E. Vesentini ( E d.)
Geometry of Homogeneous Bounded Domains Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Urbino (Pesaro), Italy, July 3-13, 1967
C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy
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ISBN 978-3-642-11059-7 e-ISBN: 978-3-642-11060-3 DOI:10.1007/978-3-642-11060-3 Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg 2011 st Reprint of the 1 Ed. C.I.M.E., Ed. Cremonese, Roma 1968 With kind permission of C.I.M.E.
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CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO (C.1.M.E.) 3" Ciclo - Urbino 5-13 Luglio 1967
GEOMETRY OF HOMOGENEOUS BOUNDED DOMAINS
Coordinatore
EdDardo Vesentini
. " v GINDIKIN,S. G.-, PJATECCKII-SAPIRO 1. 1., VINBERG E. B. :Homogeneous Kilhler manifolds pag. 1
GREENFIELD S. J. KAUP W. KORANYI A. KOSZUL J. L. MURAKAMI S. STEIN E. M.
: Extel1dibility properties of real Submanifolds of (n . pag.
89
: Holomorphie Abbildungen in Hyperbolische RauIl)e. pag.
109
: Holomorphic and harmonic functions on bounded :3ymmetric domains. pag.
125
Formes harmoniques vectorielles sur les espaces localement symetriques
pag.
197
Plongements holomorphes de domaines symetriques. pag.
261
The analogues of Fatous's theorem and estimates for maximal functions pag.
287
CENTRO INTERNAzIONALE MA TEMATICO ESTIVO (C.l.M.E.)
v ..
S.G. GINDIKIN,LI. PJATECCKII-SAPIRO, E.B.VINBERG
If
HOMOGENEOUS Kl{HLER MANIFOLDS"
Corso tenuto ad Urbino dal 5 al 13 luglio 1967
HOMOGENEOUS KAHLER MANIFOLDS (1)
by
.,
.. S, G, GINDIKIN, 1. 1. PJATECCKII-SAPIRO, E, B, YINBERG
Introduction, R e c all 0 f c e r t a i n res u It s --------------------------------~---
,
I, Definition of homogeneous K!Ihler manifolds,
Let h
=g +
i
be a positive definite Hermitian differential form on
the complex manifold M, Then differential form and
g
is a positive definite symmetric
is a non-degenerate skew-symmetric differen-
."
tial form of type (1,1), and (1)
g(x, y) = ." (Ix, y)
where
I
is the complex structure operator, The complex manifold
with the pos itive definite hermitian differenti:al form K:hlerian if one of
h
M
is called
the following equivalent conditions is satisfied
(Kl)
o
d
(K2) The parallel translation with respect to the riemannian metric g
(K3) form
preserves the complex structure of the
In local coordinates h
tang~nt
space,
i, e,
hll~
Zll, Zll the coe·fficients
of the
can be represented in the form
o2
(2)
logtp
ozeL oz~
where tp is a positive real function. The prof of the equivalence of conditions for example in
[13,27],
An automorphism ( 1 )English
(Kl) - (K3) can be found
of the K:hler manifold
Translation by Adam Koranyi,
M
is
an invertible
- 4 -
holomorphic map preserving the form
h. We shall denote the group of
Il
all automorphisms of the Kahler manifold M by G(M); its connected o component by G (M) • We stall also consider the group GA (M) of all M and the
invertible holomorphic transformations of the manifold ,group
GR (M)
of all isometries of
GR (M) the connected components of the respectively. In
and GR(M)
GA(M)
as a riemannian manifold. Then
0
0
We denote by GA (M) and groups
M
conditions are given in order that
G~(M)
[8, 12]
= GO(M).
some sufficient We are not going
to dlscuss these conditions here. However the connection between the groups
GA (M)
and
G(M)
Ii
The Kahler manifold
G(M)
acts
will be considered in certain cases. M
is called
homoge~~
if the group
transitively on it,
Often the homogeneity of a K~hler manifold is defined by the transitivit)' of the group [3]
and of Tits
GA (M). From the results of A. Borel - R. Remmert [22]
it follows that, if a compact KYhler manifold is
homogeneous in this sense, then there exists on it a K~hlerian structure (compatible with the given complex structure) with respect to which it is a homogeneous in our sense, In the non,.compact case, it is unlikely that the consideration of homogeneous complex manifolds carrying Kghlerian structures will lead to a significative classification. The simplest examples of homogeneous K~hler manifolds are the hermitian space
Il,
the complex torus
and.the unit disc
Tn, the complex projec-
tive space
pn
K in the complex plane.
In the follo-
wing three
paragraphs we shall describe three fundamental types of hoII
mogeneous Kahler manifolds which have an extremely important significance for the theory. n
In the following we shall abbreviate the words "homogeneous Kahler manifolds II by "h.K. m,
IV
- 5 -
These are nian metric
h.K. m.
IS
which have zero curvature in the Rieman-
g. The,) are easy to classify. First of all, every homogene-
ous locally flat
h.K. m. is isomorphic with. the hermitian space
Hn
In fact, by a known theorem of E. Cartan it is isomorphic to an Euclidean space as a riemannian manifold; from the rlhler condition (K2) it follows that
the complex structure is invariant under parallel transla-
tions. Any locally flat
Hn
h.K. m. can be obtained by factoring
by
some lattice. The group
o GA (M)
hull of the groups
for a locally flat h.K. m.
M is the complex
GO (M) • A maximal complex subgroup of GO (M) is the
group of parallel translations. It is transitive on
M.
These h.K. m. 's have been studied by several authors and have been com~letelJ classified (Lichnerowicz [11] [26] ). We note that Wang [26]
, Borel [2]
I
, Wang
found all simply connected complex n
homogeneous manifolds. Some of these do not admit any Kahlerian structure. We formulate the fundamental result concerning this type of h.K. m.
Let
M
be a simply connected compact
h.K. m. • Then the group
is a compact semi-simple Lie group with trivial center, its isotropy subgroup is connected and is the centralizer of a torus. Conversely, if
G is
lizer of a torus in
a connected compact Lie group and
K
is the centra-
G, then there exists an invariant Klthler structure
on the homogeneous space nitely many sub-groups
G/K. Every, complex Lie group has only fi-
(up to conjugation) that are centralizers of tori.
- 6 -
They can all be easily found. Every simply connected compact an algebraic manifold
h.K. m. M
can be realized as
in a complex projective space
way that the automorphisms of
M will be
pn
in such a
the restrictions of unitary
II
projective transformations. However the Kahler structure of M will in general be different from the K~hler structure induced by
pn.
As in the 1 )cally flat case, the group G~ (M) of the simply connected compact this case
h.K. m. M is the complex hull of GO(M)
subgroup
KA
subgroup
GO (M) • However in
has no non"trivial complex subgroups. The isotropy
G~(M)
of the group
(the group
is
KA
is connected and contains a Borel
not the complex hull of the group K
).
Let us look at a typical example. Let
G
be the group
L ni = n
K = K(n l , .•. , n s ) '
of n xn
unitary matrices. Let
be the subgroups consisting of all diagonal
block matrices of order
n l , ., ., ns • We call an (n l , .•. , ns) flag a sequence of subspaces of the hermitian space Hn of dimensions
+ n 2, .•. ,
n, n l
ns
+ ... + ns" l'
homogeneous manifold
G/K
contained successively in each other. The can be realized as the manifold of
(n l , ..• ,n s ) flags. In a natural way it is contained in a complex projective space ; the KRhler structure induced by this inclusion is invariant under the group
G . We should mention that the group effect i vely on group
G,
in this example acts non
The kernel of the action is the center of the
G/ K which
G
is
contained in
K. This is in agreement with the
gener"l theory, since the automorphisms group of a simply connected compact
h. K. m.
The group
always has a trivial center as we remarked before. GA
of all non-singular complex
is the complex hull of the group
G
n Xn
matrices
and acts analyt-ically on
G/ K'
but it does not preserve the Jd.hler structure. The isotropy subgroup of
GA is the the group
KA = KA(n 1, ... ,n s ) which consists of all
- 7-
triangular block matrices with blocks of order
n l , ••. , ns on the diago-
nal. This is how one describes (up to t'1e choice of the K~hlerian structure)
all simply connected complex
h.K. m.
which are connected with
the unitary group. For the other compact Lie groups there is an analogous construction. Matsushima
[14] proved that every compact h.K. m.
rect product of a simply connected compact
Let ce
D
be a bounded domain in
<en. The Bergman metric
n
Kahlerian
the
[1,5,27]
is a di-
h.K. m. ana a complex torus.
n-dimensional complex spa-
defines in
D
a canonical
structure. This structure is invariant under all analytic auto-
morphisms of the domain The domain
D
D
, that
is, now
GA(D)
= G(D) •
is said to be homogeneous if the group
GA (D)
is transitive on it. In the case of a homogeneous domains the coefficients of the Bergman ,metric can
be found on the basis of (2) where for
l(J
one has
to take the density bf the invariant measure. Beside the canonical KMhlerian structure there may exist other K8hlerian structures in a homogeneous domain which are invariant with respect to respect to some transitive subgroup of it. of the other fundamental types of D
the group GA (D)
GA (D)
,or with
Differently from the case
h.K. m. , for the bounded domains
does not contains any non-trivials complex
subgroup. In the following we shall abbreviate the word IIhomogeneous bounded domains n In
as
Dh. h. d. n
the unit disc
{Iz I < 1} .
In
([:2
there exist two non-isomorphic
- 8
h. b. d.
~
the complex ball
IS:
and the bi-cylinder
The non-isomorphy of these domains was proved by Poincare The non-existence of other
[4]
h. b. d. in
. He also found all h. b. d. 's The domain
z (: D
([:3
is called
D C ([:4
was shown by E. Cartan
([:2
in
sy~tric
if for every point
there exists an involutive analytic automorphism
z
for which
1S
[15].
b
z
of
D
a unique fixed point,
Every symmetric bounded domain is homogeneous and is a symmetric space, Using the classification of symmetric spaces, E, Cartan enumerated all bounded symmetric domains that for
n -:S
3
all h, b. d,
IS
[4],
In the same work he established
are symmetr lC. In connectlOn with this
he posed the problem: are all
h, b. d. 's symmetric? And if not, how
can O'1e construct them? A. Borel acted upon by
[2]
and K03zul
semi~simple
[9] showed that if a h. b. d, there is
group of analytic automorphisms
then
their domain is symmetric, The same result, with still weaker hypotheses, was proved by Hano [7) , In [16]
Pjatecckii - S~piro obtained a negati ve answer to the first
part of E, Cartanls problem. He constructed an example of a non-sym4 metric h.b.d. in ([: (We shall describe it in § 6. ) It turned out later that the symmetric domains are in a certain 9
sense exceptional among the h, b, d, 's in ([:
n
,while for every
re are only finitely many bounded symmetri-c domains is interesiin;.; that the non-symmetric
it. b. d,
IS
in
n thea
(Cn, It
arise naturally in con-
nection with the study of homogeneous fiberings of symmetric
- 9
domains In h. b. d.
IS
[ 17, 21]
~
. Pjatecckii-S~plro
[13, 19, 20]
studied in detail those
which 80mit a transitive solvable group of automorphisms
acting without fixed points. He proved that every such h. b. d.
is iso-
morphic with a non-bounded homogeneous domain, which is lJ8mogeneous under a group of affine transformations (a descript ion of these domains, so-called Siegel domains of type 1
and II, will be given in the firsT
part of these lectures) ; In the joint work
[25] bv Vinber'J Glndikin and PJ'atecckii-Sapiro ,b'
the same result was obtained without aay restrictive hypothesis. It turned out a posteriori that the condition imposed by Pjatecckii-S~piro is not really a restriction, For every h. b. d.
K~hleriall structure) in the group ble solvable subgroup exists a realization of
(wHh any homogeneous
CO (D) there is a transitive splitta-
T(D)) acting on D
D
D without fixed points. There
as a convex unbounded dom3.i n suchI that the
elements of the group T(D) are affine transformations. The group o C (D) has no center. The isotropy subgroup is a maximal compact subgroup in
CO{D), All these results were obtained in
[25]
. In these lectures we
prove SOIPe theorems about h.K. m. 's from which the result of Pjatecckii-Sapiro follows under the hypothesis that the domain
D ad-
rpits a transitive splittable solvable group of automorphisms.
5. IE~_~.!E.~::'!..~r:.~..?!_!:E.!::~~33J'__~0_~~~ ~ ~~~~ ~ _IS!~~~r:. manifolds. Every h. K m.
which has a transitive semi-simple group of automor-
ph isms admits a holomorphic fibering with a simply connected h.K. m. as its fiber,
the base of which is analitycally isomorphic with a symmetric
- 10 -
bounded domain (;£breI [2] ,Matsuhima
[14]). Every h.K.m. which
has a transitive reductive group of automorphisms decomposes into the direct product of an
h.K. m. admitting a transitive semi-simple group
of automorphisms and of a locally flat h.K. m. (Matsushima [14]) . These theorems and several other results, some of which will be discussed below, gave us the basis for the following conjecture. Fundamental conjecture.
Every homogeneous KAhler manifold
admits a holomorphic fibering, the base of which is analytically isomorphic with a homogeneous bounded domain, and the fiber, with the induced
~hler h. K. m.
structure, is isomorphic with the direct product of a locally flat and a simply connected compact h.K. m.
.
Besides the cases mentioned. above (results of
A. Borel and
Matsushima) this conjecture is essentially proved, even though this is not explicitely mentioned, in our article
[25]
for
h.K.m.
IS,
which admit
a transitive group of automorphisms on which the pre-image of the differential form TJ = 1m h (d. § 1) is the differential of some left-invariant form. In this case there is no locally flat factor in the fiber. A considerable part of these lectures will deal with the proof of the fundamental conjecture for K!ahler manifolds which admit a transitive splittable solvable group of automorphisms. This result is due to Vinberg and Gindikin. Let us make some remarks in connection with the fundamental cpnjecture. The fibering about which we have spoken is unique, since its fibers can by characterized as the maximal sets on which all bounded holomorphic functions are constant. Therefore it is preserved by all analytic automorphisms of the manifold. Furthermore the base of the fibering, being' a h. b. d. , is homeomorphic with an affine space. Consequently this fibering is topologically trivial. Its structure group is the group of invertible holomorphic map of the fiber, and is a complex Lie group (cf. § § 2 and 3). According to a theorem of Grauert phic fiberings are trivial.
[6] such holomor·
- 11 -
Therefore if the fundamental conjecture is true, then every h.K. m. is, as a complex manifold, isomorphic with the direct product of h.K. m's of the three fundamental types described in § § 2 - 4 •
PART 1- Siegel
~omains
We have already spoken in the introduction about the important role played in the theory of
h. b. d. 's by their affine homogeneous realiza-
tions. In the case of symmetric domains we usually consider their realization as
"disc".
In these realizations the isotropy group of some point of the domain consists of linear transformations.; Here we shall consider other realizations of the type of the "upper half planen in which there is a transitive group of affine transformations (this group can be interpreted as the isotropy group of a point of the boundary of the domain). In the course of this, we shall consider certain special classes of affine homogeneous domains: the Siegel domains of type I and II . In this paragraph we shall talk about the following simplest generalization of the upper half plane to the case of several complex variables. Let
V be an open convex cone in the
lR n (i.e. if
x,yt:.V, then
AX+
/lyEV
n dimensional real space for A~ O,/l~ 0, A+/lfO)
not containing any straightline The domain in
D{V) = lRn
(I)
is
[;n
called
a
Siegel domain
of
+
i
type
V I
associated
with
the
- 12 -
cone Proposition 1. Every Siegel domain of type
I
is isomorphic with
a bounded domain. Proof. The convex cone ned in some
V, containing no straight line, is contai-
n - sided angle, Making a linear transformation of
this angle can be transformed into the positive octant of : yk
VI
> 0 (k
= 1, ••• , n).
the same formulas to the domain
mn ,
IR n
If this transformation is continued by
(8n, then the domain
D(V'). The domain
D(V) becomes a part of
D(V') , being a direct product of upper
half planes, is analytically isomorphic with the n-dimensional circular poly-cylinder {IZkl
< 1, k=l, ... ,n}.
That is the domain D(V)
can
be mapped into a subset of this
p~ly-cylinder •
We introduce an auxiliary notion. We call skeleton of the domain n DC (8 a set r2D such that: every functiOh
a)
of
D)
f(z)
which is holomorphic on
and assuming its maximum modulus
ximum modulus in some point of ~ ) for every point
z0
(
in
D,
D
(the closure
reaches its ma-
r2D r2D
there exists a function,holomorphic
in D whose modulus assumes its maximum in the point
Zo
and only
there. It is clear that the skeleton
exists, D
and
that
is the set ( 1 )For
is uniquely defined,
if it
it is preserved by all automorphisms of the domain
which are holomorphic on Lemma 1.
r2D
D
The skeleton of the Siegel domain of type
D(V)
mn
Siegel dqmain of type I a more widespread name is "radiaied tube domains" •
- 13 -
Proof: a). Let the maximum modulus of the function morphic in
D
(. D , 1m z = O. o 0 Re z, = 0 • The function of one variable
be assumed at
We may assume that
the point
also
for
A
~)
z
o
will be holomorphic in the half plane reaches its maximum
f(z) holo-
for
A= 1
tRe
A
> o} and its modulus
But then it assumes its maximum
0
=0 •
From the considerations of the proof of proposition 1, it is
clear that, without restriction of generality, we may assume that the domain
D(Y) is contained in a direct product of upper half planes
zk
{1m Then for the point
f(z)
x
{1m
Let
Xo
~
zk
n}
1
will satisfy the condition
domain
= 1, ... ,
o _ 0 0 n - (xl"'" xn)(. IR ,the function
=
(The, point
0, k
>
in the definition of. the skeleton
~)
will be the unique maximum of 0,
\f(z)\
in the
k = 1, ••• , n} and therefore also in
V be a cone in IRn
0
D(V)).
having the properties mentioned abo-
ve o We denote by tions of if
lR n which preserve
V
0
A cone
V
is called homogeneous
G(V) acts transitively on V. Analogously to Siegel domain of
type
I
tions of on
G(V) the group of non singular linear transforma-
D,
Here
n
D c<e <en
,we denote by
which
then we shall
G (D) the group of affine transformaa preserve D • If the group G (D) acts transitively a
call
D a homogeneous Siegel domain of type
we shall not explain more precisely the term
lI affine
I.
homogene-
ous Siegel domai n IIsince in the case of Siegel domains one always talks
- 14 -
about homogeneity with respect to an affine group. Proposition 2. The Siegel domain of type I D(V) is homogeneous ~,
and only if,
th~
V is homogeneous.
V is
a homogeneous cone and
Proof 1. If ve group
of
automorphisms on
G(V) is a transiti-
it, then the maps of
(Cn
of the
form
z--A
(2)
where
A E:: G(V)
z+a
(more exactly, we have to take the complex continua-
tion of the linear transformations of lR n ), a E:: lR n , form a transitive group in
D(V).
2. We show the converse. Let domain of type
I,
ving
is a non-singular complex linear transformation, a E:: (Cn).
D(V) (A
Under such a map
and let
(2)
D(V) = D be a homogeneous Siegel
the skeleton
be an affine transformation preser-
f2D must be preserved • This follows
from the general fact that the skeleton is preserved by maps which are analytic on the closure of the domains. However in our case it is also enough to mention that the skeleton of the boundary of
D
containing
f2D
o ')ince the skeleton 0
preserved, the linear transformation real • Then for our automorphisms, sformation
A and the
cone
is a maximal flat component
V
A
.
1S
and the vector a must be
must be preserved. From this it A entering in (2) form a transi-
tive group of linear automorphisms of the cone examples of homogeneous
= lR n
y = 1m z is acted upon by the tran-
follows that the linear transformations
struct
f2D
V • In order to con-
Siegel domains of type I, it is suffi-
cient to construct examples of homogeneous convex cones not containing straight lines. Example 1. Consider the cone matrices
g
of
order
lines, in the space
1R
V
of symmetric positive definite
e . This is a convex cone, n
,n
=t(e+l) 2,
containing no straight
..
of symmetr1c matnces of
- 15 -
e.
order
The automorphisms of the cone are the mappings
(3)
y
where
is
g
a
~
g Y gt
non-singular matrix of order
transposed. This is
e,
and
g'
is its
the formula for the change of the matrix of a
qua-
dratic form under a change of variables. The transitivity of the group G(V)
follows form the possibility of reducing every positive definite qua-
dratic form to a sum of squares. We mention that the transitivity is preserved if we restrict ourselves in (3) to triangular (for example upper triangular) matrices with positive diagonal elements • The corresponding Siegel domain of type is usually called
I
D(V)
(in this case it
"Siegel upper half plane") consists of the complex sym-
metric matrices of order
e
with
positive definite imaginary part.
Example 2. As another example we consider the cone plex hermitian positive definite matrices of order a cone in the real space
mn
e
, n = 2
e
V of com-
,considered as
of hermitian matrices. In it
there acts transitively the group of non-singular complex matrices
g:
(4)
where
g* =
ve diagonal
gt
here the group of triangular matrices with real positi-
elements acts on
V transitively without fixed points.
The corresponding Siegel doma.in of type set
of these complex matrices
hermitian
matrix
1
2T
.
(z - z*)
z
of order
z=i E
by (5)
z
~
e
can be realized as the for which the
is positive definite.
The Siegel domains of example 1; 2, the symmetry at the point
I
- z
-1
are symmetric. In both cases,
(E is the identity matrix) is given
- 16 -
It is clear that it is sufficient to give the involution at one point. It turns out that a Siegel domain of type
I
is symmetric if, and
only if, the cone
V is self-adjoint with respect to some scalar product
(The adjoint cone
V" consists of these
<x
, y
> >
0 for
y( V , y
all
=1=
x for which the inner product 0 ) • We are not going to prove this
result here • ~xample
3. In order to construct non-symmetric homogeneous Siegel
domains of type
I, one has to construct homogeneous non-self-adjoint
cones (with respect to any scalar product) • Such cones appear first in
1R 5
0
Consider the cone in 1R 5
(6)
Yll
Y33 -
Y22
Y33 -
2 Y13 2 Y23 Y33
Its
> 0 > 0
.
0
adjoint cone is the cone of symmetric positive definite matrices
of the form
This cone is not linearly equivalent with the cone (6) • Correspondingly there exist homogeneous non-symmetric Siegel domains of type
I
in
the following paragraph) n
~
4
0
(Cn
for n ~ 5 • Let
us recall (cf. also
that non-symmetric h. b o d. 's exist in (Cn for
Let us mention also that there exist an analytic continuum of
non-isomorphic Siegel domains of type of all h. b o d. 's
I
in cen
, for
n ~ 11 (in the class
there is a continuum of non isomorphic ones for
n~
7) •
- 17 -
2. Siegel domains of type II, From the concluding remarks of the previous paragraph one can infer that not all h, b, d, IS
are isomorphic with Siegel domains of type 1.
One can get to the same conclusion from simplex considerations too, For the complex ball
with
n .::: 2 , there exists no realization as a Siegel domain of type I ,
This fact will in full
be a consequence of the results of the following
part, but let us show right now that the complex ball cannot be mapped onto a Siegel domain of type
I by a mapping which is holomorphic on
the closed ball, For the proof it is sufficient to remark that the skeleton of a Siegel domain of type
I has real dimension
n
=dim<e D,
while
the skeleton of the ball coincides' with its boundary, i. e. has real dimens ion function
2n-l 1 1
This follows from the fact that the modulus of the reaches its maximum only at the point (1,0,.",0)
zl-2 in the closed ball, and the group of unitary linear transformations acts transitively on the boundary of the ball. The ball can be mapped onto an affine homogeneous domain by setting
z = 2
"
•• 0
z
J
n
=
We obtain as image the domain 2
1m z-
(7)
>0 ,
We describe now a transitive group of affine transformations of the domain
(7), We consider the maps z .... z
(8)
u .... u
+a + +c
2i
I
Uk
~k + i I
ck 2 ,
- 18 -
where
aElR, CE
n.1
It easy to check that the domain
(7) is preserved by the mappings
(8). Besides we have the automorphisms z (9)
point
2
___ Au
u
The mappings
(y
---,\
z ( .il
>
0) •
(8), (9) generate a transitive group. In fact any
(z, u) satisfying (7) can be mapped by (8) onto a point
> 0)
(iy, 0)
, and this point can be mapped by (9) onto (i, 0) • For the
proof of the transitivity of the group of automorphisms it is enough to prove that an arbitrary point of the domain can be carried onto some given point.
(7) admits the following generalization.
The construction Let
V be a convex cone in m m n
F:
X
-+
will
Rn
not containing straight lines. The
be called a V - hermitian form if
(11)
F(u, v) =Frv,uy,
(12)
F(u, u) E V ,where
V is the closure of the cone
(13)
F(u, ul " 0 only if
u
In the case where
V ,
=0
V is
the positive half line, a
V -hermitian
form is a usual positive definite hermitian form , Siegel domain of type II to the points
V-hermitian form n
F
(z, ul, z E
D(V, F) associated to the cone V and n+m is the domain in
m
satisfying the condition
1m z - F(u, u) E V
(14)
For
n = 1 we obtain the domain (7) . The Siegel domains of type
can be considered as special cases of Siegel domains of type (m = 0) .
II
- 19 -
First of all we prove following Proposition 3 • Every Siegel domain
of tyPe
II, D(V, F) , is ana-
lytically isomorphic will some bounded domain . Proof. Since the
V-hermitian form
form if we change the cone domain
D(V, F)
V to a cone
in a domain
F becomes
D(VI, F) ,where
Viis an octant. So
V is such an octant, and
we can assume that it is the positive octant of 1R
... ,
F
n
n
(by making a linear
if necessary). In this case all components of
F: F l'
will be non-negative definite hermitian forms,
We represent each of the forms duli
VI -hermitian
VI:=J V , we can include the
we can restrict ourselves to the case where
transformation
a
Fk
as a sum of squares of mo-
of linear forms
(15)
Fk(u, u) =
L ILjk (u) 12 J
From the set of all forms
Ljk we choose a maximal set S
of linearly independent forms is equal to Ljk
m
since by (13) the forms
have a unique common zero, In the sums (15) we replace by
all forms which do
not occur in the system S ,
We denote the resulting hermitian forms by D(V, F) contains the domain variables in
<em
Fk . The
D(V, F). Choosing the forms in
we obtain that the domain
is the direct product of
n
D(V, F) in
domains of the form
cally isomorphic with the direct product of The proof
o
domain S
as new
these variables
(7), that is analyti-
n balls.
is complete.
Now we consider the question of the automorphisms of a Siegel n+m domain of type II. A Siegel domain of type II D c <e is called homogeneous if the group n+m <e which preserve D of type
II
we
G (D) of these affine transformations of a acts transitively on D In a Siegel domain
have always the analogues of the transformations (8):
- 20 -
~u z
(16)
z+a+2i F(u,c)+ i(F(c,c) , n m .... u + c (aElR ,c E: C ). -+
Before determining the general form of affine automorphisms , we study the skeleton of Siegel domains of type II • Lemma 2, The skeleton
riD
D = D(V, F) ,;onsists of those points Proof.
Let
f(z) be
dulus assume;. its maximum
of the Siegel domain of type
II
(z, u) for which 1m z = F(u, u).
a holomorphic function in
at the point
D whose
(zo' uo)E.D ,1m Zo
Using the mappings (16) , we can assume that
Uo
=0 ,
t
mo-
F(uO' uo).
Re zo = 0, i, e.
Zo = i Yo' YoEV , Yo F O. Then, just as in the proof of Lemma 1, the function (p (A) = f( A z ,0) will be holomorphic in the upper half-plane, o and its modulus assume its maximum in the point .1= 1. Using the mapping (16) it is enough to prove property
(~)
for
u = 0 • In that case, if (z, O)ED{V7F} , then zoE.TIlVf, and the funco 0 tion constructed in the proof of lemma 1 will satisfy the required conditions (it is essential that when (z, u)ED(V, F) ,then and
1m z
t
0 for
z E.D(V) ,
tOby (13) ) •
u
Now we shall study the general form of the automorphisms if a Siegel domain
of type
Proposition 4. domain
D(V, F)
II.
Every affine automorphism preserving the Siegel
of type II
is of the form
z
... A
z + a + 2i F(B u, c) +
u
... B
u+c
j
F(c, c)
(17)
where
n
aElR ,cE
m
a linear transformation of (18)
A is an automorphism of the cone ([;m
such that
AF(u, u) = F(Bu, B u) ,
v,
B is
- 21 -
Proof,
Every mapping (17) is composed of a map (16) and of
a map
p
(19)
-+ A z
~u-+Bu A
where
B satisfy condition (18) . It is clear that under this
a.nd
condition, (19) is an automorphism of the domain
D(V, F) .
Suppose that we have an affine automorphism of the domain
D(V, F)
(20)
z -+ Lll
Z
+
L12 u
+
u -+ L21
Z
+
L22 u
+ b2
Combining (20) Re
bi
=0,
with a mapping (16), we can arrange that
Therefore the point the skeleton, i, e,
b1
--+
b 2=0 ,
We shall consider mappings of this form,
Furthermore the map
(0,0)
b1 '
must preserve the skeleton
r2D '
(0,0) must be transformed into a point of
z = F(u, u) , It follows that
1m
(b 1,0) , Re b i
(20)
=
0 • So we may assume
that
b 1 = 0, since
in
(20)
=b 2 = 0 • Consider
the points
(x, 0) , x ElR
therefore their image belongs 1m L
11
to
= F(L
n
,They belong to
r2D , and
r2D ' i. e,
21
x, L x ) • 21
Since the left-hand side is a linear form, and the right-hand side is a quadratic form,
we have
Consider the points (iy, u), y = F(u, u) • This images must belong to the skele ton; i. e, (21)
- 22 -
1m L12 e
Whence
i
is independent
u
of
i. e, L12 u = 0 ,
and , since this is true for every u, then
By (21), furthermore Lll Y i. e.
our map
= LU
F(u, u)
= F(L 22
u, L22 u) ,
has the form (19), and A, B satisfy (18) . The proof is
complete, The V-hermitian form a transitive group
G
F
is called homogeneous,
of automorphisms of
g E G there exists a linear transformation
if there exists
V such that for every g
of the space
",m II.-
such that (22)
g F(u, v) = F(g u, g v) Corollary
ous if F
The Siegel domain of type
I,
and only if
II
D(V, F) is homogene-
V is a homogeneous cone and the ·V - hermitian form
is homogeneous , For this it is enough to remark that, by a map (16) the point
(z, u) E D(V, F) can be transformed into
a point (i y, 0), yE V by proposi-
tion 4 • These points must be transformable into each' other by maps of the form
(19)
Proposition
4
has the following generalization :
Proposition 5, Every non singular affine transformation mapping the Siegel domain of type II D(V, F) out a Siegel domain of type ni +m I D(V l' F 1) C ([; is of the form
II
)lU ... Az + a + 2i F 1(Bu, c) +
I where
...
Bu
+ c, is a linear transformation of the cone
- 23 -
m
V onto
V
1 '
a linear transformation
is
B
a:;
of
1 such
that A(F(u, u)) = F 1 (Bu, Bu) The proof is analogous to
4.
the proof of proposition
n+m The Siegel domains of type II D(V, F) C (C and ---n 1+ml D(V l' F 1) C a:; are affine equivalent if, and only if, n = n l' Corollary.
m =m A :
h IR
and there exist ->
IR
n1
isomorphic linear transformations
B :
ffi1
A V = V A F(u, u) =
such that
1
F 1 (Bu, Bu)
The study of homogeneous Siegel domains of type the study of homogeneous
V
II
is reduced to
V - hermitian forms for homogeneous cones
• The classification of these forms up to linear equivalence for con-
crete cones is an interesting problem of linear algebra. Let us consider the Siegel domains associated with the cones of example
and 2.
Example 4. Let V be the cone of symmetric positive definite matrices of order
e
It can always be assumed
(ef. the following para-
graph) that the map (3), where g is upper triangular matrix with positive diagonal elements, can be continued to Let us first consider the case where
e
X q
matrices
,in the sense of
a:;m is
(22)
a space of rectangular
u
We set
(23)
F(u, v) = It is clear that
F
'12
is a
(u v*
+
V
ul)
•
V -hermitian form. Let us show that
homogeneous. We consider the maps of
the cone
V :
is
- 24 -
y .... g(y) where For
is an u E ~m
= t g t
upper triangular matrix with positive diagonal elements,
we set g (u)
(24)
=t u ,
For these maps conditions (22) is satisfied, We obtain other examples of homogeneous V-hermitian forms if mo m we restrict the form (23) to subspaces ([: of the space ([; which are invariant under left multiplication by upper-triangular matrices t (mappings (24)) , For this it is necessary that the rows u 1;, •• , ue q1 q of the matrix u belong to subspaces ([; , ••• , ([; of some q1 q chain of subspaces ([; > • " > ~ of the space ([: q , It is clear that one can
([;q
choose a basis in
so that the subspace
([;
rna
consists
of step matrices of some type. (The first elements in each row vanish, and the number of these elements does not decrease when we go from one row to the following ) • It turns out that the domains
D(V, F) associated to the form (23)
are non-symmetric unless D(V, F) is a Siegel domain of type 1. We prove this in the simplest case, Let
e=
2, q = 1, let
Then we obtain the domain in
u be matrices with the second row equal to zero. ([:4 given by the following conditions:
> O'Y11-lul
(25)
2
> 0 (y .. =Im IJ
z .. ). IJ
This is the first example of a non-symmetric homogeneous bounded domain constructed by 1. I. Pjatecckii-S~piro in [16] • We give a proof that it is not symmetric.
First of all !:emIE~
we prove the following lemma:
3, The symmetry of .the Siegel domain of type II D(V, F)
- 25 -
at the point
(zo' 0) , if it exists, (z, u)
where
z
-+
-+ (
is of the form
tp (z),
t/J
(z)u)
tp (z) is the symmetry of the Siegel domain
at the point z6 ' and
t/J
depending analytically
on
(z)
is
z
a
of type
linear transformation of
I D(V) {:m
•
We mention first. that the symmetry is unique at every point
~,
(it must be the reflection in the geodesies with respect to the Bergman metric). Let the symmetry at the point (zo' 0) be
(z, u) .... ( tp(z, u), t/J (z, u»
(26)
•
It must commute with every automorphism of
D(V, F) which pre-
serves the point (zo' 0) (because of uniqueness), in particular with (z, u) .... (s, e
i8
u).
Hence tp(z, e
i8
t/J(z, e Because
i8
u) = tp (z, u) ,
u) = e
i8
t/J (z, u)
of the analyticity of tp and t/J in
a
neighborhood
of
0
with respect to u we obtain that the symmetry has the form (26). Setting
u = 0,
domain
we obtain that
D(V)
Lemma 4, ~.
at the point
z Zo
The domain
-+
tp (z)
is the symmetry of the
• (25) is non-symmetric.
The symmetry at the point
(z
= iE,
form
(27)
( z, u) .... (- z
-1
,t/J (z)u)
,
u
= 0)
must be of the
- 26 -
Under an analytic automorphism a point of the skeleton go into another point of the skeleton or to infinity, i
However under the map (27) the point of the skeleton z = ( 1 u = 1 goes into the point z = '12
(1+' _1_1i
must
S1 D
1) 1
'
-I-i) _1+iwhich does not belong to the
skeleton, Therefore there exists no symmetry at the point
(i E, 0) •
Remark, It would be possible to compute the volume element for the Bergman metric of the domain (25), and check that it is not invariant under maps of the form (27) • Example. 5 • Let V be the cone of hermitian positive definite mam , trices of orfier • We realize the space
e
complex rectangular matrices
(e X r) ,
u(l) of
type
eX q
and u(2)
of type
We set F(u, v} :: u(1) v (1)*
(28) Let
t
+
v
(2)
be an upper triangular
To the automorphisms of the cone v
--->
(2)
(complex~
matrix of order
e,
V
g(y)
we ma k e correspond the map o'f
u
t Y t*
mm \L-
(29) Condition (22) is satisfied , The corresponding domains are symmetric if one of the number q, r We denote by
uk the pair
rows of the matrices chain of subspaces a;mo
of pairs
sl
is equal
to zero,
(U~I) ,u~2)
uk E
u(I), u(2)
~. ,. ~ II::
s
) consisting of the k -th
+
a;q r , and consider the space u = (u(l) , )2) ) for which UkE,
in
• If we choose a
,t) ,
- 27 -
then
we obtain a subspace which is invariant under the maps (29). The mo restriction of the form (28) to a:: gives a homogeneous V - hermi-
tian form. Among the forms so obtained there are families of non-equivalent forms depending on certain parameters. As a result, by proposition 5 we obtain a continuum of affinely non-isomorphic homogeneous Siegel domains
of type II , By the results of the next part, they are al-
so ailalytically non - isomorphic. The simplest continuous family of non isomorphic domains is obtai-
e = 2,
ned for
of domains in
q
=:
r
f,7
=:
1,
s1
=:
2, s 2 = 1 • In this way we get a family
(ef. Part II) ,Besides this family, in (£7
there
are only finitely many analitically non isomorphic h. b. d, 's' •• We state now one of h. b. d, 's
the fundamental theorems of the theory of
in f,m
Theorem. Every homogeneous bounded domain in ly isomorphic with a homogeneous Siegel domain
<en is analytical-
of type II.
This theorem in its final form was proved by E. B. Vinberg, S. G, Gindikin and I. I, Pjatecckii-Sapiro [25] In these lectures we will
give a proof of it under certain hypothe-
ses concerning the group of automorphisms of the domain.
Let D
=:
IT(V, F)
be a homogeneous Siegel domain
Cn+m
,let
clear
that
gular
affine transformations of the space
of type II
in
G (D) be the group of affine automorphisms of D • It is a Ga{D) is a closed subgroup in the group of all non-sinf,n+m
we found the general form of the transformations
• In proposition in
4
G (D) • First of a
- 28 all we select the subgroup
N(D)
of transformations
(16) of the group
G (D) • It is immediately verified that N(D) is a normal nilpotent a subgroup (of step 2) of G (D) • The maps of the form (19) , I. e, the a automorphisms of the cone V which can be continued to a;m in the spnse
of
(22)
form a complementary sub-group
P(D) of
N(D)
in
C (D) a (30)
W,' seh'ct which
w ill
Part
in
play
= P{D)
Ga (D)
the one-parameter subgroup
• N{D)
II)
1 t
bt : (z, u) ... (e z, e
The maps
bt
their commutations
(32)
T
CfI
{bt } C P(D)
an important role in our later considerations (ef.
(31 )
(33)
G (D) a
2t
u )
commute with the maps in with
(a): (z, u)
the elements (z+a, u) ,
of
aEffi
(c): (z, u) ... (z+2iF(u, u)+iF(c, c), u
P(D). We compute
N(D) , We set n
c E
+ c),
Then
(34) Now we study the isotropy sub-group of the point Lemma 5, The isotropy sub-group G (D) a
is
a
Proof. dered to
maximal
compact subgroup
of
The compactness of the group
be a linear group, follows from
riemmanian metric Let now
at
K
Kl
p
E.
any point
D P cD in
G (D) • a K,
which can be consi-
the fad that it preserves a
(the Bergman metric; ef.
introduction) •
be some maximal compact subgroup, and
a bounded open set contained in
D • We consider the set
Kl
M be M,
which is the result of the application of the transformations from Kl to the points of M
; K1 M is a
bounded set. Its center of grav.ity
- 29 -
p
belongs
to the domain
xed point of
Kl
Kl
D
is convex, and
is a fi-
' since the center of gravity is an affine invariant"
Since the isotropy subgroup with
D because
of the point
p
is compact, it coincides
. It remains only to note that the isotropy
subgroup of the
various points are nonjugate, and the subgroups conjugate to
Kl
are
maximal compact, From lemma 5
it follows in particular
that the number of conDee",
ted components of the group
G (D) is finite, In fact a connected component of the identity in Ga (D) - must act transitively on D • Therefore every connected component of non-empty intersection with
G (D)
will
a
and hence the number
K
have of components
of
G (D) cannot be larger than the number of the components of the a compact group K. that is finite,
GO(D) coincides with the connected com~ a ponent of the identity of its own normalizer in the group G (
Proof. Let
N contains
N
be
G (D) ,Let a
cienny little from
the normalizer in question, It is clear p ED
and suppose that
the identity map, so that
that
g E. N differs suW-
g p ED. Then
g D i. e,
gEGa(D), Consequently in some neighborhood of the identity of
Ga (([;n+m)
the subgroups
identity components The subgroup
Ga(D)
and
N
coincide and therefore their
coincide. G C G (C n+m ) a
parts of the transformations in
is called
triangular if the linear
G
can be expressed by triangular man+m trices in some basis, The affine group in C can be considered in IJ'n+m+l a natural way as a linear group in \LUnder this inclusion the triangularity of an affine group is equivalent with the triangularity of the corresponding linear group,
- 30 .
It is known
[23]
that the maximal connected triangular (;ubgroups
of any linear Lie group
are conjugate. By the remark just made this
theorem is also true for affine groups. An affine group is said to be algebraic if a subgroup
it can be described as
of the full affine group whose coefficients satisfy certain
polynomial equations. Again the algebraicity of an affine groups is equivalent to the algebraicity of the linear group corresponfin to it in the sense indicated above. Let
be the comiected component of the identity of a lin
G
linear algebraic group. Then [23] (35 )
it admits the factorization formula
G = K
where
K
triangular
is
a
T
connected compact subgroup and
subgroup of the group
T
is a connected
G•
By the remarks made before the factorization
(35) is valid also
for affine groups, Let us study this factorization, First of all the subgroup
K and
T
intersect only in the identity element
(K consists
of semi- simple elements with eigen-values of modulus one, while the elements of
T
have positive eigen-values) • From this considerations it
also follows that
K is a
maximal compact subgroup and
T
is a
maximal connected triangular subgroup, Finally it can be immediately verified that
K and
T
in
(35)
may be replaced by any subgroup co-
njugate to them, i. e. we can take any maximal compact subgroup and any connected triangular subgroup, Proposition 6, domain
D
of
type
The group II
G (D) for a homogeneous Siegel a coincides with the connected component of
the identity of some affine algebraic group, Proof. We prove that the normalizer is an algebraic group . Then proposition 6 . The elements of the group
N of the group 6 will
follow from
N are given by the condition
G (D)
a lemma
• 31 -
g Taking the adjoint representation a Ad
of the group
in the Lie algebra of the corresponding affine group, this condition can be transcribed as (Ad
(36)
where
is
G (D) a
g) G (D) C G (D) a a the Lie algebra of the group
verify that the representation tion (36)
is
equivalent to
Ad
G (D) • This is to a
is rational and therefore condi-
a system of polynomial relations, From all
this we obtain Proposition 7.
Let
G (D) be the group of affine automorphisms a of a homogeneous Siegel domain of type II D C (Cn+m • Let K (D) a
be the isotropy group
of some point
p ED.
T (D) be a maximal connected triangular sub-group of the a group of the group G (D) • Then a Let
G (D) = K (D) a a
(37)
K (D) a fixed points on
where
n
T (D) = a 1;he domain
T (D) a
{ e } • The group
T (D) a
acts without
D
This last statement follows from
the fact that
T (D) a
intersects
the isotropy sub-groups only in the identity element. Remark. T (D) a
(3~)
The nilpotent normal sub-group
From proposition
4
T (D) a
wheI'€ the sub-group
it follows
N(D) is contained in
(ef. 30) that
T(V) • N(D)
T(V) is obtained by the extension in the sense
(22) of some triangular group
of automorphisms of the cone
V
•
- 32 -
The construction of the S.iegel domains of type wing generalization:
We consider the domain in (39)
U c mn
Let
D(U)
I admits the follo-
be an affine homogeneous domain.
([!n
= lRn
+
u
It is clear that the affine automorphisms of the domain
continued to
en
and together
with the translations of
rate a transitive group of affine automorphisms of an affine homogeneous domain It is usual to call
base
in
U can be
Rn they gene-
D(U). So
D(U) is
([!n
the domains of the form (39) tube domains with
U. By a theorem of Bochner the tube domain
holomorphy if and only if its base
D(U) is a domain of
U is convex.
PropositioE- 8. The tube domain
D(U) is analytically isomorphic
with a bounded domain if and only if the convex hull
U
of its base
does not contain any straight line • The sufficiency of the condition is proved in the same way as in proposition
1 . The necessity follows from the fact that
tains a straight
line, the domain
D(U)
if
'" U
con-
will contain a complex line.
Then, by Lewy's theorem, every bounded holomorphic function will have to be constant on this line. From proposition
8 it follows
Proposition 9. If
U C Rn
that the domain then
U is
D(U)
imm~diately
that
is an affine homogeneous domain such
is analytically isomorphic with a bounded domain,
a convex domain not containing any straight line.
In particular, if
U is
an
affine homogeneous domai n in
the convex hull of which contains no straight line, then
Rn
U is a convex
- 33 -
domain, To prove the sufficiency, it is enough to note that an H. B, D. is a domain of holomorphy and to use the theorem of Bochner quoted in proposition
8
Now we formulate a result concerning the conditions that a homogeneous Siegel domain of tyPe II be analytically isomorphic with a tube domain, Proposition 10. The homogeneous Siegel domain of type II D(V, F) is analytically isomorphic with a tube domain an appropriate choice of a real sub-space F(u, v) Em
(40)
n
for
all
D(U) if and only if for Vm
mm in
u, v 6m m n m in lR +
In this case one can take for
U the domain
y - F(u, u) E V
(y ElR , u Em ) •
(41)
we have
n
m
We shall not show here the necessity of these conditions, since this would require a long analysis of the group of analytic automorphisms of the domain
D(V, F) •
The sufficiency is very easy to prove. We make the analytic transformation
z ---. z - i F(u,
ii) ,
and as a result we obtain the tube domain with basis (41), We mention that it can be deduced from proposition 10 that every real convex affine homogeneous domain containing no
striaght line is
affinely equivalent with a domain of the form (41) , However this result can also be obtained directly [10, 24] , The domain of example 4 satisfies the condition (40), and therefore is analytically isomorphic with a tube domain. The domain of example 5 cannot be mapped onto a tube domain.
- 34 -
PART II
~
Linearization.
1. E!t~0.: E_~].~~!?~!: ~. The fundamental method in the theory
of h.K. m. 's
theory of Lie groups and of homogeneous spacE»
IS
algebras. We shall show here that to every h.K. m.
(just as in the
the reduction to Lie M
it is possible
to make correspond a Lie algebras with certaiL additional structures, which determines
M
uniquely up to local isomorphisms. Let
a connected transitive group of automorphisms of the manifold
M
K be the isotropy sub-group at the point
.r;;
Lie algebra of the group to
G
and by
p EM • We denote by
X its
be
G
Let the
sub-algebra corresponding
K. We consider the map n:
G-M
defined by the formula n (g) = g(p) •
(1 )
Its diffe rential
dn
tion of the Lie algebra M
at the point
operator
on
I!J onto
the point e EGis a linear the tangent space
p. The kernel of ~
where
is .3
dn
is)(.
transforma~
T of the manifold We define a linear
such that dn(jx) "I
(2)
tor
at
dn (x)
the operator of the complex structure in
determined in this
way only modulo
T . The
opera~
TC
We define furthermore the antisymmetric bilinear form
;>
on
;J
by
the formula (3)
where 71 "1m h
g (x, y)
" 71 (d n (x), d n (y))
(see § 1 of the introduction) • The quadruple {~,
defined in this way has the following properties :
TC,
j, C:f}
- 35
(K AI)
.2
- 1 (mod K
J
J K eX,
~
[k jx 1== j [k, x] (modX)
(KA 2)
+j
[jx, jyj ==) [jx, y]
(K A 3) (K A 4)
.? (k, x) = 0
(K A 5)
g (jx, jy)
(K A 6)
for
+ [x, y 1 (mod JC)
[x, jy] for
k EX;
= §l (x, y) ;
J (jx, x) > 0 for x
1. X;
,x) + g( [z, x] ,y)
J ( [x, y] , 2)+J( [ y, z]
(K A 7)
k EX;
=0 •
The property
(K AI) follows immediately from the definition of 2 and from the fact that j = - 1 • The property (K A 2) expresses He invariants of the complex structure on group. To prove it If
one has to
use
T
with respect to
the isotropy
the following well known fact:
d k is the differential of the map
k EK
at
the point
p,
then dk(dn (x) ) = dn((Ad k) x) •
(4)
The property (K A 3) structure
~m
is the integrability condition of the complex
M (see e, g. [9] ) .
The property
(K A 4) follows immediately from the definition of
5, The properties
(K A 5) and
(K A 6) express the fact that
is
Tf
the imaginary part of a positive definite hermitian form, Finally property (K A 7) means, in virtue of a known formula, that the left invariant differential form on
G which is equal to j> at
the identity, is closed, which fact in turn is equivalent to of the form iJ • In
this
the closure
way (K A 7) is the K~hler condition
(K 1)
(see § 1 of the introduction) transcribed in terms of the Lie algebra The quadruple sub-algebra X,
{ca'
n, j,
5'1
consisting of a Lie algebra
a Imear operator
j
,
;if ,
;g a
2.nd a antisymmetric bilinear
- 36 -
l'
form g on
will
K~hler algebra if it has properties
be called a
(K A 1) - (K A 7) , By abuse of language we' will often call the Lie algebra
W
a
K!hler algebra.
By construction the group
G acts effectively on
M. This however
need not always be so, If it is true, then the KHhler algebra {~,.n;, also has the following property;
(9) E
J£
j'Jl
contains no non-zero ideal of G,
A KHhler algebra satisfying this condition will be called effective. with respect to
From the invariance of the function
the isotropy
group, it follows that
5' ([k, xl , y) + J (x, [k, yl ) = 0
(5)
Vk E
)(, •
However this condition is not new • It is a consequence of (K A 7) if we set
z = k and use
(K A 4) •
In this way to every h.K. m, of automorphisms
G acting on
M
together with a transitive group
it, we have associated an effective
dIller algebra;
It is easy to see that this KRhler algebra determines the KHhler manifold
and
M
the group
G uniquely up to local isomorphisms.
Conversely, let {~, X, j, j} be an arbitrary KHhler algebra and let
G be any connected Lie group having
assume that the connected sub-group
X c
bra
~
j
as its Lie algebra. We
KeG corresponding to
, is closed. Then on the homogeneous space
there is a uniquely determined invariant to
'J
K~hler
the sub-algeM = G/K
structure associated
and J by the relations (2) and (3) • The Kllhler algebra constru-
cted from
{M, G}
will coincide with the original {~, X,
j·'.d.
2.KMhler algebras correspondin,g'to fhe three funda-----------------------------~~----------------
~_e_nJ~.!._!1..:e..~~_(!.L_h_°Y.:.,~.[~,r.:~9.~~_::!f!E1.~E.'_~_a_nJ.!9.!.~~·
For each of the three fundamental types' of (h.K. m.) (see Introduc-
- 37 -
hon) it is possible to establish conditions which characterizE' the K'\hler algebras corresponding to the h.Ko m,
of that type •
We give here these conditions and provE' their neress.ity; The ciency, except for the trivial case of a locally flat obtained from the classification
suffi~
h.K, m. , can be
of (h. K. m.) of the corresponding types
[ 14, 25]
Let group
M
be a locally flat
h.K. m., and let
G be an arbitrary
of its automorphisms containmg the group I of parallel translations.
The group
G can be factored into a semidirect product G = K,I
and its Lie algebra
~
into a semidirect sum
(6) where
J
is a commutative ideal corresponding to
I .
Conversely if the K~hler algE'bra ~ corresponding to the h.K. m. M
admits a factoring (6), then
commutative group
M is a locally flat h.K. m. since the
corresponding to the ideal
J
acts
on it
transitively. Let us consider
now
the other two fundamental t:'pes of h. K. m,
The Kahler algebras corresponding to these types of
h. K. m. have one
common property qJ(X,y)
(ND) where
"W
([x,yj)
w is some linear function on
~.
The ~hler algebras having this property will be called .9.0n"~egen~ rate (in [17 - 21, 25]
they were called
j - algebras) .
The non-degeneracy means that the form zero
in
qJ
is cohomologous to
the sense of the cohomology of the Lie algebra
J!.
In this
case it is automatically closed, i. E'. the condition (K A 7) is satisfied.
- 38 -
If
2
Wis a semi-simple Lie algebra then H (G)
= O. Therefore all semi-
-simple K~hler algebras are non-degenerate. In Koszul's work hermitian form are of
h
[9] it is shown that if the coefficients of the
, defining the Kahler structure on the h.K. m. ,
the form dlog If!
where If! is the density of the invariant measure, then the corresponding n
Kahler algebra is non-degenerate. In particular this is true for h. b. d. 's (using the classification of
having a Bergman metric
h. b. d. IS [25) one can show the non-degenera-
cy of the K'ahler algebra corresponding to a h. b. d, with any ture,
cf~
lemma 1 in § 1 of part
K~hler struc-
3 ).
n
Thus the Kahler algebras corresponding to simply connected compact h,K. m, 's and also those corresponding to h, b. d,ts are non-degenerate, On the other hand from the decomposition (6) one sees that in a Ifahler algebra corresponding to a locally flat bilinear function
J
having properties
h,K, m, , every skew symmetric (NL) and (K A 4) , is zero.
In this sense these .Klhler algebras are ftmaximally degenerateil, In order to get a property which distinguishes the corresponding to simply connected compact algebras corresponding to
K~hler
algebras
h, k. m, 's from the K£hler
h. b, d, 's we introduce the notion of a K~hler
sub-algebra. Let
{~,)("
r;;
algebra of
j
's } be j
On the space x
.£<:"ahler algebra and let
~1 be a sub-
satisfying the condition
(7)
j x == j 1
a
G C G 1 1
~1
(mod ')(,) for
+
one can all
x
K
define an
E~l
operator
jl
such that
- 39 -
We define the bilinear
J
the form
and we set
{~l'Xl' \' {~,
X.
j
rd
form
Xl = X n ~1
S1on0 1 UJ
as the restriction of
• It is easy to see that
is a I&hler algebra. It is called a
J.}
The Kl:ihler algebra
Ij, f} is called
{~I X
K~hler sub-algebra
prOper if it has the fol-
lowing property: II
(R) Every compact semi-simple Kahler sub-algebra of CJ is contained
in
K. II
It is clear that the Kahler algebra corresponding to a simply connec-
ted compact
h. K. m. is
II maximally
non-proper ll since it is itself semi-
simple and compact. II
On the other hand the Kahler algebras corresponding to
h. b. d. 's
are proper. In fact h. b. d.
let
{w' X,
j,S}
be the Id.hler algebra corresponding to the
D_ with the tranSiti:e group of automorphisms
G, and
let
~1
be a compact semi-simple Kahler sub-algebra in it. We denote sponding
to
by
~1
(the isotropy group
G1 the connected compact sub-group of G corre-
and we consider the o;rbit of which is
G1p of the point
K). This orbit is a compact complex
sub-manifold of D. The coordinate functions restricted to be constant,
therefore
G1P
=
p ED
{pl. This means that
G1P
must
G1 e K and
~ 1 eX which has to be proved.
§3. ~~~~~~~~~~_~!~~E£~~~~~_~~~~g~~~0~~~2~~~j£~~~~_ Let us
consider the structure of a Kihler algebra corresponding
to a homogeneous Siegel domain vex cone
V e ffin and to the
of type II D(V, F)
V-hermitian
form
associatErl to the con. m • F m q:;
- 40 -
First of all
we describe the infinitesimal affine transformations
corresponding to the affine automorphisms of the domain
D(V, F) •
This can be obtained from the formula (17) of the first part by differentiating with respect to the parameters of the group • We obtain the following infinitesimal affine transformations (x+iy, u)
(8 )
(x+iy, u)
(9 )
-+
(x+iy, u)
(10)
Where in (8) A group of the cone to A
n
(a, 0)
(a ElR )
(2i F(u, c), c)
is an and
V
(Ax+iAy, Bu) ,
-+
(CE::([:
m
)
element of the Lie algebra of the automorphism B
is a linear transformation of
by the condition F(Bu, v) + F(u, Bv)
AF(u, v)
To the one-parameter group {b t} (see the first part) there corresponds the infinitesimal transformation g : (x+iy, u) o
( 12)
->
(x+iy,
2:. u) , 2
so that b Let now G domain
t
=
exp
t g
0
be any transitive group of affine automorphisms of the
D(V, F) containing all maps
(16) of the first part and also the
one parameter group {b t }. We denote bY:J its Lie algebra and by ~0
'
~ 1 and ~
2:. the
sub~spaces
2 the transformations (8), (9) and It is clear that
if + ~. = ~o + (11
of the algebra ~ (10) respectively.
corresponding to
- 41 -
and
g E. o
(}J
{/o
,Taking the commutator
of
g
with the infinitesimal
0
transformations (8), (9) and (10) we see that· ~A( "0 )1)1/ 2) is the eigenspace of particular
ad
corresponding to the eigenvalue A • Whence in
~0
it follows that
(14) In order to define to operator pick a base point
~0
the algebra:f we have to
p D(V, F) • We choose p
where
in
j
= (i ~o'
0 )
is sorre point of the cone V
Computing
'by the formula (2)
we find
that
(15 ) (more exactly one can arrange this to be true using the arbitra'riness in the definition
of
j) "
Since all affine automorphisms of p
are of the form
{19} of part
Lemma. The element
b0'
O
I, of the
D(V, F)
we
have II
preserving the point Ii
C C!J 0
Kahler algebra
~
is uniquely
determined by the following conditions 1) The operator
ad g
o
is semi-simpl~ and its eigenvalues are
0,1,1/ 2 2) value /\;
Let egA ,be the eigenspace of
ad go corresponding to the eigen-
then,for appropriate choice of
3) Proof.
Let go be another element possessing the properties listed
- 42 -
in the lemma. From these properties it follows that if the subspace W C ~ is invariant under will
be invariant
ad
'If 1
then also the subspace
(for any choice of
We keep the notation subspace
go
the eigenspaces of ad
iff
0
is invariant under
go=-~o'
[go,goJEWo' Hence it follows that
goj
=j
Since j,
which
j)
go' By (14) the
is an ideal and consequently it is invariant under
Therefore also
[ go' j
j V'fEJi,
ad
ad
go'
go' In particular
jgoE~l'
but then
i.e.
go .
go and
go can be interchanged except for the choice of
depends on
[g, j g 1== j g
g, we have o
Now we write down
0
0
the integrability condition
(mod X
0
(K A 3)
)
for
the
elements
whence
The precise equality of
and iJ' follows from o °0 their eigenvalues. The lemma is proved. g
the reality of
We have (16)
go The element
algebra
is
=j
s
SE9J 1
where
•
called the principal idempotent of the K~hler
'/J'
The corresponding infinitesimal transformation (17)
s: (x
~roposition
+i
y, u)
-+
(y
o
is of the
form
,0) •
1: The homogeneous Siegel domain of type II D(V, F)
is uniquely determined by its
KRhler algebra
which
is constructed
from a transitive group of affine automorphisms containing all transformations (16) of part
1 and
the one-parameter group
j
bt } •
- 43 -
Proof: We show how the cone structed from
the algebra
'!J • To
infinitesimal transformation
V and the form F
each
a E ERn we make correspond the
(9) •
In this way we get an isomorphic map of)R
We denote it by SAB
T ,
can be recon-
Formula (17) shows that
T
11
onto the space
lJ l'
(y ) = S , We denote by o
the infinitesimal transformation (8) , An immedi.ate computation
shows that (18)
therefore the diagram
(19)
is commutative. The cone
V is
the orbit of
y
o
for
rated by the, infinitesimal transformations Consequently the cone
T
(V)
the action of the group geneA,
is the
orbit of the point
under the' group generated by the infinitesimal transformations This determines a Cone
T
(V)
T
(Yo)
ad
and therefore also the cone
=S SAB'
V
r.; ,
isomorphic with it in terms of the algebra Making correspond to m every c E(C the infinitesimal transformation (10) we obtain an isomorphic map
of the vector space
m
C
onto the space
te it by'll, Computing some commutators we immediately see that
From
the hermiticity of the form
F
it follows that
• We
deno-
- 44 -
F(c, c) = 1m
(21 )
F(ic, c)
so that (22)
T
We
have
a
(F(c, c) )
l-
=
[j«p (c), «p(c)]
commutative diagram
(23)
where
the map
at the bottom is defined
(24)
-->"41
u
This
determines
Proposition 2 : an element
s
with
by the formula
[. JU,ul.
F
in terms of the algebra
Let
~.
n
be a non-degenerate Kahler algebra having
the following properties
1)
I)e s 1 =[.ft', js 1 = 0;
2)
[ js, s 1 = s ;
3) The operator ad js
is
semi-simple and has eigenvalues 0, 1 and
4)
j ()J C (/J where Uf is the eigenspace of ad 0). 0 1-). 0 "). to the eigenvalue A •
Then
c.;;
the K~hler algebra corresponding to some Stegel
is
domain of type
js corresponding
II
D = D(V, F) with a' transitive group of affine automora
phisms satisfying the conditions of propositions 1 The domain the operator ad
D js
is
a Siegel domain of type 1 does not have 2 eigenvalue,
if
and
only if
We mention that condition 2) follows from condition 4) since jSEjo'
- 45 -
;;r
The condition of non-degeneracy of the algebra
can also
be omit-
ted, For this s,ee lemma 1 in § 1 of part
3
The way of constructing the Siegel domain its
D(V, F) which has
K%hler algebra is clear from the proof of proposition 1. As
has to take the orbit of the point
s E:.
the infinitesimal transformations defined
V
one
1 under the group generated by
X
,
XEd 0
•
The form
F
is
by formula (24) •
It is necessary to
is a V
ad
~
Jj as
~
prove also
that
V is a convex cone and F
hermitian form . This is the hardest part of the proof of
pro~
positiO'1 2 • We will not carry it out here • It can be found in paragraph of
our article
[ 25 1 •
We mention, however, that if it is known a priori that Il
Kahler algebra of some h, b. d.
then
the V - hermiticity of the form
F
derations
UJ
is the
{}
the convexity of the cone V and
can
be obtained from analytic consi-
(cf. part 1) •
The verification
that
CJ
is in fact the
It
Kahler algebra of the Siegel
domain just constructed is not difficult. The last statement of proposition
In certain cases
2 is obvious.
an h,K, m, admits a transitive splittable solvable
group of automorphisms. For example all homogeneous Siegel domains of type II
have this
property (cf. part 1) • One 'can make use of this
in
order to remove the ambiguity in
It
the construction of the Kahler algebra corresponding to a given h.K, m., which is there because of the several possibilities to choose the tran-
- 46 -
sitive group of
automorphisms.
Proposition 3: of automorpphisms
Let
T
of the
be
a
transitive splittable solvable group
h. K. m. M. Then T acts
on
M without
fixed points. All transitive splittable solvable groups of automorphisms of
M are conjugate
of some fixed point Proof.
to
T by an element of the isotropy group
K
p E: M .
By hypothesis
We show that
• We denote by byc.X and
g the
7J (M)
the Lie algebra of the group
lie algebras of the groups
K and
o G (M) , and
T
respectively.
We have:
n T • The operator induced by ad k in the space ~(M) /)(= 0/?:nJ( preserves the positive definite hermitian form (ef. (4)). Let
keK
Consequently it is semi-simple and its eigenvalues have modulus 1 • On the other hand from the splittability of the group
T it follows
that its eigenvalues are positive. From here it follows that ad k
X'
trivially on
CJ
part
(ef. (4)) •
at
p
Since
k
Thus
K
(M)/
is
i. e.
the map
an isometry,
nT = Ie f
•
If
T'
the group G(M) containing then,
k
of
M
acts
has trivial linear
k =e is
a splittable solvable sub-group of
by the same argument,
This is possible only if T = T • Consequently mal splittable solvable sub-group of the group . G(M)
T
KnT' is
a
= {e f •
maxi-
- 47 -
For the proof of the second assertion of proposition 3 it suffices '" T
to show that every transitive splitt able solvable group phisms
of
M
is
conjugate to
In fact the element
g
T
of automor-
by some element g EGo (M) •
can. be represented in the form
g=k t
(ke:K, t
= T ) ,
and then T "gTg
-1
=ktTt
-1 -1
k
=kTk
-1
We denote by
Ad adjoint representation of the group o Its kernel is the center Z of the group G (M) •
GO(M).
It is obvious that every maximal splittable solvable sub-group of the group GO (M)
must contain the connected component
Therefore in order to show that GO(M) ad
it suffices to show that
ad
T
and T'"
T and
ad
ZO of
Z •
are conjugate in .., T are conjugate in
GO(M) • We make use
now of the following theorem
In every linear Lie
(cf.e.g. [23] ).
group all maximal connected triangular sub-
groups are conjugate. In orde.r to apply this theorem we have to show that the group Ad T
is triangular, i. e. that the operators
genvalues not
9
only on the algebra
0"
Ad t, tEO T
but also on
have real ei-
the whole algebra
(M) • From the fact that
in
G(M)
T
is a maximal splittable solvable. sub-group
(see above) it will then follc;>ws that
Ad T is a maximal
triangular sub-group in Ad GO(M) • So it remains to prove that acting on the space
5
(M) is triangular.
We do this only for the case when
M is
a h. b. d.
neral case the proof can be reduced to the case of uses theorem If
M is
Ad T
a h. b. d.
In the ge;' if one
2, § 7 • a
h. b. d.
then the group
Ad GO (M) is the connnected
- 48 -
component
of the identity of some algebraic linear group, This is pro-
ved in § 3
of our article [25] ,
From this it is easy to deduce the algebraicity of the group ad T , In fact
.
ad T
is a maximal splittable solvable
0
Ad G (M).
(Ad T)
(i. e, the smallest algebraic group conta~ a is also a splittable solvable group . Consequently (A~ T)a =
But its algebraic hull ining Ad T)
in
sub~group
= Ad T • Like every connected solvable algebraic linear group Ad T
can
be factored into a semi-direct product Ad T = (Ad T)
where (Ad T)R
group containing all
tive eigenvalues on
In other
r
(Ad T)r
is a commutative
sub~
semi-simple elements whose eigenvalues have modu-
1 , Since all linear
sformations
• (Ad T)
R
is a normal sub-group containing all elements of Ad T
which have positive eigenvalues and
lus
the group
'6
transformations contained in
(by the fact that
contained in (Ad T)I words
all
Let us find the center of
t E. T
have posi-
is splittable) the linear tran-
are equal to the identity on
elements
belong to the center of the group
T
Ad T
for which
G.
Ad iE-(Ad T)I
T,
T
Since
T
is splitt able
it follows
that it is connected, On the other hand the center of the algebra is trivial, since
for any IV (
and hence [jx, x]
f
x
E.
G,
x
f
0,
[jx, xl) = !? (jx, x) > 0 0, Therefore
(Ad T)r = {e f,
and
Ad T = (Ad T)R'
which we had to prove , Corollary 1: If the h.K, m, ble group of automorphisms n
Kahler
algebras,
M
admits a transitive splittable solva-
, then all such groups determine isomorphic
- 49 -
In fact all
~ (M) •
these
By proposition K, which
Ad(M) morphic
as
K~hler subMalgebras of
K:hler algebras are 3
they are conjugate with respect to the group
contains
j
and ."l ,
and
consequently they are iso-
n
Kahler algebras.
Corollary 2: Let Biegel domains of type 1) The domains
D= D(V, F)
and
D1 =D(V 1,F 1) be homogeneous
n.
Then the following conditions are equivalent:
D
and
D1 are affine isomorphic.
and
D1 are analytically isomorphic.
2) The domains D n
3) The Kahler algebras constructed using transitive triangular groups of automorphisms for The implication lary
1,
and
D and
1)~2)
3) ~ 1)
is
D1
are isomorphic.
obvious. 2)
follows from
~3)
proposition
follows
from corol-
1.
Now we introduce the following definition: A K%.hler manifold will be called
if it
~l
admits a
automorphisms. The K:hler algebra
transitive splittable solvable group of
{y )(,
,
"l } will be called normal
j,
is
a
splittable solvable
h.I<. m. and
G
is
a
){=
0
and
transitive group
II
the Kahler algebra constructed from if and only if
G
is
By Corollary
1
Lie
a
algebra. If
of
its
if
is a
M
automorphisms, then
{M, G f will
the pair
be normal
splittable solvable group. of
proposition 3,
to
every
normal
KMhler
II
manifold there corresponds uniquley a normal Kahler algebra. n
For normal Kahler algebras the axioms
(K A 2) and
void. The remaining axioms have the following form:
(K AN 1)
(K AN 2) (K AN 5 )
·2
J =- 1
[jx,jyl= j [jx,y]
+ j [x,jy] + [x,Y]
"l (jx, jy) = q (x, y) ;
(K A
4) are
lor
(K AN 6)
x If
U;
q([x,yj,Z)+ q([y,zl,x)+ q([z,xl,y)
(K AN 7)
The simplest example of a ve KRhler algebra. It
normal Id.hler algebl1a is a commutati-
corresponds
A simple example of
o •
to
a locally flat
h.K. m.
a non-degenerate normal K'!l.hler algebra is
the algebra
if
(24)
s
where
is
=
+
(js)
U
+
(s)
the principal idempotent
is a hermi-
2 tian vector space, The commutator of the elements u, v E UiS given by the formula [u, vl
(25)
The form q is = 0 on
(js) +
= (1m (u, v))
defined as
Uand positive
By construction the
s •
dw on
where
w is a linear form
s .
II
Kahler algebra
(24) has a principal idempo-
tent and therefore corresponds to some homogeneous Siegel domain of type II
D(V, F) •
Recalling the proof of proposition 2, As V (s)
=
it is easy to find V
and F.
we can take the positive half-line in the one dimensional space
Y 1
then the form
F
will
be given
on UJin
way 1, 1 F(u, ul = '4 [Ju, u 1='4
1
1m (( ju, u) ) s = '4 (u, u)s
the following
- 51 -
(see formulas (23) and (25) ) , In this way the domain of the pairs
(z, u)
(z a complex number, u , 1m z (u, u) > 0
EO
D(V, F) consists
U) satisfying
the condition
i
As
we mentioned in the first part, the Siegel domain is analytical-
ly isomorphic with a complex ball. The K'ahler algebras of type (24) are called ~~men~ary normal II
Kahler algebras.
It can be shown
n
normal
Kahler algebra
of elementary
can
[19,20 1 that every non-degenerate
be decomposed into
a semi-direct sum
ones, i. e.
(26)
Je(i)
where
is
n
a Kahler sub-algebra which is an
elementary normal
n
Kahler algebra , and [y(i)
,y(j)]
cf(j)
for
i>j .
This pecomposition is unique . The number m is called the rank of the algebra We gebras sum
f'
describe the structure of of
rank
non~degenerate
normal
Ktihler al-
2 ,Every such algebra decomposes into a direct
of sub-spaces
J
= ( js 1)
(27)
where
X
is
a
+ (SI) + jX + X + U1 + (js 2) + (s2) + U2
Euclidean vector space, Ul' U2
are hermitian vector
spaces, The operators ad jSl and ad jS2 are scalar on each the decomposition
sub~space
(27). Their eigenvalues on these sub-spaces ,are
given by the following table
in
- 52 -
0, °
1,
1 , 2
°
(28)
1
1 1
'2'2
2,0
0, °
0, 1
°'2
One sees from it that the element potent
1
-'2
1
s" sl
+ s2
is a principal idem-
and
jo =( 91
js l)
+ (js 2) + jX
,
= (sl) + (S2)+X,
e "U
(/1 1 2 Furthermore we have
+112 the relations
[jX, jX 1 = [ X, Xl" 0,
[jx,yl (29)
= (x,y)sl
. (x,y eX) ,
[uk' v k 1 = (Im(uk , vk))sk
(uk' vke.Uk, k:: 1,2), (XEX) •
[jX, U2J and[U1 ' ~]
The other commutators, except for
ahout which we shall speak below, are zero
(this follows from the con-
sideration of weights, cf. table (28» • The decomposition (26) has the form
The form f? is
(jsI)
+
(s ) 1
+
= (js ) 2
+
(s )
+ 11
defined as
2
dw,
jX + X
2
+U1
.
where
w is a linear form equal
to zero on each term of the decomposition (27) except
(s 1) and (s2)
- 53-
and is positive on
s 1 and s2
If until this moment the structure of the
mined only by the dimension sitive Ort
ue
numbers w(sl) ,_ W(s2) , jX
U2
X&2
111, ~
X,
f
was deter-
and by the po-
now in the definition of the commutators
there is a great arbitrariness. Namely for
(30)
[jx, u 1 = x'ou
where
Kl!hler algebra
of the spaces
U1X 112 •
jXxU2-and
Finally we describe the commutators on
x EX and
E.1i1_ ,
U U1
the circle denotes an arbitrary bilinear map
X X 2 -+
having the following property (xou, xou)
(31 )
Commutation
1
=2"
U{~
on
(x, x) (u, u) •
is uniquely determined by the formula
(32)
From (31) it follows that since for any tr ivial
x X, x
f
if
X
f
0 then dim
0 the linear map
u
-+
x
U1 :) dim U2 0
u (uE.
U2) has
a
kernel. The proof of these statements can be found in [201 •
n § 6 • ~_xJ~E~2~~_~!._~c!"~IE~l._~~El.~;:_~!.[~~~~~,
In a normal K~ler algebra
there is
a canonical positive defini-
te scalar product (33)
J'
(x, y) = ~ (jx, y) Lemma 2 • Let
J be
a K:hler ideal of the normal KMhler algebra
Then. the oru,ogonal complement
nical scalar product
(33) is
J{ to J
with respect to the cano-
a Kihler sub-algebra.
- 54 -
Proof, Since
J
is an invariant sub-space under
J
with the orthogonal complement to is also invariant under
j,]{ coincides
with respect to the form
and
'?
j , In order to prove that ]{ is a sub-algebra
it suffices now to write down the condition (K A 7) for
x, y e.
J{,
z
An analogous statement holds for arbitrary I&hler algebras, Let now h
-
A(h)
J
J
of the algebras
J1
li be arbitrary normal K~hler algebras and
let
be a representation of the Lie algebra }{ by derivations of
the algebra
On
and
f
. We form the semi-direct sum
J and]i,
so
that for
[h, xl
he. 1(, x E:
J
A(h)x
we define the operator
so that its restriction to
J and
should coincide with the corresponding operators given there, and we
define
'?
so that on
J
and
7<
it coincides with the forms given there,
and
o, It is easy to see that in order that the algebra II
J
constructed in
this way should be a Kahler algebra (With)( =0) it is necessary and sufficient that the following conditions be satisfied (34)
[A(jh) -
(35)
,?(A)x,y)
for all hE.
JrC,
x,yE.
+
A(h), j 1
0,
,?(x,A(h) y)
=
J,
For the normality of the algebra
j
it is necessary and sufficient thai
in addition the eigen-values of the operators A representation A
of a normal
0,
A(h) ,
K~hler
transformations of a hermitian vector space
J
he.ll{,
be real.
algebra}{ by real linear will be called symplectic
- 55 -
if it satisfies condition (34) and (35), where we now take for
c)
plex structure operator in
and as
q
the imaginary part of the
hermitian scalar product, The symplectic representation
A
the the eigen values of the operators
is called normal if in addition
A(h) , h E
11 are real,
It is clear that the study of symplectic representation has to play
an important role in the theory of K~hler algebras. We now find all normal symplectic representation of the two simplest normal K~hler algebras, Lemma 3, tative
K~hle-;
Every normal simplectic representation A of a commu"
algebra
Jt
is trivial, i, e. A(h) =
a for
h E
Ji,
Proof, It is enough to prove the lemma for the two-dimensional algebra
'J1. ' In this
case
JtL= Us} and
[js, s 1 =
a • We p
set
= A(js)
and represent the operators
where
p l' q1
+ (s)
p
are linear' and
,
q
and
=A(s) q
p 2' q2
to the complex structure of the space
in
the form
anti-linear operators with respect
J . The condition (34) for h
means that p
(36)
From the relation
2
=
[js, s 1
=a
it follows that
=a
= s
- 56 -
Separating the· real and imaginary parts of fPI' qlJ
+
[P2' q2]
0
rPI' q2]
+
[P2,ql]
0
p,q
From the first of these equalities we obtain,
we obtain
subtracting
(36) ,
o, whence (37) Condition (35) means that the linear part of the op erator
A(h)
is skew-symmetric and its anti-linear part is symmetric with respect to
.J
the real part of the hermitial scalar product- on From (37) it follows that S q2" 0 atId p 2
.1
eref
ore q2"
0
.
S
0
q= ql
is a skew - symmetric operator. However by normality its eigen-values are real and therefore q = O. Analogously p = 0 . It is a little more difficult to prove the following
Lemma 4. ---sion 2 i.e.
Let
1i be
'J( = (js) + (s)
and elementary normal Klthler algebra of dimen-
,where
(js, s1 = s.
If Ais a normal symplectic representation of the algebra
hermitian space ; ; , then the operator . 1 elgen values O'~"2 . We denote by value
J;
the eigen spaces of
j:1"J A(s)x =
t 0
on the
p = A(js) is semi-simple and has
p
corresponding to the eigen
.A
Then
'J;C
).
for
XE~l '2"
for x
E.J 1 '2'
- 57 -
The proof of this lemma can be found in Lemmas 3
and
4
will be used
§ 7. S t r u c t u reo f nor mal
§ 2
of
[ 20]
in the third part of these
.
lectures~
K~ hIe r man i f old s •
---------------------------------~-
In the third part of these lectures we shall prove the IIfundamental theorem ll about the structure of normal rdhler algebras which is based on the following two assertions:
(*)
Every non-degenerate normal rdhler algebra has a principal
idempotent,
(**)
Every normal K"ahler algebra
decomposes into a semiadirect
sum of a commutative K~hler ideal and a nonedegenerate normal K'I.hler sub-algebra. Here we consider
some consequences of this algebraic theorem.
Putting the assertion
(*
) together with proposition 2
we obtain
the following theorem:
Theorem 1. Every homogeneous bounded domain admitting a transitive splittable solvable group of automorphisms is analytically isomorphic with some homogeneous Siegel domain of type II . The assumption about the existence of a transitive splittable solvable group of automorphisms is essentially superfluous. This is shown in our article
[25]
From the assertion
Theorem 2.
(**) we deduce the following theorem: 11
Every normal Kahler manifold M
admits a hoI om orphic
fibering with the following properties : 1)
the base is a homogeneous bounded domain;
- 58 -
the fibre is locally flat (with respect to the induced KMhler
2)
structure) ; 3)
the group of those automorphisms
of
M
which preserve eve-
ry fibre, acts transitively on the fibre.
Proof. Let
G be an arbitrary transitive splittable solvable group
of automorphisms of
K~hler a Lie
M , and let
f
be
the corresponding normal
f
algebra. We realize the Lie algebra algebra of hoI om orphic vector fields on
[f]
We denote by
the complexification
Lie group
Furthermore
f
which has
[j)
let
M •
of
Yin
the Lie algebra
M, by (G 1 the simply connected
of all holomorphic vector fields on complex
in the usual way as
as its
Lie algebra.
be the sub-algebra of
se vector fields which are zero at some point
f
formed by tho[G Jo
p e:M and let
be the corresponding connected sub-group of [G] •
I G J in
The group
general does not act on
inbedded in the complex manifold the vector fields belonging to fields on [G . Let now
j
1/
!
[G] • "1 0 7J
=cJ + I'~ be
rG J/ [G Jo
The orbits of tativity of
that
[J J become the corresponding vector the decomposition of the
K~hler
connected normal sub-group of
M can be
as a domain so
into a semi-direct sum of a commutative
a non-degenerate normal
M, but
sub-algebra
~hler
J(
K~hler
ideal
J
algebra and of
We denote by
G corresponding to the ideal
the
;r
I are complex sub-manifolds of M • From the commuI
it
follows that the
KKhler structure induced on these
sub-manifolds is locally flat. The complexification [J] of the ideal algebra
J
is an ideal in the Lie
[J].
We denote -group of
IG].
by
[I] the corresponding connected normal subd
- 59 -
Since every orbit of
I as
a complex manifold is the quotient of
a complex vector space by some lattice (see paragraph
2
of the introduc-
tion), the group of its analytic automorphisms is a complex Lie group. Consequently every vector field in
[J 1determines
a one parameter group
of mappings of the manifold M • This means that
( G] / [G]
M
is contained in
as a domain invariant under the action of [I J •
0
The fibering of
M
by the orbits of
I
is
a restriction of the
holomorphic fibering
[G1 / [Gl o-- [GJ / [I]. [G1
0
and therefore is itself holomorphic. From the preceeding if follows that this fibering satisfies conditions 2) and
3) of the theurem. Its base is the complex manifold
complex structure of which is determined by the operator We show now that the group
G
Let
G
on
decomposes into the semi-direct product
of the normal sub-group I and of the sub-group SUb-algebra;:C
H corresponding to the
§.
be the universal covering of
G. It is obvious that it
decomposes tnto a semi-direct product of the normal sub-group corresponding to the ideal }( • Like
J and
G, the group
G
the sub-group
H
It is not difficult to prove that the center of
morphic
G
--+
3).
G is in I
'" I
corresponding to
is also a splittable solvable Lie group
and therefore its center is connected.
Lemma 6 , § 1 of part
JI J .
, the
Mil
~onsequEmtly
j
is
in;;;
(see
the kernel of the natural holo-
• This means that the group
G
decom-
poses into the semi-direct product of
I and B, and the group
is simply connected. We see now that
H acts transitively and without
fixed points on space of
Mil
Mil. with
H
Under the natural identification of the tangent the Lie algebra
U,
the tangent space is given by the operator
the complex structure on on
'}i[.
Therefore
Mil
- 60 -
n
is exactly the algebra
h. b. do corresponding to the non-degenerate normal Kahler
J{
Theorem ted in
• This proves statement
2 is a special case of the nfundamental hypothesis" formula-
§ 5 of the introduction.
PART III - Structure of normal
§
1) of the theorem.
K~hler
algebras.
1. Statement of the fundamental theorem and its co----------------------------------------------;:,~03iJ~~.! As it was shown in part II of these lectures, the study of normal
KHhler manIfolds reduces to the study of normal K'ahler algebras (ef. §4, 7) The structure of normal K!!hler algebras is described by the fo11owing theorem:
Fundamental theorem : Every normal KHhler algebra can be decomposed into a semi-direct sum (1)
where
is a commutative K!!hler ideal and
Ji
is
a non-degenerate
K!!hler sub-algebra. Every- non-degenerate normal K!hler algebra has an element
s
s
[ js, s]
1) 2)
with the following properties
The operator
ad}( js
is semi-simple and has eigenvalues
0,1,1/2
3)
If
]r{ A
denotes the eigen space of
ad}(, js
corresponding
- 61 -
to the eigen-value .A
then
'[JIG,
We also note that
j;(=
Xl.
ft
XI'1 c Jt),~'"" .
The elements is called the principal idempotent of the K!!hler
f .
gebra
The outline of the proof of this theorem will show that
f
s
be the following: we
admits a decomposition (1) where:l
K%hler ideal and
J1
is
al-
is a commutative
a KHhler sub-algebra possessing an element
with properties l), 2), 3) • In order to obtain from here the second statement of the theorem it
is enough to show that in a non-degenerate normal K%.hler algebra there are no
non-zero commutative KBhler sub-algebras and this follows from
rjx, x]
the fact that in such an algebra
= 0
(see. § 4
of part II) •
The first statement of the theorem will follows from the following lemma.
Lemma l: ment
s
71
K~hler algebra '}C
there exists an ele-
having the properties listed in the theorem, then the algebra
is non-degenerate. Proof.
for
If in a normal
Transcribing the condition of the form
--:-:.- 11 A , Y
(2)
E.
'}{I<' , z = j
s
!?
being closed
we obtain
(A+ f.l) !? (x,y) = (js, [x,y]) • From this equality it follows that
!? (x, y)
=
0 for
x,y€.
and therefore (by the condition of hermiticity (K A 5)) also for It is clear now that for every
x, y
EO:
J1
!? (x, y) = w( [x,y] ) ,
where
J( x, yE
71
0 •
- 62 -
1 A
w(u) =
y (js, u)
o
for
u
for
u
E;
J( A
AF 0 ,
,
J1o
E:.
2. We indicate'some consequences of the properties of the decom-
pos ition (1) listed in the theorem. We note first that [s, x] = j
(3)
x
for all
x E.
7{
"0
This follows immediately from the integrability condition (K A 3) • The other consequences we formulate as lemmas (of which only lemma 3 will not be used in the sequal) • Lemma 2: The operator 1 values 0, :!:. '2 ,and
is semi-simple and has eigen-
(4)
[s,x]= jx
(5)
for
--"2
Proof. We consider the representation of the twa-dimensional Kllhler algebra generated by
sand
js
the adjoint representation of the algebra
on the ideal
f .
It
d
induced by
is easy to see that
it is symplectic in the sense of the definition given in the second part. Therefore lemma
2
is an immediate consequence of the lemma
on symplectic representation
Lemma 3: gonal with
J1) and
~
respect to the canonical scalar product
Proof. true all
The sub-spaces
We set x
€
f)
fo
=
~
Y E j~
.
+
')1; .
are mutually orthoon
J
The formula (2)
It shows in particular
is clearly
that if [x, y] =0
- 63 -
and
f
+P
A
0 , then
IS (x, y) = 0 •
Now we distinguish a) tion 3) and
of
IS
VE:.J1",
the theorem )
we see that
(u, v)
= )S
(x,y)
u E
c;;,
,v
Let
- IS
+
a
>
0
=0
j uE
since
J1.
1-
5' (condi-
1-IS+a>1
= ju
,
y
=v
0
J:r, IS <
€
• Then
ad js • Applying (2) to x
=
(J
and from the commutativity of Since
[ju, v]
and
and eigenvalues' of
is not
b)
Ji r ,
ue.
Let
three cases
• Then
a
it
by
x
= ju,
ju
[ju, v J
follows that
applying (2) to
(4)
y
=v,
J _? = 0 •
we obtain that
(u, v) = 0 • c) that
It remains to show
~ ) =0 •
IS (js,
jw
IS (
Any element
2 the form
where
J, ')() = 0 • We prove first UI can be represented in
that
w E:
of
J
2 1
• We have
2 IS (s, w) ,
lS(js,jw) and on the ather hand, by
(5) IS (js,jw) = IS (js, [s,w1)
= IS ( [js, s] ,w)
consequently Let
IS (js, jw)
now
[ju, v]£dl
=
- IS
I
IS (s, Us, wJ ) = '2 g (s, w)
J
0
u Ed's, V€ ](,and
since
+
+a >0
IS
• Then
ju E:
~~
and
•
-2
Applying lS(js,dl ) and
2
=0
(2) to
xt ju, Y = v
we find that
(u,v)
= 0,
and considering that
If
IS
>
a
,then
-IS
< I_a
- 64 -
(;;; , 71~) = (j 0 ,j J(cr) Lemma 4. Proof. Since [ that
A
[J(o ,J] 0
=
0 •
We consider the operator
Xl' do J = 0
commutes
o •
A = ad;;o
h
where hE:".
,using the integrability condition it follows
with
j . By the closure of the form
g
and the
3 ,
commutativity of
s>
(Ax, y)
This shows that
+
s>
(x,Ay) =,. 0
the operator
A
(x,YE.
dJo '
is skew-symmetric with respect
to the canonical scalar product. Since its eigenvalues must- be real we have
A=0 ,
Lemma
5;
.1+,u+ y =0,
Let
hE '}{ ~commutes with
If the element so
JJ;
with
Proof.
Let
a 6 0 • Then
[h, a 1 , x) = 0 for all
s> (
Since,
, we have
and therefore
j [h, a]
h,a
0
J
Lemma 6. The ideal for which
g= x
s>
coincides with the set of all
[jg, g] = 0
+h
where
x
E;r,
o = [jg, gJ
hE:
U.
[jh, h]
[ jg, g 1 Then
(mod
and
s>
x E.
~.
([h, a] , j Ih, a]) = 0
.
Proof. It is enough to prove that if Let
it commutes al-
by the closure of the form
by the condition of the lemma, by lemma 2,
'J; then
J) .
o then
- 65 -
Since which
;Jt(
is
a
non-degenerate
was to be shown Lemma 7, The center
and is
z-;;;o
Lemma
From lemma that
Z
Z
1
of the algebra
h
is contained in
0,
do
j - invariant
ProoL
that
K!!hler algebra, we have
shows that
6
4
and the fact
Z:) J , Furthermore
it is clear
, it follows
that
coincides with the centralizer of
Let
-
2
From the integrability condition we find that Cja, jh] = j [ja, h]
for
a11
h E..
J( 2
so that S'(j [ja, h] , tja, h] ) :: S(Cja, jh], tja, hJ ) , Using the condition account
that
(K A 7) of the closure of
[ja, [ja, h} ]
:: 0
and
[jh, fja, h1 ]
(j [Ja, h] , [ja, hJ
This proves that
Ua,h]
s:
and
taking into
0 we obtain
) = 0
o and
ja E.
Z
The purpose of the remaining paragraphs of this part is the proof of the fundamental theorem, Here is the logical scheme of this proof.
- 66 -
ILemmas
8,9 (punto 2,
2)
§
Lemmas 10,11 (punti 1,2,
~ Lemma 12
I
(punto 3
§
3)
I
!!)]
Lemmas 13,18 (punti 4,6
§3)
---.------~.
/
I
/r---------'+-Proposition 3(punti 1, ;--~~)l
________________~J
//
Fundamental theorem 2nd its consequences - lemmas 2 - 7 ( §1, punto 2
§4)
---~-.-------~---~
The theorem will be proved by induction with respect to the dimension of the Ktthler algebra. The dotted arrow indicates that in the proof of lemmas
13 - 18 we use the induction hypothesis.
If it is known in advance that the algebra is non-degenerate, then
the proof becomes considerably simpler. In this case the whole right-hand side of the scheme ( § 3 lemmas
2 - 7 from
in the text) turns out to be unnecessary and so do
§ 1 •
1
1. The fundamental theorem will s ion of the algebra Proposition 1.
be proved by induction on the dimen-
together with the following assertion:
In every normal Id.hleralgebra there is either
- 67 -
an elementary or a commutative non-zero id.hler ideal. Assuming that the fundamental theorem holds for agebras of dimenn ,in
sion less than paragraph 2.
§
§ §
2 and 3 we prove proposition 1 , and in
4 we prove the theorem for algebras of dimension
f
Since the Lie algebra
n
•
is solvable and splittable, there
is a one-dimensional ideal in it. In this ideal we choose a non-zero element
r
so that
[jr, rJ
= r
or
we show that in the first case the element mentary KRhler ideal, and in
§
=0
[jr, rJ r
• In this paragraph
is contained in some ele-
3 we show that in the second case it is
contained in a commutative K'!hler ideal. To start we prove two lemmas connected with these two situations.
g
II:
f
Lemma 8.
Let
gu be the sub-space conSisting of those elements
for which
[g, r ] = [jg, r] = 0 •
(6)
tJD is
Then rator
invariant under
ad gP jr Proof.
and
ad
jr
moreover the ope-
commutes with
The invariance
of!fJ with respect
to
j. is
immediate
from the definitions. From the integrability condition we have [jr, jpJ = j [jr, p]
(7)
for
all
p
E.
£p.
Using the Jacobi identity and
(7)
we find
[[jr,p] ,r] = 0, [j Ijr,p1,r]= L[jr,jPJ ,r]
so that
[jr,p] E
follows.
ffJ.
From (7)
0
the commutativity of
ad jr
and
- 68 -
Lemma 9, For all
u, v
E.
f
d~ ~(etadjr u, etadjr u, etadjr v)
(8)
= S'(jr,etadjr
[u,v]) .
Proof. d ill - 0 -
J (
(tadjr
r"Jr"
e
e
u, e
tadjr)
v
=
tadjr u J, .etadjr v) + O( tadjr J e
- 0 (" -.) Jr, [tadjr e u, e tadjr v
J) -_ 0J ( Jr, "
U"
["Jr e tadjr J
v] ) =
etadjr [ u ,J) v.
(In the second step we used the fact that the form 3, In this point we prove proposition
follows in the case where
[jr, rJ
=
r
2
q
is closed).
from which proposition 1
•
For this we will not use the induction hypothesis concerning the fundamental theorem. Proposition 2. If in the normal dimensional ideal
(r) (1) such that
n
Kahler algebra [jr,r J
=r
there is a one-
, then the algebra
~
can
be decomposed into a direct sum of sub-spaces (9)
where
vf = (jr) + (r) + U is an elementary KlIhler ideal and " 1 [Jr,u] ="2 u for u U; 2) ~ is a KMhler sub-algebra orthogonal to fand commuting 1)
E:
I
with
jr
and
r
.
(1) By (r) we denote the one-dimensional ideal spanned by the vector
r .
- 69 -
Let
Proof, the relation a
and
b
[jr, r in
if be the sub-space constructed in lemma 8,
J =
r
1
g - ajr
Using lemma If in
9
- br
f1J,
E.
decomposes into a direct sum of sub-spaces
~ = (jr)
(10)
Using
gE. ~ numbers
one can find for any element
a unique way, so that
This means that
a)
By
tP,
+ (r) +
we shall study the eigenvalues of
(8) we set
fJ0 then
u;: r , vE
ad
flJ.
jr on
the right hand side
is
that tadjr e r
(11 )
t
=er
we obtain
o (r etadjr v)
( 12)
),
According to lemma on
!JO.
=
a e -t
for
the operator
8
9 (jr,
e
Finally' for any
tadjr ) v gE
= ~
Jr, e tadjr) g
0('
(14)
J
a e
-t
q (e tadjr u,
(15 )
u EO
fP,
product using that
commutes with
by
Let
p E.
E:
(10) and (II)
a e
e tadjr) v
v
-t
+
=
e
b
u, v e a e- t
C7U d , we obtain
t
~
+ bet + c
then in this equality we can change over to a scalar ad.,? jr
tadjr r (e tad "u, e -I) b)
ad j r
for all
Formula (8) now gives for all
( 16)
flO
v E.
Therefore
( 13)
If
all
= ae -t +
(1Y be
commute, We obtain
and
be
t
+ c
for
all
an eigenvector for ad jr
pE.
[f,
VE
1
corresponding to
0,
- 70 -
eigen-value A
Then
By formula e Since
2
e
A(t)
(p, p) = ae
Uj ). Let
-b
can
(p,p)fo,
[ jr,p ] 0Ap,
p
=e
,\ t
p.
(16)
check that the operator also on
tadjr
+ be
t
+ c
and
[jr,z]=f\q+p. Then
.
We
therefore
:;u such that
q be vectors
1
0, :. '2
g:> (and
ad jr is semi-simple on
p
1
have only the values
e
in tadjr
q=e
At + >.t q te p
and
by formula (16) (e tadjrp, etadjr q) = e 2 ~ t (p, q)+(te 2 At (p, p) =ae
-t
t
+be +c.
Being (p, p) = 0, this equality is impossible. c) Since the operator a =0
adg>jr
has no eigen-value -1, we have
in (12) . Looking at the formulas that were deduced from (12), we
see that in each of them lue of
a = 0 , and therefore -
'12
is not an eigen-va-
ad;? jr. We denote by
the eigen-space of
to the eigen-value
• The algebra
~
:;
ad ff-' jr
corresponding
decomposes into a direct
sum of subspaces
UJ =
(17)
(jr)
{I
ad~
Since invariant
under
orthogonal to Since
jr
+ (r) + !fl. + 1 2
commutes with
go 0
the subspaces
j
J?
j • Formulas (12) • (13) and (16) show that
(jr) + (r)
[Po C jJ
+~
2 we have
eigen-values shows that
N
[ r,
= (jr) +
J ]=0 . 0
(r) +
~
2
are
[j0 o
Consideration of the is an ideal.
is
- 71 -
~o,
The sub-space is a sub-algebra (see
§
being the orthogonal complement of
6, part 2) setting
U/?1 ()(}JI = fP
0
N
,
, we obtain
2 the decomposition (9), which has the required properties,
§ 3, Commutative K!lhler ideal. I,
In this paragraph we consider the case where
[jr, r] = O.
Under the hypothesis that the fundamental theorem holds for algebras of lower dimension then
7J
we prove
Proposition 3 : If the normal K1!hler algebra dimensional ideal
(r) such that
ned in a commutative K!lhler ideal
!.J
contains
a one-
[jr, r] = 0, then this ideal is contai-
K.
In all the lemmas of this paragraph the hypothesis of proposition 3
is assumed. By
{fJ we
denote the sub-space constructed in lemma
8
Lemma 10 Proof,
For any
g E
1 from the Jacobi identity and from
the
integrability condition we have [rjr,g], rJ = [jr, [g,rJ] = 0 [j [jr,g] ,rJ = [rjr,jg] ,r]= 0 i. e.
[jr, g]
E~
2, Lemma 11:
Proof: to
(jr)
(ad jr)2 = 0 ,
a) We show that the sub-space
+ (r) • Let p
E:.
r.
[jr,
d1 is orthogonal
By the closure of the form if we have
- 72 -
g (r,
= S'(jr, [r
[jr,p] )
o.
,pJ)
Using lemma 8 we also obtain S>(jr, [jr,p] ) = -
~(jr,j
b) From lemma 10 and
[jr,jp1 ) = - g(r, [jr,jp]) = 0 . a) it follows that
(18)
Formula (8) ·shows then that for any
u, v
rg
£
d 3 O( tadjr tadjr) d 2 D(· tadjr 1) = --3- .) e u, e v =.) Jr, e . [u, v 2 dt dt
=g(jr,
[jr, Ur, etadjr [u, vJJJ )
=
0
I
i. e.
(e
(19)
Whence for
p
(e
eigenvalues on
=p [JJ
tadjr
ZE d'such
tadjr
2 v) = at +bt+c
P, e
tadjr
2 v) = at +bt +c •
we obtain that the operator adjr (and therefore on
c) We assume that x, y,
u, e
EtJ
(20) Setting v
tadjr
(adjr)2
has no non-zero
f! ).
F o. Then tl).ere exist elements
that: [jr,xl
[jr,y] =
=0 ,
0
o.
[jr, zl
,
It is clear that tadjr
e
x
=x
I
By lemma 10 We get
e
tadjr
y
=y+tx,
e
tadjr
t2 z = z+ty+ - x
Y E. &Pa.nd we can substitute
2
p
=y,
v
= z in (20) •
- 73 -
t2
(y+tx, z+ty + -2- x) Since
(x, x)
f
:=
at
2
+ bt + c
0 , this is impossible.
3, With the aid of the operator ad jr
filtration of the Lie
f!
algebra
be the natural projection of ;) We denote by
A
we construct a
U = r;/ (jr) + (r) onto U . 0
Let
the operator induced
the integrability condition it follows that
n
and let
by adjr on commutes
A
j-invariant
U . From
with
j on
U
We set
~ (-1) ()J (0)
:=
1'
If
:=
n
-1
~ (1)
:=
n
-1
~ (2)
(Ker A) , (1m A) ,
(jr)
+ (r)
Lemma 12 : The sub-spaces of the Lie algebra
form a .i-invariant filtration
~(i)
, Furthermore
(21 )
o Proof:
a)
From lemma 11 it follows that
A 2 := 0
Therefore
Ker A :::J 1m A , and we have the inclusions ;; (-1) :::J ~(o) :::J J(l) :::J fj(2)
ud (i)
The invariance of the sub-spaces from the commutativity of
A
with
J'
u 2 = [jr, g2
J ,Since
follows
j.
b) We prove the commutativity of u 1 := [jr, gl
with respect to
Y
(1)
0
Let
(adjr)2:= 0 ,
gl' g2 c
~
and
- 74 -
Therefore
[[jr, GjJ ,[jr,
V]] = 0
Since
~(1) =[jr,~] + it remains to prove
(jr)t(r) ,
that [Or)
+ (r) , [jr, U/J] = 0 (}
but this follows from lemmas 10 and and 11, c)
It follows immediately from the definitions that
[~(-1)
r2) ] c
,
~(1) ,[~(o)
,
~(2)] c ~ (2)
d) We show that [
~(o)
kE.~(O),
Let
,
~(1) ]
u E~(I)
c
r~(O) , ~(2)J c ~(I) , it
. Since
is enough to consider the case where ( jr, [k,g]]
~(1)
= [k, uJ
u = jr, g
and
glC.:r
We have
+ [[jr, k] ,gJ
Since
r· [k, gJ] ,[[jr, kJ , pr,
gJE: ~1)
we also have
[k, uJ e)
We prove
g£r:! '
~(1) •
that
[1 (-1) Let
E.
u
6.
C;; (1)
,zr(I)] , Since
c 1(0) (jr, uJ
=0
and [jr,g]E:1 I )
,
- 75 -
we obtain using the Jacobi identity and the already proved commutativity of
~(1) f)
gl' g2 E
that
r jr,
f;j (0)
It remains to prove that
:! (0)
[g,UJE(:;t).
[g,uJ1= O. Therefore
• Since
[?(2) ,
1
(O)J
11 =[[jr, gJ
[jr, [gl' g2
J
Consequ~ntly
(0)
[gl' g2~?J
The proof of lemma 12
is
c r;;(2) we have
+ [gl' [jr, g2 J]E: ~2) •
,g2l
•
is finished.
4. If in the filtration constructed above
is a commutative
a sub-algebra. Let
Kahler ideal in
V and
CJ (o) = ;; then f proposition
Therefore in the following we shall assume that
fJ (
0)
= ;/(2)
3 is proved.
t
fj .
The induction hypothesis may be applied to the K1ihler algebra Cj(O) • Let
(22) be the decomposition correspondipg to this theorem. Lemma 13 : Proof: lows that
g
Let
[jg, g]
Lemma 14 : Proof: ma 14
r;(1)
E.
C;;(l) . Then
= 0 • By lemma 6
[fJ ,J'J
Let
[[jr, g1
cd .
,0
g E.
jg E
~ (1)
• From (21) it fo1-
this means that
g
E.!;J.
c ~(O)
Cf· ,
x£
J . Then. [jr, gJE:Jl)
and by lem-
= 0 • Therefore [jr, [g,xJ] = [g,[jr,xJ]
o.
Thereby the lemma is proved. From lemma 14
~(or
it follows in particular that if the sub-algebra
is commutative, then it is an ideal in;;
so that in this case
- 76 -
proposition 3 is proved,
71
In the following we assume that
(0) is not commutative, i.e,
o,
tjA.
We denote by
s
the principal Idem potent of H, We denote by
the subspace formed by the vectors which ar{J annihilated by so-
me power of the operator
Lemma 15 ; If
ad
A + p.
js - ;\ , It is clear that
> 0 or A = p. = 0,
I[~, J;l :LJ Proof,
Let
g E:.
[rg,x],J.()+J4+ II)J = Let
= 0 ,
Cjv' XE..(h , Then by lemma
According to lemma 5
then
it is sufficient to
14
prove that
0 ,
YEU.() +,.u + y) , By the commutativity of [rg,X] ,y
J = [[g,y] ,x]
=
d
0 ,
We have
[g,Y]e0'(O) 1
d
_(A+f<-)
eX
-( + )
+JJ.
> 0 then [g, yJEJ since the operator adxjs has only non-negative eigen-~alues, Consequently in this case _ [g, yJ ,x J = 0, If If A + P.
~ ma
=).J. = 0 then [g,y}E..
J( +d, x£~
r
and
4, 5. We
consider the graded Lie algebra
[[g,y], x]
=0
by lem·
- 77 -
1
~=
(23)
+ 1(0) + ~(1) + r(2)
(-I)
which i~ associated to the filtered Lie algebra g
61
(1)
g
we denote by
:J . For every, element
the corresponding element
Of:! (1)
(I) .
From (21) it follows that
o ,
(24)
We define
fj (-1)
on
a trilinear operation
the formula
(25)
abc =
nar,
a] ' b
(a, b, c)
~a
b c
by
J ' c1 .
We established some properties of this operation, Lemma
1) The operation (25) is commutative.
16:
pr, abc] = [(Ifr, a]
2)
Proof:
1)
= [[Jr, [ 01(-1) rJ
analogously that 2)
1]
By the Jacobi identity
[[fIr,a] ,bl ,c] since
' b] , Ifr, c
, 0(-1) ] __
If
ra, bJ] , c]
-[[Ifr,b] =
,aJ ' cl
0
0 • This shows
that abc
= bac. One proves
abc = acb
We take the commutator of both sides of equation (25) with
jr
and use the Jacobi identity. From the properties of the graduation and from (24) it follows that
[Ir,
[Jr,
of the three terms on the right
IJr, 1 cJ
which
we
aJ]= 0,
[(jr,al,
[Jr,
bJJ= 0 , so that,
hand side, there remains only [[
B;:~ aJ,
b]
have to prove
~}If we regard the same element g E. ~(i) as an element of ~(i-1) then g = 0 , However in the following it will always be clear which of the subspaces ;; (i) we have in mind ,
- 78 -
The following lemma is fundamental for ·this paragraph Lemma 17 : Proof.
abc = 0
-(-1) a,b,cE.;j
for all
.
Tq the decomposition (22) of the algebra
1
(0)
,
there
corresponds the decomposition
(26)
!(O)
of the algebra
[1(1)
(27) and
from lemma 14
J
[ s, r = jr
so
1J
Jl =
0
that
r e.
or
±
a=O
E;t,
d . Since
t'
where
ad jr
maps
~ (-1)
a= 0
has eigenvalues
(30)
aE
-12
or
it is clear that the
isomorphically onto
operator ad js is semi-simple on 1 + -2 on . it. Relation (29) shows that it is
Let
2
d_ cV • Going over to the al-
then. jr E
From the definition of the sub-spaces fJ{i) operator
(r) is an ideal 1 then and if a= _ -
we obtain the following relation
[ 3S, Jr] = - ajr
(29)
,
that this case is impossible.
Furthermore, if r gebra
,
if follows
.By lemma 2
t (r),
0
if follows that
From lemma 13 [js,r] =ar
,JJ
+ 1(2)
[~(-1)
(28)
1~
. By lemma
fj (
1)
:f (1)
and has eigenvalues
semi-simple also on
1 + - + a there. - 2
a,
~ (-1) A
J
abc
b"" ~ (-1) ~
J
c
E
• The
:fi(-1) , so that V
,
= [fUr,a1 ' bJ c] 1=
o.
~ (-1)
0
and
- 79 -
Then also
[fr, abc l
(31 )
[rUr,al ,b1 ,[fr,c]]
=
f
0
•
From what it has been said above it is clear that (32)
1
+ - +a
or
ct
-
2
Furthermore
1 or 2
1
then
which is imps sible in view of (28)
and (30) •
Using the symmetry of abc ,
we obtain now
A+,u,,u +v, v +
(33)
[ [1(-1) -
-(1)
[jr, aJE$A
'
sfied only in the case where
=
jr, c
or ~
9(1)J
,
[_
I
a , 1 + a or -2
g +b >0
Lemma 15 shows that if
Since
A-
]
,~
E;;; (1) 1'_Cl'
A+v
=
b
(1)J
+a
o,
.
then
= o.
,condition (31) can be sati-
- 2 a < O! A - a ,v
- a are not simul
taneously zero • Using the symmetry of abc
A+,u,,u +v,v +A <
(34) where at (35 )
we obtain
2a,
most one of the three numbers is equal to
a
It is easy to see that, just as in the case
a
0, also in the case
- 80 -
='2 the conditions 1
a
(32). (35) cannot be simultaneously satisfied,
The statement of lemma 17 can be rephrased as follows
6,
~(1)
From the commutativity of
and the Jacobi identity it follows
that (37) Condition
(36)
means that
By (37) this is equivalent to
1)J c
[[~(1) ,~J,
(38) From (37)
1
~(2)
and from the closure of the form
~
it also follows
that
Comparing this with (38) we finally obtain the relation
[[~(1) ,~],r(I)J = O.
(39)
Lemma 18 : The centralizer in
;J
is
a K!hler
Proof,
ideal.
Note that
Z( 1(1))
(40)
since
jrE
~ (1)
c ~(o)
and Z(jr) c 1(0)
Equality (39) means that (41 )
Z (~(1) ) of the sub-algebra Z/ (1)
- 81 -
From the Jacobi identity
,~J]
[ jr,Li( y(l))
(here we use the fact that
[Z(
Consequently
C
[Z(~(l) ) '1(1) ]
jr E Cj(l)
i
1) )
and therefore
,:;]c
[[Z(i ')) -1]. g
(42)
=
0
[jr,
Z ( ~(1))] = 0).
,and
r;(0)
c g(l)
(1)]
Furthermore from (41) it follows that
[rZ( ~(1)), ~J,
~(1)]
c
[E( ~(1)) ,Z (~(1) )
and
~([rZ ~1)) , ~J.' ~(l)J
(43)
C
g
dZ( ~(1) , lu/ 1) ) J,
Comparing (42)
DZ(~(l)),~]. Therefore
zE.
~(1)
) C
~(l))
Z:
0 •
with (43) we ffuU that
0/(1)] = 0,
Z( r;;(1))
Z(~(l)),
is an ideal in
Then
9
Cf .
Z (~(1))
is invariant under
and
A the
= ad7 (1)
A
jz
commutes with
commutativity of
that it is skew-symmetric with respect to ;> Therefore the operator
j •
jz£ Z(jO)) • From the integrability con-
ditions it follows that the operator and from the closure of
[Z(~(l)), ~] CZ(~(l)
i.e
We show' now that the ideal Let
'
1 (1)
j,
it follows
•
is skew-symmetric with respect to the
canonical scalar product and since it has real eigen-values we have A
=0 • 7.
This means that
jz
E
Z(ff(l)) • The .lemma is proved.
Now we can prove proposition 3.
We denote by
N the center of the ideal
a commutative ideal in
3'
Z (~(1))
• This will be
and by lemma 7 applied to the K'!hler algebra
- 82 -
Z( {:f(1)) NF
o.
and ideal
N is
j-oinvariant. Since
N~
~(1)
we have
So proposition is proved under the induction hypothesis of the
fundamental theorem • From propositions
2 and 3 , which we have proved
in
§ §
2
and 3 respectively, proposition 1 follows.
§ 4. Proof of the fundamental theorem. Let
r; be a normal
KAhler algebra and assume that the statement
of the theorem is true for all normal KAhler algebras of dimension lower than that
ff . Then. we can apply propositions
of
Let
N be
of these propositions and let Then ~ t
is
Cj
a Klhler ideal in
!! '
(47)
c-r' }(l ;f" =dl + v
(cf.
§
6, part 2) . By the
can be decomposed into a semi-direct sum
is a commutative
Klhler ideal and
sub-algebra having a principal idem potent
Jt,
is
a
N+
d'
KBhler
s'
Applying the lemma on symplectic representations (cf. 2) to the KBhler algebra
fJ .
satisfying the conditions of one
Cf
a KlIhler sub-algebra of
l'
J'
3 to
be the orthogonal complement of N.
induction hypothesis
where
2 and
§
6 ,part
we see that
(48)
We consider separately the two cases corresponding to the two possible types of the ideal a)
N
N is an elementary Klhler algebra . We set
J =J' , )( =
N
+)(' ,
s = r + s'
,
- 83 -
where
r EN
~ = d +Jt
We show that the decomposition theorem and From
2.
is an element fulfilling the conditions of proposition
s
(48)
commutes with
is
a
principal idem potent for the algebra
d
it follows that jr
is of the kind needed for the
and
r
is an ideal in:t
}t
• Since
fI'
we have
[js,s] =[jr,r]
+
r + s' = s
[js',s']
By the same considerations ad js'
on
ad jr
on
ad js = (jr)
It follows now that the operator ad js
+ (r)
.
is semi-simple on the
sub-space
W= to check how
js
acts on
Consideration of that
~
0, 1,
and has eigen-values
(jr)
+
(r)
+)(, eX
there, and
the sub-space
the eigen-values
of
U
j
ad jr
by s'
on ; ; shows
of the two-dimensional
The representation and
js'
It remains
(ef. proposition 2) .
[elf' ,U]e U .
algebra generated
W= U{?c .
K§hler
will be sumplectic.
By the lemma about the symplectic representations of such algebra,;; ( §
6
,part
2)
u=z(
1
-y
where
ad JS' =
jUre .1
~
U'.,1
onU,;l
- 84 -
Since
ad U jr
setting
2
1 we obtain 2
UJ1+U+U 011 ad js
2
=)
on
This finishes the proof in the case b)
N is
a commutative
U). a)
KHhler algebra.
We set
From (48) . it follows that the ideal decomposition
1 =(f+ Jt
J
is commutative so that the
satisfies the requirements of the theorem
S'Jme problem In
§
5
of the introduction to our lectures
about the structure of the homogeneous
the Ilmain conjecture II
KHhlerian manifolds (h.K. m. )
was formulated, We give some corollaries of it here, It is possible that some of them are valid
for a wider class of manifolds. It should be very
interesting to find direct proofs for these statements.
1.
Each
h. K m, whose points are separated by bounded holompr-
phic functions is a bounded domain. 2.
If the points of a
h. K. m.
are separated by holomorphic
tions but all bounded holomorphic functions all locally flat,
func-
it are constant then it is
- 85 -
4.
Each simply connected h. K. m. is holomorphically convex.
5.
If a h. K. m.
is a
Stein manifold then it may be holomorphical-
ly fibered in locally flat h. K. m. and the base is a bounded domain. For arbitrary hOlomorphically convex complex manifolds the Remmert fibration is known, its fibres being defined as maximal sets on which all holomorphic functions are· constant. The base of this fibration is a Stein manifold and the fibres are compact. This result could be applied for the proof of the main conjecture if the propositions
4
and
5
were proved.
The similar construction for bounded holomorphic functions, about which we do not know any general results, gives immediately the fibration mentioned in the main conjecture. 6.
Each h. K. m.
with
negative Ricci curvature is a bounded do-
main. 7.
Each
h. K. m. with zero Ricci curvature is locally flat.
Hano and Kobayashi considered one canonical fibering of an arbitrary homogeneous complex manifold with invariant measure. It seems to be not difficult to p1'ove that for h. K. m. Ricci curvature and the base
is a
the fibres of this fibering have zero h. K. m. with non-degenerate
Ricci
curvature. It is possible that a further investigation of the base and a direct proof of the statement 7 will lead to a differential-geometrical proof of the main conjecture. 8.
Each compact group of automorphisms of a
h. K. m. has an
orbit which is a complex submanifold. In particular, each one-parametric compact group of automorphisms has a fixed point.
- 86 -
REFERENCES 1.
BERGMANN S. , RUber die Kernfunctionen eines Bereiches und ihr Verhalten am Randell, J. reine und angew, Math., 1933, 169, 1934, 172, 89-128 •
2.
BOREL A., nK'!thlerian coset spaces of semisimple Lie groups" , Proc. Nat. Acad. Sci. U.S.A. , 1954, 40 , 12, 1147-1151.
3.
BOREL A., REMMERT R. , RUber kompakte homogene KMhlersche Mannigfaltlgkeiten ll , Math. Ann. , 1962, 145, 5, 429-439 .
4.
CARTAN E. , IISur les domaines bornes homogenes de llespace de n variables complexes II I' Abh. Math. Sem. Hamb. Univ., 1935, II, 116-162.
5.
FUKS B. A. , II pecial chapters in the theory of analytic functions of several complex variables", Moscow, 1963 (Russian), (English Translation published by the American Mathematical Society in 1965) •
6.
GRAUERT H., nAnalytischen Faserungen tiber holomorf-vollstandigen Raumen ll , Math. Ann. , 135, 3, 263-273.
7.
HANO J. , nOn Kahlerian homogeneous spaces of unimodular Lie groups", Amer. J. Math., 1957 , 79, 4, 885-900.
8.
KOBAYASHI S. , NOMIZU K. , "On automorphisms of a KHhlerian structure II , Nagoya Math. J., 1957 , II, 115-124.
9.
KOSZUL J. L. , II Sur la forme hermitienne canonique des espaces homogenes complexes", Canad. Journ. Math. 7,4, 1955, 562-576.
10.
KOSZUL J. L. , "Ouverts convexes des espaces affines ll , Math. Zeitschr. 79, 1962, 254-259.
11.
LICHNEROWICZ A. , "Espaces homogenes kHhleriennes", Colloque de geometrie differentielle", Strasbourg, 1953, 171-184.
12.
LICHNEROWICZ A. , "Sur les groupes d'automorfismes de certaines variHes KHhleriennes II , C. r. Acad. Sci. Paris, 1954, 239, 21 , 1344-1346.
13.
LICHNEROWICZ A. , IITheorie globale des connexions et des groups d'holonomie", Rome, 1955.
14.
MATSUSHIMA Y. Sur les espaces homogenes KHhleriens dlun groupe de Lie reductif" , Nagoya Math. J. , 1957, 53-60.
- 87 -
15.
16.
POINCARE H. , nLes fonctions analytiques de deux variables et la representation conformeD, Rend. Circolo Mat. di Palermo, 1907, 23, 185.220.
, " PJATECCKII,SAPIRO I. I. , nOn a problem proposed by E. Cartan", Dokl. Akad. Nauk. SSSR. 113 (1957) , 980-983 (Russian) • v
·v
17.
PJATECCKII, SAPIRO I•• , IIGeometry of classical domains and theory of automorphic functionsl!, Moscow 1961 {Russian~ (an enlarged english edition is being prepared for print·) .
18.
PJATECCKII, SAPIRO 1. I., DClassification of bounded homogeneous domains in n-dimensional complex space n, aokl. Akad. Nauk. SSSR, 141, 2 (1961) , 316-319 (Russian) = Soviet Math. Dokl. 2(1961), 1460-1463.
19.
PJATECCKII, SAPIRO 1.1. ,liOn bounded homogeneous domains in an n-dimensional complex space", Izv. Akad. Nauk SSSR, Ser. Mat. 26 (1962), 107 - 124 (Russian) .
20.
PJATECCKII, SAPIRO 1. I. , The structure of 26 (1962) , 453-484.
21.
PJATECCKII," SAPIRO 1. I, nThe geometry and classification of bounded homogeneous domains n, Uspehi Mat~ Nauk. 20 (1965), no. 2, 3 - 51 (Russian) = Russian Math. Surveys, 20 (1965) , no. 2,1 -48.
22.
TITS J. , nEspaces homogenes complexes compacts n , Comm. Math. Relv., .1962, 37, 2, 111 - 120 •
23.
VINBERG E. B., 'The Morozov - Borel theorem for real Lie groupsn , Dokl. Akad. Nauk. SSR 141, 2 (1961), 270-273 (Russian - Soviet Math. Dokl. ~ (1961), 1416 - 1419 •
24.
VINBERG E. B., liThe theory of convex homogeneous cones n , Trady Moskov. Math. Obshch. 12 (1963), 303-358 (Russian) = Trans. Moscow Math. Soc. 13 (1964), 340-403.
25.
VINBERGE.B., GINDIKINS.G., PJATECCKII, SAPIROI.I.,"Onthe classification and canonical realization of complex homogeneous bounded domains', ibd., 359-388 (Russian) = ibd. , 404-437.
26.
WANG H. C. ,nClosed manifolds with homogeneous complex structure" Amer. J. Math. , 1954, 76, I, 1-32.
27.
WElL A. , "Introduction 1958.
v
.. ,
v
~
v'
j algebras",
ibd.
"
a l'etude
des
varieh~s
kllhlei'iennes n , Paris,
CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO (C.L M.E.)
Stephen J. GREENFIELD
EXTENDIBILITY PROPER TIES OF REAL SUB MANIFOLDS OF ([n
Corso tenuto ad Urbino dal 5 al 13 luglio 1967
EXTENDIBILITY PROPER TIES OF REAL SUB MANIFOLDS OF en by Stephen J. Greenfield (Massachusetts Institute of Technology Cambridge, Mass) A
History In 1906 F. Hartogs [4] discovered that
,,2
a neighborhood of the bicyc1inder in
\I..
a function analytic in
could always be extended to an
analytic function defined in a neighborhood of all the bicyclinder. Not much later h: (
(1910) E. E.Levi [6] 2
found a local analogue of Hartogs' result: let
--;.. R be differentiable and suppose that
M
=h
-1
(0) is a submanifold
of (2 . If h (1)
det
h h
at
then
p €M
-
h
-
h
zl z l zl z 2 z
h 1
-
z2 z 1
hzl
z2 z 2
hz2
f
0
0
z2
functions analytic in any neighborhood of M extend to
be analytic in a fixed open set on one side of M . The boundary of the fixed open set includes a neighborhood of
p
in
M
In the early 1940' s, Bochner and Martinelli
that if
11
M is
of
.
[21
[9] showed
a compact differentiable hypersurface bounding an open set
([n, then any function analytic in a neithborhood of M has an
extension analytic in
71..
(Their result was actually better, for they re~
quired only that the function be defined on
M and satisfy certain appro-
priate partial differential equations.) In 1960, H. Lewy nal manifold
in
[3
[8]
published an example of a four-dimensio-
having the property that functions analytic in a
neighborhood of it extend to be analytic in preceding, Lewy just needed the function satisfying
differential
equation).
a fixed open
set. (As in the
defined on the manifold and
Lewyls
investigation of extension had
- 92 -
S. J. Grenfield
an important byproduct
[7]
. The first example of what he calls 'atypi-
cal' partial differential equations was discovered; these equations have no solution in any open set. E. Bishop
(1965)
r1]
four-dimensional manifolds in
gave a general method of considering /i'3 ~ and discovering when they had the
'extension property' to open subsets of by B. Weinstock
( 3 • His work was generalized
~3] to submanifolds of real codimension 2 in en
(1966) • Other recent work on extendibility has been done by H. Rossi
and
t12]
B
R. O. Wells
Some definitions from several complex variables. Let
functions K
[14-16]
f
K
be a subset
of
([n . H(K) is the collection of all
defined and analytic in a neighborhood of K . We say that
is extendible to a connected subset
res:
H(K') If
H( '11.) hull of
A subset
L
~
H(K)
K
K' of
en (coritaining K) if
(the natural restriction map) is onto.
is a subset of
11.,
open in
(. n ,then
the
K'It ' is tp E. U Ilf (p)1 ~ sup IfI) f < H(1l)} . K U of is H(j,L) - convex when: K compact subset K,
of
L . ~ K 1.(. compact subset of L . A open set it is
C
1L of
(n
is
a
domain of holomorphy only when
H( 1.1) convex.
Statement of the problem, and an investigation of the hypersurface case . A problem of
several complex variables, of importance also to
function algebras and partial differential equations, is : n given two subsets of t ,K and L (with K C L), when is K
extendi-
ble to L? Here we investigate this problem when K and L are submanifolds
- 93 -
S. J. Greenfield
of
I[
n
. How big (dimension) can L be in terms of K ? For K of high
codimension, can L
be open? In what sense is
K part of the boundary
of L ?
n+l
Bishop has written: "It is thought that a manifold M
C (
n
has, in general, the property that holomorphic functions in a neighborhood of
M
extend to be holomorphic in some fixed open set.
prove this, and discover what
"in general" means.
First we examine the, case of a hypersurface fully. Since
We will
tr'.,n
M of
II.,
care~
{' n has a complex structure, the complexified tangent
bundle splits as the direct sum of equal-dimensional subbundles :
T( (n) ~ (
= H( ()
+ A( d: n) .
At P f tn, H( ([n) (the holomorphic tangent bundle) is generated by 1~ j
,
(b!.)
~
n . The antiholomorphic tangent bundle, A ( c"') is gene-
J p
ted by
?l
(~)
u z.
J p
We know:
n
~_
~'H(
C) ,
-
(If
rV will
V is a vector bundle,
Consider now T(M) 12
tI:
C
rH(
I
T( (, n)
M®
t.
(2)
H(M)
nA(M)
Since the Lie bracket is [fA(M),
= 0,
)1 C oJ
n
rH( ~ ) .
V.)
Define the holomorphic n
(resp. antiholomorphic) tangent bundles of M by
t n ) n T(M) €' tC ) •
nl
CfYJ sections of
denote the
.
(resp. A(M) = A(
IC
H(M) = H( f, )
n T(M) ® {:
Then we have these properties:
[rH(M), rH(M)]
- - invariant in
C
rH(M) .
T(M) @ C, we know also that
rA(M)] C fA(M) . But H(M) + A(M) is not all of
In fact, H(M) + A(M) has codimension
1
in
T(M)Qa ([; .
T( M) ® Q: •
It is not too difficult to show that condition (1) of E. E. Levi
- 94 -
S. J. Greenfield
is equivalent to [rH(M),
(3)
In fact, M is
fA(M)]
¢ r(H(M) + A(M)) .
extendible (to an open set of
(3) is true . This gives us ample encouragement to
/I'll
11.) only when
initiate a theory for
higher codimension. D
C - R
Definition:
manifolds
A C - R
manifold is a pair
differentiable manifold of dimension n + k (n nal complex subbundle of T(M) ®C H(M)
(4)
nA(M)
= 0,
(M, H(M)) where ~
dle of
a real
k) and H(M) is a k-dimensio-
so that (if A(M) = fI(M)) [rH(M),
fH(Mil
These conditions are adapted to imitate (2), we shall call
M is
C
f'H(M)
and of course
H(M)) the holomorphic (resp. antiholomorphic) tangent bun-
M. There are numerous equivalences for the conditions of (4) • In particular we can get:
Proposition: If M is
a
real differentiable manifold of dimension n+k,
then the first conditions is the same as : There is a (2 k)-dimensional subbundle R of
T(M) so that R
is a complex vector bundle (that is, there is a real bundle map 2 J R ~ R so that J = - Id R ) . The second condtion is equivalent to : [a, b] + J [Ja, h] + J la, Jbl
~
[ra, Jbl = 0 for
any a, bE rR . The
vani~
shing of the Nijenhuis tensor for C-R structure. ) Proof: Put
R = re
(H(M) + A(M)) , and J can be obtained easily.
The second equivalence is a computation. The
C-R codimension of M is the fiber
dimlR(T(M)/R). (We
- 95 -
S, J. Greenfield
shall assume Examples: that
M
M
a) If C-R codim M = is
° , then the first condition of (4) says
almost-complex. The second, by the Newlander-Niremberg
h01
theorem
connected, so this number is well-defined) .
, implies that
M
is
a
complex manifold,
n
b) Suppose P on
n·n \I.
with C
0()
=
I:1 a.J
_0_ is a partial differential operator ~Xj
complex-valued coefficients, and that
always linearly independent (over structure
u: ) .
Then P
P
and
Pare
determines a C-R
~ n ,with
C-R codim n - 2 • 4 c) (a non-example) S has no C-R structure, in any on
C-R
codim. The proof uses equipment of algebraic topology. d) Contact manifolds are C-R manifolds with
C-R
codim = 1
There are many other examples. In the case of complex manifolds, the
~ complex, with its ac-
companying Dolbeault cohomology groups, is well-known. We won't go into details, but merely mention the following: let M
be' a
C-R manifold. If
J1 p, q
r (/I. PH(Mt@AgA(M))oK. ) ,
=
then: :;'. J:l,o,g~ 1(y0,g+1
Theorem: Theorem:
CJ
_2
?J
- / rIJ
JJ
is well-defined.
= 0, and this statement is equivalent
to the second condi-
tion of (4) . The cohomology groups defined implicitly above have been studied by J. J. Kohn in the case that He has obtained
M
is compact and
some very interesting results
Guided by (3) we want to consider keting
rH(M)
and
rA(M)
I
C-R
codim m = 1 .
[5]
cross-terms! obtained by brac-
. So we define the
Levi algebra of M ,ll!YI),
- 96 -
S. J. Greenfield
to be the Lie
r A(M)
subalgebra of
rT(M) ~ (;
~~ that
. We will
point of M. It is clearly
generated by
rH(M) and
i(M) has constant dimension
a - -invariant
subalgebra of
V of
so there is a complex vector subbundle
at each
rT(M)
T(M) ~4: with rV
= ~(M).
We define the excess dimension of Z(M) , ex dim £(M) , to be the fiher dim
t
I
+ A(M) )) .
M
N )( T, where
(V (H(M)
If
is a real manifold, and The converse is true
L.
theorem of
H(M)
is
a
= H(N)
iI<
locally by the
Niremberg
Theorem: Suppose If
if
N
0, then ex dim ::l(M)
= O.
(non-trivial) complex Frobenius
[11] :
(M, H(M))
p E M, there is an
complex manifold, and T
is a C-R
manifold with ex
open neighborhood of
morphic to an open neighborhood of
0
in
diml(M) =0.
p in M which is C-R iso-
/R h-k /T'k >< ~
(with the natural
C-R structure on the latter) • (Note: Since we used the phrase
'C-R isomorphism' above the concept of
a C-R map. should be mentioned. If (M, H(M)) and manifolds, a
C-R map f: H(N).
Equivalently, df
commutes with
the notion
of
C-R
with map
C-R
M ---+ N is a differentiable map so that
df@ l(H(M)) C
algebras ) commutes
(N, H(N) ) are
,~,
'J'
or
j\f (the natural map of exterior
When
M and
N are manifolds,
coincides with that of complex analytic map-f
is required to satisfy the appropriate Cauchy-Riemann equations.)
E
Embedded Suppose
C - R manifolds, and generic manifolds M is
a sub manifold of
not necessarily a vector bundle. If H(M) is H(M))
is
a
C-R submanifold
of
(
n
. Then
H(M)
is
a vector bundle, then (M,
n
n
( !C ,H( ([ )) . But how can we
- 97 -
S. J. Greenfield
guarantee
that
T(M) H(M)
p
ce of
p
H(M) is a vector bundle ? n
(by affine translation) is
a real linear subspace of (. .
is canonically' isomorphic to the largest complex vector subspa-
(n contained in
T(M) • With a short digression to
linear alge-
p
bra, we can obtain a large supply of 'good' M's . Linear algebra: Let sion k
W be a complex vector
space of complex dimen-
and V be a real vector subspace of
n
W of real dimension
m(V) denote the maximal complex subspace of
• Let
W in
V.
Then : max(O, k-n) Let
~
.
dIm
a:.
(m(V))
~
'k2
G fR, p(W) (resp. G (p(W)) be the p-dimensional real
(resp; complex) Grassmanians of
W. Put
.
0
G(((W) = G G: (W)
+ G ('
1
(W)
n
+ ... + G({W)
•
k
If V € GIR, (W) ,then m(V) t G G:: (W) Theorem: If P = max (0, k-n) , then
k m: G IK (W) ~ G t (W) has the
following properties : is
a dense open subset of G ~ a
Proof:
00
C
k
(W)
map.
Take coordinates, and use an argument based on ranks of matri-
ces. Elements dimension
k
of
m
and other
-1 (G (; P(W)) are called generic subspaces of elements of
G
~
k(W) are called exceptio-
nal subspaces. If T(M)
point of M M consists
p If p
of
is is
a generic subspace of generic,
generic points,
an and
open
Ii: n, we call neighborhood
(N,H(N)) is
a
p N of
a generic p
in
C-R manifold,
- 98 -
S. J. Greenfield
If H(N)
called a generic C-R submariifold of N = codim
C-R codim
f
0, observe that
in q:n .
f\
We give a general example of non-trivial generic submanifolds of ~.,.n
=n + k
,1' be C00 real-valued functions 1 n-k on en. Suppose p~. [\1'.-1(0). If dr 1 (p)A ... f\.dr k(P) f , there J J nis an open neighborhood 1), of p in en so that 1'.-1(0) =M j J \l
with dim iR
. Let
l' , •.•
·V.n
is
C00 submanifold
a
of
~ 1'1 (p) .1\ ... A~r n-k(P) f p) then fold
of
1£
can
..... n of codimension
If.
(the rj are holomorphically transverse at
0
be chosen so that
submanifold, for
~r 1 A. ~ l' 2 = 0 ) .
Theorem: If
is a
the above manner then
f: M
~
is
a generic
gives on example of a non-generic
C-R submanifold
generic
C-R submani-
of
If
If.-
n
given in
by transverse, holomorphically transverse functions,
C
is a C-R
map
~fl\ ~r111 is
M n
en. (1'1 = xl' 1'2 =Y1 in
M
n-k. If, in addition,
said
to
be
only when
A ~r n-k
...
O.
"relatively holomorphic" and
the partial
differential equations above are usually called the "induced"
or "tangen-
tial" Cauchy-Riemann equations. Note that the restriction to
M of any
function holomorphic in a neighborhood of equations, and so is
a
C-R
map.
All real hypersurfaces of fold of codimension
M satisfies these differential
If'n I(..
are generic. A C-R submani-
2 is either or a complex analytic hypersurfacJ2.
The extendibility questions can be answered most completely for generic submanifoldS of
~
n
with non-trivial holomorphic tangent
bundle . Therefore we restrict attention to this case.
- 99 -
S. J. Greenfield
When ex
t (M) = 0,
dim
given by the theorem in Theorem: Let
f
H{M)
[31
D
M be
a
we have more information than was
generic
C-R
submanifold of
O. The following are equivalent for
p t:
ex
p
cM
so
codim
that \lis C-R
isomorphic
11
that
to
IRs
p ~ M so
n
a =
d) There is
fundamental system
>(
s = C-R
Cq ,with
that 'Uis not
B ()M
p
When
ex
U of
extendible small
open
is
H{B) convex.
dim
it (M) = 0 ,
of . holomorphy . When
teresting
of neighborhoods
S., S. domains of holomorphy . jEl J J <; I a fUndamental system of neighborhoods U of
e) For sufficiently
ex
dim
balls
M
is
B
of
analogous
(n with cen-
to
a
do-
> 0 , we will· have in-
X{M)
extendibility .
F
Bishop discs
Definition : A family of subset
11.
into
lR n
of
4:
n
for
There is Wells
analytic is
[Iz I < 15 of
the subset D
of
M.
p E M so
main
U
fundamental system of neighborhoods 1) of'
a
c) There is
ter
with
:t (1.L) = 0
dim
b) There is
n
M:.
a) There is a fundamental system of neighborhoods p € M so that
(.
[14]
any a
a
discs
map
()
in
en parameterized by a
F : U x D ---7' en {where
so that F{t, -) is
analytic as a
D is map of
t
classical
Kontinuit'assatz formulated
by R. 0 .
- 100 -
S. J. Greenfield
1&
Theorem: Let If
F :
'U x D -
rameterized is
be
a
en
is
by '[I.-
a point
simply connected domain a
so
continuous
that,
for
(a degenerate disc),
family of
F(
11 ><
IRd.
analytic discs pa-
u /- U,
some
then
in
F (
t uJ
)C
D)
oD) is extendible to
F( 'U,x D)
Bishop
used
the theorem
above to
investigate extendibili-
ty problems of submanifolds by creating families of analytic discs containing degenerate discs so that : a) The boundaries of
the discs are contained in the given sub-
manifold. b) The interior of the family an open
set
in
a) and
a
F( Ux D)) "fills up"
(that is,
manifold of higher dimension.
b) will
be called, respectively, the problems of exi-
stence and non-triviality for a family of analytic discs. Bishop investiga1'3 ,and showed that with suitable M4 lOn 'V ted these problems for an assumptions
(corresponding to generic with ex dim
existence problem and
could be solved with
the discs were
subset of
non-trivial: their
a continuous family of discs, interiors contained an open sub-
\L.
not simple. It involves the solution
with
strict boundary conditions. For
n
1[, Weinstock,
in
showed the existence of
a
2
generalizing Bishop's work,
large collection of differentiable families
of analytic discs. He used this codimension
is
equations
differential
generic submanifolds
folds of
"2) the
/i'3
The existence problem of partial
X(M)
in
to ~n
~
show extendibility from submanito
me conditions as Bishop: generic and ex
open
sets in
dim
For higher codimension, using what is actual simplification of
(
n
(under the Sa-
;t(M) " 2 ) .
in
some respects an
the Bishop-Weinstock proof, we can obtain
[3]
- 101 -
S. J. Greenfield
Theorem: Let real
dimension
p €:M at
real
p,
be
a
generic
n+k. Suppose
then
cell ter of
M
for
dimension
submanifold of
dim
ex
sufficiently small
all
there is
C-R
n+k+l
with ex
M (\ Bp ~ ~N . And N will n+k-l of a map F: JR x D -
t ;> 0
~
balls
B p C-R submanifold
generic
a
Z(M)
dini
be given
.If (n in with N
of
t(N) ~ t - 1 and
as
B P
a subset of the regular set
,..n U, so that
a) F is
analytic in the second factor. n+k-l >toD)e M. b) F( IR c) F
contains degenerate discs.
Remarks on the proof:
+ A(M)
Select
u E
r H(M) so
[u, ii] ¢H(M)
that
P
+ P
P
Then
[u,
uJ p ¢ T(M) p ~
(;,
and we
use
the Weinstock theorem,
specifying (essentially) that the centers of our discs will lie in the i
0, til
tion of
direction. These discs will be non-trivial. The generic condi-
p
M insures that some subset of the interior of the family is ge-
neric. Finally it is shown without
too much
trouble that the excess dimension
of the Levi algebra of the new generic manifold behaves correctly. A very geometric example will perhaps make the theorem (and the idea of the proof) clearer. Consider re
zk
= x k + iy k
M can be zed by
given IR
2 .
1S:
. In this
simple case
a
I
y 2 = z 1/ 2 disc
in ([;' 2 , whe-
whose boundary is
on
precisely. Indeed, a full family of discs parameteri-
- 102 -
S. J. Greenfield
e
i 6
F .,
(x 2, Y2' R, 6)
i y 2
F
is nontrivial; its
Jacobian
is
2 Y2 R
a 'typical'disc
\
'~
x2
line of degenerate discs
G
Iteration
and some theorems
If we apply the second theorem
use
in
F
repeatedly, and then
the Kontinuitatssatz, we obtain:
Theorem
IfM
is
a
e ='ex dim tiM) >0
then
a generic submanifold
N with
dim
In
sense
M is
a
certain
M
en , and
C-R sub manifold of
generic is
extendible N = dim a
to M
a
set
containing
+ e.
'boundary'
of
N
(function
algebras) We can
also
ner-Martinelli results
get
a result analogous
to
for a compact hypersurface :
the Hartogs-Boch-
- 103 -
S, J, Greenfield
2 Theorem: Let
M be a compact generic
with
H(M) fo a , Then M
fold
N with
is
extendible to a
tion
set containing a
C
submani-
dim N = dim M + 1 ,
!'roof: By the previous theorem, if assume ex
C-R submanifold of
dim
P(re(H(M)
M
is
not extendible, we must
t(M) = a, Then computation shows that the distribu-
+ A(M))) is completely integrable and its maximal inte-
gral submanifolds possess the structure of 'non-trivial complex subman nifolds of {' ,Then the result desired follows from: Lemma:
No compact subset
non-constant
of
analytic maps
(I know three
([;
n
from connected complex manifolds,
proofs
of this, all depending more or less on
the maximum principle, I leave it as A C-R imbedded
manifold
complex
is the union of the images of
is
an
called
exerclse,)
O-complex
submanifolds, a-complex
if it possesses no
manifolds
are higher co-
dimensional manifolds with properties similar to strictly pseudoconvex hypersurfaces, A result which includes E, E, Levi's original theorem follows, O-complex generic C-R submanifold' of 3 Theorem: If M is a n , wlth H(M) fo a , then M is extendible to a set containing a
e
submanHold Proof:
with dim
N
N = dim M+l ,
Easy, but needs some equipment that has not been mentioned
here , Notes:
2
and
3
are
'best possible'
ble to
more than
one
higher
In
2,
M
ction) , But we can
may not
in that
M
may not be
extendi~
dimension, be
N (there
always arrange that
)N
may be cobordism obstru-
n
M
fo
¢
- 104 -
S, J, Greenfield
H
Reinhardt manifolds If
is
K
the coordinates
a
of
subset
a differentiable map
p£1R
n
, L
n ,(It is
dimension
rank
1, .. , , log
-1
an
is
(p)
(n, let
a
n-torus,
[Z £
=
Kli!;
I z1'"
= K n[z
of maximal
L( (z1'"'' zn)) = (log Jz 1 If
o~
is
Z
of
zn
f
0
K
I
none of
J, We define
(n~Rn by
L
/zn') n generic subrnanifold of (£ of 1 1 S X", X S (n tines) topologically,
Then: Lemma: L
-1
If
is
(M)
mens ion
is
M
a
generic
a
en
sub manifold of
C-R
1\
s_ubmanifold
of
of dimension k , n ([ ,of C-R codi-
n-k, is
called
a
Reinhardt manifold.
Extendibility problems for Reinhardt manifolds are easily handIed because of the followings te the convex hull of
result. (If
n
R ,ch
KC
K
will deno-
K ) ,
Theorem: Suppose is an arcwise connected subset of IRn. K -1 -1 -1 Then L (K) is extendible to L (ch K) . And if L (K) is extendible to
'U,
1 - 3 of
if
M
is
an
all
of
then
V r;.
L- 1(ch
Using
this
G in
any range of
is
a
curve
K) , ([3]
) .
theorem, we can construct examples of theorems
with
codimension and
convex hull' all
of
(n+1) - dimensional Reinhardt submanifold ( If
~ (We M
say: the holomorphic hull
is
any curve
in
is
In particular n-1 IR, then L (M)
dimension~
which is all
of
extendible to [n".)
Rn , the coefficients
of the
- 105 -
S. J. Greenfield
'Frenet formulas' of ween the two A
M
can be
generators
used
in
E
describe
the relations bet-
~ (L -1 (M))
of
straight lirie segment
So by the theorem
to
and
IR n is convex
S in
the result
above,
L
-1
(S = ch S) .
(S) is
not ex-
tendible. L
-1
(S) is
fold. In ter
therefore a local
fact, computation
family of
strips
in
Riemann
is
surfaces
ticular, we
to
continue this
see
that a
Reinhardt
or infinite
manifold
no straight line segments.
correspondance,
and our theorems on
become simple theorems on
convex hulls. In par-
get:
From G 3
if a
straight line segments, dim N
ch M contains
= dim
an
submanifold
M of
ch M contains an open
R n contains
subset
no
of a manifold
M +1 .
G 2 becomes: if
dim
(which are either annuli
O-complex .only when M contains
holomorphic hulls will
with
shows
is not hard
We can
N
of a real and complex mani-1 that L (S) is an (n-1) parame-
the plane) . Then it
L -\M)
product
open
M
is
subset
a compact submanifold of
of
a
manifold· N with
dim
IRn , N=
M+ 1 . Problems I will mention only a few
of the numerous problems
out-
standing. a) Suppose is in a
a
C-R
M
map, is
neighborhood
of
is
a
C-R submanifold of
f the restriction to M of M?
([;n. If f: M~( a map
holomorphic
- 106 S. J. Greenfield
b) Is
theorem
c) What
is
G
1 true for non-generic
the geometric and analytic significance of the
Kohn cohomology groups mentioned in d) If fold
A
U
11
for
U :
folds
of
en? e) In
a manifold M
+e
is
an .In II...
spread over
bility
What (See
G
N
M?
open
subset of
('
n
,there is a
Stein mani-
Ilargest l set
of extendi-
~
",I IN
I
,and
can
be
[3]
is
the
said (imitating this) for
and [16J
if M
is
D?
is
dim N
real submani-
).
extendible always less
to
a
than
set or
containing equal to dim
?
f) Can every
C-R
manifold
be locally C-R
imbedded
in
{n ? (If so, we can imbed it locally as a generic C-R submanifold of some /l'n
\I..
•
Global imbedding is too much
complex manifold
then
N >" T (see
to require - if
N is a compact
D) cannot he. globally imbedded.)
REFERENCES
1. Bishop, E. , "Differentiable manifolds in complex Euclidean space", Duke Math. J. n (1965), 1-22 2. Bochner, S., "Analutic and meromorphic continuation by means of Green's formula", Ann. Math. 39 (1938), 14-19 . 3.
Greenfield, S., Cauchy-Riemann Equations in Several Variables (Bradeis Univ. thesis, 19S7).
4.
Hartogs, F. , "Einlge Folgerungen aus Cauchyschen Intergra1formel bei Funktionen mehrer Ver'anderlichen"Silzb Munchener Akad. , 36 (1906) , 223.
5. Kohn, J.J., "Boundaries of complex manifolds", Proceedings of the Conference on Complex Analysis (Springer-Verlag New York Inc. , 1965). 6. Levi, E. E. , "Studii sui punti singolari essenziali delle funzioni di due o pili variabili complesse", Annali di Mat. Pura ed appl., 3(1910) 61-87 . 7. Lewy, H., "On the local character of the solution of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables", Ann. Math., 64(1956), 514-522. 8. ------, "On h,ulls of holomorphy", Comm. Pure Appl. Math. , 13(1960), 587-591.
9. Martinelli, E., "Alcuni teoremi integrali per Ie funzioni analitiche di pili variabili complesse" , Rend. Accad. Italia, 9(1939), 269-300; 10. Newlander, A., and Nirenberg, L. , "Complex analytic coordinates in almost complex manifolds", Ann. Math. , 65(1957), 391-404 . 11. Niremberg, L., "A complex Frobenius theorem", Seminars on Analytic Functions (Institute for Advanced Study- United States Air Force Office of Scientific Research, 1957) . 12. Rossi, H. , report to appear in the Proceedings of the international Congress of Mathematicians (Moscow, 1966). 13. Weinstock, B. , On Holomorphic Extension from Real Submanifolds of Complex Euclidean Space (M.1. T. thesis, 1966). 14. Wells, R. O. , "On the local holomorphic hull of a real submanifold in several complex variables", Comm. Pure Appl. Math. , 19(1966, 145- 165.
- 108 -
15. ----------, "Holomorphic approximation on real-analytic submanifolds of a complex manifold", Proc. A. M. S. , 17(1966), 1272-1275. 16. ----------, "Holomorphic hulls and holomorphic convexity of differentiable submanifolds'~, to appear Trans. A. M. S.
CENTRO INTERNAZIONALE MA TEMA TICO ESTIVO
(C. 1. M. E.)
W. KAUP
HOLOMORPHE ABBILDUNGEN IN HYPERBOLISCHE RAUME
Corso tenuto ad Urbino dal 5 al 13 luglio 1967
HOLOMORPHE ABBILDUNGEN IN HYPERBOLISCHE RAUME. W. Kaup ( ERLANGEN) Es sei (vergl.
7<. die Kategorie aller reduzierten komplexen Ir::iume
[7]) und HoI (X, Y) die Menge aller holomorphen Abbildungen eines
komplexen Raumes X scher Raum und
d
in
einen komplexen
eine
Raum
Y. 1st T ein topologi-
stetige reelle Flmktion auf TXT, so heisst
d
eine stetige Pseudometrik auf T ,wenn
und
d(x, z) ~ d(x, y)
d(x, y)
+ d(y, z) fur aHe x, y, z
~
0,
d(x, y)
=
d(y, x)
E: T gilt. Zur Verein-
fachung der Sprechweise wollen wir fUr jede Unterkategorie ~c.7< vereinbaren: Definition 1 : Eine invariante Pseudometrik auf
1<. ist
=======
d, die
jedem komplexen Raum
auf
so
X
dung
zuordnet, dass
fur
X C Reine stetige Pseudometrik Y ~ "A. und
Jedes
dass
nat.i.irlich
Die von
jede holomorphe Abbil-
"invariant" erfiihrt eine gewisse Rechtfertigung dadurch,
CARATHEODORY definierte invai'iante Pseudometrik rasst die
der invarianten
Metrik
= {z £ C:
Kategorie 'J<..
ausdehnen. Dazu gehen
I I
D(a, b) = a'5a -_ b1
Izl < I} und setzen
fur alle
CX(x, y) : = sup fEF wobei sei
biholomorph ist.
dy(fx, fy) = dX (x, y) gilt, wenn
unmittelbar auf
F:
X£
im Einheitskreis E: =
7<. und
x, y
E:
X (vergl.
If I <
1 . Damit gilt
[5, 1 ~
D(fx, fy) ,
= HoI (X, E) die Menge aller holomorphen Funktionen
mit
-X
ist eine Metrik
f auf
dann
genaL1 dann, wenn
F die
Punkte von
X
trennt ; (3) ~X(x, y) = 1 gilt genau dann, wenn
sich
wir aus von
~::;ne ~kung 1: . :.(. . ;1),--c_ _ is_t_e_in_e__in_v_a_r_i_a_nt_e_P_s_e_u_d_o_m_e_t_r_ik__a_u_f_~_--,-_
(2) c
dx
f: X .... Y aus i< gilt
Die Bezeichnung
X
eine Vorschrift
x
und
y
in verschiedenen
- 112 -
W.Kaup
Zusammenhangskomponenten (4) 1st c:X(x, y)
If
(y)
I
von X liegen;
< 1 , so gibt es ein f
F
E:
mit f(x) = 0 und
= cX(x, y) .
Der Beweis ergibt sich
leicht mit Hilfe des Satzes von MONTEL, der fur
beliebige komplexe Raume gilt, und mit Hilfe des Maximumprinzips flIT holomorphe Funktionen. Wir wollen noch einige weitere Eigenschaften von
C
Bemerkung 2:
£: X die Kugel K (x)::::
f
~ y
e:
Liegt fur jedes r
X : cX(x, y)
<
r}
.
< 1 und jedes x
x
angeben und zeigen r--
relativ-kompakt in X, so ist
X holomorph-
konvex. !3eweis :Wir d'urfen
X
als
K C X ein Kompaktum und f e:
r
}
zusammennangend voraussetzen. Es sei dann
K: =(Y £' X
die F -konvexe Hulle von K.
: If(y) \
:s
K
f E: F mit
ein
Also kann
K
und
y C " K gibt es
f(x) = 0 und
mit endlich
If(Yl\ $ sup
I
f(Z)\ < d. z E; K .. vielen Kugeln mit einem Radius
CX(x, y) = 1\
Iur alle
besitzt ewen Durchmesser
d < 1 • Zu beliebigvorgegebenen Punkten x E K stets
I f(x)1
sup x.E: K
" werden, d. h. Kist kompakt. Also ist X
< 1 uberdeckt
F-kQnvex und damit insbesondere
holomorph-konvex. Q.e.d.
Weiter
gilt
Bemerkung 3:
Es seien Y C X komplexe Ra'ume mit der Eigenschaft, dass
jede beschr'ankte holomorphe morphen Funktion auf c
-Y
auf
auf
Y
eindeutig
X fortgesetzt werden kann. Dann
zu einer holostimmen c
und
-------X--
Y uberein.
Beweis : Es sei
T : X~C
Funktion
f: Y .-. E eine beschrahkte holomorphe Funktion auf Y und
die eindeutig bestimmte
Fortsetzung. Es geriugt offenbar, f(X)C
C E zu beweisen. Das folgt jedoch mit Denn w'are f(Xo)~E
fu'r ein Xo aus
einen bekannten Trick (vergl. [2] ): X, so w'are im Widerspruch zur Vor-
- 113 -
W,Kaup
(f - f(x))
aussetzung die auf Y holomorphe Funktion
-1
o
holomorph fortsetzbar,
Q,e,d,
Wegen des Riemannsehen Hebbarkeitssatzes (vergl. 3 z. B,
stets dann anwendbar,
komplexer Raum ist und plement
x
nieht auf ganz
wenn
[9J ) ist
Bemerkung
X ein normaler zusammennangender
Y ZAR1SK1-offen in X ist (d, h, wenn das Kom-
von Y analytisch in
X
ist),
Wir wollen einen komplexen Raum X
besehr'ankt
K-vollsfandig,~
zu jeder nieht-diskreten Teilmenge A C X eine beschr'ankte
nennel'l, wenn
holomorphe Funktion auf
X existiert, die auf. 6. nicht
konstant ist.
Jedes besehrankte Gebiet uber dem en ist. z, B. beschra'nkt K-vollste:naig . Fur alle X sei Z (x) r
<
r} ;
die fur
r
J
x f:X und alle r mit 0< r~)
> 1 werde Z (x): = X gesetzt, Dureh r
{r >0: x EZr(y) und
erhalten wir eine Pseudometrik (1) f
~
x- Zusammenhangskomponente der Kugel {y € X : cX(x, y)
~X(x,y): '" inf
Satz 1:
e:
~X auf X
ist eine invariante Pseudometrik
X
~X
gilt
normal
und
auf X konstant, so gilt
YC X
eine
ZAR1SK1-offene Teilmenge, so
'3
folgt
aus
von
f(Z (x) ) C Z (fx) gilt, 1st jedem
eine Metrik, die die To-
~
l!0;
X-
~y:: S'x I Y ,
Beweis : Die Invarianz r
und es gilt
erzeugt;
(3) 1st jede beschrankte holomorphe Funktion (4) 1st X
[13]), auf 1<;
(vergl.
(2) 1st X beschrankt K-vollst'andig, so ist pologie von
y€Zr(X)}
r
x E: X
eX(x, y) > 0 fi.ir die Topologie
ein
von
y ~ Z i. (x) mit X
Tatsaehe, dass stets
X besehr'ankt K-vollsiandig, so existiert zu
£ > 0 , so dass
alle
der
Z€ (x) relativ-kompakt in X liegt und y
f
erzeugt, Es sei nun
ZARISK1-offen, Wir du'rfen annehmen, dass
x gilt, Daraus folgt , dass ~X X
normal und Y C X
Y jede Komponente von
X
sehneidet, d. h, jede besehrankte holomorphe Funktion auf Y
ist eindeutig
auf
X
holomorph fortsetzbar, Wegen Bemerkung 3
- 114 -
V, Kaup
gilt somit
cXIY" c Y (was iibrigens mit dem Maximumprinzip auch direkt
folgt) , Andererseits ist fur jedes offene zusammenliimgende U nY
die Menge
Es seien Y C X komplexe
R'aume und f: Y
Abbildung, Notwendig daftir, dass f
T:
X -+ Z gestattet, ist
9y
I
~X Y "
zusammenh'angend, d, h,
~
U C X auch
,
Z eine holomorphe
eine holomorphe Fortsetzung
- falls Y dicht
in
X liegt
- die folgende Be-
dingung: ( l\'t)
Zu jedem
x e:: X gibt
(Yn ) in Y,
so
dass
es eine gegen x konvergente Punktfolge
die Bildfolge (fy ) in Z einen Ifaufungspunkt 11
besitzt, 1: allgemeinen ist Satz 2 : Es sei
diese Bedingung nicht hinreichend, es gilt
X ein normaler komplexer Raum und
Kl- offene Teilmenge, 1st und
f: Y ---. Z
so existiert fur Beweis : Sei eine Folge Sei
einer
Y C X eine ZAR1S-
Zein beschrankt K-vollst'andiger komplexer Raum
eine holomorphe Abbildung, die die Bedingung (.) erIullt, f
genau eine holomorphe Fortsetzung
x ein
beliebiger Punkt in X, Wir
(y) in Y mit
lim
X"
n
V C Z eine
pakt ist
jedoch:
(-Kugel
y
V
von
X ~ Z,
wahlen entsprechend (_)
und q:" lim f(y) Eo Z ,
n
n
(bez, f Z) um
und eine Umgebung
T:
q
so
klein,
biholomorph
dass
V kom-
aquivalent ist zu
beschr'ankten lokal-analytischen Menge in einem Cm, Wegen Satz 1
existiert eine Umgebung U von
x
mit
Riemannschen Fortsetzungssatz ist f bar ,Diese Fortsetzung Die Bedingung ne Iftille fry) C Z
ist
in (*) ist wegen Satz 1 (4) wegen Satz bezuglich
1 (1) ist dann ~
nY) C
in den Punkt
V, d, h, nach dem x holomorph fortsetz-
wegen ( ~) naturlich eindeutig,
(lit) ist
vollsfandig
f(U
z, B, dann eritillt, wenn beziiglich
die abgeschlosse-
S' Z ist, Denn die Folge (y n)
eine CAUCHY-Folge bezuglich auch
die
Q,e,d,
Bildfolge
~Y'
und
(fy) eine CAUCHY-Folge n
Z' Insgesamt erhalten wir daraus
Satz 3: Es seien
Y C X wie in Satz 2 und Zein komplexer Raum, fUr den
- 115 -
W. Kaup
'?z
eine vollst'andige Metrikist
und
homogen
§lZ
T:
Mannigfaltigkeit) ist, ist
~Z
Z ; dann
kann
Radius
z
folgt sodann, Z
liegt,
d. h. Es
und
o
o dass
wenn
r
in
Z
jede Kugel yom
Z
erzeugt. Es
liegt. Aus der
Radius
Romogeniiat
< r relativ-kompkt in
jetzt
jedem
komplexen
Raum
'r:X ~ X
von
X
und
~X
setzen -1
X
wir
eine weitere ste-
die universelle
fur
alle
x, y ~ X
-1
('t' (x), 't' (y))
dann
Satz 4 : (1)' rT ist eine invariante Pseudometrik 1st die universelle Uberlagerung so ist (3)
Automorphismen von
ist insbesondere vollstandig.
~Z
(jX(x,y) :"
(2)
O"x
eine
auf ~ ;
X von X beschr'ankt K-vollsiandig,
Metrik, die die
Topologie von X erzeugt;
1st jede beschranlde holomorphe Funktion ·auf
X
konstant (z. B. wenn
X eine zusammenh'angende komplexe Liegruppe ist) , so ist (4)
K-voll-
> 0 , so daBs die Kugel mit
tige Pseudometrik zugeordnet. Dazu betrachten
dafUr gilt
beschrankt
Topologie von
ein
relativ-kompakt
werde
Uberlagerung
Z
ist Z automatisch eine komplexe
eine Metrik, die die Punkt z E: Z
urn
zu
folgendermassen eingesehen werden: Wegen
gibt also einen r
ist,
(d.h, die Gruppe Aut (Z) aller
operiert transitiv auf
Satz 1
beschr'anktK-vollst'andig
X -+ Z fortgesetzt werden.
eine vollstandige Metrik
standig und homogen Z
Z
ist). Dann kann jede holomorphe Abbilpung f: Y -+ Z
einer holomorphen Abbildung Dass
(z.B. wenn
CTX :;"X :)
(5)
Fur X" E
(6)
1st
C
x gilt
X normal
und
vX~
"S~X
konvergiert .
eine
=
~
;
(l"E" S'E " c E " D ; und
Y
C
X
endlicher Fundamentalgruppe reits dann
G"'x
eine
ZARISKl-offene Teilmenge mit
71'1 (Y), so ist eine
CA DCRY -Folge beziiglich
Folge (y ) in Y be-
n (j y' wenn sie in
X
- 116 -
W. Kaup
Beweis : Jede Decktransformation (d, i, ein Automorphismus g mit
gilt
X
t
xt
wobei
-1
le('l
-1
die
Topologie von
entsprechenden Eigenschaft Abbildung
von
~
von (j
und der
folgt
Tatsache,
f: X -+ Y eine holomorphe
X beschrankt
einfach
dass
aus der
zu jeder ho-
Abbildung
Y mit kommutativen Diagramm
~
~
y
X~
Y
X
~
1;".j, f eine
X
existiert ,
zusammenh'angende komplexe Liegruppe, so ist
eine komplexe Liegruppe, und wegen rametriger Untergruppen Funktion auf
C --., X
X konstant, (5) ist
schen Lemmas,
der
w:
einer
(i. a, verzweigten) analytischen
setzt
werden, wobei
"X
ein
der
und C' A. auf Y
X
X, so kann
gefunden
werden
Y iiberein,
7f
Uberlagerung komplexer
ZARISK1-offene Teilmenge enth'alt (vergl. '"
des Schwarz-
Nachweis von (6) : Die
kann wegen
normaler
X
beschr'ankte holomorphe
eine einfache Konsequenz
Y~Y
auch
Existenz geniigend vieler einpa-
ist jede
und es fehlt nur noch
universelle Uberlagerung
Y als
X erzeugt, wenn
Die Invarianz
K-vollstandig ist;
f: X
(x, ty) ,
(x), y ~ 't' (y) zwei belie big gew'ahltePunkte sind, und aus
(i) folgt, dass
lomorphen
S X'"
= inf
(5' (x, y)
1st
X, Deshalb
operiert eigentlich diskontinuierlich auf
auch
(i)
€.
X
'f= 't g) ist eine Isometrie bezuglich ~ X ' und die Gruppe", r aller
Dec]{transformationen
(j
von
(Y) endlich zu 1 1\ 'f: X ~ X fortgeRaum
(9] ) , Also
ist,
der
stimmen
1st nun (y n) eine Folge in Y mit lim "
Folge (Y ) in mit 'f y = y und lim y €' X ",n n n n Nun ist yn eine CA U CRY - Folge beziiglich 0' und
Y
eine
X
damit auch bezuglich IJ '" , d, h, die Bildfolge y ist eine CA UCRY -Folge
Y
bez'tiglich Vy .
Yn
n
Q.e,d,
-117 -
W,Kaup
Als Anwendung l'asst sich zeigen Satz,5.:Es sei X ein normaler kOIIlplex.e.r Raum und N ex eine analytische Teilmenge, die
folgender Bedingung genugt :
(* *) Zu jedem
x € N gibt es eine zusammenhangende Umgebung
x
in X, so dass die' Fundamentalgruppe 71'1 (U - N) er:v;ilich ist,
Dann gilt
fUrY: = X - N und jeden komplexen Raum
Ie Uberlagerung beschrankt K-vollstandig ist: f: Y
~
U von
Z, die die Bedingung ( .... ) von Seite
Z,
dessen universel-
Jede holomorphe Abbildung erfilllt, ist zu einer holo-
morphen Abbildung X ~Z fortsetzbar , Beweis :Wir d'Ltrfen
w;. (Y)
eine Punktfolge in
Y mit
CAUCHY-Folge beziiglich lich der Metrik
O"z .
endlich annehmen , Sei dann lim
y
n
= x, Wegen Satz
cry, d,h, (f(Yn)) ist
x
~
N und (y n)
4 (6) ist (y) eine n
eine CAUCHY-Folge bezug-
Wegen ( .. ) durfen wir annehmen, dass die Folge
fry ) wenigstens einen Ha'ufungspunkt besitzt, d, h, die Folge f(y) ist bereits n n in Z konvergent,. Q,e.d, 1st itir
X eine komplexe Mannigfaltigkeit, so ist die Bedingung (!II. *)
N z, B, dann
erfUllt,
wenn N eine Codimension
> 1 in X
besitzt, Nen'nen wir jetzt einen komplexen Raum X K-hyperbolisch, wenn
crX
eine vollst'andige
Satz 5
Metrik auf
X ist, so liefert
der Beweis von
speziell
Satz 5': Es seien X, Y und N
wie in Satz 5, 1st Zein K-hyperbolischer
komplexer Raum, so ist jede holomorphe Abbildung f: Y holomorphen Abbildung Bevor solI
--+
Z zu einer
X -+ Z fortsetzbar ,
wir auf die Notwendigkeit der Bedingung (fit lit) eingehen,
der Begriff "K-hyper;bolisch"
n'aher untersucht werden: Da jeder
lokal-kompakte metrische' Raum vollst'andig ist, auf dem eine Gruppe von lsometrien transitiv operiert (vergl. Satz 3), gilt zun'achst : Bemerkung 4:
Jede homogene komplexe Mannigfaltigkeit, deren universel-
Ie Dberlagerung beschr'ankt K-vollstandig
ist, ist K-hyperbolisch.
- 118 -
W. Kaup
Durch einfache Rechnung folgt weiter Bemerkung 5 : Ein komplexer Raum
-
X 1st genau dann K~hyperbolisc\1 wenn
---------~---~--.---......-------.---
die universelle tJberlagerung K~hyperbolisch ist. Speziell ist also ein komplexer Raum K~hyperbolisch, wenn er Uberlagerung eines kompakten Raumes ist und eine beschrankt K~vollsr:1ndige Dberlagerung bes itzt, Da die hyperbolischen Riemannschen Fl'achen gerade den homogenen Einheitskreis als universelle Uberlagerung haben, sind also speziell alle hyperbolischen Riemannschen FEichen K-hyperbolisch, und der Begriff
1I~.::hyperbolisc_h~1
erfahrt dadurch eine gewisse Rechtfertigung, Weiter
gilt Beme£kung 6 : Sind X ~
XY
und
Y K-hyperbolisch, so auch das direkte Produkt
und jede analytische Teilmenge A eX,
Bemerkung 7 :Ist X
ein
K-hyperbo~ischer
komplexer Raum und N C X
die Nullstellenmenge ein..:'E.2-2lf
X
so
X - N K-hyperbolisch,
ist auch
das Komplement
Beweis : Es sei (jy'
giert
Dann
ist
beschr'ankten holomorphen Funktion,
Y: = X - N und (y ) auch n
somit gegen ein
x
eine
(y ) eine CAUCHY-Folge in Y bezliglich n
CAUCHY -Folge bez'ciglich
E X , Nach
O"x
und konver-
Voraussetzung existiert eine holo-
morphe Abbildung f : X
-+ Emit
f -1 (0) = N ,E * : = {tEE : t
f
ist K-hvperbolisch . f
bildet
in
CAUCHY-
Folge beziiglich
'J" E
* ' d, h,
Y
f(x) C
E
.. ab,
E*
d. h, f(y ) ist eine n
und somit
x
e
y, Also ist U"'y
vollst'a'ndig,
Q,e. d,
Jedem komplexen Raum X lisierung
X .... zusammen
Abbildung
0}
~-: X lI
komplexer Raum,
-
mit
ist
in eindeutiger Weise die Norma-
einer diskreten eigentlichen holomorphen
X zugeordnet
(vergl. l~1), X· ist ein normaler
und man zeigt leicht
Bemerkung~: Mit X ist auch die Normalisierung Xi( K-hyperbolisch. Betrachten
wir nun fur
jedes
n >0
das
folgende
((lJ)
- 119 -
W,Kaup
:Es
Bei~pi~l
faltigkeit
sei
der
A
eine kompakte
Dimension
projektiven Raum
komplexe Mannig-
n, die singularit'atenfrei in einen komplex-
P N eingebettet
von kompakten hyp,erbolischen netes. N stets
K-hyperbolische
sei
(ist z, B, A
Riemannschen
erreichbar), A bestimmt im
ein direktes Produkt
Frachen, so ist das f'ur geeig_ N+1 Vektorraum C einen, N+1
analytischen Kegel X, dessen einzige Singularit'at der Nullpunkt 0 C C ist.
{o}
Y: = X -
ist also
eine komplexe Mannigfaltigkeit, und man hat
eine holomorphe Abbildung f : Y -+ A, die nicht holomorph fortsetzbar ist, da f(U Bedingung
n Y)
(* ,) ist also
=A in
auf
f'ur jede Umgebung U von
Satz
5 wie in
Satz
X
0 gilt, Die
51 notwendig und kann
nicht durch eine Bedingung an die Codimension von N
ersetzt werden,
Fur komplexe Mannigfaltigkeiten ist die Bedingung (tHi) jedoch vermutlich
(10J ) .
'uberfliissig (vergl. Es sei
X
ein
..
komplexer Raum, dessen universelle Uberlagerung
X beschr'ankt K-vollstandig ist, Dann ist die Gruppe morphismen von trik
O"x
(1 il
).
X als Gruppe aller biholomorphen Isometrien der Me-
eine reelle Liegruppe,
die eigentlich auf
X
operiert (vergl.
eX) die Gruppe aller Decktransformationen, so Aut(X) ~ N( l")/ r, wobei N(,,) = {g E Aut(X): gr= r g}
1st'
bekanntlich
r
Aut (X) aller Auto-
Normalisator
C
Aut
von
Automorphismen
r mit
in
Aut(X) ist. Versehen wir
ist der
jede Gruppe von
der KO-Topologie (=Kompakt-Offen-Topologie), so
gilt Satz 6 : Es
sei
komplexer Raum menge
X ein und
Y
zusammenhangender normaler K-hyperbolischer C
X das Komplement einer analytischen Teil-
N ex, die die Bedingung
(iA,4jl') aus Satz
5 erfullt, Dann liefert
die Zuordnung g ..... g \ Y einen topologischen Isomorphismus der pe {g 6 Aut (X) : g(N) = N~ Y nicht
auf die Gruppe Aut(Y) . 1st
homogen sein, und ist N zus'atzlich
kompakt, 1st
Grup-
N 1= ~, so kann
kompakt, so ist auch Aut(Y)
N die Singularitatenmenge von X, so gilt Aut(X) = Aut(Y) .
- 120 -
W,Kaup
Be~eis
: Die Beschr'ankungsabbildung
'fJ
von G:=
h
e: Aut(X): g(N)
=NJ
in Aut(Y) ist injektiv und stetig , Wegen Satz 5 I ist If- bijektiv , Da G und Aut (Y)
Liegruppen
mit abz'ahlbarer Topologie sind, ist If' ein topolo-
gischer
Isomorphismus, Die folgenden Aussagen ergeben sich unmittelbar
aus der Tatsache, dass X
operiert (
unter
[I:D)
G = Aut (Y) als lsometriengruppe eigentlich auf
bzw, dass
die Singularifatenmenge von X invariant
Aut(X) ist ,
Daraus ergibt sich nun Satz 7 : Fur jeden kompakten K-hyperbolischen komplexen Raum X ist die
==e --
Automorphismengruppe Aut(X) endlich, Beweis: fiir
Nach
einem
Satz, der von BOCHNER und MONTGOMERY (
kompakte komplexe Mannigfaltigke iten und von KERNER ([14)) Ilir
kompakte komplexe R'aume bewiesen worden ist, ist
Aut(X) eine komple-
xe Liegruppe, die holomorph auf X operiert. 1st G die
1-Komponente
von Aut(X), so ist also speziell fur jedes
durch
finierte Abbildung so
(3J )
ist
G=
peebenfalls Da auf
i II
G -+ X
holomorph und wegen Satz
,Da X
kompakt
x E X die
([13]
kompakt ist, ist
Aut(X)
g~
gx de-
4 (3) konstant, Alals
lsometriengrup-
) und deshalb endlich,
einem kompakten komplexen Raum jedes holomorphe Vektorfeld
integrierbar ist
([121), ergibt sich insbesondere die
Folgerung:
X
1st
ein
so existiert ausser D
kompakter
o kein
K-hyperbolischer komplexer Raum,
holomorphes Vektorfeld auf
X ,
Wir wollen den Begriff "K-hyperbolisch" noch etwas erweitern und setzen Definition: wenn auf
Ein zusammenh'angender komplexer Raum X heisst hyperbolisch, X
eine vollst'andige stetige d(fw,fz)
Iur
alle w, z £ E
heisst
und
hyperbolisch,
Metl'ik ~
d
existiert
mit
D(w,z)
fe Ho 1 (E, X), Ein beliebiger komplexer Raum wenn jede Zusammenhangskomponente hypel'boli.sch ist
Offensichtlich nangt
diese Definition nicht ab von del' speziellen
- 121 -
W.Kaup
Wahl der
invarianten
Metrik
D auf
zusammenh'angenden komplexen Raum so dass
E. Betrachten wir nun auf Y
die
jedem
gr'osste Pseudometrik ky'
stets ky(fw, fz) ~ D(w, z)
gilt, Diese
existiert;
man setze n'amlich 11 k (x, y) : = inf Y
wobei inf
tiber alle endlichen Teilmengen
strecken ist, ren mit ky auch
Ek=1
fur die holomorphe
fl (z) ::: x, fn(zn)::: y
und
wir
nun mit
7
r.Y
D(zk' zk_l)'
[z o,z 1"'"
Abbildungen
z 1 von E zu ernr f 1"." fn E Hol(E, Y) existie-
fk(zk) = fk+1 fti'r 1
die KOBAYASHI-Pseudometrik
Bezeichnen
fur x, y
auf
Y
Sk < n ,
nennen
(vergl.
Wir wo1len
(1~) .
Kategorie aller zusammenh'angenden kom-
plexen R'aume, so l'asst sich leicht zeigen : Bemerkung (2) X
g:
ill kist eine
ist hyperbolisch
d ist auch
invariante Pseudometrik auf
genau daMl , wenn kX vollsfandig ist (denn mit
kX ~ d vollsi:indig),
Daraus folgt nun unmittelbar fur alle komplexen R:iume Bemerkung 10: 1st Topologie und die Satz 8:
kz j
Y
hyperbolisch, so stimmen auf
X
und
Y
Hol(X, Y) die KO-
Topologie der punktweisen Konvergenz 'liberein .
1st Y kompakt und hyperbolisch, so ist auch Hol(X, Y) kompakt ,
11
Y ist kompakt in der Produktxt:X topologie; es kann aufgefasst werden als Menge aller Abbildungen von X in Beweis: Das
direkte Produkt
yX =
y versehen mit der Topologie der punktweisen Konvergenz, Wegen BemerX kung 10 liegt HoI (X, Y) in Y abgeschlossen und damit kompakt (denn ein gleichm'assiger Limes holomorpher Abbildungen ist wieder eine holomorphe Abbildung) , Satz 8
ist
Soezialfall des folgenden allgemeineren Satzes (vergl.
[1 i] ) :
1st X zusammenh'angend und Y hyperbolisch, so ist die durch (f, x) -+ (fx, x) definierte Abbildung
p: Hol(X, Y))c X ...., Y xX
eigentlich (d. h. fist stetig
- 122 -
W,Kaup
~
::t. 't'-1 (K) ist kompakt, wenn K kompakt ist) , Dieser Satz kann als
eine Art MONTELscher Satz den (vergl.
cp
-1
(K)
(8J ),
denn fur
(ur
jedes Kompaktum K = K >< KeY J< X ist
genau dann kompakt,
K1
n f(K 2 ) of
Y)
gibt es
¢}
ein
n K1
(d, h, die Folge
o
1
mit
nf(K 2) =
¢ ftir
n
> no
(f k ) konvergiert gegen den idealen Rand von Y) ,1st
([6])
X zu-
HoI (X;Y) ein komplexer
~ eine holomorphe Abbildung, Daraus wird in
(vergl. auch
2
f£Hol (X, Y) :
jeder diskreten Punkfolge (fk ) in Hol(X,
sa'tzlich kompakt, so ist nach DOUADY Raum und
t
wenn die Menge
kompakt ist , Zu
also
holomorphe Abbildungen gedeutet wer-
[1
i]
gefolgert
[4]):
Satz 9 : Es sei X
ein
zusammenh'a ngender kompakter komplexer Raum und
Y ein kompakter komplexer Raum, der eine beschr'ankt separable Uberlagerung besitzt, Dann gilt (1) Es gibt nur endlich viele Zerlegungen von X, die durch holomorphe Abbildungen f:X
-+
Y erzeugt werden (insbesondere gibt es nur endlich
viele holomorphe Abbildungen von X auf Y mit zusammenhangenden Fasern), (2) Zu
vorgegebenen Punkten
lomorphe Abbildungen
x £ X, yrc:. Y gibt es
f: X....,. Y mit f(x)
nur endlich viele ho-
y,
(3) Zu jeder holomorphen Abbildung f : Y -+ Y gibt es eine natllrliche Zahl n 2 n >0 , so dass die Iterierte g = f eine Retraktion ist (d. h, g = g) . Insbesondere ist also jede surjektive holomorphe Abbildung f: Y --t Y ein Automorphismus endlicher Ordnun,s von Y ,
- 123 -
W.Kaup LITERATUR ~11 ANDREOTTI, A. and W. STOLL: Extension of holomorphic maps, Ann. of Math. (2) 7~, 312 - 349 (1960) .
BEHNKE, H. u. P. THULLEN : Theorie del' Funktionen mehrerer komplexer Ver·anderlichen. Erg. d . Math.·~, Berlin: Springer 1934 BOCHNER; S. a. D. MONTGOMERY: Groups on analytic manifolds. Ann. of Math. ~, 659-669 (1947). BOREL, A. a. R. NARASIMHAN: Uniqueness Conditions for Certain Holomorphic Mappings. Inventiones math. ~ , 247-255(1967) . CARATHEODORY, C. : UBER das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Ver·anderlichen. Math. Ann. ~,
[6J
76-98 (1927) .
DOUADY, A. : Le probleme des modules pour les sous-espaces analytiques compacts d lun espace analytique donne . Ann. Inst. Fourier 16
,1 - 95 (1!'66) .
GRA UER T, H. : Ein Theorem der analytischen Gacbentheorie. Publ. Math. ~ , 233-292 (1960).
[8] -
u. H. RECKZIEGEL : Hermitesche Metriken und normale Familien holomorpher Abbildungen . Math. Zeitschr. 89, 108 - 125 (1965)
[9J - , u. R. REMMERT: Komplexe Riwme. Math. Ann .
..!1~, 245-318 (1958)
'[10 HUBER,
H. : Uber analytische Abbildungen Riemannscher Frachen in sleh. Comment. Math. Helv. ~, 1-72 (1953)
[1 UKA UP, W : Endlichkeitss'atze
fur Systeme holomorpher Abbildungen in hyperbolische R"aume. In Vorbereitung
[r ~ KA UP,
W. : Infinitesimale Transformationsgruppen komplexer Raume. Math. Ann. ~, 72 - 92 (1965) .
[131 KA UP,
W.: Reelle Transformationsgruppen und invariante Metriken auf komplexen Raumen. Inventiones math. ~, 43 - 70 (1967) .
[r~ KERNER, H.: Uber die Automorphismengruppen kompakter komplexer R'aume. Arch. Math.
Q,
282 - 288 (1960).
[1~ KOBAYASHI, S. : Intrinsic metrics on complex manifolds. Bull. Amer. Math. Soc.
E '
347-349 (1967) .
(16) REIFFEN, H. J. : Die Caratheodorysche Distanz und ihre zugehorige Differentialmetrik. Math. Ann. !..~, 315-324 (1965)
(1 ~ REMMERT,
R. : Holomorphe und meromorphe Abbildungen komplexer R"aume. Math. Ann. ~, 328-370 (1957)
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
A. KORANYI
"HOLOMORPHIC AND-HARMONIC FUNCTIONS ON BOUNDED SYMMETRIC DOMAINS"
Corso tenuto ad Urbino dal 5 al 13 luglio 1967
- 127 -
HOLOMORPHIC AND HARMONIC FUNCTIONS ON BOUNDED SYMMETRIC DOMAINS by
Yeshiva University (New York)
The main purpose of these lectures is to study questions of elementary analysis on bounded symmetric domains, namely the realization of these domains as generalizations of the unit disc and the upper halfplane, the study of the structure of their
boundary and
the boundary behaviour of holomorphic functions. This is done in sections
3 to
6 which contain material otherwise available only
in journals (mainly
~.4]
, [161
and [24] ). Some slight simplifications
and improvements have been made here; it will, by the way, be apparent that the subject still has plenty of open problems. A second purpose is to make all this more accessible to the analyst who is not an expert in Riemannian geometry. Our study uses parts of the theory of Riemannian symmetric spaces. For this standard treatise is that of Helgason
[8J ; another
very clear and con-
cise introduction can be found in chapters 1, 2, 8 of the book of Wolf
02J.
These books contain all that is needed here, but they also
contain much more. In our section I we outline how the part of the theory actually needed can be built up with minimum effort. In particular, we do not need locally symmetric spaces and we can avoid most of the topological difficulties.
In this section most proofs are
omitted, but they should be easy to fill in with the help of the references. Section
2 is a slight rearrangement of material contained in
[8J.
- 128 -
A. Koranyi
Because it is the basis of all that comes later, we present this material with proofs.
(This does not mean all proofs; here and also
later some the proofs, preferably the lengthy and uninstructive ones, will be omitted. In these cases easily traceable references will be given). The prerequisites for (i)
the reading of these notes are the following.
The fundamental facts about Lie groups and their homo-
[51
Ch. IV, or , in a nutshell,
[22J
geneous spaces
([8] Ch. II, 'or
Ch. I sec. 5).
For this, in turn, . one needs the basic definitions
about differentiable manifolds (same references) and some facts about covering spaces and covering groups (
[5]
[22J
Ch. I sec. 8, or
Ch. II; a good reference for all the above is also
[ID ).
The fundamentals of the theory of semisimple Lie groups
(ii)
and Lie algebras. (Best exposition' for our purpose in Also to be found in (iii)
Li 1] , [13],
[8]
Ch. III.
etc.)
Some of the elementary facts of Riemanpian geometry:
The existence of geodesics, normal neighborhoods, the notion of completeness.
[8]
and
b2].).
(A few pages in any book on the subject; also in
- 129 -
A. Koranyi
1.
Preliminaries.
Generalities on complex domains. mean a connected open subset main and
f: D-+- D'
we say that
f
G(D}.
of
C.
By a domain we
D'
If
is another do-
is holomorphic with holomorphic inverse,
is an isomorphism.
automorphism of denoted
D
n
D
D' = D ,f
If
The automorphisms
0:
D is said to be homogeneous
is called an
D form a group, if
G(D}
is tran-
sitive on D . If main, then
D is bounded, or isomorphic with a bounded do-
D has an invariant Hermitian metric, the Bergman
metric. More precisely, to everyz E. D there is attached a po-
i gjk(Z}}
sitive definite Hermitian matrix
(j, k = 1, ••. , n) ; the
length of a tangent vector
v
~
)'
= L.J (a
c)
j~ J
is defined by
L
)v/ ~.
- J
+a.---..,J (j z.
J
gjk(z) a j a k
vector can be written in this form);
if
f: D --;,. D'
e. g.
[8] p.293-300).
We denote by D
is an isomor-
I(df}v ID' = Iv ID
phism, then it is also an isometry, i. e. (For the proofs see
(every real tangent
I(D}
the group of all isometries of
as a Riemannian space, and give
By the Mayers-Steenrod theorem
I(D}
I(D}
the compact-open topology.
is a Lie group (a proof in
- 130 -
A. Koranyi
a special case, still sufficiently general for r 1 p. L8 j
170-172).
closed subgroup of
our later use, is in
By the theorem of Weierstrass,
I(D);
G(D)
in particular it is a Lie group.
is
a
- 131 -
A. Koranyi
Symmetric domains and symmetric spaces
Definition:
A bounded domain
D
is symmetric
if for every zeD there exists s C'G(D) such that (i) z -1 and (ii) z is an isolated fixed point of s . s s z z z If we equip D with its Bergman metric, it becomes a member of the following class of spaces: ~efinition:
symmetric if for every M
onto itself such that
of
s. P
A connected Riemannian space
of
L
n
(complex) di-
mensions we can talk about its underlying real manifold
point
is
s of PEM there exists an isometry p -1 s sand p is an isolated fixed point p P
Given a complex manifold
mensional).
M
Using a local coordinate system
PEL, every (real) tangent vector at
zl' ...• ' zn
p
(2n-dl
M
at a
can be written
in
the form
)'
J
v = LJ(a.~ J u Z. J
)
( j = 1, ... , n )
The mapping
a.--7'- ia. J J transformation J on M
+
.
induces a (real) linear
(for each p )
which is easily seen to 2 be independent of the coordinate system and is such that J = -I. P In this situation we say that J is a complex structure on M; p
L fold
P
is reconstructible from with complex structure
tiable map
f: M---7M'
M
and
J.
If
M'
is another mani-
J', it is immediate that a differen-
is holomorphic if and only if
dfoJ
=
J'o df
- 132 -
A. Koranyi
at every point of
(Cauchy - Riemann equations).
M
We say that
is a Hermitian space if
M
Riemannian space with a complex structure for every vector
J
such
M
is a
I
that
\ Jvl = v \
v. Clearly every domain
with Bergman metric is a
Hermitian space; every bounded symmetric domain is a Hermitian symmetric space, a notion defined as follows. Definition: mitian space phic isometry point of
A Hermitian symmetric space
M such that for sp
every -1
such that
sp = sp
P
Eo
and
is a Her-
M there exists a holomorp
is an isolated fixed
sp' In order to study bounded symmetric domains one looks at
Riemannian symmetric spaces first, then singles out the Hermitian ones, and finally the bounded domains.
Riemannian symmetric spaces mannian symmetric, let
r
transformation on Hence if small
M; p
Let
M
be Rie-
then since
is a geodesic with
t, at the moment).
r p
(ds) is an involutive linear pp is an isolated fixed point, (dSp)p=-I.
O) = p,
then
It is immediate that
sp r(t) =
r(
-t)
(for
Tt = s ((t/2)s r(O)
,
is a one-parameter (local) group of isometries, called transvections based at rallel
p, which
has
r
as one
translation of vectors along
of its orbits, (and induces pa-
y).
Now a simple process of
"continuatfon" shows that every geodesic can be continued indefinitely, so
M is complete.
- 133 -
A. Koranyi
Let
p, q E. M.
sic segment joining them; carries
p
to
q
This shows that on
let
r
By completenes s there is a geodebe its midpoint.
Then
g
=
s s
q r
and is on a one-parameter group of transvections. I (M) (the identity component of I(M)) is transitive
o
M. (These remarks show that Riemannian or Hermitian
symmetric spaces and bounded symmetric domains may be defined in new equivalent ways.
We formulate the new definition here for
domains: A bounded domain if it is homogeneous and for some
such that
s
-1
and
s
Let We fix a point (i. e.
K =
0 If:.
1g E
subgroup of
Z
is symmetric
point
if and only
to D there exists
Z0
is an isolated fixed point of
M,
and
and
denote by
J).
s t:: G(D)
s.)
M be Riemannian symmetric and let
GIg
G,
o
D
G
I (M). o K the isotropy group at 0 =
o =
0
M
can be identified with the coset space
K
is compact, contains no normal
G/K. g
~
s og So
is an involutive automorphism of
G;
it is easy to see that its fixed point set has the same identity component
as
K.
{!
of
It induces an involutive automorphism
G.
Denoting by ~
The fixed point set of G' is the
(-1) -eigenspace of
as a vector space direct sum.
(5'
(Q
k, ,
exp t Y}
that
with
G=KP =P K
Ye .
fl·
the Lie algebra of
we have
K
!if =n + ce
It is now not hard to check that the
one -parameter groups of transvections based at
t
of· the Lie algebra
Denoting
P =
0
1exp
are exactly the
Y lYE
~J
it follows
- 134 -
A. Koranyi
r
It is also easy to see that d
1/ ( Y ) = dt
(exp t Y ) • Oft = 0
commuting with the action of for all to
11'
k 6 K).
by
P , we
dratic form
Q
K
defined by
:~~Mo
is a vector space isomorphism
(i. e.
'If
ad(k) 115' = (dk) 0
0
f
~
Transporting the Riemannian structure from get an
M
o - invariant positive definite qua-
ad(K)
oni? Definition : An ortogonal involutive Lie algebra (oiLa)
is a triple
(f!, 6',
f!
(i)
such that
Q)
is a real Lie algebra
(ii) G' is an involutive automorphism of always denote its eigenspac~5 for then have
[If)
1f'J C 1e
JB
(iv)
Q
quadratic form on
and
-1
[f( 11{] C?
I
(iii)
1
contains nc, is an
by
Ie
#".
and
(We shall
ff;
we
[1f, ~?] ctt)~
I
non- zero ideal of
-;;!'
ad(;& ) -invariant positive definite
%" .
We have associated an oiLa to every Riemannian symmetric space. Conversely, given any oiLa reconstruct a
simply connected
K
G
corresponding to
define a Riemannian metric on
we can always
R.iemannian symmetric space from
it by taking the simply connected group the analytic subgroup
(:f/,6') Q)
·C/K = M.
but this causes no difficulty. Details are
with Lie algebra
18 '
and using
(It may be that
ff7, Q to
Gf
I (M), o
e. g. in [22J p. 242).
In general there are several symmetric spaces corresponding to an oiLa of
(since non-connected
groups may sometimes be used instead
K), but in the cases that interest us it will turn out that the
spaces are automatically simply connected.
- 135 -
A. Koranyi
Orthogonal involutive Lie algebras. is called Euclidean
[111 ff] = 0
if
space is Euclidean). It is called and
ad( Ie)
J
Q)
(the corresponding simply connected irreducible
acts irreducibly on
1fJ r5
The oiLa
if it is not Euclidean
--r.
By rather simple algebraic reasoning
(e. g. [22]
p. 235-237) the following decomposition theorem can be proved:
(if, 6")
Given any oiLa
17 = 1:$-- -~!J;
where each
Euc'lidean for
j= 0,
irreducible for
is semisimple,
1tj = [ft j' ~j]
Q) ,~
(~,G'lfti,
QI9j) For
j:> O.
and
is a direct
Q I~j
sum
is an oiLa,
j'7 0, ' j
is a scalar multiple
of the restriction of the Killing form to 4'.. Let defInes its
dual
transporting that one
~f "
e'
(f!,
(f!*.
and
1I'if is
l3J <0' , Q) be an irreducible oiLa.
e/~ Q~) Q to
by
cf'*
11/<
=
.It' + i ~
and by
in an obvious way.
compact, the other
One
It is clear
non-compact.
In general, if the decomposition of
(1,
6"', Q)
contains no Euclidean factor and each factor is compact (resp. noncompact). we say that it is of If
compact
(respt. non compact) type.
G/K is Riemannian symmetric and the corresponding
oiLa is of non-compact type, then
G/K must be simply connected;
this is shown by a straightforward argument in In the irriducible compact case,
[8J .
II mflY be simple or
non-simple. In the latter case it is easy to show ( [22J; p. 238) that
y
=
1 & 1,
with 1 compact simple, and
g- is
fe
'V
1 is the
diagonal subgroup. The dual of
such a
compler simple algebra.
is compact simple, then
If
ff
the corresponding
<0
may
- 136 -
A. Koranyi
be any involutive automorphism of it. (The classification of the latter amounts exactly to the classification of all real simple Lie algebras (e. g.
22, p. 238); this remark
is irrelevant for our purposes since
in the Hermitian case the classification is considerably easier).
OiLa's of bounded symmetric domains.
First we have
to single out the oiLa's which are associated to Hermitian symmetric spaces.
M is Hermitian symmetric,
If
ture operator
commuting with the action of
J0
to a complex structure is easy to see
Mo has a complex struc-
that
~ ,
J
on
each
.Y;
11'
K.
commuting with
carries a d £.
J0
It
in the decomposition into Euclidean
j
J
an irreducible factors of ~ is invariant under
Now we consider only the irreducible case. By Schur's lemma
a dlf.
irreducible.
extended to
(f;
(the complexification of
must be
1f. is
lity, } of
is still reductive
is the center and
p.223)
[1e -:K?] 1
fl' ~J
(e. g. the Killing form of
;e ),
so
.;t =}. $ [li3
is semisimple.
I
By irreducibi-
is one-dimensional and the corresponding analytic subgroup
ad (K)
ticular,
[131
not semisimple. It follows immediately that
can be seen to be negative definite on
J-
is not
simple .
-I? where
1{f )
By a property of semisimple algebras (e. g,
this implies that
!
7f
If
acts as
J E ad(:t)
j}e
and
~ (cos J
e
())I + (sin
ad(exp }
P )J}.
In pa£
) \~ •
Furthermore, still in the irreducible case, one sees
- 137 -
A. Koranyi
je
that
Ie
and since adJe
acts irreducibly
cfl
(this follows since
G = Io(M)
M= G/K, Then
be Hermitian sym-
K is contained in the centralizer
But the centralizer of a torus in a compact connected
group is always connected, so
K is connected.
It follows that K
contains a maximal torus, hence also the center of ter of is
simpl;y connected
[8] p. 214-216.
Suppose now that in with the Bergman metric.
D
Let
is a bounded symmetric doma-
D
be its universal covering
it is still Hermitian symmetric
x.... x Mt
with
G ; hence the cen
By some further argument it follows that M
G is trivial.
space;
has no center
ff ).
on
metric of compact type.
J'.
in?" ' since
is a maximal proper subalgebra
Let now
of exp
8'
is the centralizer of
Mo
15' = MoxM1 x
So we have
Euclidean, the other
Mj irreducible Hermi-
On D we have the bounded holomorphic functions
tian symmetric.
z1' ... , .zn; their differentials span the cotangent space at any point. These functions
lift to
D
and have still the same properties. So
they can not all be constant on any of the
,
Mj .
Now
Liouville's
theorem shows that M does not occur, and the maximum principle o
shows that all the
Mj's are non compact.
pact type, in particular
So
D is of non-com-
D is automatically simply connected.
Cartan subalgebras . Let
M = G/K,
G = Io(M) be
any Riemannian symmetric space; we have the corresponding decomposition
tf = Ie +1:'.
is abelian, called a
by
[t,f]c!e.
Every subalgebra of /
contained in ~
A maximal such subalgebra
Cartan subalgebra of the pair
(1I.1t).
a
is
- 138 -
A. Koranyi
Similary as in the case of Cartan subalgebras of Lie algebras one can prove (e. g. [22J p. 252-253) that (i)
every (/:,
elements) such that "Or; (ii)
exists
k E K
contains elements
equals the centralizer of
a
if
such that
I
(called regular
X
ff
in
is another Cartan subalgebra, then there I
ad(k)-t:lI
tt = k l) € K
(iii)
X
=(/:5
;
ad(k) c/o
(immediate from (ii) and
Zorn's lemma). From Cl,
=
exp z:z, •
(iii) it follows that
Using that
a fundamental fact.
G
(E. g. if
=
P K G/K
P = exp
it follows is the
¥,c K A K
that
G
=
, where
KAK,
2-sphere, this amounts
to decomposing an arbitrary rotation in terms of the Euler-angles). (t ad( -a ) is a commutative algebra of semisimple linear transformations on
r'
so one can introduce roots and root spaces
with respect to it. It is known that one obtains in this way one of the standard root systems (a proofs is in the Appendix of
[19}).
The
Weyl group of this root system (called the small Weyl group) is realized by the subgroup of K which normalizes symmetric space
is realized as
G/K
(0]
a ,
if the corresponding
p.246). These last remarks
will not be used very much, a large part of what we do is independent of them.
- 139 -
A. Koranyi
2.
Imbedding Theorems of Borel and Harish-Chandra.
We start now with an irreducible oiLa =
Iff' + y;
(f'
':c! , Q),
;Jf=
such that./t' is not semisimple, and we want to show
that there exists a corresponding bounded symmetric domain. (Note that we do not know yet whether there is a corresponding Hermitian symmetric space; we surely can construct a Riemannian symmetric G/K
and carryover the complex structure
J
of
f
to the tangent
space at every point, but it is not clear yet that the 'almost complex structure" obtained in this way really derives from a complex structure. This question will be settled without using the NewlanderNirenberg integrability condition). c Let? be the complexification of
dI/J' c,
-£If c = ~~ d' IG
we have
'f! c
?: be the conjugation of
L e t-g .:Iv
+ cr t,7c
with respect to
~? has a one-dimensional center:f' (0
such that
J
2
ad(H)
(H f:
f
f )
definite Hermitian from checking).
~ IX
' E_
B 7' (X, Y) = -B(X,::r Y) on
i
o(] = H~
:rEo(
Each E
IG
1et
We know that Z
is skew-Hermitian with respect to the positive
c
and
ad(H)
11
c
is a Cartan sub algebra of
and
- E_ c(
follows.
i
'
be any maximal abelian subalgebra of K.
It follows that
so as to satisfy
dnce
1v'
+ 1• r&
spanned by an element
With respect to this we take a standard basis such that
XC'
= f Ll
extends to
= - I for J = ad(Z). Let
Each
1"
(H Ii:
H 0{
, ••• ,
1!' c. EO(; •••
fixing the lengths of the for \( E
f)
¢'
EO(
which is defined as
preserves
[1&, Ie]c fe, l1e, -rJc.r. Hence
(trivial
fl? c and -;rc
for every
- 140 -
A. Koranyi
D( ,
root case
either
or
Eo(
6
zr
<:X. is called a compact, in the second a
15
We denote by
the set of positive ..j.
b
c
In the first non-compact root.
non-compact roots.
We define
.=;Z(E~ O{E~
4. - _ ~ b - c(ei
fLE-«
1 c ·1+ +1- (vector space direct sum). It is now clear that E _ ~ ( IX <E f) can be arranged. Since f C t'! we have Hoe E i f for each 0( • g.- I- since 1: is the centralizer of f}-. We Then
=
'yE
E:
order
the dual of
possible),
;ftc. = "!: i.
f
i
=
0(
-i 0( (Z)
so that
>-
For o(E
p
JEO( = \:f(Z) Eo(;
we have
By the choice of ordering,
and we have
X
Eo( + E_O(
Y
HEo( -E_q)
JX 0/ = Y1:( Lemma
11 '
0(
> 0 (this is trivially
J = ad(Z) extends to a complex linear transformation on
Q' (Z)
= i.
We also introduce a basis for
c
0 implies
stable under Proof.
ad
by. J2 = -I, C{(Z) =
Similarly
If'
JE_c:i = -iE_ 0( •
defining, for all £y€
JY 0{ = - Xc{ 1.
le c
Z('t
and ~ - are abelian subalgebras of
.
ad(Z) has eigenvalues
corresponding eigenspaces
r.::t ,;lec.
~ i, 0 on
By a standard
-er c
with
arg~ment
Jacobi identity), if X, Yare eigenvectors for the eigenvalues
(the
PI
- 141 -
A, Koranyi
J, f<-'
[XyJ
then
is either
0 or an eigenvector for
[rc+) g+] = [~-) "It-}
Hence
Now let
0,
[letr,-g:i:] c~j:,
GCbe the simply connected group with Lie
algebra ;5'c, and let
' K , KC , p+ , P-
G, GV
lytic subgroups for the corresponding subalgebras of G v ' G, KC , K, P
LEMMA 2,
+
GC ,
groups of
+
±
is
Gv
'J
ff ad
works for
(-g'+)
1f+,
KC and
q.
G,
adGc
ad (%"0) is an alge-
+
nected group
adGc(P)'
phism; also
p+ is closed,
LEMMA of GC with Lie algebra Proof, invariant and (trivial),
faithful repre-
"8'+homeomorphically
(p+) of the adjoint group of
P +, being the preimage under ad Hence
3,
o-"*:
K is closed for it is
as a subalgebra of
Exponentiation maps ad
the closed' subgroup
c
are closed sub-
and conjugation with
bra of nilpotent linear transformations, and it is a
y
,
the identity component of the fixed point set of
the identity component of KC()
in
;J' c
induces an involutive automorphism T-If
hence closed, The same argument with CO
sentation of
be the ana-
exp; r-~ P- are homeomorphisms, Proof.
respect to
;;}+)<,
onto
GC , Now
is a covering of the simply conexp:
KC P-
Ie c + f-:
Kc P-
1S
,
t-, t
C+
15' + ~P+
is a closed analytic subgroup
. a group smce
1-
The normalizer of ~
subgroup with Lie algebraffl c +
"1f-
must be a homeomor-
1;- is
ad
is the normalizer in
(lei of~-
GC is a closed
The identity component
- 142 -
A. Koranyi
of this must be
KC r PHEMAHK.
KC
sees that
n
P-
= [e
Looking at the adjoint group of
J.
KC •
Hence
P-
is
GC
one
a semi -direct
produCt. VVe denote the homogeneous space M"*
(the reason for this notation will be clear soon). For brevity
we denote the identity coset by
Ie c + t' -
Since
M*- has a natural complex structure and
subalgebra, by
x.
holomorphic maps
tangent space
(standard fact, see
Mx is isomorphic with ~
+
is a complex GC acts on M'*
e. g. [22] p. 258).
The
as a complex vector space.
In the following three theorems we examine the orbit of x
'*' under
in
M
G U' G,
THEOHEM space.
and
1.
P+
GV /K is a compact Hermitian symmetric
v: Gt! /K .~ M* defined by
i(gK) = gx
riant holomorphic diffeomorphism onto Proof .
f/'v " (ffC-+ 1)-)=18,
-equivariance, which is clear, this
image of
GlJ /K
which shows that eK
onto an open set.
By the remarks above,
t.
(since
a By
is true at every gK , hence the
G V is.
Then its image
compact, hence closed. By connectedness, l- maps
•
t: is
is open.
Gv /K is compact, since
connected
Gv-equiva-
M'*.
homeomorphism of an open neighborhood of GV
is a
G
l-
onto
is a covering. But
is also
M~ M* is simply
simply connected and K • P connected ), so > -1 'If is a diffeomorphism. t carries the complex structure of M back
to
Gv/K. and the action of
So
Gv/K
C
G tr
is Hermitian symmetric.
becomes holomorphic by equivariance.
- 143 -
A. Koranyi
THEOREM space.
2.
M= G/K
is a Hermitian symmetric
j: M ~M~defined by
The map
j(gK) = gx
is a
G-equi
variant holomorphic diffeomorphism onto an open subset of Proof.
G-equivariance is trivial.
to-one it is enough to show for this is enough to show Suppose
Gn KCp- = K. PI) KCp- =
whence
p=p -k- 1 ,
and therefore
+
P-
,-..;* T-If p
G = KP(P = exp~),
GC induced by
(since
'*
-
'r.
G ,
We have
-
ro/ 1 = '" ~ (k p ) = k (p )-
if
'0
p2 = (p -)2.
p2 = e
Now to show one-
p = kp- (P<E: P, k E KC, P E P-).
be the involutive automorphisms of p- 1 =
1e }.
Since
M.,If.
Applying
Ad Gc
Z:~p-)2 € p+,
'Z""to this, p -2 =
is a faithful representation
onto upper resp. lower triangular matrices ), and hence
of
p=e.
is a diffeomorphism onto an open set for the same reason as
was in theorem 1. Also similarly to
z..
brings a complex structure to holomorphically; so see
M
M
with respect to which
J
on
U'
/K =
M~
acts
t .
In the following we do not write out G
G
j-£
is Hermitian symmetric. It is also easy to
that this complex structure induces
consider
i-I,
and
t, and j, but
ME M*.
Note that, in the special case of
G C = SL(2, C), what
we have done so far amounts to having constructed the Riemann sphere S'? and imbedded the unit disc into it as the lower hemisphere.
Next
')
we imbed the complex plane into
THEOREM 3. is a
KC -equivariant
SO
by stereographic projection.
M
k
defined by
S(X)=(exp X)x
holomorphic diffeomorphism of the complex vector
- 144 -
A. Koranyi
7:+
space
onto an opeh subset of Proof
p+ r"l KCp- = ~ e
show Let Let
~+
X€
a.
t"
M
ad{g) E_{3
t.
such that
E_{3
is
Assume that
(mod+
+cf-l H(3 (mod
e
one-to-one
#
we must
g E P+(i KCp-.
c(3E(3 ((1> (-= ¢ J C~+ 0 . Denoting 1G'~ l.
exp X = g.
[;C £_{?ll =C(3}{~ =:
"5
To prove that
X =L
Then
be the lowest root such that
we have
*
,,'
?'(;+)
16).
But
""" .... 0
and, exponentiating, KC p-
normalizes
'"
<0-
ad(g)E_(3 (;;' ~-, which is a contradiction. . + + exp : ~+ ~ P is a diffeomorphism. p+ ~ p • x c = .v+ ( nc + -) Its image is everywhere regular, since (7 + 'e so we must have
t .
'5l
is open by a dimension count. Finally, let
KC .
k f
.~ (ad(k) X)
We have
=
k{exp X) k -1 x = k (exp X)x = k ~ ) (X), which shows
= exp (ad(k)X) x
KC-equivariance. We proceed to show that
'S-l{M)
is a realization of
+
G/K as a bounded domain in ~ .
We say that the roots denoted 0{ and
.:l. ;3
are strongly orthogonal,
if they are not linearly dependent and if 0( +
cr-!3
(3
are not roots.
;
LEMMA
4.
There exists a set
orthogonal roots in ."f". such that
Y!
(g';/e).
gebra of the pair
One constructs lowest root in
C(, ~
cp'
a ~ C(EA ~ R X .:x.
6
b
of strongly
is a Cartan subal-
inductively, always choosing the
orthogonal to the ones already chosen. We omit
the proof which is a direct
(though not short) algebraic reasoning to be
found in [ 7] p. 582-583
[8J p. 314-315 .
or
- 145 -
A. Koranyi
by
H, E+, E
!
Let
LEMMA 5.
c
be the Lie algebra over
and the relations [E+, E-]"
[n,
H,
E+]
2E+,[H,E-]"-2E-
t
In any (real) Lie group with Lie algebra
we have, for all t E. R,
exp t ( E+ + E-) " " exp (tanh t) E+" e~p (log cosh t) H. exp (tanh t) EProof. The simply connected group with Lie algebra
C ),
with
H" (0
-1.)
SL (2,
1 0
E+ " (
00
o0 i0 )
E-"(iO)
It is enough to check the identity in this group; this is a trivial com-
putation. LEMMA 6.
A(x) "
Let
5110(e.1 L to( Eo{ /1 ~ I <1 } . Proof.
Let
~ tv< }.
for some real
A" exp ,:;;r, • Then we have
g E. A.
k p - E: K c P - .
So
g x " p + x,
-<& +.
notes the center of
G);
...,...
"exp~!J.
~
(tanh t "< ) EO(
and the adsertion follows.
F
THEOREM 4. a bounded domain in
t -X X
Using strong orthogonality of band
+ . + Lemma 5 . we have g" p k P WIth P and
g" exp (J.t;:fJ.
Then
+
Me) ( ~ ).
D"
Go (D)" S-l(G/ ([.)
~-1
)
(where
(M) C
is de-
every element of this group extends to a
holomorphic map on a neighborhood of
D.
The isotropy group at
o is a d ( K / C ) . Proof. KAK (x)
We have
KA(x). By Lemma 6,
G " KA K, hence
M" G(x) "
r~(X):1o(~ ~EO(II ~ J<1J.
- 146 -
A Koranyi Since
i
~
')
ad (k)
is
r:-1 ) (M)
K-equivariant,
o.'fi
bO(
Ed
II
b<>(
l·d ,
=
"¢'-l
>
K
KA (x)
'f- 1A(x)
k E K }.
The other assertions are trivial, Note.
One can study the orbit of exp i {/[,cG u
a way similar to lemma that
~ (1:+)
6.
in
With the help of this one can show
in dense in
M* (
rJ 16
p. 286-287).
Some remarks on classification.
It is clear now that
in order to classify bounded symmetric domains (up to isomorphi.sm) it suffices to classify all pairs (Y,G") where
b
is an involutive automorphism, and
A;,
is not semisimple. The classification of all dition on
ft?
is known
out those for which
Ie
(e. g.
[8
JCh.
[23].
is compact simple,
the fixed point set of G' ,
({f, <5) without any con-
v IX) and one can simply select
is not semisimple; the problem can be solved,
however. whh considerably less work. due to J.A.Wolf
[;f
Since
a Cartan subalgebra of both
te
/!if
The simplest way is the following,
is one-dimensional and! is and
Ie ,
it follows
that;;r has
exactly one simple non compact root; this simple root occurs with coefficient pact roots. coefficient last
in the roots in
cP
,with coefficient
0 in the com-
In other words there is a simple root which occurs with in the maximal root.
It is also easy to see that this
condition is sufficient to guarantee a splitting
under an involution, with
*
not semisimple.
7}t
=
If, + ~7
So the problem is
reduced to the computation of the maximal roots of the simple algebras. Carrying out the computation (done in
[23J ) one
finds
that there are four infinite classes of bounded symmetric domains corresponding to certain classical groups (they are called "classical domains ");
- 147 -
A. Koranyi
and two "exceptional domains" corresponding to exceptional groups. Explicit descriptions of all these domains are known. The classical domains have been much studied, they are described e. g. in
E7
[l2J
and
[18J.
The exceptional domain for the group
is discussed in detail in [9], cf. also
ceptional domain (corresponding to tion in
[18], § 18,
[3].
The other ex-
E 6 ) is given an explicit realiza-
and another one in
[10].
A table of the most
important constants characterizing all these domains is given in [15
J.
All this is rather irrelevant from our point of wiew; in these lectures we shall never have to make use of the classification.
- 148 -
A. Koranyi
3, Boundary structure of
D.
The Cayley transform and some lemmas . the notation of sec. 2.
We use
We also assume that we are in the irre-
ducible case ; the extensions to the general case are trivial. For
c jj
0(
reb.
and for
we define
'it 1'X 0( expt;
CO(
=
TI
0(€1'
c
0/
c They are elements of For each
Lemma 1.
G
also of
GU'
o(EfJ
[x.~
ad (cct')
Yo<' /----?>
XI{
HD(
~ r----'" -yo(
Proof. algebra as in Lemma
l.-
i ),
computation. Define
.6),
and
with respect to
1+
f-
Yrf/'
Ho(
span a 3-dimensional
the elemnt corresponding to
.2-4;
is
SL (2, ([' )
(C(E:
XC>(,
and acts as stated ,
CC(
by a simple
as the real space spanned by all
as the orthogonal complement of
in
1- I
iH I)(
in
B. Lemma 2.
it acts trivially on
J+.
ad(c)
interchanges
i ; - and
Ja
- 149 -
A
Proof.
ex (H) = 0
for all c{ <E
II
Corollary.
ad (c) to
y([
(or projection)
H~ C I~
since
1++ J ct
i
j-) ct
to
with respect to of the ordinary
J{)u~
J U'Ccr is /
Hence
HE
I
t-
DI, x~1 0 =
is a Cartan subalgebra
JiZ([ carrying
is an automorphism of
(r)+JCI(J(£,
system of
For the second note that
(O(Ei1) ,
LJ
off·
rlranyi
The first statement follows from Lemma
1 and strong orthogonality of implies
T7
It follows that the root
the same as the restriction system onto
-root
~-
(;-)
.
The following three lemmas allow a complete description of this root system.
(They will be used only at two points, most of what we do
is independent of them.
Still they clearly contain information of fun-
damental importance).
D(+
r'Cf-(,
Lemma 3.
are not both roots.
Lemma 4. -root is either
If:X E
0, or
LJ
and
l/ is any root,
In particular,
then
c{l. ( implies r{-dh(
The restriction to
± 0(, or + ()! + l~ - 2 -
2
The proofs are rather long arguments based on the ordering and on standard properties of roots. very clearly in [7]
on pp.585-588
They are presented
which can be read without any
preparation. Lemma 5.
The small Weyl group, acting on
consists of all signed permutations of
D.
i;-
- 150 -
A, Koranyi
Proof,
~2
is shown that
-,'- _ex + t'-/.) -2t:J2 and
are
-
On the pages referred to above of
42
(0(
r3 EL1)
(f or a 11 com b'ma t'lOns
f
T -/f' are (r -)-roots, (-- t'.( (3,1'» ,hence ~ -
() Il (tv;{;)ELJ) are
such that
+
is an
II
[7] it
(/l-root if and only if
') 0 fsIgns.
Suppose
CX _ _ ,_b &. 22
Then their inner product is negative
r
is an
(/-)-root.
i 1-
It follows that either all combinations :!:. :!:. I~}\ AI II ( )-roots, or /..J. is the union of subsets Lj ,D ex) [XII 0(1 /'..' /I N' -2 + .:.-. ( E t-Jo I IX t'5 LJ) is never a root. The
- 2 latter case is impossible by irreducibility .
(This is the first point
where we are really making use of irreducibility). It follows that the system of restricted roots is one
of the classical systems
:!:. ~ actually occur).
or
BCt
(depending on wheter the
Ct
The Weyl group of both of these consists
of all signed permutations of
.6 ,
Boundary components.
The m<;l.p
is a (real) linear isomorphism -Z:~'~+ such that jX~
for
0( E
ad K,
~
=
jYO(
EO(
(trivial computation).
since
J
does.
Let
j
j =
1- ( I 2
- jJ)
iEcx
=
commutes with the action of
=O(~'JREo(;
Of)+
it is clear then that
jct=ut'+. Under the form complex Euclidean space. mation that
T { Ef.>
B",
The norm
111
([
liT
has the structure of a
Ii
of any linear transfor-
will mean the norm with respect to this structure, (Note
J ((3 E rj)
is an orthonormal
system, and that
ad(G U )'
- 151 -
A. Koranyi
in particular
ad(c)
and
ad(K),
Proof.
D is invariant under
the norm on the right, since and
ad(k)
is unitary.
By Lemma
r 1(E)
Now
~ bo( HO(, since
ad (c)
and and
ad(r 1 (ad(k)E)) = ad(k) ad (j-1(E))ad(kr 1
ad(r 1(E))
E E A(O) ,
+
E E: D('\ or; .
0
bo( EO«( lb ol )<.1).
()(ED
ad(c).
J
has the same norm as
are unitary.
J
E =
is transformed by
the eigenvalues of the latter are
By Lemma
;
:!:. bC(:!:. bt1
and it is clear that all eigenvalues are for all C( E
ad(K) , and so is
Hence it suffices to consider
this means
2-6
act by unitary transformations).
L1 .
~
ad( :r,bo( Hol ), we know that
4
I
if and only if
2
let I-denote the real space_spanned by
); ~-t
(ex, ~ €:!J),
and possibly :!:.b<¥
We have to make some definitions now.
The centralizer of
into
L\ -F
in
I b« I< 1
For a subset
iR« (CXEr).
~t£
is clearly
'Z ([ f(b .
by
HD{ , EO(, E_ 0(
(
{3.lA-r
<X E L1 ), the centralizer of
f-tt)tf. (1'.-)(( D d:£ II + l7 +foJl.4-r ~
L: or
Ll-f dd.
is
Both centralizers are clearly reductive, and by Lemma 3 have the same derived L~lgebra, which we denote by We define intersections of
1r
~
with
.q(,F_
dp 1 n'.1f}
1e..1'..(I)
%ff/r) Ie
)
etc.
u+-) a.T' br '(;r We denote by
(£
'#T . as
Gr
, ~V-'
- 152 -
A. Koranyi etc. the corresponding analytic subgroups of G l . ~
Let
~
r
Mr
= G
'f Mr
Lemma 7.
where
jr'
Sf'
G
r'
r ,
G
d:
is commutative.
>-t )- M"'-<
r
jr
realization of Gr /KT'
tiation in G {;.
2-2, 2-3
construc-
~+
Sj,
~+r
D =
Dn
+-
r
1f1'
is the standard
as a bounded domain. The algebras
Proof.
by construction;
J
they are iso-
The following diagram,
G
~
In particular,
and
respectively;
r /Kr .
and
~
M ).
and the vertical arrows denote inclusions
I
Mr
(orbits in
are totally geodesic Hermi-
M , M
U/ K
M
r;S
, Mr
denote the maps of theorems
ted relative to
under
M = G (x)
'*
tian· symmetric subspaces of morphic with
(x),
x ' flr)u
are invariant
the same is true after exponen-
This proves the first statement;
the rest is tedious
but trivial checking. As an abbreviation, we write Also, for
E € ~+
,
we write
(This has nothing to do with
Lemma 8. In particular,
cA-l'. 0
Proof. so it commutes with
cr'
E E Dr'
iE a _r
c 4 _ r is Gr'
instead
ad(c.r )E. )
For =
E
Er
. c 6-r' E
.
in the exponential
of
= iE4 -
&~ ~
r
+E.
cr ,
This reduces the first assertion to the
- 153 -
A. Koranyi
second.
The second is reduced by strong orthogonality to the obvious
identity in
SL(2,
i)ti O)(~ 1
(whence
cLl - fI (x) =
1
0(2
1 (iE.6-r ),
D
,
D
with
stronger
is
c tl-t' • Dr
(real
However,
necessarily
c 6-]:' •
or complex)
UE
tfu,
yO( (o(E~-n,
and
A~ W0(
=
0,
, W]
following
[ul' W]
LU l , H.1-r]
and
i
[U l , = O.
affine
intersection
of
hyperplane passing through
U = U + U •
_l~ 2
~ ad V II ~ 2 , then
and
where
U2 - A-r all( Yo<,
ad(V)W = 2W+
U2 ...L Y /:; _ i ' )
11 ad
is the
Ul
is orthogonal
Let
W II( = Xo( + iH\J(
One computes then (using Lemma 1 ) that
2 WO(
from
Dr'
YA-:r l.u,
+ U E ~,
to each
[Y~
the
+?r+
in"!( + .
We can write
W
we want
iEtI_f'
By Lemma 6 the statement amounts to showing
Proof.
V=Y"-_r
9.
iE 11-I'
orthogonal to
that if
is on the boundary
result.
with the
and
etc.).
and that the intersection of the affine subspace
Lemma D
1
c ,1-]7. Dr
It is now clear that
of
0)
1
vl\~ 2 HA- r So
and
Ul
with
(since fy2' W] =
WJ
[ Ul' W]..L
now implies
1+ T,
~l'
w.
[Ul' W] T
e Ie ,
=
O.
But
and hence
is in the centralizer of
H Ll _ r
.
- 154 -
A. Koranyi
Application of Lemma be in the centralizer of each
is called a boundary component
---v D
of
now shows that
U1 must
(o(ELl-fl, finishing the proof.
H 0(
ad(k) c 4 _ r
A set of the form
in the boundary
4
of
D
.
'D
J'
Obviously it is contained
D
'u D
(il
Theorem 1,
is the union of mutually disjoint
boundary components. ~.r' .Dr ,= ad(k)cA_r'D r
(iiI
if and only if rand
r'
have
the same number of elements. (iii)
If
~:
~D
U
I
U
is the complex unit disc ) and
f(U) () F
=
¢
implies
is a holomorphic curve (here
F
is a boundary component then
c
F.
G
on the boundary are
((U)
Each
F
is minimal
with respect to this property.
k
(iv)
V
The orbits of
.
K ad(k)cLl_rD r
There are
Proof. D
iE ~ _ f1 +
c
ad(k
E EUD,
~ -1
)c .1-
ad(k)E .1.- r
A(O)
( any
t= rank of point
E
and also into
is onUiA(Oj, so of the form
( b 0(
This shows that
r· D['
(some
same number of
into
I I< 1 )
bot Eo{
(ii) =
ad(k)E
such orbits
D = KA(O) ,
can be transformed by some k€ K
iA(O); ,if
E
Since
(i)
t
G(iE A- t' )=
For this k € K)
elements.
we have to see only that if and only if
Using
this reduces exactly to Lemma 5.
the
r
'1"
Ell-r= have the
K-equivariant map
D). in
· 155 -
A. Koranyi
Vie may assume
(iii)
intersects it.
~
Let iE t.1-f
direction
'
hyperplane through
r(U)
Now
~ Dr .
f
Suppose
CU)
be the complex coordinate function. in the
I~ {I
therefore constant.
F=c t._{l
assumes then its maximum,
0
·fnJ)
It follows that
iELl_r
/J ct._l'
D,."
Dr
cl1 _P
because then
would have to stay entirely in a boundary component of The statement on minimality is clear,
is
is contained in the
' and, by Lemma 9, in
cannot intersect
and
C
it
tJ-r Dr
for any two points of
F
can be joined by a holomorphic curve. (iv)
Lemma
L:f bo( Eo() I b«/ <: 1J
+
{ iELj._r
is also the orbit under ad(K)c A- r
. Dr .
exp
as one sees at once from
A. )
It follows that
given in
(iii)
Corollary.
The union of all
S = G (iE Do
which is also an orbit of K
note
that
every automorphism maps boundary
is equal to the number of elements in
boundary of
contains
r
G(iE L\ -
components isomorphically onto boundary components.
components is
is
To see that it contains nothing else,
by the caracterization
D
aT'
(In fact one can show easily that this
and Lemma 7.
2-6
under
iE Ll- r
The orbit
The rank of
r , and must be preserved. a-dimensional
) = K(iE A );
boundary
it is the only orbit of
G
and it is equal to the Bergman -Shilov
D The last statement means that every continuous fm:lCtion
f :
D----7"
which is holomorphic in
(in modulus)
on
function
by
f(z)
S
D
assumes its maximum
; this is easy to see by approximating the
functions
fIr z)
principle on the boundary components.
(0 -< r <. 1), and using the maximum It can also be seen without
knowing about boundary components, as it will turn out later.
- 156 -
A. Koranyi
S
as a homogeneous space
not make use the machinery of Lemmas G(iE 1...)
K (il1.l,
"
the isotropy groups
F,
'6
for
RJ ''''
S ,
F
and
and F
L
of
G
,cy6 Ll)
and
iX, iY
Zl'
Xc JI'
~
ad
Z
.
IS
K,
respectively, D, since
2
or
or ~
IF
K/L
stands
1 span a 3-dimensiona1 simple real Lie
#1- ,
c€
exp
([ (exp
tvv)
can
exp
referring to
9' ~.
be regarded as a representation The element of
SU(2).
1 (1i I i)' '('!.
I, -I J;
at
"Z - Z
't h as or d er
SU(2) corres-
8•
1
Every irreducible representation of
{1J
G
(as usual, Hf,
of the simply conn!cted group c
We know that
we now want to study
are transitive on
o
G = FK = F K. o We define Zl
algebra #f. C
· pon dmg to
3 to 9 •
this set is called
and even
implies that
This section does
accordingly the image of
has kernel
SU(2)
c
has order
8
4
fO ad(c) \5 "
Note that " ad(exp i ad(c)
4
~
7
I5'(X) ) = ad(exp (-i
commutes with
the
(2:
"
= [~
ad(
ad (c) ex
-1
~
.
7t
1) -eigenspaces of
J
the centralizer
1f'1+'
of
?i
•
We define
ad(c)4 in
~
in
Ie
in
etc. , in the obvious way.
"
Hence
•
,I'\:.
-«"J
~:;. ~+~
6* c)
p;{£
II
tl,~
~f,
X ))
since it preserves
~1~2
C;
ad(c) (2;'-1"
so it preserves
((j
It also preserves
1fl
- 157 -
A, Koranyi
It,
+ 11
is
G'-invariant;
it is a sub-oiLa
with the exception that it need not satisfy sec, 1.
This, however is irrelevant
changes is
that to the sum
vi
summand, on which
t2
=
4' ()( 111-'
and
10)7
Lemma 1
ad(Zl) =
~2'
2
fl q;; ,., (/;) 1t
/tJ 1= I'v
V2
+ /'VI -ILJ
10.
On
~1'
I
of the
Q)
definition in
for the decomposition, all it there is added another
·te +t:t1 I
acts trivially.
It follows that
ad (Z).
On
ad (Z) 2Z 1
Proof.
fW
tomorphism of
(iii)
(1, ci,
of
is the image of
iX
under an au-
By the remark made on the kernels of the
representations of
SU (2),
exp
(1
Zl)
and
exp ( ': iX)
=
c
have the same order on any irreducible representation space in The sum
ad1flt(exp #y )', have order
8
is
~2
of the irreducible spaces where they
12
~
+
by definition .
t! 2 + t~ Hence ad(Zl) +
sented by diagonal matrices, so on ad exp (
]t.,
1 Z 1)
=
e
t4
II
,
using the ordering of roots it follows that
1!2~ +It on ad(c)
~ 2'
~
f2
This
Zl €
.art
implies,
ad(Zl)
in particular,
f
:ve
is reprehave
2" 1
there, and i + on - 2 1 ad(Zl) =2 ad(Z)
The same argument with the irreducible spaces where
has order
4· gives
the statement about
We denote by corresponding
to
G l' K1
the analytic subgroups
of
if G
f1' leI
Lemma
11
G1 (x)
Hermitian symmetric subspace of bounded domain is
-%1'
D ()
~~
,
1f',
G1 /K 1
is a totally geodesic
Its standard realization as a
158
A. Koranyi
By the preceding
Proof.
and is invariant under simple direct
11
splits under CO
ad(Z).
J
checking.
'1.
Note that it may happen that~ 1 = then that
is of tube type
GlK
fi? 1 ' ~,
t
~1
9'1 =
~1
the
11
M 11 ad(c)(?e C 'f'V
ad(c) have which
=
ad(c)
L l'
-1
)
cx ,
and since
I).
We define: (ad(c)2
L
Kl
t
in;e
is at
,is
iE
the Lie algebra of
L
M r"C-C?,U'
Since
ifu
(feC + -!(-) (\
This means those
V t::'
=
is since
K,
we
K
for
c5
V, this
V = V
ef ad(c) <0'
Using condition means
4
Denote it b y ' .
rU'
ad(c)
ad(c)
The Lie algebra of
if (I ad(c) 1e <5
~1
the isotropy group of
+ ~).
preserves
I'
12.
"> (iE.D
Since
D
(+ 1) -eigenspaces oJ
t2
+
Lemma The Lie algebra of
say
is a tube domain). 2 are preserved by ad(c) , by an argu-
(note that on
ment seen before
We
(it will be seen later that it is in
this case that the Cayley transform of
Proof.
The rest is
2 ad(c) V
=
V,
i. e.
= ad(c)
-1
VEt.
and
V =
For
LIthe proof
is similar.
Remarks.
1::1; space
ad(c)
2
is an involutive automorphism of
it is easy to see that there is a corresponding (real) symmetric Kl (iE Ll
) =
Kl /L l' totally geodesic in
M~ .
This means
- 1'i9 -
A. Koranyi
that if
G/K
is of tube type
then
S
is a compact)
space.
In the general case one shows easily
K(c 2x)
K/K'
Lemma 13 ----._-
+
(iii) ad(c)
2
carries
ad(c)
2
interchanges
i~2
with
(-1)-eigenspace of
interchanges the
11
(:t: 1)-eigenspaces of
interchanges the
J
7f1
in
Proof.
= ad(c)
S
K1/L1
(i)
ad(c) with
and
1;
with
~; in
M*,
+
(ii) ad(c)2
Th. 4.9) that
is a Hermitian symmetric subspace of
is a fiber space over it with fiber
12'
(r15J
symmetric
- 2,..1 \.?
i
(i)
V
- ad(cr 2 V
~2
into
The second half of roots so that
Let
-i C( (Z1)
= - (ad c)6V = ad(c)2V,
amounts to a convection:
>0
with the former convention that
(iii) note that if
ad(c)2 V
implies
D( >- 0
-i Q;' (Z»
=:t: V,
then
= "+ JV (where we used ad(c)2 Z1 from the definitions). Now the last definitions:
.+
hence
ad(c)
0
We may order
=
2
a 1+ n
ad(c)
-Z l'
the
(this is in accordance
implies
is proved in the same way as
%1
2 0'(ad(c) V)
Then
The proof of the converse is similar.
(i)
(ii)
V EO i ~2'
~
(i)
ad(c)2 JV
> 0). To show
= -J ad(c)2V =
which is easy to check
- 160 -
':\.. Koranyi
+ $2
(~+ o/~ . 2 2
+
14~
;U-
fel~ =
t+
Lemma
14
) rl ad(c)(!
tJ,{,~
+
2
i% 1(.*= 11 + i ifl 1 eigenvalues -1 AJ ad(c) 'vv 2
O,
~
+
1, _ + 2
(i)'
is semisimple on
an d correspon d· lng · elgenspaces
/fi 2 -+
"
"
[4; }-U-;]c a;
Proof.
ad(Y)
I
#-+
is semisimple
Hence its eigenspaces on C We know that spaces on
Cf
+ 1
ad(Zl) ~ +
are
0,
+
+ I- 2- , f 1 - '
+ 2 with
,
"
so
"
ad(c)-I~1 TV
I/l~ are real forms of VI
(ii)
eigenvalues of idlC + f\. ,~-
with
~
ad(cr 1 11-1
(iii)
ad(Y)
eigenspaces
13
flC
on
and preserves
are real forms of its eigen-1 i ad(c) ZI = "2 Y , and that the i + + i with eigenspaces
-
2'
Hence ad(Y) -1 C ad(c) I , etc.
has eigenvalues
te
R.emains to check that that is clear from Lemma
is step 2 nilpotent.
(ii).
ad(c) -
1
~I
C
We also see
n
fj!
dL? 17f= II.- , but
that ad(c)
-1 i f + ·vv1-
- 161 -
A. Koranyi
are real forms of obvious
-1
111 -+ ,
in
(ii)
follows.
(iii) is (of
ad(Y)
case). Theorem ad(c)
=
whence
by the general rules about brackets of eigenspaces
in this
rI
ad(c)
-1
Ije' *"
+
ff.
k' *" ,
are closed
in
F
o
The Lie algebra of
(i)
2. -1 ad(c)
#+
It is the normalizer
is of ad(c)
-1
fA 'Yf/
+
Jr',
~+ the analytic subgroups for -1 1;-1 N+ . . a d( c ) IS a semlad(c)
+ N, =
F
K'
direct product.
f
Proof. =
are in
ad(c)(
f!;
~C
/v
+
The Lie algebra of -
1: ) n
~.
ad(c)
F
is
-1/jp*"
and
'/1/ '
if we can show that they are also in
ad( c)
-1 A-J
'YV
ad(c)(;Ie C +
+
~ -),
-1 1/7 ~ then the first statement follows by a dimension count. Now ad(c) n:- ' = hC -1 b -1 -£/+ = ad(c) c ad(c) ad(c) 'n'1 + C ad(c) b1 = ad(c)
Je' *"
since
IV
-t61
ad(c) 2~' 1 = -(/+ 01 . .
Finally
ad(c)
-1
/lA
rf/(.,2
+
c ad(c)
-1
(1z+ +
+
-
0/2) c ad(c)( t.t2 + ~
by Lemma 13 (i). The statement about the normalizer follows from Lemma 14 (i). 2-3 .
(ii) is proved by a standard kind of
argument, the same as Lemma
- 162 -
A •. Koranyi
4.
Generalized
halfplanes.
This section is entirely independent of the three ding
ones and has an elementary character. Let
J2 c W
A subset open" and
prece-
W
be a finite-dimensional real vector space.
is call ed a regular cone
a 0:/::.
convex and such that
~
-y
n.
The dual cone
JJ.'
It is not hard to show that
Let sion
n1
f- a
now n2
~: cp
is a function (i\
:1 > a imply fly E 12. by SZI ",[tYiw1
s:l'", n .
is also a regular cone, arid
V l' V 2 be complex vector spaces of dimenrespectively;
let
W be a
in W .
An
Q. -
real (orm of Hermitian form
is complex linear in the first argument,
V (v, u) q>(u, v)
(ii)
12 ,
y €
is defined
be a regular cone
V l' and let
if it is non-empty,
(iii)
~
(u, u) E
I2.
(iv)
rP
(u, u) '"
a
(conjugation with respect to
for all implies
The generalized halfplane
W),
u c, V 2' u '"
a
corresponding to
Sl
and
is then defined by
(This notion is due to Pyatetskii-Shapiro; of type
I
there is no
or
II
rJ>,
he calls D
a
Siegel
or not
If
depending on whether
n 2 '" 0
and D
'tube domain'
is simply the
over
domain n2
D. )
=
O.
- 163 -
A. Koranyi
The distinguished boundary B =
i (z1'
and holomorphic
on
f
then sup
D ,
is a minimal subset of
D
is
z2) E V 1 X V 21 1m z1 -
It is not hard to show directly that if D
of
D
(z2' z2)
o} .
is bounded continuous on
If I =
I
f I ' and B B This will, however
~up
with this property.
D
P
be an immediate consequence of some later results.
n
admits a group
D
morphism, whose elements are pairs
of affine holomorphic auto-
g= (a, c) E W X V 2' acting by
~(c,c)
+a+2ip(z2,c)+i
g
+c It is easy to check that
of step
2
(if
n
n2 = 0
is simply transitive on
B,
it is even Abelian).
It is of interest to consider the group
and the group
~ D
is called
G (Q))
It and
G(Q,
g1
affine-homogeneous
is transitive on
~)
and nilpotent
f2. ;
¢ (u, v)
if
G(Q)
G(0.,P)
=
= P(g2 u , g2 v ),
pr 1G(n,~)
(the
l1~,v J .
projection onto
in this case the group generated by
is tr.ansitive on D.
All we do in this section is valid for any generalized halfplane.
Let us note that
one-paramet er group
G(n,
I(e 2t 1 ,
~)
etI)
always contains at least the
ItE
JR' S '(
In the following we assume that with their standard complex Euclidean structure. unimportant, Haar measures.
V1 = a;nl , V 2 = a;n2 This is actually
quite
all it is good for is to fix the normalization of certain
- 164 -
A. Koranyi
We define the measure
J WXV
g
~
on
B
B
(z2' z2)' z2) dX l dX 2 dY2
It ---;;. B
is a diffeomorphism
0
action of
n.
(3
Lifting
16
to
,equivariant
For
0
:=
<: p
n
(both left and right
For a function D
we denote by LP(B) the LP - space
00
~
with respect to
ft
for
under this map , it is immediate
to check that we get a Haar measure on .1+ invariant, of course, since / () is nilpotent).
define
S f(u) d(3 (u)
by
2
g.
the left
f
f (xl + i
~
~
([;
D --;>
f
by
ft(zl' z2) For
0 <: p
of holomorphic functions of
We define
H
00
(D)
we define
<. 00
f
on
as the space
D
HP(D)
as the srace
such that
of bounded holomorphic functions
on D . The following is a variant of a theorem and plays a fundamental role in this section. we denote
B;:.\ (u, v) =
4<;l,
Hermitian quadratic form on Theorem let
1.
p
(u, v»
V2 For
z
For fixed
B~
of Gindikin
[ 6]
~ E 52' is an ordinary
- 165 -
A. Koranyi
{z (l d) = (det B.::.. )
3/4
e
2Jti<~, Zl+2fi~(Z2,C()-iP(Z2'Z2)+i~(0(,D{»
Then the map
'! ~ f{z) =5 / .Q
n
>< R
is a Banach space isomorphism particular,
H2(D)
LP(;\,({,) 2 -:; ,
r(~,C()d:t do( z
n
L 2( Q X R 2) ------".. H2(D)
(In
is a Hilbert space. )
The proof is too long and technical to be given here. We should note, however, that if
D
is the upper halfplane in
then this is a classical result of Paley and Wiener. case
(n 2
=
0)
it was proved by
Bochner
[2)
case in a somewhat different form and with only a
Q;,
In the tube The general sketchy
proof
is in [6] . The present version in due to (to be published) .
E. M. Stein and the al1ihor
- 166 -
A. Koranyi Theorem 2.
f E H2(D), then
If
Sketch of proof: on By
B
by setting
Y1
=
Let
P
(z2,z2)
j
f exists in L 2(B). t B be the function defined
f
lim
,tt~R
inOthe formula
of Theorem 1 .
using the Plancherel theorem twice, one sees that
J
n2Ir(~,Cl())2 (1_e-47I(tYf~dO(
dl
.Q.' X R From here the statement follows by the Lebesgue dominated convergen('e theorem. Remark
This theorem
to compute inner products of functions in compute the inner product in
? L ~(B)
gives a manageable way H 2(D);
one only has to
of their boundary functions
The following short discussion is of a quite general nature; it is probably rather familiar to the reader who has studied the question of the existence of the Bergman kernel and Bergman metric. Let
H
be a Hilbert space of complex-valued func-
tions defined on a set E. kernel
for
H
S : E x E ---
E --7'
if, defining
(w E E fixed), we have
w
It is then automatic that
suppose that
A
Cf /--;'- 'f)
(l ' Ilz')/) w
by
S (z) = S(z, w)
all
f €
S(z, w) =
w
H,
S(w, z),
w
€
E·.
and
it contains a family of elements where
is an isomorphism =
A
<:f: E ~
/t-?H .
a:
is defined by
We claim that
S
is then a reproducing kernel for
defined by H.
S(z, w) =
- 167 -
A. Koranyi
In fact, defining
t
= S(z, w) some
rE£,
as above,
S
W
A
Any
lz) = lw(z)
w'
and we have
A
we have f <E H A
( fJ ' /t w)
(f, S ) w
S (z) = w A equals for
r
A
f
=
(w) = f(w),
proving the claim. All this applies, of course, to the situation of theorem 1, and after an easy computation we have :
Theorem called the
H 2(D)
3.
has a reproducing kernel,
Szego kernel, given by
S (z) = S(z, w) =
w
1, .n.
e _27(.<;t,
f
(z, w)
(det
B
~
)d I\.
r\,
where
Corollary. BUD
For fixed
wED,
as a bounded continuous function
analytic
on
(in fact,
S w S
extends to is even
w
D). Now we define the Poisson kernel of
P (u,z)
\ S(u, z) \ 2
(u Eo B,
D
by
ZED).
S(z, z) We shall also use the notation zED,
P
z
P (u) z ia a function on B.
Lemma function on
D
1.
Let
Then, for all
F
P(u, z) , so that, for fixed
be a bounded hoI om orphic
zED,
P ). z
- 168 -
A. Koranyi
F
Since
Proof. FS f H2(D) z By Theorem 3, we have
FS
z
is bounded and
(FIB
1 S( z, z)
u
o
For every
in
z
~u
( e)
z
(szl'
P
for all
z
u e B , z €: D .
is a continuous
for all
= 1
u E B
zED.
and every neighborhood
o
B,
5
lim
s
z E: D,
5BPz(u)d~(U)
(d)
of
P(u, z) ~ 0
F(z) .
B, vanishing at infinity . (c)
N
z sz\ B )
(FS ) (z) z
(a)
For all fixed
(b)
function on
4.
,8
B.
(FSZ)B' SzIB)
S(z, z)
Theorem
<E H2(D)'
z
has continuous boundary values on
1 S(z, z)
(Fk, P) z
S
u ¢.N
0
Denoting, for real
VS z 2) S
n
P (u) d z
(3 (u) s
we have, for all UEB, S
S
P(z , u)
Proof .
(a)
0
and
z ED,
ZED,
P(z, u)
is clear from the definition.
The first
- 169 -
A. Koranyi
part
of
(b)
follows from the corollary of Theorem 3,
part can be deduced from a similar property of
S
the second
(which, in turn,
')z
is a consequence of the simple fact that
S r::.. H~ (D) for all w w (c) follows by applying Lemma 1 to the function
we omit the proof).
F :;.
1. To prove
IF 1=:;1
(d)
we take a function
F, holomorphic on
F(u) = 1 and o (It is easy to check that F = cS . , for any no + ly Y = (y l' 0) , Y1 6.Q and appropriately chos en c
D,
such that
on
D,
m
=
sup u¢'N
IF(u) ) < 1
has these properties. )
By Lemma 1 ,
lim
=tI
z
l
5=
Using
N
F(u)P(u,
.5
ueN
z)d~(u)+
1.
J
(I-m)
lim z·~u
U{.NF(U)P(U,
P(u, z)dl3(u)+m P(u, / u,N
(c) it follows that
which proves
F(z)
o
On the other hand, )(F, P z )
lim
(F, P ) =
z·~u
0
J
P(u, z)d#(u) uEN .
z)d;3(u)/~
z)d/'~(u). 0
(d). The proof of
(e)
is a straighforward checking, based
on the formula of Theorem 3 Remark.
For
t
E:.Q
we define pt: ?{;--:'l>R
by
- 170 -
A. Koranyi
'""t . P (g) = P(0,g .. and for
f:
B~
( it,O) ) ,
£:n'7(J;
we define
by 7(g) = f(g .0).
Then it is easy to check that, denoting by of f (F t (7:) = (f, P z ) ) ,
~
Ft(g) = Ft(g. 0)
?i.
where-/( denotes convolution on
on
B,
and let
F
the Poisson integral
we have
rV
5
Theorem
F
Let
rvt
(f
*
f
be a function or a measure
=
P )(g)
f = lim F t
be its Poisson integral. Then
t~O
(a)
pointwise if
(b)
in
f
.c
(c)
in the weak-
*
(d)
in the weak- '1f
L P,
L'f
is continuous and bounded,
f
LP(B)
(1~.:p«X))
LO'
topology of
if
ff L
topology of measures if
f
DO
(B.)
is a
finite Baire measure. The proof is immediate from Theorem 4 by standard methods of functional analysis. The following generalization of Fatou's thorem is somewhat more difficult. For For C\
>
0
g
= (a, c)
and a cone
(0
E
,?j' we
define
such that
{.-L\
II= g
t!:U
Maxi' \ a
1, \ c \ 2 } •
we define the "ad-
missible domains"
ret:
Ja(O) =
r, ,\
- "I
'-V
~
g • (iy,O)
(g • 0) = g ¢
0
..
r
I y (J}' c::;
,(0)
(\IJw
(in the case of one complex variable these are _ non tangential angular domains on the upper halfplane).If
F, f
are
functions on
D, B
- 171 -
A, Koranyi
respectively, we say that
F
a. e., if for every fixed C>(
lim z -+u
to
u
EO E.
the Poisson integral of
u0
I b:) (
we have
f( u ) o
)
00
6
Theorem
f E L
If
f,
the
F
Lebesgue's
and
(E)
converges to
The proof is based on Theorem extension of
f
o
r~
o
and (J
F (z)
z E
for almost all
converges
4
f
F
is
admissibly a. e.
(e)
and on an
differentiation theorem to the group
16
it will be contained in a joint article of E. 1\1. Stein and the author in the Annali della Scuola N ormale di Pisa. We should note that for special types of D 6
can be' considerably strengthened .
with a symmetric domain of
~ube
that it is enough to assume strengthened the case
by
p=l
complex ball,
When
type,
D
is isomorphic
Norman Weiss
f E LP(E)
Theorem
[21J showed
This
(p .:;>- 1).
was still
E. M. Stein
(unpublished, as of now) to include 1 The case of f E L (E), D isomorphic with a
was settled by the author
extended these results to include all symmetric domains (for
Theorem
D
i-i. \A/eiss has recently
isomorphic with classical
f EL I (E)).
7
(1
:.= p'::: 00), then
- 172 -
A. Koranyi
F
is the Poisson integral of a function in The proof for
[14J
it is explicity given in much harder theorem);
p ::> 1
LP(B).
is a rather standard argument;
(is one variable this is called the
it is due to
E. M. Stein
[20]
F.
is the Poisson integral of
f
,then
J2
HP(D)
M. Riesz
!I F t II
p
;:.llf
II
if
isometrically
into
F
\~
f
LP(B)
F€ HP(D)
p L (B)
(immediate from the Remark after Theorem
This shows that the map
is
.
L (B)
t E
and
From this theorem it follows that,
Note .
for all
p =1
The case of
p.337 .
4).
imbeds the Banach space (if
1 ~ P ~ 00 ) .
In the case of the one-variable upper halfplane the set of Poisson integrals of all
(say, bounded)
functions
on
characterized as the set of (bounded) harmonic functions on It would be interesting to have an analogous If
D
B
can be D
result in the general case.
is symmetric, we shall show in sec.5
that every Poisson
integral is harmonic in the sense of symmetric spaces (but not every harmonic function D
is a Poisson integral).
nothing along these lines is known.
In the case of a general
- 173 -
A. Koranyi
The Cayley transform of a bounded symmetric domain.
5.
Let in
1j
+
be a bounded symmetric domain imbedded
D
c•D
We want to describe
in the standard way (sec. 2).
in the case of the unit disc this amounts to rotating the Riemann
~ around a horizontal axis,and thereby mapZ + i Z L>.. ping the unit disc onto the upper halfplane, r-r iz + 1 ,which sphere by the angle of
is equivalent to the usual Cayley transform). A priori it is not clear that
the action of
c
on
7;"
+
is defined as
S(-
so we need to know that
f
-L
c" D C '>
contains
out to be the case, and we shall see that
makes sense;
~-i
(and
eM. c. D
D
~ (M)),
=
This will turn is
always a gene-
ralized halfplane. c • D
To compute transitive on
D
(ef.
sec.
on
we use the fact that
by theorem
(By definition of
ad(c) F
0
=
K'*. N+
we have to study the action of this group on ad(c)F
is the isotropy group of
'tg' ~
ad(c)G
at
this latter point can be interpreted as a "point at infinity 11
c • iE
in
0
F
is
S as a homogeneous space), so
c • D = (ad(c)F ) (c' 0) = (ad(c)F )(iE II ) . o 3-3,
Fo
We know that
K'
'*
being a subgroup of
K
on 7j+bY the adjoint representation.
Lemma
*' T form 11! of ~i ' quadratic form
self-dual with respect to the restriction of the
B '7: Proof.
that
is a cone in the real
1 .
We show first that
E,,6, E .(,P + ; by definition of
lJ,J.
+
Ell E 'Vt--~
We know
...u.; we have to show only that
- 174 -
A. Koranyi
Ec,. ~ ad(c),.
For this we compute
ad(c)
-1
this compu-
;
ED
tation is reduced by strong ortogonality to a computation in the 3-di mensional
simple algebra, and the identity -i
shows that
ad(c)
-1
ED E
1
) ( 0
1
0
(I' ' d
Next we note that by theorem
lf~
instead of real
K{
W~,
normalizes
linear transformations.
It follows
1
-i
3-2, applied to
i. e.
a~s
on
i l c ~.
Kl"* at
The (connected) isotropy subgroup of is
1
( i
2
-v7'
by
iED
a dimension count now shows that Q is
(L 1) 0;
open 2
We have 3-1),
hence
ad(c) zl
~1
zl t.
-zl
Then
(immediate from Lemma
iZ l C
1(;,
exp ti zl E
This is the group of real homotheties; it preserves the set
D of
is a cone.
#7
is identified with its dual under
adjoini:j of the linear transformations in
DI , Ch. I, with
by some
(relatively easy)
Propositions 9, 10).
Kt,
tfJ1
corresponding to and
1ft
since
,
K~
Ere'
results of Vinberg
tl + i
i -0: lIon
symmetric, respectively.
Ojl
, so The group
([20al, Kf coincides
and the linear transformations /u vv + 1
are skew-symmetric
Therefore, to show that
Q' ==
n
it
..0.'('\0.+0) by showing e. g. that E A c Qi . that Q = ad(K~) • EA= exp (ad(i "11))· EA'
is enough to show that To see this we note For every
V
E.
i 11' exp(ad(V))
E R).
acts transitively on
The group of adjoints of =
Q
K~ (t
is positive definite on #~
, since
- 175 -
A. Koranyi
ad(V) is symmetric.
Therefore
Bor (E, E A ) > 0
i. e.
B'Z: (exp(ad(V)). Etl ' E A ) > 0 ,
E En.
for all
This means exactly that
E.1 E..Q I, finishing the proof. Lemma 2 .
a:
ad(c)2 = i ad(X).
For this we have to note that
irreducible representation spaces on which Hence
~
ad(c)2 = exp (ad (
which in turn implies that
+
7t .
e - 2'
(since
=
1
iX))
( using at
-
ad(c)
ad( 4-iX)
T
[u, x]
- [U,
=
Lemma
tp
-cc-
3
U C ~2- .
i on it
><
= -
ru, iEAl
a;
T
we denote
on ~
B "/:
+
U, V € 1f2 - Ti
E6
f: .~(~ ~;
We have
is abelian).
For
21 ad(U) ad(V)*
is a function
~
eigenvalues + i
iELJ - [u, iE-Al
* its adjoint with respect to
=
8.
~ i ), and the assertion follows
the last step that
rl\ (U, V) 't'
is the union of
has order
must have
For a linear transformation
by
-ce2<£+~
must have eigenvalues
To prove the Lemma, let ad(c)2 U =
0;
First we show that, restricted to ~~:+ ~,
Proof.
we have
[ U, ad(c) 2
-z::
(V)
J.
with the properties :
(i)
~
is complex-linear in the first argument,
(ii)
is Hermitian with respect to ~,
(iii)
1-
for all
k €
K1
*"
- 176 -
A, Koranyi
cP(ad (k) U, ad(k) V) = ad(k)
~ (U, U) ~12 for
(iv)
implies
Proof,
[~V,
= -
iEA ]
equality.
~ V), - ad( 'Z:
~ (U,
and
U) = 0
¢ (U,
so in ,u+,
V)
Property
(J,L
(ad(c) 2"'CV
we obtain
= ad(V)-it iEA
It is trivial to check that
ad(zl)'
1:-'
7: (V) E ~ ~ Lemma 2, and the
ad(V)* =
= -ad(rvV)iEA
-eigenspace of
Uc
all
U = 0,
Using
easily checked fact that
tD (U, V),
proving the first is in the (i) of
(+i)-
P
is
also trivial.
+
To show + with respect to
(ii), let Jk denote complex conjugation in ~ 1
;-
~.
Since
+ n a~c)
+
~=~
:J1
'1'
)l-
is
the restriction of the complex conjugation of with respect to -1 ~ -1 -2 ad(c) i. e, of + ad(c)6'~ ad(c) = ~ ad(c) 75 ad(c) .
11'
Now let
U, VEt;. We have -2 2 2 2 2 6"'G ad(c) -U = - r:;rr: ad(c) U = - '0" ad(c) '(; U = - ad(c) 7: U ~
'0
--Z:::ad(c)
-2
2
(ad(c) 7:V) =
.-J '0 V
=-V
and hence
fA-
J.: (U, V) =G'<;:,ad(c) -2 'J, J:. (V, U), fJ! :I! (U, V) =21 [ - ad(c) 2 ?:: U, - V] = lJ! To prove
both
trivial on
t
the
-If'
ad(K'),
if'
-Ion
it-
ad(c)2
i aI'
r
and -z; are Hence their
,which means that it commutes
with every
From the second form of the definition
of
statement is now obvious. To show
B-z;-
we note that -
and equal to
product is trivial on element of
(iii)
(cf: (U,
U), V)
~
0
ip
(U, U) E for all
J2. V IE:
it suffices to show
12 -(since
n
is self-dual),
- 177 -
A. Koranyi
*"
Given
VEIL,
we choose
Since
ad(k)
unitary, by
where
U' = ad(k)U.
is
'
E~
which is
),
~
=0 ,
(U, U)
This means that
it follows that whence
(iii) we have By(P (U, u), V)
.
= Bt' (~(U:
0
~
by the positive
B 7: . If
formula.
ad(k)V = ELI
such that
This is further equal to
~B~ (ad(U')ad(U'( ELl definiteness of
k 6 K'
[z::U,
yJ
then
ad(U)
=0
-ad( 'e'U)E£).
= O.
* E 6.
=
0
by the last rz;; U E ~;,
Since
By Lemma 3-14 then
~U = 0 ,
= 0 , finishing the proof.
U
2 + + I - ad(c) 7: : 12 --7>1112
Lemma 4 .
is a linear
isomorphism. Proof. 2 + ad(c)
Let
Zf2+ ;
V E-
?;V e ~2
then
and
+
Considering the definition of #2 ' we have to
show only that
V - ad(c)2
~V
E
ad(c)
ff
(then the map will also
be surjective, by counting dimensions).
To show this we apply the
conjugation with respect to
which was computed above
to be G"'z: ad(c) that it leaves
-2
.
ad(c)1'
The same computation as in Lemma 3
V_ad(c)2-& V
fixed.
From now on ,given EE + + projection onto and ~2. by E 1, E 2 .
if'!.
Theorem 1. g = exp(U + (I - ad(c)2?:)V) acts on
'1:
+-
Every for some
-«,+
g eo N Uc
we shall denote its
So
E = E1 + E2 .
+.is
of the form
+
#1..
Ve
2i~(E2
+
fR.'
by g. E = E + U + V +
(ii) shows
,V) + ip(V, V) .
It
U'), ED)
- 178 -
A. Koranyi
Proof . 4.
The first statement follows from
Lemma
Applying the Campbell-Hausdorff formula , we have , 2 g = exp{U + (I - ad{c) 1: )V) =
+
= exp (U) • exp
ad{c)2~v])
[V,
• exp (V). exp{-ad{c)2z- V),
since all other brackets vanish by Lemma
#, ,'* is easy to compute. (J
factor on
3-14.
The action of each 2 -Ijp a: -ad{c) 'T-'V
%
We have
so the last factor acts by the adjoint representation; exponentiating we get
exp{-ad{c)2~V).
E = E - tad(c)2'1;V, EJ E + [E2' ad{c)2-z;v]
=
since all other brackets vanish, again by Lemma other three factors in the factoring of
g
belong to
3-14.
The
p+;
therefore
~ Using now the definition of 'f!
they act on ~ + by translations we obtain the statement. Theorem domain constructed in
2.
D
If
sec. 2, then its Cayley transform,
the generalized halfplane
Proof.
1E I 1m E1
~)
=
K'*
(K'* C Go{.Q, ~)
equality can be shown by using that but is unimportant). K,;lE;
K'
if? (E 2,
E 2) E
is
·P}··
n
N+, and follows from Lemma 3 (iii);
n
is, a symmetric space,
Our generalized halfplane is affine homogeneous, since
acts transitively on
equals the orbit
-
c. D
For this generalized halfplane the role of
(in the sense of sec. 4) is played by GO{QI
is the bounded symmetric
"!
D. +
It contains
• N (iEL\. ) =
iELI
(ad(c)F o)(iE j"
at the beginning of this section, this orbit is
c ' D .
and therefore By the remarks
- 179 -
A. Koranyi
Theorem
Let
3.
be the distinguished boundary of open subset of
S
E 11m El - ~ (E 2, E 2) " -1 Then c • B is a dense
B
{
c•D .
the Bergman-Shilov boundary of
o}
D.
Proof. We have B::(N+.K'~'O:: (ad(c)Fo).O. Then c-I.B:: Fo.(-iE~, . -1 smce c • 0 " -iEA as it is easy to check. It follows by a dimension -1 count that c • B is open in S. Assume it is not dense. Then -1 there exists a point u in the inferior of S-c • B, and a function o f ho1omorphic on D, continuous on D, such that assumes f -1 its maximum at u and nowhere else. Then g"f c c is a o bounded holomorphic' function on c. D, continuous on c·D and
I I
such that
sup c•D
I g \ ';> sup I g I . B
This is impossible by the results of sec. 4.
Explicit formulas and harmonocity of
P(u, z) •
We want to find more explicit formulas than those given by Theorem
4-3
for the Szego
and Poisson kernels of
We will also determine the Bergman kernel which is necessary for the proof that Poisson integral)
c· D .
(up to constant multiple ),
P(u, z)
(and therefore every
is harmonic.
We use
the norm function
r - function of Koecher, r~s) N(E b
N:
SL-4-
which are defined by
=
N(y)s
) " 1 .
5~
e-B-z:- (y,t)N(t)s-l dt
R
and the
- 180 -
A. Koranyi
By a simple change of variable one sees g€ K'
'*
that, for
,
N(gy) = (det g) N (y) where det g
ad(g) \ 4ij
is the determinant of
The same change
.
We
of variable shows that everything is well defined . Gindikin showed,
[6 J ' r15]
d.
n1 -
where
1
(s) = (271-)
.
Mt +1
n 1 = dIm
,
t t
---.
~'*
2
1
and
Theorem
4
note that, as
n1
-1
j7J o r (T
n1 -
t
(t -1) )
s -
is the rank.
The SzegH
kernel of
c·D
is
n2
1
S( z, w)
)N(r(z,w))
Proof . We look at 4- 3.
Let
'A
det B
g €K~ ,
= (det B,.
to
(det g)
k
). (det ad(g)
for some
A./+
' YI- 1 ), since
of this form
all
hence
k = n 2/n 1 ,
k
'f~
)2,
The latter factor must be
(where det gis, as above, relative
I-dimensional representations of
(recall that
I-dimensional center).
in the formula of Theorem
then a trivial computation shows that
g
equal to
det B4\
-1-n 1
K1 , hence also det B9I
and det B;\
*'
K1
'*
K1
is reductive with
is homogeneous of degree cN(A )')"\&ht4.
are
n 2,
The constant
c
- 181 -
A. Koranyi
can be computed
n2
to be
by verifying that
2
(this depends on some identities for roots;
= 2Bz-
it is
5.2 in [14] ).
The statement follows now immediately from the definition of the norm function.
Note.
A more explicit way of computing the norm
'will be indicated in sec. 6 . Lemma
The Bergman kernel function ofe • D
5.
is
K(z, w)
Proof. kernel one knows in the first,
From the general theory of the Bergman
[8]
(e. g.
antiholo~rphic
p. 293) that
(i)
K(z, w)
is holomorphic
in the second variable, (ii)
K(gz, gz)
I ~ (1:1 l
domain.
. For a homogeneous domain these properties clearly determine
K(z, w)
up to a constant factor;
= K(z, z)
for all automorphisms
g
of the
it is even enough to consider
in some fixed transItive subgroup bf the group of all automorphisms. It is obvious that our formula fulfills (i). (ii)
for the transitive group
r (z, w) So
(ii)
ad( c )Fe
=
K'
*.
We check
N+. N+
invariant, and also has Jacobian 1 everywhere on holds for
g t. N+
equal to the determinant
of
If
ad(g)
g E. K,-If-
11"+ (cf.
leaves D.
~ i;~)
Lemma 3
(iii)).
By the computation in the proof of theorem 4 , this is equal
will be
g
- 182 -
A. Koranyi
to
(det g)
Now
N(
with
det g
referring to the action
of
on .;;1..+ 1
g
-2- "2/n" gw ))
f (g z,
-2-1l:!nt
finishing the proof:
(det g)
With a considerable amount of extra trouble one
Note. finds that c
(d.
[14]
and references given there);
p.346
of the Euclidean volume of the domain
Theorem harmonic function on by all
5.
c •D .
D.
For every fixed
u
e
B,
P(u, z)
is a
(Harmonic means that it is annihilated
G-invariant differential operators,
has the mean
this is also the reciprocal
value property with respect to
or what is the same, it the orbits of the isotropy
group at every point. )
Proof, , sketch, (see
The following proof, which we shall only
is rather roundabout, but seems to be the only one known
[14J
§
3
for more details), The crucial point is to prove
d!3 (gu) fo (u)
P (gu, gz) d
P(u, z)
that
- 183 -
A. Koranyi
for all
g
e
ad(c) G,
B
(these
is still in fixed
g).
Once
z E c • D, and u
u C: B
such that
form a dense open set in
B
this is known, OEe can deduce
gu
for every
the mean value
relation P(u, kz) dk
P(u, w)
P
w
(K
w
being the isotropy group at
(u)
w
wE c • D),
by noticing that
p w
is characterized by the properties
(i)
P
(ii)
for hE K
w
(hu)
P (u) ( d/.3 (hu)
w
(in virtue of the transitivity of
dfJ (u)
w
K
w
on
)-1
B).
Using the invariance of the Haar measure, it is trivial to check that the left hand side also satisfies To prove 4
Lemma 5,
and
not matter).
(i)
and
(ii).
(-*) we proceed as follows. k
8(z, w) :: cK(z, w)
One knows how
K
(the value of
transforms under
By theorem k
does
automorphisms;
one deduces that 8(gz, gw)A (z) A (w) :: 8(z, w) with
g
glj (
A (z) :: (~ (g~
g
definition of
)
)k
(, z
P(:u, z)
uE B
l
such that
z, w E. c • D,
U : H (c • D) g
-+
2
By
that
A (u) \ 2 :: P(u, z) g
gu
<= B .
One shows next that, for
2
g € ad(c)G,
(one chooses a holomorphic branch).
it follows
P(gu, gz) for
for all
H (c • D)
defined
g E. ad(c)G , by
(U f)(z} g
f(g
-1
z)A _l(z) g
- 184 -
A. Koranyi
is a unitary transformation.
In fact,
it is trivial to check
U 8 ) = (8 , 8 ), and the statement follows from the fact g w z w that the 8 (z e c • D) are dense in H2(c • D). Using this, we have z f E H2(c. D), for all (U 8 , g z
One sees easily that there are enough functions this can be true for all d
f
r8 (gu)
d(O (u)
finishes the proof of
=
H
2
so that
only if
I
A ( ) g u
I2
,
which
It is now not difficult to see that, given a
bounded continuous function F
in
('7f').
Note.
ction
f
harmonic on
f
on
B, there exists a
c. D and continuous on
c:15,
unique funsuch that
- 185 -
A. Koranyi
6.
The Poisson integral on the bounded domain
D.
In this section we will translate the results of sec. 5 from ~he
c • D to Szego
D;
we will also find rather
and Poisson kernels of D.
(It is interesting to note that
the only known way of getting to these results This
c • D .
explicit formulas for
[14J
is by passing through
seems to indicate that in some sense a generalized
halfplane is a more elementary object than a bounded symmetric domain in the standard
(bounded) realization .
It is also true that all bounded
homogeneous domains can be realized as halfplanes
(affine-homogeneous) generalized
(result of Vinberg, Gindikin and Pjateckij-Shapirol,
seems that they do not in general have a canonical We denote on
S,
of
S
by
f
as
S= K(iE A. )
= G(iE A );
It
we
define
(O
f : D ---+ 0: r
by
(l f
1/
Ilnd
H
(D)
LP(S)
f (z) = f (rz). r
f
~ay
just think
HP(D) on
will be the
f:D-')-O:
and
O
Generalizing a
classical
(0 -< p ""- (0)
as the
D such that
O
as the space of bounded holomorphic functions. For the sake of brevity in this section we denote
bye.
L P-space
= sup p
OQ
functions
D (in fact we
Given
definition in the unit disc, we define space of holomorphic
K-invariant measure
the results of this section will show
that it is the Bergman-Shilov boundary). with respect to
bounded realization. )
the normalized
the Bergman-Shilov boundary of
but it
We define the Poisson kernel
of 'D
by
EA
- 186 -
A. Koranyi
(JO J z (u) for
zf D ,
C/O
= t)
u € c
-1
P(eu, ez) P(cu, ie)
(u, z)
We extend its definition to
B.
•
DX S
by the following lemma .
a unique continuous extension to
yo(u,
=
z € D,
For every fixed
Lemma 1 .
S
k €
For all
z
K,
has
f1J (ku, kz)
z).
Proof.
e
-1
B
is dense open in
Sand
K
is transitive on
S; so everything is proved if we prove the second -1 statement for u E e • B For this let us use the notation e -1 e-1 k for eke ; we know that k · ie = eke • (c. 0) = e· 0 for
Using
k E: K .
(-If)
from the proof of theorem
c P(k ,. eu, k e • cz)
Jd(ku, kz) =
P(eu, ez) P(cu, ie)
P(k e • eu, k e • ie)
5- 5, we find
=
giD(u, z)
finishing the proof.
Lemma 2
For all p. (eu) Ie
is a normalized be equal to
d
l
-1
• B,
-1
in Theorem 5-5, -1 K-invariant measure on e • B
Proof.
(u).
By
Ute
(*)
P. (eu) d f).(cu) Ie IV Hence it must
ie
- 187 -
A. Koranyi
Remark .
r:;)
J
(J/t;
(gu, gz) ctj4!)u
JD (u, z)
that
Using this, one computes easily that
If =.' (u, z),
d)«)
is harmonic
which is equivalent with the fact (d. Theorem 5-5).
!JD (u, z)
one knows even without this argument that For
u
G:
c -1.
and for any
B
u € S
But, of course,
is harmonic:
this is immediate from the definition of .f/J(u, z), it follows using Lemma 1.
Theorem 1. (i)
!fJ (u, z) '/'
0
for all
(ii)
For all fixed
z f D,
(iii)
u E S , zED . ((0 v is continuous on z
for all
(iv)
For every
(v)
in S , o z·~u.~u¢}/ Defining, for functions f :S of
Uo
o
(I
r
and every neighborhood
E: S
~
u
f(k) = f(k. c)
S
18 (e, k. .j
(k) =
rV
F
(convolution on (vi)
P(u, z)
re)
(i)
Proof. S
by
showing
P ?- 0 z
r
=
0
N
O.
cr,
---,lO-
< r «:: 1 ,
(k E Kl,
F(z)
we have
S ffJz S
d/ t '
r-'
f-l(
K).
for every fixed
function on
~(,u,'Z) df1uJ= \'
and defining, for
for the Poisson integral
0}1<
zED.
D
uE S
is a harmonic
.
We show everywhere on
everywhere B.
Using
(*)
- 188 -
A. Koranyi
(Theorem 5-5) on
B
if is enough to show that
everywhere
o
le
OJ (u, z) ••. )
(in fact we should have shown this when we defined
AssuineP. (u ) = 0 .
le
for all
0
le
which by Theorem
u
o
E B .
Then
by
(~),
P.
;; 0 .
Then
S.
i . e
keK,
le
(iii)
is immediate from Theorem we use Lemma 2
le
_ 0
SB
4-4 (b)
and Lemma
and Theorem 4-4 (c) to get
P(cu, cz) P(cu, ie)
The proof of
tI
4-3 is absurd.
(ii)
To prove
for some
0
P. (k c , u) = 0
1.
p.
d IN(u) /
P(cu,cz) dj3(cu)
1.
(iv) is analogous . (v)
Is immediate by the equalities
,...,
F (h) = F (he) = F(h • re) = r
r
J
= S flu) jqu, h • re)d F(u) = = =
(vi)
r
JK
({J-l f(k) rJ (-e, k h •. re) dk
tV
(f* JO)(h) r
.
Was already proved in the remarks preceding the
statement of the theorem .
Corollaries . in place of Theorem
B 1
has to argue
of Theorems
The exact analogues 4- 5 and
(To get the analogue of a little
more:
(with
S written
4-7 follow immediately from 4-7 in the case
p = lone
In order to make sure that the boundary
- 189 -
A. Koranyi
function is a function in use
L I(B),
and not only a measure, one can
the analogous statement in Theorem
form c. )
It is also clear that, for
1
4-7
and the Cayley trans-
~ p.~
HP(D)
00)
can be
regarded as a subspace of Theorem
4-6 could also be translated to our
but this is not too interesting, since the results quoted after 4-6
seem to indicate that for the symmetric domain
D
D, Theorem
considerably
stronger theorems hold . Now a
simplification of the method
For every
f E H2(c • D)
[1~
of
suggested by
(Tf) (z) is
by Theorem (Tf
4 -7
f
D,
1/2
D
S, (cz) Ie
-1
and is defined
is defined
is holomorphic on
a. e.
on
e: K
since we know that
(ef.
S, Ie Sunder
by the transformation rule of proof of
imply
Theorem
5-5),
f(cz).
a. e.
and the mean
S
S. (cz) Ie (k c cz) =
since
1=
°
0;
in
for all
automorphisms value property would
S. (ie) = 0, which is absurd. ) Ie We also define D >< D -+ C
d:
on
B
fact, in the contrary case we would have k
N. Weiss.
by
S(ie, ie)
well defined on
making use of
we define
Tf : D v S -+ C
Tf
H 2(D)
we take a closer look at
by
) _ S(ie, ie) S(cz, cw) w - S(cz,ie) S(ie, cw) and
d
w
by
cfw
(z)
=
c1(z,
w) ,
It is immediate to check
- 190 .
A. Koranyi
J: and U10ting
that
= S(ie, ie)1j2 S(ie, cw)
d
(iJ(u, of
H (c • D)
T S
cw
D X D),
extends to
z)
Theorem
2
-1
onto
W(D)
Proof.
(a)
(Tf, Tg)L2(S)
= (f, g) .
(Tf, Tg)L 2(S)
S(ie, ie)
Js
2 . ?
T
is a Hilbert space isomorphism
cf
is the Szeg8 kernel of
For every
In fact,
is I
by definition of
H 2(e D ) we have T
and by Lemma
Z,
Sie (eu) ,-2 f(eu) g(cu) d/I..(U)
f(cu) g(cu) Pie (cu)-1 d /((u)
J
B f(v) g(v) d[3(v)
(b)
f, g E
H 2(D).
For every
(f, g)
f €
H2(c - D)
we have
if)
(Tf,
2
z L (S)
=
(Tf) (z) .
and applying
This is clear by using our previous formula for
a
z
(a).
(Tf, u 0 z) 2 @ -f 1 L (S) (Tf) (z) . To see this, we write (Tf, J ) = cJ (z, z) - ( (Tf)0 z z then note that (Tf) is the image under T of an element z ? W(e· D), of so (b) can be applied . -f -1 = C) (z, z) (Tf) (z) cf(z) Continuoing the equality we get z = (Tf) (z) •
For every
(c)
2(e · D, ) f E H
,dz ),
cf
Q
- 191 -
A. Koranyi
is
')
Tf
(d)
immediate from
(e)
is an isometric map by
since the image under on
D,
and every
such functions
(e. g. (f)
T
since
can be approximate f= lim r -?1
H 2 (D)
is now obvious from
where on
A
-
g
by
r
H 2(D):
is of the form
Since
Tf, this
One can easily show ((14J, p. 345) that
of
D
is again constant times a power of the
It is also easy to check that, for all
(z)
H 2(D))
(in
(b).
Remarks the Bergman kernel
holomorphic
f).
d is the reproducing kernel of
we know that every element of
Szego kernel .
(a), and is surjective
clearly contains all functions
f G H 2(D)
This
1 (v) .
(c) and Theorem
T
2 f E H (c· D)
for all
<=0 H'(D)
ge G,
is a power of the Jacobian determinant of
In particular, since
k € K
g
acting
acts by unitary linear
transformations,
kw)
(The last equality can also be proved directly elementary way, by noticing that of
H2 (D) . )
f !--+ f
0
k
in an
is an automorphism
- 192 -
A. Koranyi
We shall get a formula for the
Explicit formulas . Szego kernel
(and therefore also for the Poisson kernel)
of
D with
a high degree of explicitness, which makes the precise computation of We shall also indicate
this kernel in every concrete case very easy.
how one can compute such constants as the volume of
D or
B
.
(It makes good sense to talk about these volumes, since according to
the conventions and results of are uniquely determined
sec. 2 , the size and shape of
D
by the conditions that the isotropy group at
o acts by unitary linear transformations and that the largest inscribed sphere in
D
has radius
1.)
For all this we have to make some
further computations in the spirit of sec. 5.
Hence
; c.;t.
By Lemma
3-1,
ad(c)2 He( = -HO(
.-/*
,/"=
Defining
if
-
,we have
/*
(1€1~~)'
the pair
.Q.
(L 1)o H* (e), H~
Therefore
g = exp L..J
Il(~..a
t
K;- = (L 1)o H *"(L1)o
and
~
by the
KC
A typical element of
by strong orthogonality of
we have
H*.
is a Cartan subalgebra of
being a subgroup of
g. e
J
Lemma
by
.n. ).
... H",.... , we have
H'* (e)
-If
a useful fact for making computations on
adjoint representation.
~
j}j .
Fc i1/c ~
GC
We denote the corresponding analytic subgroup of (It is easy to see that
for 0( E
=
j.
lot{i y 3.
I>i,
acts on
'*
H is of the form t IX (HOI ) ~. = L...t e E = L./ e 2tO(E "
C\'E-ll
It
follows that
Eo(
I
For
,
0 .t:::. Yt:( <. 00 } .
~
tteLl.
~
• 193 -
A. Koranyi
(where
N
is the norm function of
Proof. with =
?
g = exp
det (ad(g) J "~"''1+ J
~C>(
Hot
)
=
r
By Lemma N
=
det
3-5,
n
{j(~t:.
f>
y
A
2
0(
IT
=
g
Lemma
~
E,/.!
(oJ
Yd(
r
Proof.
S(cZ o )
For - z
~
with
I ~
n1 ,
? +r~
o
1
y 6(
~
<;>=2#,,1;"'. 1:(6' LI. Since
where
is the same for each
4. yo(
g
< c1, (3...
<01, f> = nIle·
c • z0 = i
N(y)
i E(51 ((3 ~ ~ )
must be a homogeneous function of degree
we have
e
y = g
this is now trivial to compute since
g • E,g N(y)
D).
We know that
acts diagonally with respect to the basis
Hence
= rank
By the remark just made,
(log
I;
!!
[2,
it follows that
(-1< ~<1)
Eo<'
r ...
(otE!J ) .
1 - ral
A direct computation gives
(ex p '
P
n, i
4
I + r~ exp ("1' l~ 1 - rO(
where we had to use strong orthogonality
X ot ) (exp i ; ; rol Eo.') • X
,
and the identity
~)
• X
- 194 -
A. Koranyi
C iiI:) (
=
o Theorem 3
cf (z o,
we have
J(~ ~)
1 - r
= i
1- r
{2
0 A
(-1
1)
Ol
(where
z ) 0
dim a:; D).
Proof 5 - 4 and Lemma
a
Immediate from the definition of
.
To illustrate how one gets
a
precise
in concrete cases, let us consider the domain
formed by complex
Theorem
4.
Example .
formula for
J,
pX'q
matrices
(P?:' q)
z
D
I - z~ z
such that
is positive definite . This is a symmetric domain in standard realization the connected isotropy group where
u,v
are
(pxp
K
at
resp.
example is studied in detail in
consists of the maps
0
unitary. matrices .
q)< q)
[18J ;
z;.--+uzv (This
it serves very well to illustrate
all that has been done in these lectures .) and it is easy to check that by an element of
We have
n = pq,
K every
z
t
=
can be transformed
into
z
o
r
q
)
wich corresponds to the situation in Theorem
3.
q ,
The question is
- 195 -
A. Koranyi
how to rewrite the expression in such a form that it be invariant under K
and it is not difficult to notice that this can be done in a unique
cf (z,
way by setting phic
z) =
det (I - z* z )-p.
cf is
holomor-
in the first, antiholomorphic in the second variable, we must have
d (z,
w) = det (I - wltz)-p .
Remarks.
f&
K-invariant measure
We have been using the normalized on
S;
one might want to use the volume
t+ instead.
induced by the Euclidean structure of acts
Since
by unitary transformations,
these measures are
the proportionality constant being the
Lemma
5-6),
c
and then making
at, say,
K
proportional,
Euclidean volume of
It is easy to check that the Jacobian of
[14.1
Since
S
-ie
use of the precise value of the
constant in Theorem 5-4 one finds that
vol
S
The same kind of computation can be performed the Bergman kernel
of
D
Using Lemma
( [14J '
with
sec. 5) .
5- 5 one finds an expression analogous
Theorem 3, and associated with it the exact value of the volume of D
(cf. Note after Lemma 5-5).
to
- 196 -
A. Koranyi
References
[1]
W. L. Baily and A. Borel, Compactification of arithmetic quotients of
[2]
bounded symmetric domains, Ann. of Math.
i!!.
(1966), 442-528.
S. Bochner, Group invariance of Cauchy's formula in several variables, Ann. of Math. 45
(1944), 686-707 .
[3]
H. Braun and M. Koecher, Jordan-Algebren, Springer 1986 .
[4]
E. Cartan, Sur les domaines bornes homogenes de I 'espace de n variables,
[5J [6]
S. G. Gindikin, Analysis in homogeneous domains, 19
(1964),
3-92
Uspekhi
Mat.
(in Russian).
Harish-Chandra, Representations of semi-simple Lie groups VI, ArneI'. J. Math.
[8]
ll.(1935), 116-162.
C. Chevalley, Lie groups, Vol. I, Princeton University Press, 1946.
Nauk
[7]
Abh. Math. Sem. Hamburg,
78(1956), 564-628 .
S. Helgason, Differential geometry and symmetric spaces, Academic Press, 1962.
[9]
roJ
Ch. Hertneck, Positivita:tsbereiche und Jordq.n-Strukturen, Ann. 146 (1962), U. Hirzebruch,
433-455.
Uber Jordan-Algebren und beschra:nkte symme-
trische Gebiete, Math. Z. 94 G. Hochschild,
Math.
(1966), 387-390 .
The structure of Lie groups, Holden-Day, 1965.
L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc. 1963. [13J
N. Jacobson, Lie algebras, Interscience, 1962 .
[14]
A. Koranyi, The PoiS'S-an integral for generalized half-planes and bounded symmetric domains,
Ann. of Math.
~
(1965), 332-350.
- 197 -
A. Koranyi
[15]
A. Koranyi,
Analytic invariants of bounded symmetric domains,
to appear in Proc. Amer. Math. Soc.
(16J
A. Koranyi and J. A. Wolf, Realization of Hermitian symmetric domains as generalized half-planes, Ann. of Math .
.ill
(1965),
265-288.
[17J
C. C. Moore, Compactifications of symmetric spaces II. Cartan domains, Amer. J. Math.
[18J
~
The
(1964), 358-378 .
I. I. Pyateckii-Shapiro, Geometry of classical domains and automorphic functions, Fizmatgiz 1961 (in Russian),
(19J
I. Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math.
(20)
(1960), 77-110.
E. M. Stein, Note on the boundary values of holomorphic functions, Ann. of Math.
(20aJ
~
~
(1965), 351-353 .
E. B. Vinberg, The theory of convex homogeneous cones, Trudy Mosk. Mat. Obs., 12 (1963), 303-358 English translation in
(in Russian);
Trans. Moscow Math. Soc. for the
year 1963, 340-403 . (21]
N. Weiss, Almost everywhere convergence of Poisson intgrals on tube domains over cones, to appear in 'frans. Amer. Math. Soc.
(22J
J. A. Wolf, Spaces of constant curvature, Mc Graw-Hill, 1967 .
[23J
J. A. Wolf, On the classification of Hermitian symmetric spaces,
J. of Math. and Mechanics
(24J
(1964, 489-496).
J. A. Wolf and A. Koranyi, Generalized Cayley transformations
of (25]
13
bounded symmetric domains, Amer. J. Math. 87(1965), 899-939.
A. Zygmund, Trigonometr~c series, Cambridge University Press, 1959 .
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C. 1. M. E.)
J.L. KOSZUL
11
FORMES HARMONIQUES VECTORIELLES SUR LES ESPACES LOCALEMENT SYMETRIQUESII
Sommaire
--------n. 1 n. 2 n. 3 n. 4 n. 5 n. 6
Prelimmaires. • • • • • • • • • • • • • • • • Espaces riemanniens symetriques et espaces fibres vectoriels associes. • • • • • • • • • Seconde structure de fibre associe et connexion linea ire symetrique. • • • • • • • • • Formes harmoniques et Laplaciens .• • • • • • • • Decomposition du Laplacien . • • • • • • •
·
..
·
• •
I99
• •
204
2IO 212 217 Isomorphisme a,P (M, E) -+Q~ (41' ,'l1) et Theoreme de nullite 222
··
• • • • • • •
·· ·
Nombres de Betti de M.• • • • • • • • • • • • • • Le cas des domaines bornes symetriques. • • • • Decomposition du Laplacien dans Ie cas des domaines bornes symetriques. • • • • • • • n. 10 Sous~fibres holomorphes de E. • • • • • • • • • n. 11 Noyau de 6a . • • • • • • • • • • • • • • • n •. 12 Theoremes de nullite. • • • • • • • • • • • n. 7 n. 8 n. 9
· · ·· · · · · ·· ··
Corso tenuto ad Urbino dal 5 al
t3 luglio 1967
229 233 238 245 250 256
n. 1 FORMES HARMONIQUES VECTORIELLES SUR LES ESPACES LOCALEMENT SYMETRIQUES par J.L.Koszul (University of Grenoble) n. 1 P r ~ 1 irn ina ire s . ----------------Soit M une
holomorphe compacte admettant pour espa-
vari~t~
ce de revMement universel un domaine
par exemple une surface de Riemann compacte de genre Ie groupe des automorphismes holomorphes suppos~
r
de
n
n
n. Puis que
A
nest Muni
K de
M. Ceci montre que
r\A
groupe
A. On retrouve Ies
s'identifie
composante connexe neutre et
A/K et
a
r\A/K s'identifie
a
est compact, autrement dit que rest sous-
A0 de
r o --rn
de AO • Or un groupe de
A opere proprement
A. L'application canonique de r\A sur r\A/K
n
n
est un sous -groupe
zEn. Son stabilisateur est un sous-
est donc propre . Or
uniforme de
M
o
Choisissons un point
dans
-..!)
A. On sait d'autre part que
groupe compact
vement
1 . Soit
"compacts-ouverts", A est un groupe de, Lie; Ie grou-
pe des automorphismes du revMement
dans
de
>
C,
homogene, Ie groupe A opere transitivement dans n
de Ia topologie
discret
N
homogene n de
born~
Lie
A.
m~mes
circonstances dans Ia
CelIe ci opere encore transiti-
AO est un sous-groupe discret uni f orme connexe contenant un sous-groupe disc ret
uniforme est unimoduJ:aire. D'autre part, on sait qu un domaine
born~
admettant un groupe unimodulaire transitif de transformations holomorphes est un domaine de
born~ sym~trique (Th~oreme
nest donc point fixe
holomorphe
de
isol~
de Hano): tout point
(et unique) d'un automorphisme involutif
n
Montrons maintenant comment, Iorsque 1'on part d'un domaine born~ sym~trique
admettant
n
n
,on construit des
vari~t~s
pour revMemevt universel. Soit
holomorphes compactes o
A
Ia composante con-
nexe neutre du groupe des c.utomorphismes holomorphes de
n. Pui-
- 202 -
n. 1 J. L.Koszul
sque
n est
A o es t un groupe
sym~trique,
a l'~l~ment
semi-simple
de Lie
r~el
de centre
r~duit
dra,
tel groupe contient toujours des sous-groupes discrets unifor-
un
neutre. D'apres Borel et ,Harisch-Chan-
mes (dans Ie cas g~n~ral, ils s 'obtiennent par ture
un
donc un sous~groupe fini ment dans n
,c'est
r
stabilisateur dans
r\n
te raison
dans
il
n,
en
de
r
a dire
En
g~n~ral
r
ne sera pas
a
en g~n~ral
r
r.
est
n'opere pas libre-
qu'il existe des points de
n'est pas r~duit
m~me de
va de z dans
zEn , Ie stabilisateur de
tout point
A o.
sous-groupe discret uni f orme de
A0 opere proprement
Puisque Pour
r
. Soit
arithm~tique)
une construction de na-
n
dont
Ie
l'~l~ment neutre. Pour cet-
une vari~t~. Cependant rest
un groupe lin~aire (car A0 est un groupe lin~aire) et du fait que c 'est un sousgroupe discret uniforme, on
d~duit
qu'il peut "etre
engendr~
par un nombre
fini d'~l~ments. Or tout groupe lineaire engendre par un nombre fini d'el~ments contient un sous-groupe d'indice fini qui est sans torsion,c'est a dire qui ne contient pas d'autre element d'ordre fini que f, :1
I
clair que
r'
berg, ~3J) Soit est
11'
l'~lement
un sous-groupe d'indice fini sans torsion dans opere librement
dans
n
ment
et qu'il est sans torsion. D'autre part,
danEl
r
sque
r '\ n
est encore un
r' r'
neutre (Theor. de Sel-
opere proprement
sous~groupe
et
r' . n
puisqu'il opere propre-
r'
etant d'indice fini
uniforme
librement dans
AO . Pui-
de
n , il existe sur
une structure de variete holomorphe (et une seule) telle que I 'ap-
plication canonique
r'\
te variet~
n -4>
r' \
n admet ainsi
n n
tout domaine borne homogene est Ie est d 'autre part compacte car s'indentifie
a un
Soient
quotient de n
soit un
etalement
holomorphe. Cet-
pour espace de revetement universel, car hom~omorphe
a une
boule ouverte.
A0 ~tant transitif sur
r' \
n,
EI~
r\ n
A0
un domaine borne
sym~trique,
M une
variBte
holomorphe compacte admettant un revetement universel n ~ M
et
- 203 -
n.l J. L. Koszul
soit r Ie groupe des automorphismes de ce rev~tement. La cohomologie de M a ete etudiee dans differents cas et avec des buts varies. 11 s'agit soit de Ia cohomologie Ii. coefficients constants, soit de cohomologie Ii. coefficients tordus , soit de cohomologie Ii. coefficients dans Ie faisceau des sections holomorphes d'un espace fibre vectoriel holomorphe de base M. Les resultats obtenus sont des resultats ne falsant intervenir aucune hypothese sur Par
c~ntre
r .
ils mettent en jeu Ia nature des facteurs irreducibles figurant
dans une decomposition de
rl
en produit de domaines bornes symetri-
ques irreductibles. Indiquons deux enonces particulierement importants. Theoreme
. (Matsushima). Si
rl est un do maine borne symetrique ne
comportant pas de facteur irreductible ' ,omorphe Ii. une boule unite dans un espace
Cn , Ie premier nombre de Bet~.
Puis que
r e s t engendre
sous-groupe des commutateurs de
par
un
M
est nul.
nl.. Clbre fini d'elements,
r e s t donc d'indice fini dans
Iorque I'hypothese du Theoreme est verifiee. Theoreme.
de
(Calabi, Vesentini, Borel). Soit
Ie
r
e Ie faisceau des champs
de vecteurs holomorphes sur les ouverts de M . Si
est un domaine
rl
borne symetrique sans facteur irreductible isomorphe au dis que unite de 1 C , alors H (M, = (0) .
e)
D'apres un
Theoreme de Froehlicher et Nijenhuis
([8}) , il en
resulte que, dans les hypotheses du Theoreme, Ia structure holomorphe de M n'admet que des deformations continues triviaIes, contrairement Ii. ce qui se passe pour les surfaces de Riemann de genre
>2 .
Ces deux theoremes se demontrent par des techniques de formes harmoniques. Des procedes tres voisins on He utilises dans un contexte reel
pour etudier les deformations des sous-groupes discrets. Soient G
un groupe de Lie reel de
G. On sait que
famille
WI' W 2'
\'""
connexe et
r
un sous-groupe disc ret uniforme
est engendre par un nombre fini d'elements. Vne
..• , W P de
generateurs defini
tune
application injec-
- 204 -
n.l J. L. Koszul
GP
l'espace La
:
a tout
f €
F
dans
cette
injection continue est G opere dans
w .. Le groupe 1
par composition avec les automorphismes interieurs. Si tion
identique
dans G ge de
dans
r
de
tions de
r dans
~
F. Soit
dans ~.
llalgebre de
jointe). Soit en
effet
momorphismes
de
me
definit
On voit facilement
G. Par
restric-
representation li-
une
valeurs dans Ie module de la representation ad-
r
a un
f(t) une famille differentiable
parametre dlho-
dans G te11e que flO) = fO . Pour tout
ou a est une application
SI(O) est de la for-
et
G
s (r' ,
r
dans ~. En ecrivant que f(t) (SSI) = (f(t)s(f(t)sl) quels que soient s, Sl e
de
r,
-1
on voit que
de
un voisiua-
que la trivialite des deforma-
est une courbe differentiable dans
a(s) s
a l'ordre
Lie
fO est
r
de
1
r a
= f(t) s
G
fO designe llinjec-
que les deformations Gfo de
F
G est en rapport avec l'espace de cohomologie H (r, ad)
(cohomologie de
sur
G, on dit
la representation adjointe de
neaire
s(t)
dans
sont triviales lorsque la trajectoire fO
tion,
r
de
dans p
rendant
independa":te du choix des generateurs
G
(f(w 1)' f(w 2), ... f(w p)) E G .
Fest associe
topologie la moins fine sur
r
de
des homomorphismes
tive de l'ensemble F
a(ssl)
r a valeurs
= a(s) + sa(sl)
~1
s
, autrement dit a est un l-cocycle
dans Ie module de la representation
adjointe. Si u(t)
est une courbe differentiable dans G te11e que u(O) soit l'element neu-1 tre de G et si f(t) s = u(t) su(t) ,quels que soient s E et t E R ,
r
alors a(s) = u'(O) - sul(O)s
1
, ce qui signifie que a est Ie cobord de la O-co
de sur
r
dans
G
sont triviales
HI (r ,a d) ont ete
dans Ie cas OU G
(cf;
0 b tenus
1
r , ad) = a les deformations [24], [25J ). Des resultats complets
chaine u(O) .. En fait; on montre que si
H (
notamment par E. Calabi et A. Weil
est un groupe semi-simple de centre fini ([51,
t24] ).
Matsushima et Murakami ont d 'autre part donne des conditions tres generales entralnant la nullite de HP (
r, p ) lorsque
simple de centre fini
pest
et que
G
est
la restriction
a
un groupe semi-
r
dlune repre-
- 205 -
n. 1 J.L.Koszul
une technique de
formes harmoniques presentant· beaucoup d'analogies
avec celIe qui intervient dans la demonstration du Theoreme de Calabi et Vesentini. La difference principale vient
de ce que
reme concerne un espace de cohomologie
a coefficients
~e
dernier
Theo-
dans la faisceau
des sections holomorphes d'un espace fibre vectoriel holomorphe alors que
r , p ) on
dans Ie cas des espaces
H p(
un espace de cohomologie
a coefficients
se ramene en fin de compte
a
tordus sur une variete compacte.
Matsushima et Murakami ont mis en evidence les relations existant entre ces deux types de cohomologie et leur travail montre comment finalement des "Theoremes de nullite"
tres divers p,ecoulent de formules generales
donnant les operateurs Laplaciens ([15] , (16), [18], [19] ) . II faut signaler que ces
m~me
formules interviennent dans d'autres questions qui
concernent les deformations de sous-groupes discrets Lie
G tels que
r\ G soit
r
d'un groupe de
de me sure finie et en particulier les defor-
mations des sous-groupes arithmetiques des groupes algebriques semisimples
(cf.
(1J ,
[21] , [22] ) .
- 206 -
n.2 J. L. Koszul
~J__~~~~~~~_~~~~~~~~~~~ __~L~~~~~~~~~_~~:3Y35:3_~~vectoriels
br~s
On
par
d~signe
associ~s.
G
un groupe de Lie connexe semi-simple de
centre fini, par K un sous-groupe compact maximal de G et par space homogEme G/K. La dimension de
G/K sera not~e N. On commen-
cera par un rappel des principales propri~t~s de est
aR
diff~omorphe
groupe des 1utif
~l~ments
de
(!'
celui est
N
de G
algebre
-l
noyau de
[~IIIf}C
(J
A#-,
qui
sera
flJ
de
'+1;
I 'algebre de Lie de G, Ie crochet
~
dMinit un automorphisme involutif
(j
not~
(J' , •
Le noyau de
correspondant au sous-groupe c'est un
,MI,.. ) :
~tant
a gau-
-1
f$'
de l'alge-
est la sous-
K. On notera
~
Ie
suppl~mentaire de ~ dans ~ . On a [~, ~J t: ~
B
la forme de Killing de
~ , de£inie negative sur ~
B( ~
K est connexe et c'est Ie sous-
[,IW / .... ] ~ t
Soit sur
. L'espace G/K
par Ie crochet des champs de vecteurs invariants
che. L'automorphisme ~
G/K
invariants par un certain automorphisme invo-
G. On notera
donn~
bre de Lie
. Le sous-groupe
rl I 'e-
~
. Elle
est
et invariante par
(0). On definit un produit scalaire
d~finie
()'; on a
( , >
sur
en posant
quels que soient
quels que soient
a, b E ~ . Pour tout
b, b' E
~
et pour tout
b, b'
~,
,b'
>-
a E:
&"
on
a
cEo AoU,
[c, btl>
0
.
(e i )i:1, 2, .... \
positive
une base orthonormale de ~ et
~
- 207 -
n.2 J. L. Koszul (a ) _ k k-l, 2, •.. n-N
R pour
une base orthonormale de
ce produit scalaire.
b, c E. .+IV, on a
Quels que soient
L/[b, [c, eil] ,e) =[i Lk([t, akl ,e;\f, eJ, a k) ~ <[b,
[c, akl
J, a k}=
f
=~, ~<[b, akl Par suite
(L
(1)
ad
L
(2)
en
pour tout
b
Soit r2
1
)b
k
E MC-- • q l'application canonique
= G/K,
et
suite
exacte
une
relations:
(el ) b
ad (a )2
k
ad (b) ad(c) = 2 Tr.#j.. ad (b)ad(c)
les
d~duit
e)
e>
= 2 Tr ~ ad(b)ad(c) . On
a~l,
Li
soit
s
~ sK
zO = q(K) . L'application (0)
~~ .~ ~ ~ T
0
de
G sur
tangente
r2 ~(O) ,
qT dMinit ou T
z·
d~signe l'espace des
fin it
un
trique
isomorphisme riemannienne
IqTb I 2 m~trique
invariante
riemannienne tout
. Pour tout
r2
d'origine T
de ""'" sur
=
complete. Pour de
vecteurs de
tout
par
z
0 r2.
La
vari~t~
bE ANt,
t ~ exp
vecteur
u
Eo
T
z est
un
relevement
paralle1e de
cette
o
r2
q
m~trique
riemannienne
(tb)zo r2
,
g~odesique
est une exp
que
est
une
r2
est
g~od~sique
(tb)u
dans
d~-
une me-
une seule telle
Cette
bE"""
sym~trique.
zO . Par suite
11 existe sur
G et
r2 T
0
z
n.2
- 208 J. L. Koszul
Si l'on identifie 1M. et
T z
o
S1
au moyen de q
de courbure de la metrique riemannienne symetrique sur R(a, b)c = -l~, b
par
J ' c Jquels
que soient
a, b, c E:
I1-f •
T
Ie tenseur
S1
est donne
utilisant cette
formule et les proprietes de la forme de Killing, on voit que la courbure riemannienne de
r
Soit S1
est
r
alors
f.
0
sur
toute
direction de plan.
un sous-groupe discret de G ; il opere proprement dans
Par consequent
ment dans
ne
S1
S1
si
r
est sans torsion (ef. n. 1), il opere libre-
. Reciproquement, si
est
sans torsion. En
r
opere librement dans S1
effet,
etant une variete riemannien-
complete simplement connexe et de courbure riemannienne
isometrie de
S1
qui
S1
est d 'ordre hni laisse fixe
f: 0, toute
au moins un point de
S1 . Dans la suite on supposera que sans torsion et uniforme de
r \ G/ K
M
librement
dans
=
S1
r\ S1
G. Puisque
r
est
un sous-groupe discret,
r \G
est compact, Ie quotient
est compact. Puisque
, M est
r
une variete differentiable. 11 existe sur
une metrique riemannienne et une seule telle que I !application canonique que,
M est
variete
une
S1
opere proprement et
.~
M
l'application tangente
M
a
soitisometrique. Muni de cette metri-
variete riemannienne localement symetrique. crest une
orientable car
r
est
un sous-groupe de
G
qui
a
ete suppose
connexe. On peut considerer
S1
comme un espace fibre principal differen-
(* JOn conviendra d'identifier une variete differentiable Mala section nulle de TM des vecteurs de M. Si G est un groupe de Lie, TG est un groupe de Lie admettant alors G comme sous-groupe. Si G opere differentiablement a gauche dans une variete M, Ie groupe TG et par suite G operent a gauche dans TM. Toutes ces lois de composition seront notees multiplicativement.
- 209 -
n.2 J. L. Koszul
tiable de base M et
r . On designera
de groupe
dans la suite par E
un espace fibre vectoriel differentiable de
base
fibre principal
n
de la structure de
representation
linea ire
et une
application
. Feront
donc partie
p
r
de
M associe
dans un espace
differentiable
pr
: nXV
a
l'espace
E
une
vectoriel reel
--. E
V
ayant les proprie-
tes suivantes : (a)
quels que soient
v~Pr
de l'espace vectoriel L'isomorphisme
r ~ r,
v~ ,- f
(z, v) est un isomorphisme
sur
V
Q(z, r v)
la fibre de E au point rz E: M .
est appele Ie repere ----- associe
z .
n
existe dans
E
une connexion lineaire
Ie que, pour toute courbe differentiable les vecteurs courbe Si
z)
z ~ n , v E V et
(b) pour tout zEn,
a
p(r)
PI" (z, v) = PI" (rz,
PI'"
r o/(t)
fest une
('¥ (t), v)
section de
n
~
est
vecteur u Soit
riante est
param~le
M sur un ouvert
une
ferentielle covariante est nulle; donc nulle. On notera
dans
une seule tel-
n, et tout
v
E
V,
soient des vecteurs param~les Ie long de la
ou encore un relevement
v E V, l{'H Pr (f(x), v)
a un
1V
D et
section
de
E
de U sur
r 'V(t) . dans E
C
M, pour tout
U dont
la dif-
la coubure de cette connexion lineaire est
Du (resp. DX) la derivation covariante
par rapport
E TM (resp. un champ de vecteurs X sur M).
E Ie faisceau des sections de
nulle .D'apres Ie TMoreme de
E
dont Ia differentielle cova-
Rham (cas de coefficients tor-
dus), les espaces de cohomologie HP(M, ~ s 'obtiennent comme espaces de cohomologie du complexe
exo (M,E) ou
a P(M, E) designe
a
dIp d p+l (M,E) ... 0. (M,E) ~~ (M,E) ...
~
l'espace des formes differentielles alternees de de-
- 210 -
n.2 J.L. Koszul
gre p
sur
a valeurs
M
dans
E
et
ou
d
est
l'operateur
de dif-
ferentiation exterieure dMini par la· connexion D . Si Cf
fa (M, E), 0
41
c 'est a dire si
est
une section differentiable de E,
'f' .
la differentielle covariante de Si w ~ Q P (M, E) et si i=r est egale a w w. If. OU les w.1 sont des p-formes scalai1 1 i=l res et les If i des sections de E, alors
d If
est
I:
1\
(d If.) . On peut egalement caracteriser dw par la condition 1
(d w)(X 1 ,X2' ... Xpt1 ) = ~
(~1)
itl
1\
DX. w (Xl'" .Xi '· .. Xp+1)
1
1
"" ... c.x" ... X ), + ~ (-1) i+j W([X"X J , ... X" .<. 1 J 1 J p+ 1 1 J . pour toute suite de
p
champs de vecteurs
X. sur M. 1
TMoreme
I(S. Eilenberg) . 11 existe pour tout
nique
HP (M,!£) sur l'espace
r
de a
tielles
HP (r,
coefficients dans l'espace
p)
V de
p
isomorphisme cano-
des classes de cohomologie de
la representation linea ire
p .
Pour tout
p
, soit
a
PUt, V)
l'espace des formes differen-
de
p
sur
a
valeurs
dans
degre
presentation
r
lineaire
de
n
r
dans
Q p (n, V) en _1
)
A
V. On dMinit
_1
( P (r) w) (wI' w2' ... W P = p(r) w (r wI' r pour tout
r E.
r,
toute forme
w1,w 2, ... wp E Tn ayant lui qui
un
sera explicite
sous-espace
posant
W 2"
_1) .. r wp
WE o..P (n, V) et toute suite de vecteurs
m~me
au n.6
a ~fn, V)
origine. Parunproceae analogue ace-
a p (M, E) au aP(n, V) invariants par r .
, on peut identifier l'espace
des elements de
Apres cette identification, la differentiation exterieure definie sur & p (M, E) devient
une re-
la restriction
a
a;
plus haut
(n, V) de l'operateur de
differentiation exterieure des p-formes a valeurs dans V. sur n . Pour tout couple d'entiers
(p, q), soit
Cp , q l'espace des p-cochalnes sur
r
- 211 -
n.2 J.L.Koszul
a valeurs r~
dans
r
Ie
q
-module
q(rl, V)
On a
deux
op~rateul'S'
de ear-
nul cp,q
drl
cp,q
dl'1
CP,q+1
----;
Cp+ 1,q
~
provenant respeetivement de la diff~rentiation exterieure
-1
A '-'1
q+1(n, V) "
r a
eoeha'ines sur Comme
r
et
de
valeurs dans
bord du eomplexe des
l'op~rateur
Ie
r
a q(rl, V)
d:
a q(rl, V)
-module
Q q(rl, V) --"lO q+1 (rl, V) est un homomorphisme de
d:
-modules
on
a
= dr
drl d r
drl
. On
done un
d~finit
eomplexe DO Dr
en posant
d
cp,q
@
D"
d = d r + (-l)Pd
et
p+q=r
r+1 D
d
1"
D1
--1
rl
Cp , q .
sur
Les suites exactes d
a~(rl, V) ----7 Co, p ~ C1, q
(0) ----7
(q=o, 1,.... ) (p,
(0) ---?
(O°l! dans
CPr
r , V)
V)
d~signe l'~spaee des p-eoehatnes sur Hq(M,,!) ----t Hq(D""')
que soient
q
rl
1, ... )
valeurs
montrent respeetivement qu'il existe des homomorphismes in-
jectifs canoniques
dans
r a
=0,
p
et
r~sulte
espace contractile
. Du fait
que
les
r~sulte
On notera que
Ie
d~monstration.
Le
un groupe discret une
vari~t~
que
r
et
et
librement
surjectifs. Du fait que
rl
est un
les seconds sont surjectifs.
groupe transitif r~sultat
G n'intervient pas dans
est vrai toutes les fois que
op~rant djff~rentiablement
contractile
P)~HP(D~) quels
opere proprement
premiers sont que
HP ( r,
rl
la
rest
et proprement dans
- 212 -
n. 3 J. L. Koszul
n.3 Seconde structure de fibre
assode et connexion line-
aire symetrique. On conserve les hypotheses et les notations du
p
suppose que la representation
a r
restriction
note egalement
d 'une
r
opere librement
p:
que
bre vectoriel
E
G
une
l'espace fibre principal ble
de
PK
PK( soient
r" G
que soient
rs
Er\G,
fibre de
E
f a
riel
K
associe
r
point
r
definissent
a
On de fin it
une
application
l'espace
~
est
l'espace
s .n
. En
effet,
est stable par
-1
existe
dans
E
r
= PK(
E
de
V sur la
et la restriction
r '\G .
T'3
les operations
qui verifie la condition suivante:
s, v)
une structure d'espace vecto-
principal
(r'G) des vecteurs dlorigi-
.5 '& de
S tangents a
une
on a
part, quel que soit
M . L'application
donc dans
des vecteurs dlorigine
differentia-
Pr(sK, f (s)v) ne r E K,
F (sr ) p(r)v)
somme directe des sous-espace
composition
a
associe
P (s)v)
s, v) est un isomorphisme
fibre
droite).
en posant
s. Pour tout
sK de
a
definissent dans l'espace fi-
v E. V . D'autre
r
est un
K (operant
vectoriel
= Pp (sK,
s E G et
r,G
fibre
v~V
s de
= G/K,
d 'espace
En tout point ~E.r\G, l'espace ne
G dans V que l'on
de groupe
Pr
E
et
s E G
v r--; PK(
au
S1
s, v) = Pr (sK,
r
_1
quels
et
M
structure
sr , p(r)v)
~2)
dans
GI(V) et
~
depend que de la classe
de
base
(r\G) X V dans
(1 )
quels que
Vest definie comme la
representation lineaire de
fibr~ principal de
On va voir
dans
~
Puisque espace
de r
n.2, mais on
et
K. L 'espace la
trajectoire
connelX:ion linea ire flour tout
SAN\.'. Cette
5~
de-
est
5K
D S et une seule
v E V et toute courbe diffe-
de
- 212 -
n.3 .T. L. Koszul
rentiable
t'
t i----t PK(
0/ (t), v) est un relevement parallele de la courbe
r'G telle
dans
que \jf (t)K dans
s
Ie fibre
E. Cette connexion D sera appelee la connexionsymetr ique s s On notera D u (resp. DXl la derivation covariante par rapport a u E. TM
mn vecteur
(resp. un champ de vecteurs X sur
M) definie par
la connexion
symetrique. On definit comme au n.2 1'ope rateur de difs ferentiation exterieure d relatif aD. On sait que si R designe la.
s
forme de
courbure
de DS
,
pour toute forme
W Eo
0. P (M, E) ,on a
2
(d ) w = RAW • La forme de cOllrbure R n'est pas nulle et par suite s 2 (d) f O. s Puisque K est compact, il existe sur V des produits scalaires
invariants
par
p (r)
tel
produit scalaire
pour
sera choisi
un
alors
chaque fibre de
dans
que , pour tout V
sur
bres de
5E. r \G,
E
riel de par allele
E
® M
base
s'interprete
E
de
.5
M. On voit
a
r € K . Dans que
la
suite,
on suppo-
l'on notera
<, 'I
. 11 existe
produit Ii
K.
comme
M XR
dans
(relativement
un
E
Ie repere
la fibre au dessus de
tout
U;l
scalaire et ur, seul tel
t4 PK( I; ,I) soit une isomMrie de Ce produit scalaire dans
les fi-
homomorphisme diff4rentiable t'"
considere
comme espace fibre vecto-
facilement quiil est invariant par transport s
D'I
- 214 -
n.4 J. L. Koszul
N.4Formes harmoniques et Laplaciens. Dans ce R nexe orientee sur
a dire
N. On
la forme
v(e 1 , e 2, ..• eN) = 1 lorsque positive de vecteurs en
alternees de
tielle
a a
de
e 1, e 2, .•. en
X . On
de
par
X
p
il
p
sur
sur
X
degre
notera
Pour tout
de degre
N telle
1
F
1talgebre
a
M
on
M,
que la
base
M. Pour
F-module des formes differentielvaleurs notera m~trique
dans
E. Pour tout
If X la forme differenriemannienne associe
la multiplication
(resp. La multiplication exterieure existe
que
est une suite orthonormale
(X) (resp.E ('If X))
du F-rmdule ces
la forme volume positive
M.
Q P(M, E) Ie
degre
vecteurs
scalaire
gauche
notera v
riemannienne con-
un point de M. On designera par
tout entier p, on notera
champ
variet~
un espace fibrevectoriel differentiable de
.E
les
une
diff~rentielle
des fonctions differentiables sur Soit
par M
d~signera
de dimension
c test
M,
on
interieure
a gauche
par 't X).
homomorphisme canonique w I--'t ~ w N-p dans Ie F-module CI (M, E) . La famille des un
homomorphismes est caracterisee
(b) pour tout champ de vecteurs
par les proprietes suivantes :
X sur
M et
to ute forme
wELl. P(M, E)
'* E( 't X) w = (-l)P ~L.(X) * On verifie
que
**w p,our toute forme donc
w •
w
de
= (_l)P(N-p) w
degre
p.
Les applications w ~
,*w
sont
bijectives. Supposons maintenant dMini
~
M XR
dans les fibres
de
un produit scalaires E.
p:E(8lE M
- 215 -
n. 4 J. L. Koszul
Soient E
de
w,
et
W
degres
des
respeetifs p
WI\ w' a
(p+q)-forme
formes et
differentielles sur]\l[
q. Leur
valeurs
dans
a
valeurs dans
produit exterieur est une
®
E
E.
En eomposant
eette
M
fA
forme avec
on obtient
ra
notee
(WI\W').
de
degre
N;
E
F. On
c'est done
a
CI p (M, E) a
ainsi
prenant
(p+q)-forme scala ire sur
w, W'E:QP(M,E), Ie
produit
dans
v
M qui se-
(WI\(*W'
alors
de
obtenu une forme
valeurs
par
)est
une fonction
bilineaire sur
Ie
(w, WI)
F-module
F. On verifie facilement que cette forme
p = 0 , elle associe
a
deux sections la fonction ob-
dans chaque fibre de
E
Ie produit scalaire des
est symetrique. Si tenue en
Si
une
vale-
urs .. Soit
(X.). une base orthonormale de champs de vecteurs 11=1,2 ... N sur un ouvert U de M. Si w, w, € (M, E) et si p > 0 , la restric-
a?
tion
OU
de
(w, WI)
parcourt
"{j
[1, pJ
dans
a
U
est donneepar
I 'ensemble des
la formule
applications strictement croissantes de
[1, N] . Il en resulte que
(w, w)
> 0 en tout point ou
wiD. On suppose telle que Ie
maintenant donnee dans E une connexion lineaire D
produit scalaire dans les fibres soit invariant par
parallele. Cette condition est equivalente (1 )
pour
X.
tout champ
(r
s
,f) = (D X
de vecteurs
X
sur
a la
M et
tout
s
,Dx
'P )
couple de sections
Cette connexion dMinit pour tout p
un
d :CjP(M,E) --7 q P+1(M,E) s On definit , pour tout p, un operateur differentiel ';) : qP (M, E) s ('\ p-1 ...., I..A (M, E) en pOiXmt
teur de differentiation
exterieure
transport
condition
4' ' '+' ) + ( If
s
opera-
- 216 -
n,4
.To L Koszul
d
s
w = (_l)p{N-p+l)
*
d .j. w s
w~ 0. P(M, E) . 11 resulte de
pour toute forme
(I) que
,0 s w')v
{d w ,w')v - {w
(2)
s
queUes
que soient Pour
w de degre p et
expliciter,
sur
une connexion
degre
TM
qui est
definie par
'V et
M. Cette connexion
p+l .
d s
localement, 1'operateur
la connexion linea ire 'V dans mannienne
w, de
la metrique rieDS definissent
la connexion
lin~aire dans 1'espace fibre vectoriel
, on introduira
1\ p {TMf ®
Eo
M
Les sections differentiables de ce fibre etant les elements de on definit ainsi la un champ DS
X
derivation covariante
de vecteurs
X sur
M
C! p (M, E) ,
dans (;I p (M, E) par rapport
a
Cette derivation covariante sera notee
0
{pour p=O, on retrouve la derivation covariante des sections de E
relative X, Yl'
s
a la connexion D)o Quels que soient
000
Yp sur
s (DXw)(Y 1, Y2,
000
les champs de
vecteurs
M, on a
s Yp) = DX{w{Y 1, Y2,
~
0
0
0
Yp)) - '-. w{Y 1,
0
0
o'VXY i ,
0
0
0
Yp
1
pour toute forme
w £ QP (M, E)
0
Cela etant , si
(X.). =1 2 1 1
,
Nest une J
••
base orthonormale de champs de vecteurs sur un ouvett U de M et ~.
1
est la base duale de 1-.formes, d
s
Os
on a sur
= L £ (~i) =
-2:'
si
U:
s
D X
i
L (Xi)D~. 1
Soit maintenant
D
une seconde connexion lineaire dans
dont on supposera
la courbure nulle.
de degre
M
par
1 sur
a
valeurs dans
s A(u) = D -D pour tout u u
u
Soit
E
A la forme diff6rentielle
1'espace fibre End{E) definie
E TM o On notera
AI: la forme
- 217 -
J.L.Koszul
transposee : Dour de
me canonique
®
End(E)
et un
-1
que
avec Ies
A(X)
if' ) .
d = d-d trans forme une forme a s de A 1\ w et de llhomomorphis-
E ----i E . 11 existe un
tWt
seul
operateui\
aa
de
que
p WE ~ (M, E) et
soient
mt!mes
notations que d
a
plus haut
,
=L, E( d".)A(X,) 1 1
,
=l:, 1
1
un
0
+
a
sa
a a Laplaciens.
formes
a
1
tJ., 0.. s
=
(M, E) • Localement,
l..(X,) A*(X.)
~ + ~ d . Les operateurs
d
Les formes annulees
0 +~ d ssss
fJ.s
=d
et
6 a sont A
par
appeles
sont appele-
harmoniques.
Supposons p
on
= d d + ~ d,
a des operateurs des
p+1
,A
On posera
11 = da
E. Q
WI
1
da
tout
'P,
couple
(d a w, WI) = (w, daWI)
queUes
es
sur M et tout
a (A(X)" Ip , 0/) = (
,on
forme composee
(3)
et
X
fr,~iIement que Ilop~rateur
(A P (M, E) en Ia
degre
champ de vecteurs
If, IV E QO(M, E)
sections
On verifie W€
tout
produit
que
soit
M
scalaire
compacte. On definit
,>
(
~, wr)
=
l
sur
\w,WI)V
=
Qp
L4.
, (3) que
11 resulte des relations (2) et
(d s W,
WI)
=~, d),I>
< >=~, daw,w l
'
'0 a WI) ,
ce qui donne
(d w queUes que soient Ies formes
, WI)
=~ ,d WI)
w, WI . Par consequent
alors
pour
(M, E) en posant
I'd
WI)
.
- 218 n.4
J. L Koszul
ce
qui
montre que les
(a)
o
(b)
o, dw
(c)
dW
conditions suivantes sont
~quivalentes
:
= 0 .
On a donc une application mes harmoniques de degre
p
lin~aire
dans
injective de l'espace des for-
Ie p-H:me espace de cohomologie
du complexe d ~ I
D'apres
d 2 d Q l (M, E) ---1 0... (M, E) -----7
Ie Theoreme de Hodge Kodaira
(a coefficients tordus) cette ap-
plication est un isomorphisme et ces espaces sont de dimension1;3 finies (cf.
[2J). On obtient donc en definitive un isomorph is me canonique de harmonigue_s de degr~
l'espace des formes
ou
E
designe
la faisceau
p sur
des sections de E
l'espac.e ayant
HP(M, E)
,
une differen-
tielle covariante nulle. Onvoitde
meme que
(t.SW
'W>=~sw,dsW>
et
de
lion vient
A
et de 6.. des caract~risations s a de voir pour les z~ros de 6.. .
+(6sW,osW)
qui donne pour analogues
a
les ze-
celles que
- 219 -
n.5 J. L. Koszul
T'P,5 Decomposition du Laplacien. Les hypotheses et M
nlest
pas supposee
les notations sont celles du
a
~
s
,aa
on
obtient
A = 6.. +
fj. + d ~ a s a
s
On que
6.
(A s
w, w)
dira
= {:j.
s
qu'il
y
a
les operateurs d , s
+~ d + d d + d ~ as
sa
decomposition
du
as
Laplacien
6
lors-
+6. . Si cette relation est verifiee, puisque a
(D.. aw,
, 0 et
toute forme harmonique
w) ). 0 pour
toute forme
appartient au noyau de
un operateur differentiel d'ordre O. Lorsqu'il
A ,
Laplacien
precedent
compacte.
Lorsque lIon explicite Ie Laplacien D,. avec d ,
-rf
on
que" qui devront
obtiendra ainsi
~tre
verifiees
Lemme 1 .
C::.
=
. Or tJ. est a a a decomposition du
II y
des conditions de
par
toute
6.. s + 6. a
w ,. on voit que
forme
il faut
nature "algebri-
harmonique.
et il suffit qUe les con-
ditions suivantes soient verifiees : A.,y. = A
(a) pou~
(b)
toute geodesique Cf> dans M
quel que soit Si
q>
t (
est
tion
(b)
signifie
autres par
m~me
A( cp I (t)) que
transport
sique !p . Puisque
R
une geodesique
un endomorphisme
dans la
fibre au point
d +d
a
tp(t). La condi-
uns des
paralleIe relativement a D Ie long de la geodes D = D + A(u) pour tout u C TM , il revient au u
u
de dire que ces endomorphismes se deduisent
+'0
t E. R on
tout
ces endomorphismes se deduisent les
La demonstration du
a
M, pour
dans
par transport parallele relativement "d
d
,[D\f'(t) , A(
4 + C)
s a a s a s operateurs donnees au
Lemme
d utilisant s a n.4.
DS
les uns des autres
•
se fait par un calcuI de les expressions locales de ces
- 220 -
n.5 J.L. Koszul
On detaillera dans ce qUi parlant que Ie cas
general.
8upposons donc que toriel sur
trivial
de
P
un
champ
.e
m
que
E
soit
l'espace fibre vec-
le champ de vecteurs
de la met rique riemannienne
euclidienne qui fait
t E. :R que
l'on notera
de E (resp. les formes differentielles Ii valeurs
a des
identifiees
supposera que section
de
~
carree
fonctions
(resp. des formes)
la connexion E. On
-t
d'ordre
dans E) seront
valeurs
dans IR I!. • On
q,
esttelle que If = P. If pour toute s Dp If = P.\fI - A ou A est une matrice
D
posera fonction
a
8. Les sec-
Ie
differentiable de t. 8i
produit
dans
les fibres est invariant par transport parallele relatif
ion
Ds
on a,
en
t
notant
t
t
P. ('f 8 ~ ) = (P.
sant
'f a
t
valeurs
scalaire
a la
8(P. 'tj1
-
dans
m-t . En
toute fonction d
s
'-;?
sur
'f =
(P.
+ A + A* R
'f
)dt -
d~=
(P.
~s q> =
-
p2.
)dt
d (If> dt) a
* )'P,
If + P. ( A f) + A (P. If) - A2 ~
- p2 . 'f
+ (A + A* )(P.<:"f) .
, on a
d
dt) = - p. fp + (A + A
~ (.p= A*A'P , a
6q> =
if'
Re
dans
(A rp)dt
0s(fdt) =-P.~+A'P'
a( 'P
= O.
a valeurs
A
X
0/)
A = 8 (A)8, la relation precedente s 'ecrit §1(P.8.)
connex-
d'une .matrice
-1 t
(1)
Pour
X la transposee
If - A 'P )8 IjJ + If
que1les que soient les fonctions'f et -If-
d/ dt
une me.trice carree symetrique dMinie positive d 'or-
fonction differentiable de
tions
et
de vecteurs unitaires. Le produit scalaire dans les
fibres sera defini par dre
M"
. On desigre par P
IR X IRt
et on munit IR
IR
suit Ie cas OU dim M = 1 qui est plus
r" )dt
po-
:
- 221 -
n.5 J. L. Koszul
Si
6
6.
+
s
l:J.
,on doit avoir
a
A* A
(P.A)IfI+ 2A(P.1f )-(A+A*")(P. If) - A2\f + pour
toute fonction
Ceci
implique ~s
«J
sur
A = All'
D P Ar - AD P
Ia condition p. A. " matrice
A S
xion
D
T
,2
= A* et
A =6
sur
facilel!. nt
+ 6..
s
qu relle est
3 . On
a
done
un
a
associe
alors
les
formes de degre verifiee
~galement
aux notations
d rune representation
nulle
et
linea ires
conne-
sur
0
et
les formes
definie
quels que soient duit
scalaire
tout
~
E.. r\G,
v, v' E.
et Ie
ce produit sclaire,
un
E
de
de base
p
de
la
linea ire E: la
p
de
r
M=
dans
G dans
V V. On
connexion D dont la s D . On suppose choisi
V, invariant par
r (r) pour
representation lineaire
tout
de l'alge-
on a
~
(v,
+
~ (a)v r)
= 0
V et a E~ . On a vu qu ril existe un pro-
seul
repere
la
~
par
«(> (a) v , v ')
(2)
dans
sur
r E. K. En designant encore par
~
aux hypotheses des n.2
la connexion symetrique
produit scalaire
b.re de Lie
et
G/K, la representation lineaire
la restriction
courbure est
Ies
a
a la
p.A. = 0, Ie calcul pre-
si
espace fibre veetoriel
a defini deux connexions
un
If'
1 . On revient mairftenant
etant
2 + A =(P.A)
If)
Ia famille d 'endomorphismes ayant pour
R~eiproquement, s1 A
on voit
= r\G/K
1R.t.
est invariante par transport paralleIe relativement
montre que
et
dans
P. (Alp )-A If> :-A(P.
'"
signifie que
c~dent
de degre
valeurs
et P.A = O. Puis que
sit>
\0
a
IR
=0
dans
les
fibres
de
v"--; PK (~ ,v) soit
fnetrique
connexions lineaires dans
E
E
tel
que
pour
isometrique. Utilisant
riemannienne symetrique
sur
M et
on definit comme au n.4 les Laplaciens
- 222 -
n.5 J. L. Koszul
D. = D. s + 6 a ,
Lemme 2. Pour que
v'
E
'f
une
s~(tb) K
OU
quels que soient
V,
Soit en effet
'f (t) = r
reperes Ie long de \f> t € lR et tout
~
V
~ (b)
VI) =
geodesique de s E. G et
definie
definis
par
par
r(t)v = 1'1 (t)
1'1 (t)
P (s)v
sont
sulte que
Elle est
de la forme la famille de
Soit 1'1 (t)
= Pr(s~(tb)K, v) pour tout
V
des
D . On
a
1'(t)
G/K. Les reperes r(t)
I'
s
lP
sont donc egalement
"¥
(t) =
'(t) E: l.fJ(t) n\
(t) definis par
param~les ~ (~(tb))
~ (s)v)
D. La courbe
verifie la condition
reperes
s
dans
= Pr (8£lSP(tb)K,
t E. R . Par consequent, les reperes
symetrique
Em.
b
pour la connexion
r\G
sm(tb) dans
(tb), v)
M,
v E V . Ces reperes sont paralleles pour la connexion D
des reperes paralleles
r
0
b E. rtl .
sexp (tb)K est un relevement de lp
car
=
- <::.'
<~(b) v, VI)
(3)
il faut et il suffit que
I'
pour tout
s
(t)v=p
K
(s~
Ie long de
rs(t) pour
tout t . 11 en re-
s S - D )1' (t)v = - D r(t)v (DID r ' (t) 'f I (t ) If' (t ) 0 0 0
pour tout = r(t)
~
(b) pour
Ies uns
If que de
f E IR . La forme A o tout
t. Ceci
des aut1'es
. On les
a
de
voit
est
equivalente
v, v'
E
V.
donc
montre que
transport
telle les
parallele
Ie
que A( \fll(t)) r(t)= A( 'P Itt) ) se
deduisent
long de la geodesique
A( ~ '(t))r (t) = I' (1) ~ (b) pour tout t. Puiss s (t) sont des isometries de V sur les fibres
m~me
reperes
E, on
par
est
I'
s
que
a
la relation
A( r.p '(t)) = A* (1fJ'(t)) pour
Compte tenu
du
Lemme
tout
t ~1R
~ (b) v') quels que soient 1,
ceci acheve la demonstration.
- 223 -
n.5 J.L. Koszul I
?
Pour toute representation duit scala ire On Ie =
(,)
voit
1- +
V:]'
sur V qui
en construisant Ia
l'
dans VC
G dans V, il existe un pro-
de
v~rifie
les conditions
repr~sentation
= V + V--:J.
.s.
des
(2) et (3)
lin~aire
V d~duite de p
par
extension
scalaires. La sous-algebre + V-:]' I'\. est une sous-algebre semic simple compacte de 1- . Il existe donc une forme hermitienne d~finie
positive
h
sur
que soient r~elle
tions
de
h
est
(2) et (3).
h( ~ (x)w, Wi) + h(w, f (X)WI) = 0 quels c et w, Wi E V . La restriction a V de la partie telle
un
que
produit
scalaire sur
V qui
v~rifie
les condi-
- 224 n.6 J. L. Koszul
QP(M,E)-7CQI(\II~1r)
n.6. Isomorphismes
et
Theoreme de
n u.11 it e . Pour etudier les formes operateurs Laplaciens
a
tifier
ces
par les applications Pour
tout vecteur
a
l'application tangente
formes,
E:
est
au
p
me alternee
w~
de
il est commode
a
valeurs
T( r '-G) notons
wK l'image
r'G
dans
~ K. On definit degre
p
dans
valeurs
sur Wt
V.
de
w par
sur M = r\G/K. de
'1M
sur
TM. Si wE. ~ (M, E) ,
sK dans
*- a
les
dans l'espace 1T
de
r
et
de les iden-
b -7 ( ~ b) K est un isomorphisme
sur
dessus du point
E
differentiables
l'espace des vecteurs d 'origine
me alternee de degre
valeurs dans
l'application canonique de r"G
5 = r s,
Pour tout point
w
a
M
formes altern~es sur '"l
des
constitue
sur
sur
fibre de
la
done pour
a
E qui
tout ~ EI'\G une for-
valeurs dans
en po-
V
sant
pour toute suite
b l' b 2, ... b p €: M..
b1,b2, .. ·bpE:~
I 'application
tion Soit
differentiable
d
p( Kl .?J)
lJ
t-1 w de
-1
lJ
soit la
suite
H W", (b ,b , ... b ) est une applica-
"G dans
S
1
V; on
2
P
la notera
w(b 1, b , ... b ).
l'espace des formes alternees de degre
C:I p (M, E)
tes de PK on voit (1 )
r
que
2
p
sur
p
~
a
dalls l.T . On a obtenu une application lineaire canonique injec-
valeurs tive
de
~
Quelle
5 r(r
quels que
dans
Q p ('M.) '1J ) . Compte tenu des proprie-
que
-1 -1 -1 b1 r,r b 2r .... r bpr) = f(r)w~(b1'b2' .. ·bp)
soient ~ E. r"G, r
a.t
t. K
et
b l' b 2 , ... bP E:
.w.
On verifie que l'image de l'application canonique a..P(M, E)---j est Ie sous-espace
(4ft, 7Y) constitue par
les formes
O! (-ut w
7.J)
qui sa-
- 225 -
n.6 .J. L. Koszul
tis font
a
la condition
QP (M, E)
(I). On identifiera dans la suite les elements de
Pour
tout
est
de vecteurs invariant
a
de
G. On
On
on note L Ie champ de vecteurs
xEI}
dont la valeur en ~
neutre
a P(1tf , If)
et leurs images dans
~ x. C'est la projection
r (x)w
dans
a
valeur en l'element
(~, 'lr)
(x , w) E })< a.?
a
Ia
definie par
quels que soient ~ E r\G et (x, w)
du champ
] = [L ,L ] quels que soient x, y E: ~ • x, y . x Y d Q p{11'\, ()) deux structures de , modules.
(f (x) w) 5 (b l , b 2, ... bp ) = f
(2)
r'G
r'G
L[
La premiere fait passer de forme
sur
gauche sur G ayant x pour
a donc
definit
sur
x
L w definie
(x) (W~ (b l , b 2, ... b p ) )
b I' b 2, ... b p E Itt. La seconde fait passer de
par
x
(3)
ou
L. w(b , b 2, ... b ) designe la derivee de Lie de Ia fonction x 1 p i3(b 1, b , ... b ) par rapport au champ de vecteurs L . 2 p x Compte tenu de l'inclusion [~,~ C 1t( ,on definit enfin dans
C{ p (:"'t-.
(a, w) €. ~ )( (4) quels
une structure
u)
o..P (itt
I
V)
a
la
(e(a)W)(b 1,b 2, ... b p )
{. -module qui
forme =
-
fait passer
11 est clair que,
e(a)w definie par
L
Gi(b 1, ... [a,bJ,
quels
~ (x), Ly et
que
soient
e(a) de
deux. Par suite, Ies endomorphismes une representation connexite de K, on
lineaire
de {.
x, y e
... bp }
~ et a E ~, les en-
Q p (1lt 7.T) commutent deux a
p (a) + La + e(a)
dans
a
P (#t
ou
a f: ~ deiinissent
if). Compte tenu
voit facilement que Ia condition (I) est equivalente
la condition (5)
de
b 1,b 2, ... bp E14t.
que soient
domorphismes
de
o pour tout a E
h
de la
a
- 226 -
n.6 J. L. Koszul
On a ainsi une nouvelle maniere de caracteriser Ie sous-espace
Soit
(e.). _ une base orthonormale de i't\. pour Ie pro11-1,2, ... N u duit scalaire sur ~ deduit de la forme de Killing (ef. N 2) et soit ("l. i) la base duale. On choisit d'autre part sur V un produit scalaire verifiant les conditions (2) et (3) du N.5 . Avec l'irlentification faite plus haut, on a
L l.t(1l, .)L Ie.
d = s
(6)
d
(7)
~L.(e)L
=-
s
6- s
(8)
1
_L
L2
d
e.
1
L
e.
i, .i
1
/\
u.
(9)
.
1
V 2 Lp(e)+
=
a
i\
~ £(,.l .' 1
J
a
LE(l) f
a
~L(e')f(e) . 1 1
d
1
1,
(e i )
1
E(1 i) L(e )L [e i , eJ] L (e)
P ([e,e] )
J I
J
J
1
Ces formules sont valables sur Ie sous-espace Gt~ (-1fI u)
de
a.P(11(, u)
teurs
pour tout
p . Elles seront prises pour definitlOn des opera-
a
!J., a
d, d, ,d ,6., sa s.a s Soit (a) _
k k-1, 1, ... n-N
duit scalaire sur} dans (8) (10)
(11)
Lemme
et
b.
(9)
s
6a
En on
sur Ie complementaire. une base orthonormale de r1
rempIa(:ant
2k <0.,1
[e i' e j] par
obtient
L
L
2 e.
~
Lf(e/
1. Pour tout p, on a
L
k
1
+
ak
L f k
e (a k ) (a ) k
l
,
e (a k)
pour Ie pro-
0J,akJ) ak
- 227 -
n.6 J. L. Koszul
L'~galite
verifie
facilement
egalite
On
(10), (11) et
Casimir associes
aux
D'apres
ctivite
Pour tout gre te
a
Ll
de
p
p,
a soit
un
laire
f<2'
parcourt dans
valeurs
)
queUes
voit
= (0) On va
en
aPr 1tI
et
f
(x) et
V) . On definit sur
traduire l'inje-
V-) peut Mre consi-
j
r . . . . G a valeurs era) pour x E:. 7- et a Eo ~
a p (1ft, V)
un produit sca-
posant
applications strictement
facilement
e
= (w,
(f
= -(w, ~ (a)
(a)w, Wi)
a
~ {,.
(bl Wi)
soient
~, p]
,
w'),
= - (w, e(a)w') ,
bE 1tl ,
w, w, E.
une forme bilineaire symetrique
ger) (w, w')
croissantes de
que
(, (b) W ,Wi)
soient
que
f)
est injectif sur
V. 11 s'identifie de maniere eviden-
dans
operateurs
(e(a) w, w')
sant
tJ. a
si
(eLl' e-r:2 , ... e-r-p) , W'(e'Cl,e'L2,· .. e Cp ); oU"(.
[1, N] . On
defir..it
n.5, toute forme harmonique est
fonctions differentiables sur
1'ensemble des
quels que
au
a. p(+tt, 1J)
de
aP ( 111,
(
(w, Wi)
ete vu
l'espace des
laissent stable
x --7 \' (x) et
lineaires
HP(M,E) = HP(r ,
Q p (#1. , V) • Les
dans
la difference des operateurs de
V) ]'espace des formes alternees de de-
sous-espace
dere comme
. Elle se
en termes de formes quadratiques.
a
sur 1f'l
5
/
est
Par consequent,
alors
aP(-ttI, 1T),
a
n.
(6), (7) . La seconde
a... p ( 1ft , zr) .
ce qui
A. a
par
annul~e
expressions
repr~sentations
~ Lx de ~. dans
x
au
de (5) .
6
remarquera que
a ~t~ vue
s
en utilisant les
des
r~sulte
+ IJ..
a
=
w, Wi E.
(~ r (e /
a..P
o! (-nr) V)
'Jet)
sur
+ ~ e{a k )
(-ffl V) .
. Cela etant, on
a.,P ("fit, V) en
r (a k ) )w, Wi )
po-
- 228 -
n.6 J. L. Koszul
Theoreme
'X,(p) est
1 (Matsushima-Murakami) . Si la forme
positive,
HP (M,E) "HP(r,
alors
r )"
(0) . (11) de
C 'est une consequence imm~diate de l'express ion Remarque.
Lorsque
';Je (p) est
est
un
sans torsion,
n~cessairement
si
r
d~finie 6..
a G, non
sous-groupe discret uniforme de p
on a encore
H
f ) --
(r,
definie positive. On se ramene en effet au cas
sans torsion en utilisant Ie Theoreme de Selberg cite
au
(0)
rest
ou
n. 1 et
la sui-
te spectrale de Hochschild-Serre. la(p)
Des condit ions suffisantes pour que ont
~t~ donn~es
ou
f"
par
A. Weil
par
i
r
discret
'8
~
'I
d'ideal de
~~fil1ie ..l:'.:.::.~jtive.
'1
Puisque
a (I'll '1) Ces
sur
-8)
Hom( -»t,
tive sur "
a (6). On
en pas
ceci entralne
Pour
ne contient pas d'idea1 de rang _0) different d~ (0) , 1a forme 'J{ ad
somme directe de {
.6. .
a ce
de
et
Hom (111,
de
ou,
qui
-k ) et
ceci implique deduit
d'ideal w(e,) = 0 1
-'Je(I) ad
[e i '
de
'1
different
pour
w E Hom ( i1f,
WI )
I (w, w) = '2
(e)1 ,,0
L i, j
2 w" 1J
l'S
tout
de et
+
posons
par
2
positive
est injeca "0, alors i, j d'apres
. Puis que ~ ne
consequent
w(e) = 1
i, j, s,' r
ils
(0).
est definie positive sur
,
t defillie
(J
tout
1
~1
que
(J
est
Hom( -1tI ,..".,).
'}((;~ ('~I'
a quels que soient
ad(w(e,)) ~ "(D) pour
1{
maintenant tout
que
de
ad revient au meme ,que f1
w tHorn (-ttl, -k) et si l':,
l~
l'espace
11\
-:Je (I)
On montrera d 'abord que
Hom(1I1,') . Si
a et
contient
trons
P" I avait ~te traite
deux sous-espaces sont orthogonaux pour 1a forme par
(J
est
est somme directe
sont stables
d
ou
cas p "I . Le cas
Go
C
(A. Weill .
2
ne contiellt pas
I
G) et
positive
d~finie
; c 'est Ie cas interessant pOL~r 1a d~formation du
24
Lemme si
Raghunathan, notamment dans 1e
ad (representation adjointe de
sous-groupe
soit
('
R
isjr
w"
Hom (It! . I'll).
:£ W,'1J
e, . On
w
w
j
ri
a . Mon-
J
js
a
alors
- 229 -
n.6 J.L. Koszul
ou
R.. =/1Te ,el ces coefficients R.. sont les lSJr 'tL r J el~ ,e.); 1 lSJr composantes du tenseur de courbure de la metrique riemanniennesymetrique
de
consituee
par
G/K au pOint
K par
rapport
e .. On a
les images des
1
a la
base
de vecteurs
les relations suivantes :
(12)
R.. - - R.. lrJs rlJs
(13)
R.. + R.. + R. .::. 0 (Jacobi-Bianchi) lrJs IJsr lsrJ
=R
jsir
--21~ ..
(14)
IJ
La relation (14) resulte de la for mule (1) du n. 2 . On verifieen utilisant les relations (12) que Ie sous-espace des w E: Hom (t'fI, 1'\'1) qui sont symetriques (c. a. d. tels que orthogonal pOClr la forme
a
( 1)
2f ad
w.. = w.. quels que soient i et j) est IJ Jl l'espace des w antisymetriques. La re-
lation (13) permet de voir que, sur l'espace des w antisym{>triques, la ford
L
me quadratique
i,j,r,s ~ ) est definie positive. ad Supposons maintenant que
R. lrJs
(1
w . w. SI Jr
est positive et par suite que .
E Hom ( '" ,1ft) soit symetrique. On peut chois ir
W
la base orthonormale (e.) de 111 de sorte que chaque e. soit un vecteur pro1
pre de w. que soient
1
Posons w(e.)=A.e. et r .. =R .... =<J[e"e.] ,[e.,e.J)quels 1 1 1 IJ IJJ 1 1 J 1 J i'J' Si II w = 0, on a A'~+ 2 ~ r .. A,. = 0 quel que a 1 i IJ J
soit i .
~ (A.+ A.)r .. = O. Par consequent, les r ..
Compte tenu de (14) ceci donne etant
:). 0 , s i
On deduit
de
+ ~ I) , et si ideal produit donc
f A.I 1
cela
>.. f
J IJ IJ si A. + A f. 0 , on a r. = O. 1 S IS ) que, si rM A= Ker (w - AI) et h1 _xKer (w + 1
IAJ.1 et
= >. = Sup j
(cf.
24
0 , Ie sous~espace
direct d'ideaux
1'1'\ + tn_A + [111 >..' "'!J de 3 est un
simples
1e.~~
w = 0 ce qui montre que
de
rang 1. Ceci etant
exclu,
>. =0,
est definie positivp sur l'espace
des elements symetriques de Hom (111, 111) • Theoreme 2. (A. Weill . SoH
G
une
groupe de Lie semi-simple connexe
- 230 -
n.6 J.
de centre Hni. Si dim
4 Koszul
G ne contient ni sous-groupe invariant compact de
.> 0 , ni sous-groupe invariant simple de rang 1, alors pour tout
sous-groupe discret uniforme
1
reG, (1) H ( r, ad)
crest une consequence directe du Ainsi qu lil a ete dit au ne la trivialite des deformations TMoreme (cf. [24]).
n. 1, de
r
= (0) .
Lemme
2 et du TMoreme 1. 1 la relation H (r', ad) = 0 entra1sous-les
hypotheses du
- 2::11 -
n.7 J. L. Koszul
n.7 Nombres de Betti On conserve
Ies
de
lVI
hypotheses et
r
pose que V = R, Ia representation
Ies notations du
de
G dans
V etant alors neces-
sairement Ia representation triviale. L 'espace fibre fibre vectoriel isomorphe au fibre fie au faisceau
des fonctions
lVI x fR
numeriques
ouverts de M . Par consequent, pour
a
isomorphe ire
des
formes
V etant
on
a
tout
Ie
harmoniques,
nulle,
de on
on
E est alors un espace
faisceau
p
HP(lVI, ~ ) est
I'espace
espaces par I'intermedia-
choisira
dans
V = fR Ie produit
p
La representation linea ire
fR
a
b..
a
s'identi-
E
Iocalement constantes sur Ies
HP( lVI, IR) • Pour determiner ces
scalaire multiplication dans
et
n.6. On sup-
de
= 0 • D'apres Ie Lemme
du
n. 6,
sur
(1 )
Par consequent,
pour. tout
'J p I 'espace
Soit
sont de
degre
sociant
a toute
des
p
et
formes qui
forme
p,
sont
w~ {TP
differentielles invariantes
par
~ p sur
G. On sait
a:.( a
sa restriction
I 'espace
m,
un
definitive
une application lineaire injective canonique de des formes
differentielles
noyau
(scalaires) sur
on definit
I'espace
isomorphis me de
aI (1'l1, IR ) appartient au
harmoniques
A .
rI qui qu'en as-
I11K C TrI.,
R) . On
de
de
a done en
gp degre
dans p
sur lVI.
Cette application peut s 'obtenir plus directement comme suit. Puisque rI = G/K Ie
sur
etant
est un
espace riemannien symetrique toute forme differentiel-
G/K qui
invariante
sur lVI =r\G/K
est par
et
de
invariante par G elle est
composee
l'etalement
est une isometrie riemannienne,
G est
une
forme harmonique;
d 'une forme differentielle w
G/K~lVI. Puisque cet etalement
west
aussi
harmonique.
Les resultats de lVIatsushima exposes dans la suite donnent une condition suffisante pour que toute forme harmonique de degre
p
sur lVI
- 232 -
n.7 J. L. Koszul
soit
image
Lemme
d 'une forme differentielle invariante par
1. Soit
ad(a)
- Tr
111
Inf
A
G sur
Q
•
2
(a , a)
aE: ~
Il existe une base orthornormale
Tr til ad (a k ) ad(a1) La forme forme
quels
f
definie
Ie
0.
centre
de mt=me
o
de
coincide avec o existe un nombre
soient
a, bEll .. On 1
orthonormale directe
different par
de
fidele;
Lemme 2.
Soit
la
dans
w
une
soient
x E
donc
ne
forme
~ ~
(L w, w')w x
contient
adjointe
>
en
pas
a
dans
!
de
nulle d'apres
x
une
re-
sur
r,G
invariante
G.
par
w, E. (1 P( m , if) , on a
+ ( (w, L WI)W = 0 )P\G x
derivee
. Son integrale sur la formule
3
m definie
est
~
x
L
somme
d'ideal de
(L w, WI) + (w, L WI) = L . (w, WI) (derivee x x x rapport au champ L). Puisque west la
base
0 .
volume W,
.f.
que
Lemme.
linea ire de
A
de
effet
(w, WI) par
rapport
1
condition du si ~
quels
f(a,b) =c.(a,b)
la decomposition
ce cas
G, (LxW' WI)W + (w, Lx WI)W est par
a
la representation
j r\G tion
A = Inf(l, c l ' c 2 ' ... c s) . Par suite toute
(0) , la representation
restriction
En
a
adaptee
de ( ,
i
~
0
que
()
simple
c.
1
s
La restriction de
ideal
tout
tel
0. 2' ... a
a, 1
a deux orthogonaux pour f
(,)
Pour
remarquera que,
de
que
<,)
1
presentation
Quels
deux
est une
elle est invariante par
K. Les ideaux simples
sont donc
Cl.. verifie
des On
~
de
f-
et
-k
sur
que pour Ie produit scalaire
Q
il
k
f(a, b) = - Trm ad (a) ad(b)
positive
de
i
soient
par
bilineaire symetrique
la representation adjointe et
que
de
de
Lie
de
la fonc-
invariant
la forme
la variete compacte
Stokes.
de
r \G
par
(w, WI)W est
- 233 -
n. 7 J. L. Koszul
Lemme 3 . Soit soient
w
un
i, jE[l,N]
2:
r'G {;
On deduit
L ) ., 1,
on
,
J
pose
1J
Lemme
(B .. , B .. ) w = -
1J
J
1J
1J
R
.. 1Jrs
(m, 1)). Quels que
L (e .)!,;,i J
ei
~
+2
1J
Q~
dans
B.. = L
(B", B .. )
.' J
1,
du
de Cl
zero
• On a --
,,(B., B.
srJl
1r
JS
~w
(0 .
2 que
fL
., 1, J
(L 2
E (1 .) L(e) w, w)
e.
J
1
f ~ (L!
w
J
w, w) w . 2 2 Puisque west harmonique 2: L w = L L w. On obtient i ei k ak en appliquant de nouveau Ie Lemme 2 puis Ie Lemme =- p
1
done
(en faisant
un
ehoix
JL ij
,L
a w) ~ P = -A
= 1:
A
eonvenable
(B .. , B .. )w = p
1J
1J
f~ k, t
p -7:
., 1, J
(L
Tr
m
termes
k
[e.,e.J 1 J
[e i , ej ]
i, j . En remplas;ant dans L Ii ] lei,e j
r, s Ie Lemme
~
1
w, L
a k ):
w, L w)w = prJ:. fa , a..\ (IL w, ak ak ) k,:e \ k il a k
,[a1,e.])(L
[e"e.J 1 J
ak
w, L w) w ak a!
w,L
at
w)w
w) w •
w = - () ([e ..' e.J) w quels 1 J
l'integrale preeedente l'un des
w par - () ([e., e.] ) w = 1 J
L (er'[~i' e j]
nouveau
(L
base
ad(a )ad(a ll)(L
L
soient
la
k
Puisque que
de
fL
f~. (fk,e.] k, 1, f 1
fL
1
2,
, e s]\E(1J r) L(e s ) w,
Y'
on
la
met
sous
la
et
en
forme
appliquant de
- 234 -
n.7 J.L.Koszul
H~)
Theoreme 1. (Matsushima) . Soit definie
sur m t9 tn
par
pour tout ferent
Ia forme quadratique
g=
')'
o
et 1 , Jsi
de
. .
~..
k
e. (!il e. ,Si
1J
1
J
H(P)
ne
est dMinie
:1
harmonique (scalaire) de degre p
contient positive,
pas d'ideal de a
d
alors
tQute
dif-
forme
~
M est image d'une forme differentielle,
en
effet
invariante par G sur rI Le lemme monique L
e. poJr
de
3
degre
montre p,
et
si
H(P)
8
que,
si
est definie
west
une forme har-
positive,
alors
L (e.)w = 0 quels que soient i et j. n en resulte que L w = 0 J e. tout i E: (1, N] . Ona done Lbw = 0 pour tout b E 1tf 1, et par
suite L w = 0 pour tout a € @I. m] . Puis que i ne contient pas d'ideal de a different de 0, -{ = [lt1/tn] et par consequent la forme w prend ses
3
YaUeurs dans les que
W E:
tIl
Corollaire. Si
applications constantes
de r\G dans IR , ceci montre
(m ,IR) .
-l
definie positive,
ne contient pas d'ideal de ~ different de (0) et si 1 alors H (M, R) = (0) .
est
En 'effet, toute forme differentielle de degre 1 sur rI qui est invariante par G est nulle parce
que
m = [i, m] .
L 'espace des formes differentielles de degre p invariantes
par
G est canoniquement isomorphe
ferentielles de degre
sur rI qui sont
a l'espace
des formes dif-
p invariantes sur une forme compacte de l'espace sy-
met rique rI . Par suite lorsque les hypotheses du theoreme sont verifiees, Ie p-ieme nombre
de
Betti de
M est
egal
au
p-ieme nombre de Betti
d 'une forme compacte de rI . Les valeurs
de
p
pour Iesquelles
ete determine.es par Matsushima
( [12J
H~)
, [v;] ) dans
domaine borne symetrique et par Kaneyuki et Nagano neral.
est dMinie positive ont Ie cas
ou
rI est un
(liJJ,gO]) dans
Ie cas ge-
- 235 -
n.8 J. L. Koszul
On suppose comme simple connexe
de
lomorphe invariante par G. elle sera notee
q l'application 2
tegrable,
J
et
on
est
donmfe
a
Ie
ce
tenseur de
G sur
o ~
tel
que
en
exprimal1l
qui
est alors paire;
ete fait dans les n.
la structure holomorphe de
rl. II
existe un endomorphisme J de I'll
est
J
rl et
T . On et q T. 0 J = Joq
Ie tenseur d 'une structure in-
obtient mod {
quels
que
soient
[m,~ C {
a, bEg
[a, jbJ
quels
que
a E f.t
soient
Supposons sur
. Compte tenu des inclusions [~, til] C rt1
cette condition est equivalente
(2)
de
= j
[a, b J
b E
3
et
plus
que
la
soit hermitienne
On
a alors
a la
condition
metrique riemannienne symetrique
o
(3) quels
que soient
a, b E.
Si
designe
forme
B.
la
e{ on = B(jb, [a, c] ) = -
et
de G/K
avait
(1 )
et
Lie semi-
rl = G/K une structure ho-
sur
O· = ( ) , J I'll C
que
groupe de
un sous-groupe compact ma·-
La dimension reelle
canonique de
seul
a j(j +1) = 0
plus
"est un
G
que K
2n (contrairement
precedents) . Soient
et un
n.2 que
de centre fini et
xi mal de G. On suppose
3
au
c
On en deduit,
a
B(
de
Killing
Cia, b] , c )
de
compte tenu de
(2)
quels que soient
~
=- B(b,
B( [a, JbJ ' c) . Par suit.e
[ja, c] )
[ia,
bJ
=- B(b, j
[a,
a, b Co
c] ) =
+ [a, jb] = 0 .
que
[a. b] =Qa. ~ + [a, jb]
(4)
quels
~
que soient
a, b ~
3 .
Autrement
dit
est
une derivation
de
~
tI1
- 236 -
n. 8 J. L. Koszul
Puis que ;) un
seul
est tel
-A. .
tre de
il existe un element
semi-simple,
que
n
o
On observera que la
ferentie11es invariantes
et
appartient au ceno met rique riemannienne symetrique est
ad (h ) = j
non seulement hermitienne,
dans ca
ho
est clair que
h
mais kaehlerienne, car toutes les formes dif-
par G sur
Q sont
fermees puis que
Q
est un
espace homogene symetrique . D 'apres E. Cartan et Harish- Chandra, Q est holomorphiquement
a
isomorphe
un domaine borne
a:; N et
de
tout
domaine borne symetrique s 'obtient ains i. Comme uniforme
type
un sous-groupe discret
G. 11 existe une structure holomorphe
dans une
seule te11e que l'application canonique
un etalement holomorphe
se decompose
complexes de
r
on supposera donne
M =r'\Q et
~ M soit
TM
n. 2,
sans torsion
sur la variete Q
au
TM c du fibre
Le complexifie
en somme directe de deux espaces fibres vectoriels
TM+ et
. Les
TM
a dire
(1,0), c'est
elements
de
TM+
sont Ies vecteurs
-v-:i
les vecteurs de la forme u
Ju ou
u E TM et ou
J designe Ie tenseur de la structure holomorphe de
M. Les
elements
TM
a dire
les co-
jugues
de
de base
est definie par
au
M
les vecteurs de type (0, 1) , c 'est
+
des elements de Comme
riel
sont
TM.
n.2,
on designe par
un
riels
aux
soit
points
de
de
complexes. 11
vectoriel un
existe
holomorphe etalement
espace
et
complexe et
r . Il et
que
eXlste
une seule
alors
f
fibre vecto-
associe 3. Q qui
r
de
Pr: Q x V - ; E. On
espace vectoriel
ulle S(l'ucture complexe
cies
r
une representation lineaire
tion linea ire complexe E
un
muni d'une structure d'espace fibre
une apolication differentiable Vest
E
dans
Gl(V) et
suppose de plus que est une representa-
dans
telle q..;e
les
chaque libre de reperes asso-
soien1
des isomorphismes d 'espaces vecto-
m~me
sur
une seule te11e
holomorphe,
mais
E
une structure d 'espace fibre que
:S/ >< V~E r cette structure n'interviendra pas. p
- 237 -
n.B J. L. Koszul
Les formes E s'identifient duit
aux
tensoriel
I\r(TMcj*®E
sections
1'espace
sur qui
lui
les
r
degr~
sur
a
M
fibr~
valeurs dans
I\r (TMt ® E (pro-
ou encore aux sections de 1'espace
fibr~
est canoniquement isomorphe. Pour tout couple
a P' q(M, E) l'espace dans
valeurs par
constitu~
de
IR)
(p, q) , soit
a
M
de
It
d'entiers sur
diff~rentielles
E. crest
formes
w
Ie
telles
des
formes
sous-espace
de type (p, q)
de
aP+q(M, E)
w(u ,u 2, ..• , u ) = 0 lots que 1 p+q
que
u
u u est une suite de vecteurs complexes ayant m~me origine l' 2"" p+q + qui comporte plus de p termes dans TM ou plus de q termes dans
des
TM
morphes
sur
w
"L cr,t
ou t1'
et t
tement sont
M,
une
forme
w
Ifcr,t. dz 4"1 1\ dz CT 2" . .. parcourent
croissantes
des sections
de
dz
type 1\ dz
crp
respectivement
de locales
de
E. 11
(p, q) 11\ dz
t
n.2
nexion D
q,!e
la
lin~aire
structure
est clair
D
de
champ
de vecteurs
n~aire
sur C . L 'op~rateur de
d~compose
re applique (0,1)
(c'est
rateurs
en
X
sur M, la
(l.P' q(M, E)
a
dire applique
supposera
u...
f .
On
a
vu
'e
r,
f1' 1:
. On a vu
E
une
con-
a
est
de
(M, E) dans
DX
est donc
ext~rieure d~fini degr~
(1,0) (c'est
ou d" est
~p,q+l L7--
par
de
liD
a didegr~
(M, E) ) . Ces ope-
dans
au
les
parallele relatif
covariante
la
suite
que
d'une representation lineaire complexe de G encore
ou
complexe des fibres. Pour tout
d'
p,q
'Cq
pour tout
dans
d~finit
GLP+ 1, q(M, E)) et ~
et
a.P' r-p (M, E)
diff~rentiation
dans
sont lin~aires sur On
[2
d~rivation
d = d' + d" ou
dz
1\
que
courbure nulle. Le transport
est compatible avec la structure
se
a
d'associ~
localement
.2'"
[1, N]
dans
(lr (M, E) est somme directe des sous-espaces
au
s'~crit
1'ensemble des applications stric-
[I', ciJ
et
locales holo-
coordonn~es
n.3
que
E
est
rest dans
la restriction
V que
alors
ar
l'on notera
canoniquement
mu-
n.8
- 238 -
J. L. Koszul
ni d June
structure d 'espace fibre
principal
r ,G.
De plus E
connexion symetrique complexe
de
plexe
V pour
i"''' G
de
points
sur
V
D
les
Puis que
tout
~
s t G, les
fibres
de
E. Il
de vecteurs sur M tiation
un
rep~res
associes
en
a
relatif
l'espace fibre
(s) est
des isomorphismes de
DS est compatible s bres. L 'operateur DX de derivation lele
a
associe
est muni d 'une seconde connexion lineaire: la
S
sont
vectoriel
est donc
que
Ie transport paral-
avec la structure complexe covariante
DS se
C'
par
d' de degre (1,0) et un operateur s operateurs sont lineaires sur C
seront
donnees
d" de s
en
La forme
a
sur M de
trois
differen-
somme d'un opera-
degre
(0,1). Ces
d' et s
d" qui
s
n. suivant.
de
valeurs
champ
°.
(d')
au
a un
rapport
2 2 = (d") = s s Cela resultera trivialement des expressions de a
des fi-
. L 'operateur de
decompose
teur
Lemme 1. On
aux differents
l'espace vectoriel com-
resulte
lineaire sm
a
exterieure relatif
automorphisme
s D qui est une
courbure de dans
compos antes
l'espace
fibre
forme de
vectoriel
R0, 2' R1 , l' R 2,0 de
types
degre
2
End(E) est somme respectifs (0,2)
(1,1) et . (2,0)
Lemme
2. On
On toute
a
a
R
en
effet
2,0
= R
une seule verifiant
=
°
R 2,01\W = (~)
forme differentielle
Lemme 3. 11 existe sur
0,2
E
sur
W
(b)
les sections holomorphes
E
~
M
est
de
E
sont les sections differentiables est Ie tenseur d 'une
rifiant les conditions
et
Ro, 2"
valeurs dans
W
= (d~)
2
pour
W
E
les deux conditions suivantes :
La projection
J
a
M
W
une structure holomcirphe (de tenseur J) et
(a)
Si
2
(a) et
holomorphe, ~M
If
sur
sur
un
ouvert
U telles que
UC M d"«P=
s
structure holomorphe sur
(b), alors
: 1) pour tout vecteur
qui est horizontal pour la connexion DS ,Ju
est horizontal pour
°.
E
ve-
u E TE DS
,
n.B
- 239 -
J. L Koszul
2) les reperes
associ~s
aux points de I"' ....... G sont des applications holomor-
phes de V dans E . Ceci demontre structure holomorphe sur E
l'unicit~.
v~rifiant
Pour prouver l'existence de la
les conditions (a) et (b) on observe
d'abord qu'il existe une structure presque complexe de tenseur E et
line
associ~s
seule
v~rifiant
,J
sur
la condition i) et la condition 3): les reperes
aux points de t"''-G
et la projection
tions presque complexes. On
E
~
ensuite que
J
champs de vecteurs )E. et
~
v~rifie
M sont
des applica-
a la condi-
satisfait
tion d 'int~grabilit~
quels que soient
les
N est un tenseur , il
sur
E. Puis que
suffit de montrer la chose pour une famille de
champs de vecteurs qui engendre Ie module des champs de vecteurs sur E. On se
limite donc, d'une part aux champs de vecteurs verticaux dont
la restriction
a chaque
fibre est un champs de vecteurs constant, d'autre
part aux relevements horizontaux de champs de vecteurs sur M. La
d~
monstration s'acheve en utilisant d'une part Ie fait que les d~rivations . s covanantes DX sont lin~aires sur ~ , d'autre part la relation RO,2 = 0 du Lemme
2.
Remarques. 1) Le Lemme 3 est un cas particulier d'un ces
fibr~s
r~sultat g~n~ral
sur
les espa-
vectoriels de base holomorphe dont les fibres possedent une
structure complexe et qui
sont munis dlune connexion
lin~aire
conservant
la structure complexe des fibres. 2) Lorsque Ie groupe G admet une E
complexifi~,
qui est dMinie par Ie Lemme 3 peut
~tre
la structure holomorphe sur
obtenu par d'autres
mettant en jeu Ie IIfacteur d'automorphie canonique ll (d. une forme compacte de
C16l, [19]),
ou
n.
3) La structure holomorphe sur E de celle que
proc~d~s
lIon obtiendrait en
d~finie
par Ie
consid~rant
Lemme 3 est
E comme
diff~rente
associ~
a n.
- 240 -
n.g J. L. Koszul
n.g '~~~~~22~j~03y__~~_~~~~a5J~E_2~E~_~~_~~~_~~~_~~~~j
E~~_~~~~~~3Y_~~~~~~~~~ On a vu au n. (,
al
ce
que
(1ft, 1T) . On
3
de
designera par
a trois
11
'tt1 c = 111 +
,t\j
[kC
+
=
9
3 c llalgebre de
est
v:Tm
somme directe
)i~=.,,+.
= "+ + 11
=E'-,
c n_ ,[11+, I1+J
complexifiee
= Kers
<
! c ,11
de
nJ =0 et I
)
i
(d. n.2) . Les sous -espaces
(11+,11+)
a
se orthonormale de
c, 11 +
= ~_,~)
+
HJ
[11+ '
la forme
C
On a
-t1
C"+ kC .
bilineaire complexe sur
g
deduit de la forme de Killing
11
sont orthogonaux pour ces
[o}.
Soit (ei)i=l, 2,.' .. 2N una ba-
e i+ N = je i pour 0 < i
telle que
f11
+ et
,
De plus [kC ,1I+J
qui prolonge Ie produit scalaire sur
formes. On
5 +V-l 3
Lie
Ker (j+ V:r)
On designera par c
Numero precedent.
-It + v:l -hKer (j- V-l)
11
3c
, l.r)
sous-espaces propres : .fvc =
L lespace
a.~ (H1
~ c, les valeurs propres 0, V-l et _ V-l de j=ad(h o ) vont cor-
. Dans
respondre
a.,P (M, E) et llespa-
va commencer par retrouver dans
la notion de type introduite au On
lIon pouvait identifier
~
N . Soit 1'{ i
la base duale. On posera
o
pour
~
constituea une
w.
Soit
c
Y to I1t
1
f;
base
la forme
\ de
constituent 11
linea ire sur
Les formes
constituent i
N . Les
w. et 1
• On
a
a
<\' x)
111 c telle que
leurs conjuguees
la base duale de la base on
une base
11+. Les
de =
~ij
xi
.
Wi(Y) = (xi' Y) pour tout W
1
x., X. de 1 J
mc . Pour
tout
- 241 -
n.9 J. L. Koszul
V-f V2
(x+x.)
e.
1
1
'It.1
\12
" . ad X. ad X. + ad x.ad x. = ~ 1 1 1 ~- c,
51:)= .
_1_
2V-l
11
1
=- -
4. 1 [ X .,
(x.-x)
V-f
"t 1'+N
(w. + w.), 1 1
V2
Lemme 1
1
1
-
(w. - w.) • 11
sur I'fI c
h0
La premiere relation s 'obtient en exprimant les e. au moyen 1 2N 2 1 des x. dans la relation :£ ad(e.) = - sur ttl (cf. n. 2) . On en deduit 1 '. 1 1 2 1= ad(x.)ad(x.) = ~ad( [x, i x.]) = -21 sur n + et que L ad ([x, x1)= que 2; .11.11 . 11 1 1 1
L
=-..!:. sur 11 • On voit d 'autre part que ad( [x., x.] ) commute avec 2 . 1 1 1 ad(a) pour tout a E. ~. 11 en resulte que ad(12:' x] .) = - ad( hJ ' ill 2V-1
pc.,
ce qui demontre la seconde relation puisque centre reduit
a (0)
3
est semi-simple donc de
•
Puis que
Vest
espace vectoriel complexe, 1'espace 1T
un
des fonctions differentiables sur r\G a
valeurs dans Vest lui m~me
un
isomorphisme canonique
espace complexe. Il existe donc un
n' r
I 'espace lA- (111,
V') des formes alternees de degre
a
~~
I'
sur
11't.
de
a
valeurs
C
I'
sur I 'espace (111 , If) des formes alternees complexes de c degre I' sur '111 a valeurs dans 1.J . On dira qu rune forme I' c w € (l. ('l11 ,?T) est de type (p, q) si I' = p+q et si w(x 1, x 2' ... x p+q )= dans
l.)
= 0 toutes les fois termes dans
que la suite
11 + ou plus de
q
xl' x , ... x
2
termes dans
p+q 11
contient plus de •
Les
p
formes de type
vn
- I exe de IA c, (p, q) constituent un sous-espace vectorie I comp \.A..p+q( ..... III . I'l p, q c,.. P . /) r c?,.. qUl sera note ~ (11'1 , v ). our tout entler r, ~ (t11 ,u ) est somme directe
des
a.P' q('h1, 1/) ou
Pour L
x
et e(x) sur
tout
x
~ ~,
Cl ('I1f ,7J)
p+q=r . on
a defini
. Par transport
au
n. 6 , des operateurs
de structure ces
peuvent ~tre consideres comme des operateurs dans
r (x),
operateurs
a( m c , if)
.
- 242 -
n.9 J.L.Koszul
On a vu au n.6 qu lil existe un homomorphisme injectif que de
a
a,r(M,E)dans
nomomorp h'lsme obtient
canoni-
. En composantcet
/}r(I11,lr)->.ll v --, l)l..r (H1 c ,lJ)on
II'lsomorp h'lsme
avec
1
r (111 , U)pour tout r U\-
un homomorphisme canonique
a. r (M, E) dans a.r ( 111 c, 7/)
de
Pour tout couple dlentiers (p, q), Pimage de ltP, q(M, E) par cet homomorphis me est Ie formes
sous-espace
telles
(J
a P ' q( 111 c, 21)
q( 111 c, v) de
f
(
(a) + La + eta))
W
=0
a E; ~.
a P' q(M, E)
Dans la suite, on identifiera tout ~l~ment de I') p, q c 7J dans \A.~ (1tI, ). Quels que soient + V-lLb
et
~tant,
les (6)
a, b £
S'
on
posera
r (a+ v:Tb) = f'(a) + v:r rib) ,ce
repr~sentation
les x, et 1
constitu~ par les
que
(1)
pour tout
at
f de
lin~aire
w,
1
S
~tant d~finis
dans
V
La+ v:Tb = La +
qui revient
a
comme au
la
a son image
a prolonger
complexifi~e
de
d~but
S
c
la
. Cela
ce n. les formu-
du n.6 donnent: d =
s
(2)
dl
(3)
d"
L t(w,)L
,IX, 1 1
s s
+~w,)
L_
,IX.
1
,
1
=L((GJ,)L
,IX, 1 1
=~E(w,)k
,IX,
1
1
d = ~~(w')f(X') + t (w.)e (i,) , a,ll,ll
2:
1
(4)
dt
a
1
= Lf.(W,)(' (x,) ,
,
1
d" = ~E(Gj,) a ,1
(5)
1
Puisque on a
1
1''-
[Lx.' Lx.J = 0 1 J
1
f' (i,) , 1
est une sous-algebre de Lie
ab~lienne
quels
NJ .
que
soient
i, j E. [1,
de
c
- 243 -
Comme
(,)
= O. On voit
s
rlemontre Ie Lemme
re
2
(d ' )
Par sutte
au
n.6, on suppose chois i
v~
+ ~, f(a)V~
(7)
(f(b)v, v>
~, f (b)v~
quels que
soient
V,
v, v')
=
0
aE
~ et
be. 111 . On suppose"a de
vr)
Utilisant
ce produit scla ire,
=
0
V. Compte tenu de
E:.
'II
E
sante
se
v,
fibre de
a
V Ull produit scala i-
=0
E V ,
'II
+ (, \Ff
quels que soienl
6
dalls
est touJours realisable, que
(V:I
et
que (d" ) =0 ce qUI s
tel que
( , (a)v,
(8)
2
rnerne
de
1 du n. 8 .
(6)
plus, ce qui
.T. L. Kos z1l1
(6) el (8)
chaque
trouve munie d'un prodllit scalaire 11ernl on definit comllle au
n.4 Ie ad,lo'nts
\ a'. , Os
Ce sont des operateurs de degre total -1, somrne dlllne compode degre (-1,0), notee aI, d
,a a
I
ei dlune cornposante de degre
I
s
(0, -1) notee d", d" , ~" respectivement. En utilisant la base s a c du 11. G les I'ornlllles de ,011 dedllit des relatIOns (7
x,,
"J'
m
d d
-~
l
S
i(x)L . x i
L t...(x) p (x)
l
a
,
1 \
1
d"a
,
I
Du fait que
[11+, I1+J
=
I1J = (0)
[~_,
dId" +d"
dId" +d"d ' s s s s Par sutte,
Ie Laplacien
s
deduil que
011
S
11 s
=d
s
¢
S
s
S
+0
s
d
o.
d I
S
S
se decompose
en
D,I + 6,,, s
s
avec
D.. Les operateurs
I
s
=d' d s
6. s I
I
S
+ ~ I d'
S8
et (j "
s
Il en est done de meme de
,6." = d" d" + d" d"
conservpm
fj.
s
S
S
Ie
S
S
des
S
formes
- 244 -
N. 9 J. L. Koszul Le produit scalaire
a r (1'11 e, if) etant
sur
defini comme au n. 6,
on a
(ct~ w,~= ~'d~0)~"sW'S>=~' d~ w) quelles que soient
les formes
pour toute forme De
m~me
we.c{(11'\ c, 7J) et
(6; w,w>~o
On verifie de
(Ll;
et
01. (11'\ c, ZT) . Par suite
W/~
w,
ou
w,w) = 0
a
= d'
a
6.,
Les operateurs
a'
A~ w = O.
a
Lemme 2. Si
WE
a
a a
b."
et
a
£::.
=
(ii)
6. a'w
Ie
c'est
a
=
= d"
d" + d "d"
a
Lia conservent
<6. ~
w,
(~~
w, w>~O et
a
a a
Ie type des
~ ~o et que
for-
(IJ. ~ w, w> =
(ll ~w, w)=
0
conditions suivantes sont equivalentes:
6.. a"
w
=D..,
s
w
=!::.."
produit scalaire dans
s
dire
w=0 .
0,
b. + Co • Par suite
sulte que pour tout
6'~
• t11
b. w
a
meme
Gi:( c, lTl ,les
(i)
Puisque
De
6."a
d ' + d' d'
On voit comme plus haut que
6 "w = 0
w = o.
a
meso
implique
implique
II ~
b., + ~"
a
6. a'
implique
w, w) = 0 implique
meme que
A
=0
!
r,
Ie noyau
6.
s
w
V
o.
verifie la
condition (7),
conserve Ie type des formes. 11
en re-
l'espace des formes harmoniques de degre r, nr c de b.. dans (.;:;Il (In , 75) , se decompose
en somme directe de ses intersections avec les sous-espaces f'lp,q C ?) \J\....f" (m , ). L'espace Hr(M,~) est done somme directe des sousespaces HP, q(M,];) avec p+q :: r constitue's par les classes de cohomologie representees par une forme harmonique de type (p, q) .
- 245 -
N.9 J. L. Koszul Pour
expliciter les Laplaciens, on dHinira pour tout
e+(a) et eJa)
endomorphismes
r
c
a ~
i.
des
G. (M1 ,VI en posant
de
X:J,..
(e (a)w)(x 1, x 2' ... x ) = - Ll4:x 1, x 2' ... [a, + r. J J
x , r
(e _(a)w)(x 1, x 2' ... x r ) = - ~xl' x 2 ' ... [a, xj xr ' J c +pour toute suite x, x , ... x ~ tn, en d~signant par x. (resp. x.) la com1 2 r J J posante de x. dans n (resp n ). On a era) = e (a)+ e (a) pour tout J + +a f: R Quels que soient a Co ~, x, y E. 3c , on a
J, '"
On notera d 'autre part
les relations
(9)
L . J. /a, \-
[x.,X.J)E(W.)L(X.) J, 1 J 1
2:: t'a, .. \ J
[X., J
.
Cela
~tant,
1,
e (a)
( 10)
1,
quels que soit
aE
l
xJl\ t (w.) L(X.)
Y
J
1
(a k )k=l, 2... n-2N
d~signant
base orthonormale de ~ , on obtient :
(11)
6,'
s
-L
L_ L x. X.
-L (12)
6.
It
s (13)
(14)
£:::.'
a
6
11
a
-L
1,
L_L x. x. 1
L
x.
1
1
L_ x.
rr(Xi
~L k
- 2:: k
1
~~(X·)r(x.) ill
~. L[X.,x.{{W j ) L{\)
+
1
1
L
J
ak
ak
1
e+{a k )
e_(a k )
+Ie(a k ) e+(a k ) k
)r CXi) +
l:~ak) k
J
eJa k )
une
- 246 -
n.g J. L. Koszul
Ces formules vont faire apparaltre de nouvelles relations entre les Laplaciens d'indice
s et les Laplaciens d'indice a
dont Ie moins
qu'on puisse dire est que rien ne permettait de les presager. Il est vraisemblable que
a voir avec la
ces relations n'ont rien
qu'on pourrait les obtenir dans un contexte analogue Lemme
3, (Lemme de la folkdance) . Pour tout
6. ' - 6"
s
s
sur
a
situation homogene et
a celui
p,
du n. 5 .
on a
a
e.X (tn c, Gr) . En effet, d'apres (11), (12), (13) et (14) , on a
6,'_A" s
s
_61
+6."
a
a
que soient a, be: ~ ,posons 13 (a+ 'V-lb) = eta) + v:1e(b) . c n P c Pour tout x E ~ ,on a alors L/ ~(x) = -e(x) sur \"""\,.k(m, lr) .
Quels
Compte tenu du Lemme 1, il en resulte que
6's
6. "
(:).,
a
s
+.6."
a
Or e(h ) W = (q_p) \'-1w pour toute forme o ') 1 ad(a k )" etant egal a -"2 sur ~ ,
L k
pour toute forme Theoreme
W~
f)p,q ~
c~G
(111, (.) ) •
1. Pour tout entier r, on 9-
WE
on
aP' q( trI c, Vl . en
deduit
D'autre part,
que
- 247 -
n.g J. L. Koszul
A
2(6.'
s
+ 6,") a
+8 ")
= 2(60,
a
s
~ ~t(""'c, If). crest une consequence immediate du Lemme
A
A +A
=
a
Corolla ire
s
1.J) ,
1. Si W~~ (me,
(a) (b) (c)
Ll w = 0, A's w 0
6." w
la relation
les conditions suivantes sont equivalentes:
60"a w = 0, A ar w = o.
et
0 et
s
3 et de
crest une consequence immediate du Lemme 1 et du TMoreme. Corollaire 2.
w E ~ q(l1',c, V) , les conditions suivantes sont equiva-
Si
lentes :
(b) (c)
0,
~ w
(a)
x ~1'I +'
L w = 0 pour tout x
w = 0 et
En effet, si
x~tI
west de type (0, q) , alors
A a' w
=
~~(x.) ~ (x.)1 w • . 1 1
b) ==7> a) . Compte tenu des proprietes du produit scalaire
Ceci montre que dans
r (x)w = 0 pour tout
w = 0 et
A~ 6"a
V. on a
(~r a w,w) ce qui montre que COI"ollaire 3, Si
:L(p(x.)w, p(x.)w, . \ 1 \ 1 1
\a)~(b).
On voit
de
m~me
que
(a)~(c)
w est une forme differentielle de type (p, 0) sur M
valeurs dans E , les conditions suivantes sont equivalentes :
,
(a)
Aw
(b)
!J.'w=O et r(x) w s A ,w = 0 et L_w
(c)
.
= 0
o pour tout X (:'11 o pour tout x E:~.
a
- 248 -
n.10 J.L.Koszul
So it
o V un sous-espace complexe de V stable par
~
(a) pour tout
, autrement dit un sous-module de V considere comme -ltc-module. Soit
E
°=
11 est stable par
transport
parallele pour la connexion lineaire symetri-
que. 11 en resulte que EO est un de E.
de E qui correspond a V 0 .
o .
Pk(r'G, V ) Ie sous-espace hbre
Dans l'isomorphisme canonique de
a.P,q(M,E) sur
a. P' q(M, EO) des formes differentielles de type
l'espace
dans Eo/est appliqve'sur Ie sous espacetit q(\)1 c, VOl de l'espace des fonctions differentiables
rateurs
d'indice a tels que phismes
a. p~ q (
\ (x)
a
/1'
a Puisque
stables par les ope-
pour des
a
ho
voir qu'il y aura cependant stabiV O con venables.
sous-espaces
de ~
appartient au centre
, ~ (h o ) commute
. Par consequent
f
(h ) laiso P'(M C,'Cf) quels que soient
a
q
Lemme 2.
Quels que
Posons
C
soient
p, q, on a
=2. (P(x.)\,(X.) +~(]1:.)~(x.)) . 1
\
1
1
1
Crest l'operateur de
f
commute avec tre
sont
V~.
car ceux-ci font intervenir des endomor-
x E.lI\c . On va
b.."
et
a
L + era) + F (a) quel que soit a ~ ~ a se stable Ie sous-espace QP( q( ttl c, If) de et
valeurs dans
, l.f0 ) n 'est pas stable par les operateurs
Ll"
Ll ' et
avec
f11 c
avec
p
a valeurs
d', d", d' , d ", {j. , , 6" s s s s s s Crest en evidence sur les formules du n. 9 . En general
lite par
0 )
(p, q)
q(l1"c, if),
B. ~' q(offlc ,I/) ou'1Ji des igne
r" G a
sur
aPi q ( 111 c, V-
Lemme 1. Les sous-espaces
ar
sous-espace fibre vectoriel holomorphe
de ~
f
\
1
Casimir de la representation c . (x) pour tout x f. PUlsque
(h o ) commute avec
3 .
_
~F(a )2 h. k
f?
. h
o
Par suite appartient
C au cen-
- 249 -
J.L.Koszul
quel et
que soit
1
2: r(x.) r (x.)
de
. 1.
!:l
1
, ce qui
1
A"
et
I
a
Ie Lemme compte tenu des expressions
d~montre
donnees au
a
n.9
(formules (13) et
~ une valeur propre de
Soit phis me
r~sulte que f{h ) commute avec 2.'~{x.) f (x.) o . I 1
. II en
de
V
et
f(h o ) -~I dans
~ (ho) considere comme endomor-
Ie
soit
(14)).
sous-espace propre noyau de
V. C'est un sous-module de V considere
comme g,c -mo-
dule. 11 lui correspond donc un sous-fibre vectoriel holomorphe PK(r\G, V5 ) dans
E
que
l'on notera
differentiables sur et
endomorphisme
a
r'\G
ctp,q (mc,~)
q,
E~. On note ~
valeurs dans
V~
Lemme 3. Pour toute valeur propre
a.
~
. Quels que
~ (ho) -
est Ie noyau de
a. p, q( 111 c, if) .
de
l'espace des fonctions
f_
de
tI
soient
p
considere cornme
(ho) dans
V, Ie
sous-
P' q{WI c, U) est stable par espace tJ.' et tJ. n 1aa C'est une consequence immediate du Lemme 2. Puisque K est un sous-groupe compact de G, toutes les
--
valeurs propres de ,ad(h o )
= v:1
pour tout
x
~
(ho ) sont imaginaires pures. D'autre part, puisque
sur t1 +
~ 11
et
ad(h 0)
+ et ,(X)Vt
I""
Lemme 4 . Soient
V~
C
= - \Fi _ \(:f
p(x)V~ c
on a
pour tout x
(resp. ~ ) la valeur propre de
V.t + 'V-I
~ t\ _ •
~ (h o ) telle que
~
_l-"_ soit maximum (resp. _te_I_Ie---,q!....u_e - - - soit minimum). Si ~ est
V-i
une representation simple de v E V tels que
~
G,
est
VI'"
1'espace des
(x)v = 0 pour tout x E: n _ •
La demonstration
est
une variante de celle qui donne les poids
d'une representation simple d'une Compte tenu obtient
V-1
alors
alg~bre
des Cor. 2
et
3
de Lie simple de 1
rang 1.
du
TMor~me
du n.9 , on
.9
dans V etant supposee
Ie resultat suivant :
TMor~me
1.
simple, soit
La representation ~
(resp."
f
de
) la valeur propre
de
r (h
0) telle que
- 250 -
n.10 J. L, Koszul
A. / \f1
soit maximum (resp. telle que tout entier
llespace des z~ros de
q
d
soit
minimum). Pour
a
Vi
dans ~ q( 111 c, coincide o,q c . { (It! .' ~); lIespace de z~ros
a
avec llespace des zeros de
6.
de
,1J) coincide avec llespace des z~ros de
c::.. A
dans
a. ~ 0(
dans q, 0
VI-
l'1li c
dans
II
S
~
sections holomorphes
de
de cohomologie
de
Q.0'o ME) (
~
,
un isomorphisme
/'),0,1 ME) (
~
,
dans Ie noyau
de
1.
que
If (M, EO f)
Corollaire 2.
Pour
tout
lA
~ )-~) •••
(cf.
[2] ) ,
degr~
p
entier
sur
donc
b. ;
~ps Ie faisceau des
a
M
valeurs
Hq(M, ~\ )
• Le TMoreme
1 admet
q, il existe un isomorphisme canoni-
HO, q(M;~ ) . q,
i~l_e_x_is_t~e_u_n_e_a.!..p.;..pl_i_c_at_i_o_n_l_in_~_a_i_r_e_i_n.:(.je_c_-
HP(M, r,;~) dans llespace des z~ros de \ o~ q( mc, U) .
a
nullit~
il existe
Ie noyau de
sur
tive de
tions entratnant la
llespa-
s
(,
soit
a. p, q(M, E~)
Pour tout entier
On exploitera ce
annul~es
d ll
isomorphisme canonique de
f::.
Corollaire de
.. ,
holomorphes de
II dans s donc les corollaires suivants :
sur
/lo, q M E
g~n~ralement,
existe un
il
E~
Hq(l\'I, ~~) est
q,
Hq(M, EO ~ ) sur
de
• Plus
diff~rentielles
de
q du complexe
canonique
a 0, q{M, Er)
formes
entier
du Th~oreme de Hodge-Kodaira
Compte tenu
dans
pour tout
deg~
~IA
~ (h o) ,soit ~~ Ie faisceau des
• Ce sont les sections
E~
d ~ . Par cons~quent,
ce
I
s
(1\1, VA) •
Pour toute valeur propre ~ de
par
6
c 'lr
Corolla ire au q
~
dans a--
n. suivant pour donner des condi-
0
de H (M,:g; ) •
Corollaire 3. Pour tout entier p, il existe une application lin~aire canop,o 0 p nique injective de lIespace H (M,~) dans llespace H (M,:S; >.) des formes
diff~rentielles
holomorphes de
degr~
p sur
M a valeurs dans
E>...
- 251 -
n.10 J. L.Koszul
Si P = N,
cette
application est un isomorphisme.
La premiere assertion
r~sulte
directement du
Pour demontrer la seconde on remarque que pour tout nul, sur les formes de type t')
I,.;~
N,n
C.,r(m,v).
(N,O)
. On a
done
Th~o!'eme
1, .
adL IJ+(a) est
Do,
s
sur
- 252 n.ll .T. L. Koszul
a • 0 On supposera dans ce N que (1 est une representation
dans V. L'injection canonique de V dans \t(qui associe
a tout
e.
siJllpl~
~gale a v sur rp)dMinit une injection de p, q( rI'\ c, V) dans
ct P' q(.mc, 1f}.
Si Fest l'algebre des fonctions diff~rentiables sur
¢. ,
r"V a valeurs
on obtient un homomorphisme canonique du F-module F ~(l' q(-ti, V)
dans Ie F-moduleQP,
q:
qmF,'\f).
Cet homomorphisme est visiblement un isomor-
a son
phisme. On identifiera dans la suite l'espace~P, q'11f!V)
V ) ; elle
a.
S t!
~l~ment vIa fonc-
tion constante
dans
de
etLl 11 • Le noyau de ~ dans a a est donc Ie F-module engendr~ par Ie noyau deq dans~P, q~7 V).
est stable par les
p, qmtC , 'IJ)
op~rateurs
image dansd P q(M1c,
b.'a
an va indiquer dans la suite, des r~sultats dus principalement
lll] ,[i6J),
a B. Kostant, (cf.
qui donnent des renseignements pr~cis sur ce noyau.
r
Puisque les endomorphismes (a), etta') et 8Ja l1 ) commutent c quels que soient a, a', a 11 E. 1\ ,f' + e+ et f + 8_ sont des repr~sen-
lin~aires de ~ c dans
tations
Lemme 1.
est stable
~
Soit en effet C = l'op~rateur de
C' = -
r+
k
et
par
b. ' a
i l l
un
9 C -module
k'
11
(p (a ) + 8 (a ))2 \ k + k de ~ C dans
simple, C
l'op~rateur
Cl (rn c, V).
f
f
de
k
c
6
de Casimir de la
repr~sentation
5"'r.>I(a )2 k k
+ p \
(~ .[x., x.] ) 11
f
1
+8 : de ~ c dans
11
a
On choisira une sous-algebre ab~lienne maximale" Elle contient
~
et
~
c =\ +
dans
c dans (V h1 , V).
Tout sous-module de la repr~sentation ~ + 8
G.(m c, V) est stable par
c
est une homoth~tie . Soit
1 = C - C' - - - (~(ho) + e(h o )) ce qui demontre Ie Lemme. 2 v:T
Lemme 2.
~
~_
~
D'apres la relation (13) du n. 9, on a
2 A' = 2 LD(a ) 8 (a ) + C + a Ik+k
On a un resultat analogue pour
e+ de
+
r ~(ak) 2
(f'(x.)t" (x) +f(x.)f (x.)) -
Casimir de la repr~sentatlOn
Vest
L
Ie
Tout sous-module de la repr~sentatioll
C1 (mc, V)
Puisque
a. (ttl c, V) = mci (I1"Ic , V) •
V-i 1)
est G.ne sous-algebre de
dans Cartan
de
3c.
- 253 n.11 ,J. L. Koszul
a~c
Soit R l'ensemble des racines de '$'c relatives semble des
q
que
ex tR telles que 0( (h o) = 'f-""1
. On notera
Q+ l'en-
et Q- 1'ensemble des
0< ER
telles
(h ) = - '-1 . Il est clair que /1+ est somme directe des sous -espace o + propres t~ ou 0( EQ et que H est somme directe des sous-espaces propres c
+
-
4£
ou ~~Q . On a visiblement
i'~
Q
= - Q
-
. L'ensemble Q
=
est appele 1'ensemble des racines non compactes. L 'ensemble
+
Q VQ
-
Rk = R-Q est
appele 1'ensemble des racines compactes ; c 'est l'ensemble des racines de relatives
a 'h
On choisira
c(qui est aussi une sous-algebre de C~rtan de
+
1'ensemble
dans sira
un
simples tel
un systeme de racines
R
des racines
tel
que
c
90(
~ IS
~)=
d 'autre
til(
LYo(' Yo<J) =
part,
d 'apres
h
2 v:1
Les
Soit
W Ie
w E W,
on
1'ensemble
des
un element
r
dans
dans Ie groupe de R, K
1
h
21Fi
que
soient
groupe
de
posera w
e:
Weyl
~.
,alors
,~) = -0( (h) 2: [YII( ,Y.t J c(
~w groupe
0
a Y-f ~;
on posera
,j3E:R. c Weyl de 5 relatif
(d.., 13 >
0(
(wR-)
=
n R+ .
On
1... r\ c. Pour
a
designera
Q+ . On sait C w de Weyl de ~ c (donc
W tels que Ie
[yO('
QTr:J. 'Yfl)= ~c(, f'
+ et
Ie Lemme 1 du n. 9,
appartiennent
he(
(ho( , hJa) quels
tout
,h]
ho( =
,on choi-
. Par suite
0
2: h~ £(EQ+
(1 )
h Eo ~
. Si
Q
quels que soient q' I ~ ~
11
soit contenu
tout q E. Q
on posera
+ yO( ,pourc/ E:. Q constituent une base de + c
Les
Q
+
positives. Pour
,YrI,) = 1 et
f<;c). T
que
4?
W) qui transforme
+
Rk = Rk
f:: e
+
nR
par
Vv'o
qu'il existe
a en
fortiori
-
Rk
=
nR . Puisque Q+ est l'ensemble des poids de la representation
- 254 n.11 J. L. Koszul
de ~ c dans
11 + qui s 'obtient
adjointe, on a rQ
+
l'ensemble des poids
de
F
(resp. Ie plus grand) element
de
p.
P
11 est clair
+
de
partie maux
Q
poids
A I(resp.~)
e+
les
sont
~ C( ou A est une c(f:A p elements. On montre que les poids minide
contenant
de la representation
de ~ c
que
a. p, q(M c, V)
dans
restriction de la representation
=Q
On notera Ie plus petit
par
+
la forme
des composantes isotypiques
de
- (A)= -
la
e+ dans
representation ou
Card~w=p.
sont de la forme ou A est une partie de Q+ contenant q
elements et
o...p,q('rt1 c ,V)sont
de
-(lPw)OUWf:W o et
la forme
e
De meme, les poids de la representation
etP, q~ c, V)
de" c dans
les poids minimaux des composantes isotypiques sont de la forme ou w
f
WO et ou Card
vant: Lemme 3. Soit
V
Partant de la 011 demontre les Lemme sui-
-c-----'------"-"---'-------''------
Ii representation
F
injectif
Qp,q(mc,v(~)), il faut
WO
dans
tel que
injectif
dans
tel que
r ~
Theoreme 1 il faut
d~
dans
k
I ne soit pas a qu'il existeunelement wE: ~
ci'
A I et
= WI
Card
b.
Pour que
rw"
(b)
Card Soit
> 0.
que
= w A et Card 1r'I = p . Pour que tJ. II ne soit pas '!'w a q( 111 c, V(~)) , il faut qu'il existe un element WI ~ WO
(a)
/.: ,01., ,).. )
V. Pour
s
a qu'il existe des elements
Corollaire
poids minimal ~ de
la compos ante isotypique de
(~)
r.p w
Si
En effet,
°
= WI
~w =
p,
71'.
~Wl
ne soit pas w, WI
E
injectif W
tels
dans
aP ' q( ...r V\
que
AI Card iI'w l = q .
qr
Ie
~ q
Ie seul
=q .
nombre des racines
element
1\ U
ot
+
~ Q
a est injectif
o w E: W tel
que
telles que 11 0 ,q c sur LA(11"1, V),
w
= ¢ est llelt~
- 255 -
n.11 J. L. Koszul
nlest pas injectif sur ment neutre de W . Par cons~quent, si t1 a c O Qo,q(Wl , V), il existe un WIE: W tel que r.>. ';' w I ~ I et Card cP 1= -1 w = q . Si ~ f: Q+ et si 0 alors (AI, (Wi) Puisque
r0>o.
(0( ,A> >
).. I est Ie (Wi)
-1
t
rD(
quent
poids R
-
minimal
f
de
cette
+ + rQ = Q
. Puis que
, on a
inegalit~
ro<. E.
4? . Par
«€
Q telles
donc
q = Card
,
Corollaire 2
. Soit
Ie
pp
nombre des racines alors
t1
a
entra1ne Wi
+
est injectif
conse-
que
CAY' 0 ("" c,
sur
V) .
a
Demon'stration analogue 3. Si (AI~».o
Corollaire
D.
alors
a est
injectif
Supposons (\'t\c,V). Soient et
q = Card
de
r
~
11w
et
sur
a...P' q( mc, V) •
va
Aa
que
W,wIEW o tels On
ne soit pas
que
rw.>. =w'A',
montrer
ment analogue On va
voit
tant
en jeu
pIe,
on
. On
des
Ie nombre
que
que
tirer
+
est reunion disjointe
Q
(A, ~> f
car
cp nq>, w w = r£P w Ul'l'J l"w
quelques
comme
du
voit
forme (f + ~ ou ~ ). 0
i>w
r
I
0 pour
= p et un
toute
raisonne-
•
consequences de ces corollaires metQ c . cl est Slm-
facilement qu'il existe exactement· une racine simple non
minimal
21'
p=Card
I' alors r1ewR-, donc~A)f~>= w 1 part, c(E. w'R- ,donc (w).,rCl()=(w'A',()=
donc
compacte : considere
pacte
a.P ' q
1es racines simples non compactes. L orsque
voit
Ie poids
Q
sur
N,
w
montre
maintenant
+
que
= (.~I, (wfJ) ~ 0 . Ceci est impossible
f3 . On
injectif
f
Sio(E.r~ net'
~J w.,F~(O . D'autre racine
+
0( ~ R et s i p+q
pour toute racine
en effet
celle du Cor. 1 •
{C -module,
~ c _ module de
plus
11
est
+ .
que tout
est combinaison
n+ est en effet simple
une racine simple non com-
elemE'nt
lineaire
et
de
+
Q est de
la
a coefficients entiers
racinessimples compactes. Plus generalement, si s est c des id~aux simples de 3 ' il existe exactement s racines
n.11
- 256 -
J. L. Koszul
simples
non compactes et
&'
ou
est
une
a
naison lineaire
A
t'
toute racine
suffit
donc
est
Compte tenu
du
~ R
(.\,
que
entiers
Ie plus
+
element
de
grand
. Pour
0
~
(t. ,«') ?
~> > 0 pour toute
0 = N et
est
combi-
P , ([- ,tP?>O
de
+
pour tout c/ E. Q il
0
racine simple non compacte.
Cor. 1 on a donc Ie resultat
Corollaire 4. Si
ou ~
des racines simples com-
poids
que
~ +~
Q+ est de la forme
simple non compacte et
coefficients
pactes. Puis que pour
racine
tout
suivant:
pour toute racine simple non compacte oq c est injectif sur G..' (ttl ,V) lorsque
A ~
a
O~q<.N.
Dans on peut Lemme tation
Ie cas
ou
est
donner explicitement 4.
la
g
On suppose que
la representation valeur
c
de
adjointe
f
~ c. ~i ~ est la racine simple non
adjointe de
9
c
qp .
simple et que
est
de
est
la represen-
compacte,~:
1
(a)
<~ b') I
P E:. R+
est Ie nombre des racines
q(>
(b)
-1
telles que I"' +
g
soit une racine. On peut choisir l'ordre des racines simples de telle sorte Ie plus grand poids
.\
egalement
grand
1\
+
Ie
plus
obtenue par
simple,
rA
de
ad soit poids
restriction de
est
Ie
poids
que
dans
Q+ . Le poids ). est alors oc la representation de 11. dans
de
ad. Puis que Ie plus
petit
1'1
+
est un .ft.c -module
-hc -module
du
11 + et
consequent la racine simple non compacte 'It . Pour qu June + ()..,o() > 0, il faut et il suffit racine 0( ~ Q verifie la condition + + que (~, r«) >0. Puisque rQ =Q, ceci montre que qp est + egal au nombre des Cl(eQ telles que (~, > 0, ce qui prouve crest,
par
o()
(b) . Toute racine c( son lineaire
a
+ E. Q est de la forme
coefficients
pactes. Par suite, si c(
F~ ,
entiers alors
~
c( +~
0 et
Q(
des «-
+~
ou
\3
est combinai-
racines simples com-
2g ne sont pas des ra-
- 257 -
n.11
.J. L. Koszul cines. Si 0( = ¥, la relation
on
a
, il en
(1)
done resulte
2(d.. (~ que
I
~>= 0
ou 1. Compte
1'6)
1
(~ '"") Lemme 5.
Si
tion adjointe de
S e est simple de S e ,alors Ill" >
En effet,
<~ , pres
lS
Ie
~"o
et
Lemme
il on 4.
rang
~
+
~'
1
~
r
de
- 1 •
est la representa-
1.
existe une racine a
>
tenu
Q
+
simple compaete ~ ce qui
prouve que
I
telle que
>1
d'a-
- 258 -
n.12 J.L.Koszul
n.12
de
Th~or~mes
Les
Nullit~
du
r~sultats
.
Num~ro pr~c~dent,
combin~s
donnent deux types de conditions suffisantes de
faisceau des sections holomorphes de I'espace autres portent
sur
la cohomologie de
a:
faisceau des sections de
a
dire finalement sur repr~sentation
s'ajouter
a
ceux qui
r a
ont
ces derniers ~te
obtenus
irr~ductible
coefficients dans l'espaviennent donc
n. 6 sans
supposer
que
S1
born~.
e
et si
TM,
alors
automorphismes
born~ sym~trique
est Ie faisceau des sections holomorphes du fibre q 1 H (M, e) = 0 pour 0" q <..;--> - 1 , ou "
-
-
,~,~
la racine simple non compacte de
I'alg~bre
de Lie du groupe des
de.n.
On applique
Ie
adjointe
On
sentation
covariante nulle, c 'est
r~sultats
au
E t' . Les
coefficients dans Ie
1 ( Calabi- Vesentini) . Si S1 est un domaine
Th~or~me
d~signe
p
vectoriel
diff~rentielle
la cohomologie de
ce de la
soit un domaine
E ayant une
coefficients dans Ie
fibr~
a
M
pour Ia cohomolo-
nullit~
gie . Les unes concernent Ia cohomologie de M
avec ceux du n.10
me resulte donc du
Cor. 2 du a
Cor.
alors au Th.
Th. 1, Eu \-.
n. 10 au cas
de la repre-
e = EO -~
et Ie TMore-
= TM,
du n. 11 et du Lemme 4
du n.
11.
Corolla ire
Si
S1
n 'est pas isomorphe au dis que unite
1
de C 1 H (M, e)=
= (0).
Cela resulte du Lemme On en deduit Ie d 'un domaine TMor~me
que,
2
5
TMor~me
born~ sym~trique
du n. 11. enonce au n. 1
qui concerne Ie cas
~ventuellement r~ductible.
(Matsuhima, Murakami). Soient
S1 un domame borne symetri-
G la composante connexe neutre du groupe des automorphismes holomo
phes de
S1
et
r
un sous-groupe discret uniforme de
G. Pour toute
- 259 -
n.12 JL. Koszul
repr~sentation simple de
= (0))
G, o~ RP,
pour 0 ~ p < p (resp. 0 p Cela r~sulte directement des
Cor. 3 et 2
du Th. 1 , n.9 .
° (r, p ) = (0)
< q
~
Cor. 1
q
(resp. RO, q(r, p)=
). et 2 du Th. 1, n.ll et
des
- 260 -
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Rieman~
11. B. Kostant, Lie algebra cohomology and generalized Borel- Weil theorem, Ann. of Math. (2) 74 (1961), 329-387 . 12. Y. Matsushima, On the first Betti number of compact quotient spaces of higher dimensional symmetric spaces, Ann. of Math. (2) 75 (1962), 312-330 . 13. - - - - - - - , On Betti numbers of compact, locally symmetric Riemannian manifolds, Osaka Math. J. 14 (1962), 1- 20 . 14. -
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18. S. Murakami, Cohomologies of vector valued forms on compact locally symmetric Riemann manifolds, Proc. Symp. Vol. 9, algebraic groups and discontinuous subgroups, (1966), pp. 387-393 . 19.
-, Cohomology groups of vector valued forms on symmetric spaces, Lecture Notes, Chicago (1966) .
20. M. S. Raghunathan, On the first cohomology of discrete subgroups of semi-simple Lie groups, Amer. J. Math. 78 (1965), 103-139 . 21. - - - - - A vanishing theorem for the cohomology of arithmetic subgroups of algebraic groups (to appear). 22. - - - - - - Cohomology of arithmetic subgroups of algebraic groups II (to appear). 23. A. Selberg, On discontinuous groups in higher-dimensional symmetric spa~, Contributions to Function Theory, International Colloquium on Function Theory (Bombay, 1960), pp. 147-164, Tata Institute of Fundamental Research, Bombay, 1960. 24. A. Weil, On discrete subgroups of Lie groups, II, Ann. of Maths. (2) 75 (1962) , 578-602 . 25. - - - - - - , Remarks on the cohomology of groups, Ann of Math. (2) 80 (1964), 149-157 .
CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)
E. M. STEIN
"THE ANALOGUES OF FATOUS'S THEOREM AND ESTIMATES FOR MAXIMAL FUNCTIONSII
Corso tenuto ad Urbino dal 5 al 13 luglio 1967
THE ANALOGUES OF F ATOU'S THEOREM AND ESTIMATES FOR MAXIMAL FUNCTIONS by E. M. Stein (Princeton- University) 1. Introduction. The behaviour of harmonic functions (in particular Poisson integrals) near the boundary is closely related to the differentiability properties of the boundary funcions. This was long ago recognized in the classical context of the
half~plane
or disc in Fatou's theorem,
and in this setting
it was put in more definitive form in terms of the appropriate "maximal functions" first studied by Hardy and Littlewood. The purpose of this note is to briefly review this, and later developments including those which deal with the product of half-planes or discs, together with the recent attack on the analogous problems for the general symmetric spaces. While we shall state some new results, we shall not attempt to give proofs here ; those will be published elesewhere.
The Poisson integrals are those harmonic functions
u(x, y) given
by (2.1)
with
u(x, y) =
y
>
0 ,
P
y
b (t) = -
spect to additive group sense. f belongs
to
lc
f(x-t) dt = P f 22* Y +t Y 1
7C
y+t
of
IR
an
2 2 1
the convolution
*
being with re-
. u(x, y) is harmonic in the ordinary
LP(-oo, 00)
space,
or
c;:ollld be replaced
- 292 -
E. M. Stein
by a finite Borel measure. We shall not dwell p Ie condition that characterizes those harmonic represented
here on
u(x, y) which
tioned above
is
that
lim
(2.2)
r~
As
a
simple
lim r-+o
now (Xl, y) is
I X-Xl I,;S
as
Fatouls
x,
then
These
y
f(x) ,
for
,for
of ~
f,
men-
00
almost
every x.
r
of
(2.2) there is a stronger
for
almost every
x.
r
theorem showed,
whenever
(2.2) holds
lim u(x, y) = f(x) for x; also wheney....:,- 0 a given x, then lim U(XI, y) = f(x) , where
for
allowed
c
1
consequence
Itl..:;;
L (-00,00),
jif(X-t)-f(X) \ dt = 0 ,
holds
(2.3)
f €
It I <
(2.3)
given
p
r(X-lldl
it, namely
Now
for
0
version of
ver
can be
in the form (2.1) .
The differentiability property of the integral
for a
the rather sim-
to
approch
(x,o) non-tangentially (i. e.
some constant c
matters can be put
in
) .
sharper relief by the aid
corresponding IImaximal functions ". For the process
of
of tre
differentiation
of the integral (i. e. (2.2) or (2.3)) the corresponding maximal function is
- 293 -
E. M. Stein
(3.1 )
f
* (x)
+j
sup
r>
(The importance
0
here
2
It I
is
that
f (x - t) dt
.
~ r
we
take
sup
of lim r~
or
lim sup r-+o
0
instead
>
r
0
).
For this maximal
function the following basic facts are
well- known . Theorem A. fE L P (-co, co)
If
II t \I p
(3.2)
I I
f~x) > 0<1<
x
II. /I
Here
A IIfl/ p
~
1. ) , f €.L(-co,co
If
(3.2') measure
I< p <
denotes
the
co,
then
p
then
(A/~)
Il f II! L P norm.
usual
p
The sighificance
f
*' (x) 3- If (x) I '
quality (3.2') (it
is
also is
holds, the
also
form is
we can
but
to
in
a
(3.2')
are
(2.2)
are
and
is
shows
p = 1 that
for
as
follows: (i) clearly
that the
modified is
reverse
ine-
functional
weaker
than
form. (3.2);
as a "weak-type" inequality) . However
the
and
that differentiability functions,
result
suitably
inherent .in the nature
analogues
nuous
this
the theorem
analogue
referred
(ii) (3.2) and
the fact
but
of
some
of
things
and
this
cannot be improved.
quantitative statements whose
qualitative
(2.3). More precisely, using theorem of
integrals
obviously holds
general principles of functional
deduce the differentiation theorems form
for
A, conti-
analysis,
theorem A. Thus
- 294 -
E. M. Stein
we can say, without much oversimplification, that the maximal theorem A contains the essence of the results
The connection with can also define an sup y/"o
Iu(x, y) \
Poisson
appropriate and
in
(3.3)
generally
as
we
>0
is
simple. Here we
far
as
is
have
Iu(x, y) I ~
sup
and more
integrals
maximal function, that
fact
y
(2.2) and (2.3) .
A f.f (x)
non-tangential convergence is concer"
ned (3.4)
It should
f
'9
~
I
sup u(x-t, y) J /tkcy
0 ,then
be
for
A f'* (x) c,
added,
as
Paley
c
> 0
pointed out,.
that
if
the converse implication. holds, i. e. f
~
(x)
~
A
sup y
Now by the use of analogous
has
each
estimates for Then as in
y
>0
u(x, y),
f
(3.3) , (3.4), and theorem
sup
>
0
I u(x,y) I
and
o.
~
A we
obtain
I u(x-t, y) \
sup
Itl <
cy
the case of differentiation this leads to the re-
suIts of almost every-where convergence (even non tangentially) of Poisson integrals contained in Fatou's theorem. So here
again the ba-
sic results are in fact contained in terms of the appropriate maximal functions. Proofs of all
these
results may
be found in Zygmund's book
[8] , volume 1 .
4. The n-dimensional Euclideqn case; a nilpotent variant. We shall now
consider the analogues of these matters but
- 295 -
E. M. Stein
/R n,
where the real line is replaced by ton. If In
.h group. We b egin WIt
of
or some more genera1
~ L P(
f
IRn) we define the analogue
(3.1) by ~
(4.1)
sup
f (x)
r>o where
B
r
denotes the ball
m(B ) is its Euclidean
1 m(B r)
of radius r
centered
measure. Then as in
r
at
the origin,
the case
and
of n = 1 , the
following is well-known. Theorem B. rem
A
/R
hold for This
With
was
ri
f-*' defined
as
proved
IB~f is
and
by Wiener
E
is
[ 6]
a
B l' B 2 ,·.· B n' •.. , so
m(Br ) ). c m (E) , where
who began by pro-
IIVitali type ll
c
•
measurable set of
a collection of balls that cover
sub-collection
~
Suppose
(4.1) the results of theo-
well.
ving the following covering lemma of Lemma.
as in
E.
finite measure
There is
a disjoint
that
is
an
absolute positive constant.
r=l
With
this
lemma
(3.2 ') follows immediately, and then (3.2)
can be deduced from it . The maximal function allows several variants, which
we de-
scribe in order of increasing generality. The simplest change, and this is trivial, is to replace the balls appearing in (4.1) by cubes. The next variant is as follows. We choose a fixed rectangle R, R
r
(4.2)
the rectangle obtained by dilating R sup r
>0
m(R) r
and
by the factor
JI R
r
f(x-t)
I
dt
denote by
r; we consider
- 296 -
E. M. Stein
This maximal function again satisfies the conclusion of theorem B, and what is important, with bounds independent of the original rectangle
R. This form of the maximal theorem follows from the spe-
cial case
of the cube by a linear change of variables. A fUrther exten-
sion is obtained by considering (4.2) again but where
{Rr }
is an
arbitrary "monotonic" family of rectangles; i. e. r 1 ~ r 2 . Here
again the bounds do
tangles. (This
form
17.) . This variant, wards and Hewitt
may
be found in
as
[2],
not depend on the family of rec-
well
as
Zygmund's book [8] , chapter
[4J,
others by K. Smith
are proved by adopting the. proof
and
Ed-
of the cove-
ring lemma cited above. We shall cite here another generalization, because it
is
not directly contained in those already mentioned, and is par-
ticularly useful in the context of the domains discussed in this conferenceo
We
let G be any locally compact group, with right-invariant
Haar measure t -+O(t
dm. We
as-sume that there
from the positive
t
reals
is given a mapping
to automorphism 0(
of G, so
t
=D(t t ' We assume also that there is a open 1 2 1 2 bounded neighbourhood U of the identity on which the 0( are con-
that
tracting,
0( tOO( t
i
,.e.
<Xt (U),
U'
if t ~ 1, and
t
(These conditions can be relaxed) . Define (4.3)
f(x)
=
sup m( t>o
f* (x)
~ (lJ) t
J
Theorem 1. The results of theorem in the case of
f
U
*" defined by
(4~ 3)
t
(V)
0(
>0
t
=
[c r.
by
f(y x ) dm(y) 0( t ((1) A hold equally well
.
It goes without saying that each of
the maximal theorems discus-
sed in this section implies (by the arguments mentioned in section
3) a cor-
- 297 -
E. M. Stein
responding result for the existence of limits a oeo Thus in the case just cited, we have that for each f (f LP(G)
then
J
I
lim
LX
t~o
k p..:S
00,
fry x )dm(y) = f(x) almost everywhere.
tal)
Among the applications of this theorem, which occur typically when pIe
is
G G
(xl t
is al
a
nilpotent Lie
group, we mention two. In the first exam-
n
n
still fR,
and if x = (xl' x 2' ..• x )€ lR , then (Xt(X) = a2 an n x 2t , ... xn t ), where the a? 0 , but the a i
not necessarily the same. This
type of situation occurs when one consi-
ders "mixed homogeneity"_ as in certain aspect of the theory of singular integrals. Other applications occur when group. For example, let
G
is a properly nilpotent
G
be the group of
• with th, automorphi,m,
l~l : tX2
The relevance of
all
matricies of the form
of this
Xg
to Poisson integrals will now
be explained. First of
all,
theorem
B
(valid
for
n 1R ) is intimately con-
nected
with (Euclidean ) harmonic functions in the n + I dimensional n+l n+l n half-space R + . Thus if we write (x, y)€ 7R + ,x E R , Y > 0, and define
(4.4)
u(x, y)
by
- J II)
u(x, y) - c y n
2 f(x-t)2 (y + t
11iJ.
dt,
y
> 0,
c
=
n
pn;l~
nn+I 2
- 298 -
E. M. Stein
we say that and u
is
u
is
f. Here
the Poisson integral of
harmonic
in the sense that it
f
Eo
LP ( (Rnl,
is annihilated by the La-
placean
:L. + 'J Y2
. Now as in the case of one variable
*
is
A f*(x) ,
> 0
Y
wheref
Iu(x, y) I~
sup
(4.5)
given
There is
by (4.1). a similar inequality for non-tangential behaviour as
well as the fact that
f *- (x)
~
A
sup u(x, y) ,if
f
> O.
y>O Because of
(4.5), and
the maximal theorem, theorem B, one
obtains the convergence a, e, of Poisson
t.;;;: p .;;;;
00 •
Theorem
integrals of functions
in L P(
R)
1 plays a similar, although less decisive role,
for Poisson integrals in the case of non-compact symmetric spaces and generalized half-planes case the reason
for
(i. e. "Siegel dom ains of
type
II"l .
In the first
this is indicated by the Iwasawa decomposition
G = K AN, and the fact that the boundary (the so called dary) can be essentially identified with
Furstenberg boun-
the nilpotent group
N . In
the
case of the generalized half-planes the distinghuished boundary can also be identified with a nilpotent group. The convergence theorem for generalized half-planes is stated in Koranyi's lectures given at this conference, and the proof will author. It
be published in
a joint paper of Koranyi and the
must be pointed out, however, that in both cases the results
are only for the special case of Poisson integrals
of bounded founctions.
What is probably needed for the general case is a more refined version of theorem 1. A hint of the ultimate version for the general case of nilpotent groups is given by the refinements for discuss.
/R
n
we shall now
- 299 -
E. M. Stein
5.The case of produet of half-planes, and some other domains. The different maximal functions treated so far have essentially the same real-variable character whether they be for IR 1 in section n 3, rR or G in section 4. These matters begin to change, however. as soon as we consider
the simplest product domains.
Since in the case
=1 ,
n
the differentiation of integrals is cal'-
ried out with respect to intervals, so in the product case we differentiate with respect to their products, i. e. nrectangles ll • That is , we still consider
f € L P( /Rn),
/Rn ,and
m(~) ~
(5.1)
where
R
f(x-t)dt
now
= f(x)
pose the question whether ,
almost everywhere,
runs over the family of rectangles with sides parallel to the
axes which contain the origin; to zero. To study this mal
and
R ........ 0
question
f,lf-
..y (x)
=
R tends
we consider the corresponding maxi-
function
(5,2)
means the diameter of
1
sup m(R) R
JI
f(x-t)
I dt
R
Let us state right away that the results here are different from what we
had up
(n > 1 ),
so that
mit (5.1)
to this point. In fact there exists an
f~* (x) = co
everywhere,
exists nowhere. In this
differentiation,
way we see
that this
and the previous type as
differentiation. However posrtive results do hold for
> 1 ,for
type of
balls which was discussed in section 4. We refer to
the present type as strong differentiation
p
and in particular, the li-
involving rectangles, is more hazardous that the one
involving cubes or
ry
f IE L 1( ].Rn) ,
strong differentiation .
~
LP (/R n ) ,
- 300 -
E. M. Stein
Theorem
G
I\r"* ~ 1\
(a) (b) This theorem
~ A
due to
f
P
f ELP(lR n ) p
If
is
P
1< p <
,
> 1 ,then
c;()
•
(5.1) holds.
Zygmund. One way of proving it is
by observing that the rectangular function
f*'*' is
sition of one-dimensional maximal functions each
P
n
obtained by superpotimes,
(once for
dimension). Since each one-dimensional maximal function preser-
ves the class
LP
out. However
if
,p
> 1, the process of superposition can be carried
1 f E L ,then
(3.2') takes us out of the class of inte-
ends at the end of the first step. 1 Actually, the real dividing line between L P and L , as far n-l the conclusion (5.1) is concerned is the class L(log L) but
grable functions, and so the process
as
we shall
not
discuss
this
point further.
These results are related to harmonic functions on the product of half-planes as follows. The product of n domain
T
is the first If
main
r
I
= z = x+iy, XE
"octant", i.e. f E
mn,
r=[y,
LP()Rn) its
y~
Ii
Poisson
half-planes is the tube
r where r >0 ,
is
the
cone
which
j = 1, ..• n
integral
with
respect to this do-
is
u(x,y) = I n P r (t) f(x-t) dt y /R (t) is the product of one-dimensional Poisson
'(5.3)
pr y
r
kernels
-n
P (t) = 7'f:, Y As fact that (5.4)
in
the case
of one variable,
it
is
an
elementary
- 301 E. M. Stein
with a similar
result
for
non-tangential approach. Also
if
f).o,
then f
-1r~
(xl.:S
From theorem ted
G,
A
sup yEr
u(x, y)
(5.4), and
the following corollary of theorem
the arguments may be
G
already collec-
proved.
Corollary,
.!~ (5.5)
lim
fEL P ( /R n ) ,
u(x, y) = f(x)
p > 1,
then,
almost everywhere.
y~r y~
As
0
in
the case
of strong differentiation, there are examples
which show that the conclusion of the corollary fails for the class L 1(}Rn). If one compares the notion of strong differentiation to ordinary differentia-
tion y
one
is tempted to modify the limit in (5.5) by requiring that y. of y are of equal orJ y to lie in a proper sub-
in such a way that the components
~ 0,
der of magnitude. That is,
r
cone theorem
0
of
p,
of
(i. e.
f
e:
-
r.
Let
Lp (
(5.6)
Je0 fuorigin
0
be
yero
r be a
the
first
proper
1< p <
!R n)
lim
y~
C()
r.
Tp = product of Then
if
•
u(x, y) = f(x) , almost everywhere.
0
stricted convergence the results
chapter 17).
octant, thus
sub-cone of
For obvious reasons we refer
point all
). This is the setting for the
Marcinkiewicz and Zygmund.
Theorem D. Let half-planes.
by restricting
and
(5.6) as
of this section
to the convergence (5.5) as unre-
restricted convergence. (Up to may
this
be found in Zygmund's book,
- 302 -
E. M. Stein
We come now to the general situation. While analogues of such
results might be presumed to hold in
a rather wide context, i. e.
including all non-compact symmetric spaces, and generalized half-planes (Siegel domahs of type II) ,ption
we shall content ourselves with
the descri-
of a stage of intermediate generality. This situation is probably
already indicative of the general context, and in addition can be described without going too mains
far afield. Thus we shall limit ourselves to tube do-
Tr
which
r
mains whose basis
are domains of positivity, i. e. those tube do-
is
a homogeneous self
domains represent an important sub-Class
dual cone. These tube-
of the bounded symmetric do-
mains. Thus form the
r
if
is such
Poisson integral of
u
r
f
(x, y) =
n
a cone,
j 1R
with
P n
Rand respect to
r (t)
P n f E. L (JR) we
the
cone
f(x-t)dt
y
r
pr(t) is the Poisson kernel of the tube domain T (Stein, y G. vVeiss, and M. Weiss [5], and Koranyi [3]). u (x, y) is then
where
harmonic
with
,metric domain
respect T
the invariant operators of the bounded sym-
r
We may then convergence as
to
in
pose the problems of unrestricted and restricted
(5.5) and
(5;6).
The first fact is that unrestricted convergence does not seem to be appropriate in this general context because as is shown in for
every
p <
0()
there exists a tube domain of
this type,
that unrestricted almost everywhere convergence fails for
[5
J'
T T' ' so
L P • However
for restricted convergence the results are more positive. As N. Weiss
- 303 E. M. Stein
showed in his thesis [7
J,
restricted convergence holds for these tube
Lp ,p > 1,. an d even for the class
domains for
L(log L) . L1 .
The problem that remains is, therefore, the problem for
A clearer understanding of this problem , leading to a positive solution, will be discussed in the next section.
6. The
case. We begin by going back to the special case
octant
r
y. > 0, s = 1 , •.. n; 1. e. J be a proper sub-cone of
D,
sup Y€
~
integral, then
I u(x, y) I
(6. 1)
f
~
L 1( /Rn) and u(x, y)
as is shown in the proof of theorem (3.2'), 1. e.
satisfies an estimate analogous to
I u(x, y)/ >(\I}~ ~llf III .
m { sup
YE
in the
Tr = product of half-pplanes. Let
r. Then if
o is its Poisson
r
when
fa
It would therefore be of interest to deduce this fact as a conse-
quence of the same
kind of estimate for a maximal function of the type
(4. 1) , or some variant of it. Indeed, as is easy to see if with
f~
defined
as
in
(4.1) f"'(x)
~
A
sup YE
definitely not be how f
-If-
true that
sup
y~~
I u(x, y) I ,<
r:
f
~
0,
u(x, y) ,but
then it is
*"
A f (x) . We shall now descri-
be modified.
must
It is interesting to point out that the right variant
(4. 1) for
this type of problem has been introduced some time ago in another connection, in analogy with certain singular integrals. In fact let non-negative function homogeneous of degree zero in = r2
(A x ),
;..
>
0
•
which
the unit sphere. Assume that terms of
r2
consider
is therefore determined by r2
r2(x) be a
}R n, 1. e.
r2(x)
its values on
is integrable on the unit sphere. In
the maximal function
- 304 -
E.M.Stein
*rz
When rz rz
r: J I
sup
f (x) =
(6.2)
r>O
f(x-t) I nit) dt .
/tl ~ r
is constant, then
we get
to (4.1) . For general
we are dealing with a maximal function that has certain preferential
directions, those where rz is relatively large. Now the following fact was
f~
proved about
by Calderon and Zygmund
[1].
Theorem E. Suppose
f € L P( /Rn),
The idea of the proof
of this theorem is to use the one-di-
I<
p < 00
,
then
(6,3)
mensional L P result origin, and
(3.2)
in each
integrate over all
then
L 1 result cannot be obtained in (3.2') . involves
the notion of
the case of a norm
is not
weak-type
1, which
in distinction to
sup
u(x, y)
y~.r 0
frz(x)
r
C lR n
a proper sup Y €-
The
word
r
is
a classic«l homogeneous self-
. Then there exists a positive function
geneous of degree 0, and is
an
L 1 inequality
the relation between
-It
dual cone,
r
directions. However
sub-additive. We shall return to this point
Theorem 2. Suppose
. 0
these
emenating from the
this way, because the
nomentarily, but we deal first with and
direction
integrable over the unit sphere, so t hat if
sub-cone of
ro
I
rz , homo-
u(x, y)
r,
I~
A
then
f~(X)
"classical ll indicates that this theorem has been
verified (by direct computation)
in all
but the exceptional case. But
- 305 -
E. M. Stein
there
can be no doubt that it holds also in this case. We give here two examples Example 1 .
octant,
Then we can take
Example 2.
r
We see
£,
Y~ > Y~ +... +y! '
therefore
that
the
a similar
L
1- G.' £ > L 1
1
0
y1
>0
1
•
problem for Poisson inte-
problem, but for the maxi-
~
fll(X) , (6.2).
can
those arising
solve if for a wide variety in
Suppose uniformly of
L1
not solve the
thorem
If/X}}
weak-type
problem of
11
for
which
;t-
f ll for general include all
2.
The main technique
Suppose
j = 1, ... n , in
~X1 \2'Ix)...n , xn 1;-' where
Pd _{22 Ix I
Il(x) =/
While we do we
> 0,
r=n-dimensional circular cone, i.e.
may be reduced to
mal function
11,
Il(x) =
= { (y l' ... , y n) ,where
Then we can take
grals
y.
is
jC:1
the following. is a sequence
1, i. e.
m
a. are positive constants J
rf/x}
what
of
J
> 0( ~ can
positive functions,
1;
be
for each
said
j.
of
00
f(x} .
(6.4)
med
(and
Lj=l
?
a. f. (x) J J
If the class
of functions which are of weak-type 1 could be nor-
this is
the condition
would suffice
to
not the case), then imply that
a substitute result holds.
f
is
itself
of
weak
~ a. < 00 type
J 1. However
- 306 -
E. M. Stein ao
Lemma
1. Suppose
q
< 1,
and
Whith the aid of this lemma we can
Lj=1
(a.) J
q
~
1 .
prove
Theorem 3. Under the assumptions of theorem 2, lim y€
1";:
u(x, y) = f(x)
holds almost everywhere,
Y4>O
The proof of lemma will
be
given
1 as well as theorems
in detail elsewhere.•
1,2, and 3
- 307 -
REFERENCES
A. CALDERON and A. ZYGMUND, On slngula'." integrals, Amer. Math. , 78 (1956) , 289-309 • 2
R. E.EDWARDS and E. HEWITT, Point wise limits fer 8(;quences of convolution operator$, Acta Mathematica, 113 (1965) 181-213
3
A KORANYI, The Poisson integral for generaliztcd half-planes am:! bounded symmetric domains, Annals of Math. 82 (1965) pp. 332-350.
4
K. T. SMITH, A generalization of an inequality of Hardy and littlewood, Canad. J. Math. ~ (1956), 157-170 .
5
E. M. STEIN , G. WEISS, and M. WEISS, HP classes of holomorphic functions in tube domains, Proc. Nat. Acad. Sci. USA, ~ (1964), 1035-1039 .
6
N. WIENER, The ergodic theorem, Duke Math. J. 5(1939), 1-18 Selected papers of Norbert Wiener, The M. 1. T. Press. 1964, 412-429.
7
N. WEISS, Doctoral dissertation 1966, to appear Trans. Amer. Math Soc.
8
A. ZYGMUND, 'Trigonometric series' , Cambtidge, 1959.
