n

0
0
e
-27
-
-l/n
P oposi ion
6
applies
in his
case
:
971
=
-7
72
=
1,
l1/n1
-
l2/n2
=
2/n1Zi n7~
2
.
When
I'
is
non-diagonalizable,
we
ha e
he
g oups
which
a e
he
in-
e se
image
o
he
co e ing
su jec ion
p
:
SU(2)
--->
SO(3)
o
he
g oups
o
index
2
(p ese ing o ien a ion)
o
iangula sphe ical
g oups
[MIL]
.
Re e ences
[BR]
BRJUNO,
A
.
D
.,
Analy ic o ms
o Di e en ial
Equa ions,
7yans
.
MoscowMa h
.
Soc
.
25
(1971),
131-282
.
[CA]
CARTAN,
H
.,
"Quo ien
d'un
espace analy ique
pa
un
g oupe
d'au omo phismes,"
Algeb aic
Geome y
and
Topology, P ince on
U
.P
.,
1957
.
[CHI
CHAPERON,
M
.,
"In a ian
mani olds
and
a
p epa a ion
lemma
o
local
holomo phic
lows
and
ac ions,"
Holomo phic
dynamics,
P oceedings
o
Sp inge
Ve lag,
Lec u e
No es
in
Ma h
.
1345,
1986
.
8 2

F
.
SÁNCHEZ-BRINCAS
[DU]
DULAC,
H
.,
Solu ions
d'un
sys eme
d'equa ions
di e en ielle's
dans
le
oisinage
des
aleu s
singulié es,
Bull
.
Soc
.
Ma h
.
F ance
40
(1904)
.
[HU]
HUMPHREYS,
J
.,
"In oduc ion
o
Lie
algeb as
and
ep esen a-
ion heo y,"
Sp inge
Ve lag
.
[KLE]
KLEIN,
F
.,
"Lec o es
on
he
Icosahed on
and
he
solu ion
o
he
equa ions
o
i h
deg ee,"
Teubne
1884,
Do e ,
1956
.
[MIL]
MILNOR,
J
.,
"On
he
3-dimensional
B iesko n
mani olds
M(p,
q, ),"
Ann
.
o
Ma h
.
S udies
84,
P ince on
U
.P
.,
1975
.
[SI]
SIEGEL,
C
.
L
.,
"be
die
no mal o m
analy ische
di e en ial-
gleinchungen
in
de
Nhe
eine
gleichgewich slosung,"
Nch
.
Akad
Wiss
.,
Go ingen
Ma h-Phys
.
kl,
Ma h
.-Phis-
Chem
.
Ab ,
1952,
pp
.
21-30
.
Ins i u o
de
Ma emá icas
Uni e sidad
Nacional
Au ónoma
de
México
Ciudad
Uni e si a ia
México
04510
D
.F
.
MÉXICO
P ime a
e sió
ebuda
el
30
de
Gene de
1992,
da e a
e sió
ebuda
el
13
d'Oc ub e
de
1992