Annals of Mathematics
Compactification of Arithmetic Quotients of Bounded Symmetric Domains
Author(s): W. L. Baily, Jr. and A. Borel
Reviewed work(s):
Source: The Annals of Mathematics, Second Series, Vol. 84, No. 3 (Nov., 1966), pp. 442-528
Published by: Annals of Mathematics
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quotients
of arithmetic
Compactification
domains
of boundedsymmetric
By W. L. BAILY, JR. and A. BOREL
TABLE OF CONTENTS
Introduction
PART
I.
PART
II.
PART
III.
? 0. Notation and conventions.
THE COMPACTIFICATION V* AS A TOPOLOGICAL SPACE
? 1. The natural compactificationand Cayley transformsof a bounded
symmetricdomain.
? 2. Relative root systems.
? 3. Rational boundary components.
? 4. Fundamental sets and compactification.
AUITOMORPHIC FORMS
? 5. Poincare series.
? 6. Poincar-Eisenstein series.
? 7. Poincare-Eisensteinseries on bounded symmetricdomains.
? 8. The $-operator.
THE COMPACTIFICATION AS AN ANALYTIC SPACE
? 9. An analyticitycriterion.
*10. Analyticstructureand projectiveembeddingof the compactification.
Appendix
? 11. Connected componentsof automorphismgroups.
Introduction
domainX and an
This paperis chieflyconcernedwitha boundedsymmetric
ofX. Its main
groupI of automorphisms
defineddiscontinuous
arithmetically
of
the
V= X/P, in
space
V*
quotient
a
goals are to construct compactification
whichV is openand everywheredense,to showthat V* may be endowedwith
a structureof normalanalyticspace whichextendsthe naturalone on V, and
of V* onto a normally
to establish,using automorphicforms,an isomorphism
projectivevariety,whichmaps V ontoa Zariski-opensubset of the latter.
We now proceedto a synopsisof the contentsand methodsof this paper,
the followingassumptions,which
making,forconveniencein thisintroduction,
X
are no essential loss in generality:
K\GRis the quotientby a maximal
compactsubgroupK of the groupGR of real pointsof a connectedalgebraic
matricgroupdefinedover Q, simple over Q, and 1 is an arithmeticsubgroup
of G (i.e., P is commensurablewith the groupGz of integralmatricesin G,
* Partial support by N.S.F. grants GP-91 and GP-3903 for the first-namedauthor,
by N.S.F. grant GP-2403 for the second-namedauthor.
COMPACTIFICATION
443
see 3.1). Our programmaybe roughlydividedintothreeparts:
and propertiesof the compactification
V* of V as a toI. Construction
pologicalspace.
II. Study of certainautomorphicforms,and of their behaviorundera
4I-operator.
III. Analyticstructureon V*, projectiveembedding.
Part I is coveredin ?? 1-4. The firstparagraphdeals with the natural
of X; i.e., the closureD of the Harish-Chandrarealization
compactification
of X as a boundeddomainD [22]. We recallthat D - D is the unionoflocally
closed analytic subsets of the ambientvector space, which are themselves
(equivalentto) boundedsymmetric
domainsin a smallernumberofdimensions,
calledtheboundarycomponents
g= F}
ofD. The normalizerN(F) ={g G GI IFew
ofthe boundarycomponent
F is a maximalparabolicsubgroupofthetopological
identitycomponent
GRofGR,and conversely.It containsas a normalsubgroup
the centralizerZ(F) ={g e GI Ix *g = x(x e F)} of F. To F thereis associated
a(n essentially)canonicalunboundedrealizationSF ofX, and a complexanalytic
mappingarFofX ontoF, whosefibresare affine
subspacesoftheambientvector
for
space,and are the orbitsofZ(F)0. These results,due to Pyateckii-Shapiro
theclassical domains[30], were extendedto the generalcase by Koranyi-Wolf
[27]. In ? 1 we reviewthosefactswhichare neededlater,establishsomepropand sometechnicallemmasforlater use.
ertiesof functionaldeterminants,
Our case of interestis whenX/I' is not compact. This impliesthat G has
a non-trivialmaximalQ-splittorus, and a non-trivialsystemQ(Dof Q-roots
(see 2.1). Section2 is mainlydevotedto the studyof the naturalrestriction
mapfromR-rootsto Q-roots. ThiswillshownotablythatQ4' is ofoneofthetwo
typesoccurringforthe systemsof R-rootsof irreducibleboundedsymmetric
domains(2. 9).
F bymeans
Section3 introducesthe notionofrationalboundarycomponent
of two conditions:
(i) U(F)/( U(F) nF) is compact,if U(F) is the unipotentradicalofN(F),
on F.
(ii) 1(F) = (N(F) n 17)/(Z(F) n 17)is discontinuous
is
defined
over Q. The main
The condition(i) equivalentto N(F)c being
resultof ? 3 showsthat, in our case, this in fact implies(ii), or rathermore
preciselyimpliesthat 1(F) is of arithmetictype. The mapF > N(F)c induces
then a bijectionof the set of rational boundarycomponentsontothe set of
propermaximal parabolic Q-groups (3.7). If X is not the unit disc, then
dimc F ? dimc X - 2, (3.15).
Section4 is devotedto the constructionof V*, followingthe patternof
444
BAILY AND BOREL
Satake's paper[33]: the unionX* ofX and its rationalboundarycomponentsis
endowedwith a topology,definedby means of a suitable fundamentalset in
X, suchthateach g e GQn G1 operatescontinuously
onX* and suchthat X*/l,
suppliedwiththe quotienttopology,is a compactHausdorffspace. This is the
compactification
V* of V. It is the unionof V and ofthe quotients
sought-for
ofthe different
where
runs
througha set ofrepresentatives
=
FjF1(Fi)
Vi,
Fi
(4.9, 4.11). It is shown that every
F-orbitsof rationalboundarycomponents
e
{ Us} such that each Ua n V is conx V* has a basis of open neighborhoods
nected (4.15).
Sections5-8 are devotedto automorphicforms,and in particularto those
which are called here Poincare-Eisensteinseries (P-E series for short); they
generalizesimultaneously
Poincareseriesand Eisensteinseries. They are first
introducedin ? 6 in a generalsetting,suggestedby resultsof Harish-Chandra
and Godementon Poincare series and Eisensteinseries, proved or stated in
??5, 6. In ?7 we turnto the morespecial P-E serieswhichplaya centralrolein
our paper. (On the generalizedupper half-plane,they are differentfrom,
althoughrelatedto, the series introducedby Maass [28], underthe name of
Poincare series.) They are definedas follows: let F be a rationalboundary
componentand arF be thecanonicalprojectionof X onto F (see supra). A P-E
series adaptedto F, of weightm, is a seriesof the form
E(x) = E,/,0 (((x7))J(x,7)
where p is a polynomialon F, (in the coordinatesof the canonical bounded
in the unboundedrealization
realizationof F), J the functionaldeterminant
of X associatedto F, and F. a suitable subgroupof 1. The convergenceof
theseseriesfollowsfromthe resultsof ? 6. Theirbehaviorat rationalboundary
componentsis studied in ?? 7, 8, wherean operatorsimilarto the 1D-operator
of Maass is developed,at least forP-E series. The main idea is to prove the
existenceof normal(absolute) majorants of the above series in certainsufficiently
big sets, whichare partsof Siegel domains,so thatit becomespossible
to deal with such seriestermwisein such sets. This majorant is constructed
of G, and to discussthe behavior
by meansof a suitablerationalrepresentation
of GQand some
Bruhat
an
individual
use
the
of
decomposition
mainly
term,we
(7.6, 7.8). Our main result is that a
propertiesof weightsof representations
limit'l E as
P-E series (adaptedto F) has, in a suitable sense,a holomorphic
we approachany rationalboundarycomponentF'; the image of E under4 is
the collectionof the limitsq4yE; if dimF' ? dimF and F' X!Fe1,
by definition
'IF'Ethen
0; moreover,the image of (DF containsthe moduleof all Poincare
manyweights(8.5).
seriesof F withrespectto P(F) forinfinitely
COMPACTIFICATION
445
Part III consistsof ?? 9, 10. In the latter,we endowV* with the sheaf
to the Vi's are analytic.
CCof germsof continuousfunctionswhose restrictions
theoremof analyticstructure,similarin spirit
Section9 provesa prolongation
to thoseof [2, 18, 35], which,combinedwiththeresultsof ? 8 on P-E series,and
knownfacts on Poincareseries[19],allowsus to provethat (V*, d) is an irreduciblenormalanalyticspace (10.4). The existenceofa projectiveembedding
of V* by means of automorphicforms,whose image is projectivelynormal,
followsthenin the usual manner(10.11).
Let dimG > 3. Then we have dimc(V* - V) ? dimcV - 2. Standard
function,
facts about normalspaces implythereforethat every1-automorphic
functionon
functionon V, extendsto a meromorphic
i.e., everymeromorphic
an
function
is
field,
functions
algebraic
V*; hencethe fieldof F-automorphic
each elementof which is the quotientof two automorphicformsof the same
weight(10.12). Also, an extensiontheoremof Serre[36] showsthenthatevery
cross section
automorphicformof the classical typeextendsto a holomorphic
of an algebraic coherentsheaf on V* (10.14); this generalizes Koecher's
principle.
Finally,an appendix(? 11) containssome remarkson the full groups of
of X.
isometriesand of automorphisms
The mainresultsof thispaperwereannouncedin [7],and are also described
with sketches of
in [5]. Similar theoremshave been stated independently,
some proofs,by Pyateckii-Shapiro[31]. Earlier special cases may be found
notablyin [2, 3, 30, 35]. These are mostlyconnectedwithfamiliesof abelian
gives a concreterealizaof the compactification
varieties,and the construction
tion,in manycases, of the varietyof moduliof such varieties. In this paper,
fora prowe leave untouchedthe questionof the minimalfieldof definition
jective model of V*, and of the possible connectionof V* with moduliof
algebraicstructures. For the knownresultsin thatdirection,we referto [37]
whereotherreferencesto relatedworkare also given.
0. Notation and conventions
In this paragraph,we collectsome notationto be used frequentlyin this
paperwithoutfurtherreference.
0.1. As is usual, Z, Q. R, and C denoterespectivelythe ringof integers,
and the fieldsof rational,real, and complexnumbers.If A is a commutative
in A
ring,GL (n, A) or GL,4Ais the groupof n x n matriceswithcoefficients
is a unitof A, and SL (n, A) or SLnA, thegroupof elements
whosedeterminant
of determinant
one in GLnA. The groupof unitsof a ringB is denotedbyB*.
subsetof G, thenN(M) or NG(M)
0.2. If G is a group,and Ma non-empty
446
BAILY AND BOREL
(resp. Z(M) or ZG(M)) is the normalizer(resp. centralizer)of M in G. Thus
GI
N(M) = {g e GgM-g-1
Z(M) = {geGGIgm
= M},
g-1= m(mCM)}=
nfmeMN(m)X
h - g *h g-(h e G) is denotedInt g. Often,we write
The innerautomorphism
9MforIntg(M), andMgforIntg-1(M).
0.3. As regardsalgebraic groups,we follow in general the notationof
[14]. However,our universal field is C, and so, in this paperalgebraicgroup
stands for complex linear algebraic group. An algebraic group here may
withan algebraic
alwaysbe (and willtacitlybe wheneverconvenient)identified
k
and
over
k-groupwill be
defined
group
subgroupof GL(n, C). Algebraic
used synonymously.For a subringA of C, we put GA - G n GL(n, A). The
withGc.
algebraicgroupG will be identified
will
The Lie algebra of an algebraicgroup,or of a Lie group, G, H,
G
usually be denotedby the correspondinglower case Germanletter. If is
algebraic,definedover k, theng = gk0 C, wheregkis a uniquelydetermined
Lie algebra over k. If k' is an overfieldof k, thengk gk Ok k'.
In both the algebraic and Lie group cases, Ad denotesthe adjoint repof Int g at e.
resentationof G into a, where Ad g(g e G) is the differential
The restrictionof Ad g to a subspaceb is denotedAd, g.
...
0.4. Let G be a k-group. Unless otherwisesaid, a character of G is a
rationalcharacter,i.e., a morphismof algebraic groupsof G into GL(1, C).
The charactersof G forma finitelygeneratedcommutativegroup,denoted
X(G), whichis freeif G is connected. The subgroupof elementsof X(G) which
are definedover k is denotedby X(G)k.
The value of a e X(G) on g e G will be writtena(g), or moreoften ga. In
the lattercase, it is impliedthat the groupoperationin X(G) is writtenadditively,and that usually no notationaldistinctionis made between a and its
whichis a linearformon g. In particular,we have, by convention,
differential,
ga = exp a(X) (X e g, g = exp X).
0.5. An algebraicgroupG is a torus if it is isomorphicto a productof
groupsC*; a torussplits overk, or is k-trivial,if it is moreoverdefinedoverk
over k to a productof groupsC*.
and isomorphic
Let G be a k-group. Its radical R(G) (resp. unipotentradical RU(G),
resp. split radical) is the greatestconnectednormalsolvablesubgroupof G
(resp. normalunipotentsubgroupof G, resp. the normalsubgroupgenerated
by R.(G) and the k-splittoriof R(G)). G is reductive(resp. semi-simple)if
Ru(G) = {e} (resp. R(G) = {e}). G is simple over k (resp. almost simple
447
COMPACTIFICATION
over k) if it has no (resp.connected)propernormalk-subgroup.G is an almost
directproductof normalsubgroupsG, if it is the quotient by a finitegroup
of the productof the Ge's.
0.6. The identitycomponentof a topologicalgroupH is denotedby H'.
We recallthat if G is algebraic,thenGc is connectedas a topologicalgroupif
and onlyif it is connectedas an algebraicgroup(i.e. theunderlyingalgebraic
varietyis irreducible).However,if G is connected,definedover R, the group
GR, viewedas a real Lie groupmay have morethanone connectedcomponent,
but will always have onlyfinitelymanyconnectedcomponents.
0.7. Let A be a set. A functionon A, withvalues in a locally compact
space, is boundedif its range is relativelycompact. A functionwithvalues
in the space R+ of strictlypositivereal numbersis multiplicativelybounded
if thereare strictlypositiveconstantsc, c' such that c ? f(a) ? c'(a e A).
Let u, v be functionson A withvalues in the set of positivereal numbers.
We writeu -< v if thereexists a strictlypositiveconstantc such that
u(a) ? c *v(a),
and u
>-
v (resp. u - v) if v -< u (resp. u
-<
(a e A)
v and v -< u).
I. THE COMPACTIFICATION V* AS A TOPOLOGICAL SPACE
1. Natural compactification and Cayley transforms
of a bounded symmetric domain.
1.1. The following notation will be used in this section.
G is a connectedreductivealgebraicgroupdefinedover R which has no
non-trivialcharacterdefinedover R. Thus GI is a connectedLie groupwith
reductiveLie algebra and compactcenter.' We denote by g the Lie algebra
of GR.
K is a maximalcompactsubgroupof GI. The symmetric
space X K\GI
It
is
then
an
is assumedto carry invariantcomplexstructure.
equivalent to
domain[22,24], and is hermitiansymmetric.
a boundedsymmetric
of the Lie algebra f of K in g withrespect
p is theorthogonalcomplement
of g. Since X
to the Killingform,henceg = f + p is a Cartan decomposition
we have the directsum decomposition
is hermitiansymmetric,
9c = fc (D P+ (D P-
(
(P+ffl=
PPc),
wherep' is a commutativesubalgebranormalizedby fc.
1 Essentially, it would sufficeto consider the case where G is semi-simple,without
compact factors. However, it is more convenientfor future referencesin this paper to
start from a slightlymore general assumption.
448
BAILY AND BOREL
b denotesa Cartansubalgebraof f, and thereforealso of g, in viewof our
assumptionon X, and ? = 1'(tbc,
gc) is the set of roots of gc withrespectto
kc. We let E,(y e I ) be rootvectors,and H, be elementsof tc verifying
[El., E _,] = Hy,9
v(Hv) = 2(vqpc)*(pc,It)`
(tc,v C D),
where (, ) is the restrictionof the Killing formto bc, and such that the
complexconjugationof gc with respectto g permutesE, and E__whenever
E, C pt. Let wi = ice J I E C p+}. Theelements
E,(y eCw) forma basis of
pi,andtheelements
Xv,= Ev,+ K_IY,
Yv,= i(Ev,- RS)
(ti 7r+
forma basis (over R) of p.
Two independent
rootsy,v are said to be stronglyorthogonalif neither
4a+ v nor y- v are roots. We fixonce forall a maximalset (4a1,*.., p)
, of
stronglyorthogonalrootsin w+,as in [24], and write Hi, E, E_i_Xi_Yi for
Hsiq Eyi,q &1,9 Xsi,q YAsi
1.2. The systemof R-roots. We let a be the subalgebraof p spannedby
Xt, and R(D = R(I(a, g), the set of rootsof g withrespectto a, to be
called the R-rootsof g. The algebra a is a maximal commutativesubalgebra
of p, and is maximalamongthe subalgebrasof g whichcan be diagonalizedin
the adjoint representation.g is the directsum of the centralizer3(a) of a and
of the rootspaces
X1,..
.-
{Xe g I[a, x]
-
a(a).x, ae a}
(ae RI)
e
AssumeX to be irreducible.ThenR(' is knownto be of one of two types,
to be denotedby Ct and BCt. If (vni)are coordinateswith respectto the basis
((1/2)Xi),thenCt consistsof theroots?Q(v?+a)/2,(1 ? i <j? t), ?'y(1 ? i ? t)
and BCt is the unionof Ct and of the set of elements ?vi/2 (1 ? iX t). In
both cases we always take as orderingthe lexicographicorderingdefinedby
the basis (Xi). The set RAof simpleR-rootsconsiststhenof
ti = (-hi-Ji+,)/2
(I<(1 i < t),
and of at = 'Yt (resp. art y
-t/2) if R4:) is of typeCt (resp. BCt).
The numberingof the simpleR-rootsthus definedwill be referredto as
the canonical numbering.
1.3. Maximal parabolic subgroups. A parabolic subgroupof G' is the
of G' withan algebraicsubgroupP of G whichis parabolic,(i.e.,
intersection
such that G/P is a projectivevariety)and definedover R. The descriptionof
the parabolicsubgroupsof an algebraic group is recalled,in a more general
setting, in 2.2. Here we introducethe minimalones and the maximalones,
449
COMPACTIFICATION
whichwill play a fundamentalrole in this paper.
Let u be the sum of the ga (a > 0), and A = exp a, N exp v. These
are closed subgroups,with N unipotent,normalizedby A, and A. N is maximal among the connectedsubgroupsof G? which can be put in triangular
productP
formover R. The normalizerP of N is equal to the semi-direct
Z(A) *N and Z(A) = M x A with M = Z(A) n K. The groupP is generated
by P0 and a commutative subgroup of type (2, 2,
...,
2) of M, which can be
describedas K n exp ida, as followsfrom[14, 14.4]. Every minimalparabolic
subgroupof GI is conjugateto P.
Assume, for convenience,X to be irreducible.We let ab (1 ? b ? t) be
the one-dimensional
subspaceon whichall simpleR-rootsbut ab are zero,and
Ab= expab. The space Cb is spannedby X1 + *** + Xb. We let Pb be the
subgroupgeneratedby Z(Ab) and N, and Vb be its unipotentradical. The
group Pb is the semi-direct
productof Vb by Z(Ab). The Lie algebrabob is the
sum of the rootspaces ga wherea is >0 and not zero on ab. Therefore,a runs
throughthe roots
b< j
i + yj)/2, (I :!: i :!-j :~ b),
t),
(-7i+- -zj)/29
(1I<i
and the rootsati/2(1 ? i ? b) in the case BCt. Let lb (resp. f') be the sum of
the subspacesga+ [ga, ga], wherea runsthroughthe R-rootswhichare linear
of ab+l,
These are two simple ideals
combinations
at (resp. a1, ... , ab-l).
of 3(ab), clearly normalizedby the Lie algebra m of M, and 3(ab) is the direct
sum of fb, lb, ab and of an ideal Mbof nm.The groupZ(Ab) is generatedby the
analytic groups Lb, Lb, Ab, with Lie algebras fb, fb,Cb, and by M. Let 3bl + 3b. Denote by Zb the
ab e eb
It is an ideal of Pb such that Pb lb
(Db.
MbED
analytic subgroupof GR withLie algebra 3b. We let Zb be the inverseimage
in Pb of the centralizerof (Pb/Zb)0 in Pb/Zb. It is a closednormalsubgroupof
Pb, with Lie algebra Jb, whose intersectionwith Lb is the centerof Lb. It
containsevery normalsubgroupof Pb withLie algebra b: in fact,the image
in Pb/Z? of such a subgroupis a finitenormalsubgroup,and thereforecentralizes(Pb/Zb)0. In particular,Zb contains(Zb)c n GI, whence
.
(1)
.
,
Zb - (Zb)c
i
nGIR
where(Zb)c and (Zb)c denotethe smallestalgebraicsubgroupsof G containing
Zb and Z' respectively.
By the generalconjugacytheoremson parabolicgroups(2.2 below),every
maximalproperparabolicsubgroupof GI is conjugate to one and onlyone of
of Pb
the groups Pb. It will sometimesbe convenientto extendthe definition
0, by puttingP, = LO= GI, I' - 0; thenj, = b, = 0 and Z, is the center
to b O
of GI.
BAILY AND BOREL
450
1.4. The natural compactification.Let P+ = exp p+and Kc be the analyticsubgroupof Gc with Lie algebra tc. These are closedsubgroups,and the
productKc *P+ is a parabolicsubgroupof Gc. The map (x, k, y) >
semi-direct
e k ey is a biregularmap of p- x Kc x p+ onto a Zariski-opensubset ? =
of Gc, which contains GI [24]. An elementg e 12 will often be
P-KcP+
written
(go Kc; g e Pi),
and the map g H-+log g+ of n onto p+ will be denotedC. It is known[22], [24]
of X
K\GO onto 4(G) = D, and that D is
that C induces an isomorphism
a boundeddomain in p+. This is the Harish-Chandrarealizationof X as a
boundeddomain. Its closure D is thereforecompact,and will be called the
natural compactificationof X. The action of GI on D is definedby right
translations;i.e., by
g = 9g _ go,9+
-
(1)
g eGOR)
(peD9
~~~~~p-g=(ellg)
and is knownto extendto a continuousactiononD. Then(1) is truewithp C D.
1.5. Boundary components(see [27], [29], [30]). (i) AssumefirstX to be
irreducible.We use the notationof 1.3. We have the direct sum decomposition
fb,C
=
fb,C (1
b+
(D
Pb
(Pb
=
b,c
n P ),
and the restrictionof C to
the space Xb Kb\Lbis hermitiansymmetric,
yieldsthe Harish-ChandrarealizationDb of Xb as a boundeddomain.
+ Eb) (1 b t), and put oO= o. Then
Letob = -(E1 +
D
=
UO:b:t Ob
Lb
GR
theorbitFb of Ob under
4(g). Therefore,
affine
subspaceof p+. The transforms
Lb is just Ob + Db, and is containedin an
of the Fb's by elementsof G are the boundarycomponentsof D. We allow
here b to be equal to zero, and view D itselfas a boundarycomponent(someif g e Lb, then0b
?b
Moreover,
=0b
+
times called the improper boundary component of D).
If X is not irreducible,then it is a productof irreduciblehermitiansymmetric spaces Xi correspondingto the differentsemi-simple,simple, noncompactideals of g,D is the productof the Harish-ChandrarealizationsDi
of the Xi, and D the productof the Di. A boundarycomponentis a product
of boundarycomponentsof the differentfactors. The Fb's or, if X is not
to the different
irreducible,the productsof componentsFb's corresponding
irreducible factors of X, are the standard boundary components.
is hereditary:if F is a boundarycomponentthen
The above construction
COMPACTIFICATION
451
ofF, and
withthe naturalcompactification
its closureF in D maybe identified
ofX. Morespecifically,
are also boundarycomponents
its boundarycomponents
=
F
0b
Db
and
X
and
then
+
the standardboundary
Fb,
Fb=
if is irreducible
withthe F,'s (b < c ! t); in fact F, would
componentsof Db maybe identified
have c - b as index in the canonicalnumberingforXb. The groupsL, and
P, f Lb are in the same relationshipto Lb as Lb and Pb are to GI. This is clear
fromthe construction.
We referto [30] forvarious moregeometricdefinitionsof the boundary
and to [29] for a proofof their
componentsin the natural compactification
equivalence.
For everyboundarycomponentF, we put
N(F)
-F},
{geGo Fg
Z(F) = {geG I f gf(feF)},
G(F) = N(F)/Z(F),
and let U(F) be the unipotentradicalof N(F). The group N(F) is the normalizer, and Z(F) the centralizer,of F. If X is irreducible,we have, in the
notationof 1.3,
(1)
N(Fb) - Pb,
U(Fb) = Vb.
Moreover
Z(Fb) = Zb.
(2)
In fact, Z(Fb) is a normalsubgroupof Pb withLie algebra3b, henceZ(Fb) C( Zb
by 1.3; on the otherhand,the image in G(Fb) of an elementz e Zb centralizes
the image L"' of Lb, and thereforethe maximal compactsubgroupsof L'',
henceit acts triviallyon Fb, and Zb ci Z(Fb).
Returningto the general case, we see, by applying1.3 (1) and 1.5 (2) to
each irreduciblefactorof X, that Z(F) is the intersectionof GR with an Rby 11.2, each elementof N(F) inducesa comsubgroupof G. Furthermore,
of F, hence (11.6), G(F) is connected,with
plex analytic homeomorphism
center reduced to {e}; equivalently,if X is irreducible,we have N(Fb)=
Lb *Zb-
(ii) If F and F' are two boundarycomponentssuch that F' ci F, then
there exists g e G' such that F. g and F'. g are bothstandardboundarycomponents. To see this, we may assume X to be irreducible.Let then b,c be
the indices such that F ci Fb GI F' ci F, GI and let u e G' be such that
F.u
Fb. Then F'. u and F, are both boundarycomponentsof Fb, of the
same dimension.Consequently,there exists v e L(Fb) such that F' . u v is
standard. F'. u v is then equal to F,; hence,g =u v verifiesour condition.
.
*
.
BAILY AND BOREL
452
(iii) If X is irreducible,and dimcX > 2, then dimcX ? dimcF + 2 for
everyproper boundarycomponentF of X.
To see this, we may assume that F= Fb (1 b t). If b-t, Fb is a
point,and there is nothingto prove. So assume b $ t. Then t > 2, and n
containsat least threerootspaces ga, whose sum intersectsu nfTb onlyat the
to a = (Yib ? ?t)/2, Yb, hence,
origin,namelythose corresponding
dimu - dim(Tb n u) _ 3.
On theotherhanddimRX -dim a + dimui,dimRFb dim(a n Tb)
and dima - dim(a n Tb)
1, whenceour assertion.
+
dim(Ib f u),
1.6. The Cayley transformsof X. The space X also admitscertainunboundedrealizations,introducedby Pyateckii-Shapiro[40] in the classical
cases under the name of Siegel domainsof typeI, II or III, and discussedin
generalby Koranyiand Wolf [27]. In this and the nextsection,we summarize
onlythe resultswhichare used in the sequel. We assume again X to be irreducible.
The Cayley transformCbis, by definition,
Cb =
H1,iib
b
exp (z/4)*(K-i - E), (1
t);
cO
e.
It verifies
( 1)
Ad Cb(Hi) = Xi,
Ad Cb(Xi)
=
-Hi,
(2)
Adcb(Hi) = Hi,
Ad Cb(Xi)
=
Xi
(I :!:
X
i
<
b)
(b < i < t)
into its inverseby the complexconjugationof Gc with
and is transformed
respectto GR. Moreover
(3)
( 4)
We put then Sb
Cb*g _gCb,
G
=
C(G
Cb),
cb
CP
(9
.
-KcP+
and let G act on
lb)
Sb
by
C(e-g. Cb 1. 9 .Cb)
( 5)
The map g H-> g Cb inducesthenan isomorphismb of X onto Sb; by definition
so = D, and So is just the boundedrealization. Often,we shall denotealso
by o the fixedpointof K in Sb.
In the next proposition,
we denoteby qb the subspace of p+ spannedby
the vectors E,t,(ae r+, a(Hi) # 0 for at least one i ! b). Thus p+ P+
qb1.7. PROPOSITION. We keep theprecedingnotation. We have Cb Zb * C
Kc*P+, and the action of Zb (resp. Vb) on Sb extends to an action of Zb on
lPI by means of affinetransformations(resp. affinetransformationswith
unipotentlinear homogeneousparts) which leave qb stable and induce the
e
453
COMPACTIFICATION
identityon p+/q,. The projection ab of p+ onto p+ with kernel qb maps Sb
ontoDb, and its fibresin Sb are the orbitsof Zb?(O _ b < t). Its restriction
to Sb commuteswith N(Fb).
This is containedin the morepreciseresultsof [27, ? 7]. Let z C Zb. We
have then z' = cb *z*cb = ZooZ'+ (z, e Kc, z4 e P+). The actionof z on Sb or p+
is thereforegiven by
( 1)
s *z= Ad z1(s) + log Z4
(sep+).
Since p+ is commutative,
we can replacez4by z', whence
s *z
( 1')
= Ad Cb
*z
*cb(s) + log (Cb
*z*Cb)+
,
(s C P , Z C Zb) -
If g e Lb, then it commuteswithCb, therefore1.6 (5) becomes
(2)
S*g = C(es.g)
(s C Sb, g e Lb)
-
In particular
(3)
(S CSb, g e Kb) .
g = Ad g 1(s)
REMARK. Let F be a boundarycomponent,and ge Go be such that
Fag = Fb. Then x X-> ab(xg).gis a holomorphic
map of X ontoF. If g' is
such that Fu g' = Fb, then g' = g*n (n e N(Fb)); since translation by n commutes with 6b, we get ab(xg).g= ab(xg')g'-1
(x C X). We have thus defineda canonicalholomorphic
projectionof X ontoF, equivariantwithrespect
to N(F), to be denoteda,. If F' ci F, thenwe have a factorization
s
UF'
=
UF'F
? aF
Y
whereaFF is the canonicalprojectionof F on its boundarycomponentF'. In
fact,thereexists by 1.5 an elementg e Go such that Fag Fb and F'.g = F,
(b _ c), and it is clear fromProposition1.7 that ac, c,b 0 b where6c,b is the
canonicalprojectionof Fb ontothe standardboundarycomponentFc.
The remark extends obviouslyto non-irreducible
bounded symmetric
domains.
1.8. Automorphyfactors,functional determinants. Let M be a comof M, and Q a complexLie group.
plex manifold,H a groupof automorphisms
We recall that a (holomorphic)
automorphyfactorforH on M, withvalues in
in x e M,
Q, is a map A:M x H- Q which,for fixedh e H, is holomorphic
and whichverifiesthe identity
(x e M; h, h e H),
(1)
,5(x,huh') = ,5(x,h).4a(x h, h')
to be referredto as the cocycleformula; it implies
(2)
a(x,h.h'.h") = a(x, h).*a(x.h, h').*a(x*h*h',h")
(x e M; h, h', h" C H).
BAILY AND BOREL
454
It follows immediatelyfrom (1) that the set R of elements h e H for which
a(x, h) = p(h) is independentof x is a subgroup, and that
,5(x, he-r) = ,5(x, h) p(r)
(3)
(Xe M; he H re R).
If M is a domain in Cn, then the jacobian Jac (x, h) which associates to
h e H and x e M the differentialof h at x, is an automorphy factor with
values in GL(n, C), and J(x, h) = det Jac (x, h) is an automorphyfactor with
values in C*.
X be an irreducible bounded symmetricdomain. We shall denote
Let M
by Jacb(x, g) the jacobian of g e G' at x e Sb, in the unbounded realization asand by jb(X, g) the functionaldesociatedto Fb, by Jb(x, g) its determinant,
terminant of g E Pb at x e Db (0 < b < t). Our next aim is to obtain some
informationon Jb(x, g) when g e Pb, which will be used in studying PoincareEisenstein series.
It is immediate that pb(X, g) = (excbg c-1)0 (x e Sb, g e GR)is a holomorphic automorphy factor, with values in Kc. It is called the canonical automorphy factor for the unbounded realization Sb. The automorphy factors
usually considered in the theory of holomorphicautomorphic formsare of the
form p(ab(X, g)) where p: Kc H- GL(m, C) is a holomorphicrepresentation. The
following lemma asserts that Jacb is of this type. It is well-known for the
bounded realization; the proof is essentially the same in the general case, and
is included for the sake of completeness.
1.9. LEMMA. We keep the notation of 1.8, and identify the tangent
space to a point x C p+ with p+ by translation. Then
Jacb
(x,
g) = Ad? ho
(x E
Sb,
g E GR;h
-(ex
Cb *g * C))
Let Xe P+. Then x + X goes under the differentialdg of the automorx g; we have to prove that
phism of Sb definedby g onto an element Y + f
(1)
Y=Adho1(X).
By definition
c g*c-'.
Write g' forCb
(2)
xfg=
C(ex g') = log(exg')+=
logh+,
and
Y + x g d= <d{(etx.
ex.g)
|J*
Clearly
etx*
ex.*g = ho.et*ho*.h?
But the bracket relations
(U = Ad h-'(X))
455
COMPACTIFICATION
X, modulo c E
implyreadilythat if p e p- and X e p+, thenAd expp(X)
whence
therefore
write
Z
U
=
X
+
(Ze
p-),
We may
fc(D
(X' = Ad ho1(X),Z' = Ad ho1(Z))
etxXh =
et(x'+z') h+
hhoWe have then
d (etx-h)
dt
.
=h.ho.(X'
t=O
+ Z').h+.
Since Z' also belongsto fc + p-, the image of the right-handside underd; is
X' + log h?, which,in view of (2), provesour contention.
1.10. LEMMA. Let fb (0 ? b ? t) be the sum of roots p e w+ such that
e
is a strictlypositive integer independentof i
E, pj. Then mb -/b(Hi)
(b < i _ t), and mb> m, if 0 _ b < c ? t.
(1.5),
In viewofthe "hereditary"characterofthe naturalcompactification
it is enoughto provethis when b = 0, G = Lb.
Let at = Ad cT-1(ac).It is the subalgebra of t spanned by the vectors
Hi (1 <i < t). We denote the coordinateswith respectto the basis (Hi/2)
also by vi. We choosean orderingon 1?verifyingthe followingconditions:
to at of the elementsof
The elementsof w+ are positive,the restrictions
w+ are the linearforms(yi + yj)/2 (1 < i _ j _ t), and also the formsrY/2
in
differences
(Y
the case BCt; the positive roots of fc restrictto the
yj)/2
(1 ? i < j _ t), and also to vyj2in the case BCt.
This is always possible[22, ? 6]. Let A = {Iv, ..., 9 } be the corresponding
set of simpleroots. It is knownthat we may assume 0 {=1, *.., v-4 to be
the set of simplerootsof fc,and that the elementsof w+ are the rootswhich
are congruentto v, moduloa linearcombinationof elementsin 0, [22]. Moreover, since fcnormalizesp+, its Weylgrouppermutesthe elementsof w+ and
leaves ,80invariant. In particular,,80is leftfixedby the fundamentalreflectionsr, (v e 0), whence
( 1 )
(bow9Vi) = 0
(1 _ i <
1) .
The sum of two elementsin w+ is nevera root,hence
(2)
(4,)
_O
(,e
7E+) .
We have therefore
a+)
for) > ?
(ME
(yl, M)i= (aie
(3u) oH
(80)
and each i; therefore,
But >(Hi) -=2(vqvij) (vi,-yi)-lis an integerforeveryv C AD,
456
BAILY AND BOREL
The relativeWeylgroupof Ad ct (g) withrespectto Ad ct1(a)containsthe
permutations
of the yi. But everysuch transformation
is inducedby an elementof the Weylgroupof gc withrespectto tic(see e.g. [14, 5.5]). It follows
thenthat (yi,yi)is independent
of i, whenceour firstassertion.
The difference
m0- m, (c > 1) is the sum of the numbers a(Ht)where,
runsthroughtheelementsofw+suchthatE~,X p+; thesenumbersare all > 0 by
the above. But thereis at least one such pa,forinstanceonewhichrestrictsto
forwhich4a(Ht)# 0, whichends the proof.
(O1+ yt)/2,
1.11. PROPOSITION. Let X be irreducible. Let Jb be the functional determinantfunctionfor GOacting on Sb, and jb thefunctionaldeterminant
function for Lb acting on Db. Then
(i) The function Jb(x, g) is constant along the fibres of the projection
Ub: Sb
-
Db
of 1.7 if g C Pb, is independentof x if g C Zb, and is equal to one
if g is a unipotentelementof Zb. The restriction72bof Jb to Zb is a rational
character.
(ii) If g e Lb, we have Jb(X, g)mb = ib(Ub(X), g)mo, with MO, mb as in 1.10.
PROOF OF (i). If g e Zb (resp. g e Zb and is unipotent),then g acts on Sb
by means of an affinetransformation
(resp. withunipotentlinearpart) in P+;
of x (resp. is equal to one); then 72b
therefore,Jb(X, g) = b(g) is independent
is a rationalcharacterby 1.7 (1').
Writeg = l *u (l C Lb, u C Zb), and let z e ZO. Using the cocycleformula,
we have
l
Jb(x.z,
Jb(X
*Z.
u)
=
Jb(X*Z,
1 ZU)
=
Jb(X,
=
l).*b(U)
Jb(X,
* )b(Z)
0') *b(Z')
Z*1)*Jb(X,
Z) 1*
b(U)
,
1(7b(U)
where z' = -1 z. 1. But Zbo is the semi-directproductof Vb by a reductive
groupwhichcentralizesLb (see 1.3); therefore,72b(Z) = 7b(z) and
.
Jb(X
* Z, 1 U)
=
Jb(x,
1 U)
-
Since the fibresof Ub are the orbits of ZO, this ends the proofof (i).
PROOF OF (ii). For everyelementg e Kb,c, let us put
T(g)
= det
(Ad,+ g-1)
ijA(g) =
det (Adp+ g-1).
We want to prove
(4)
T(g)mb
= *(g)mO
(g
e
Kbc)
Assume first that g = exp (Xb+lHb+l
+ ***+ XtHt). In this case, T(g) (resp.
A(g)) is the product of the numbers exp ,(- log g) where , runs through the
roots, such that E,, c P+ (resp. E,, cfip+). Using 1.10, we get
(5)
T(g) = llb<it
exp -Xi*mo,
A(g)
=
<ib<!gtexp -Xi*mb
457
COMPACTIFICATION
-whichprovesour contentionin this case. It is also clear from(5) that P and
* are not identicallyequal to one on the subgroupjust considered.The group
Kb,c is generatedby its derivedgroup,on whichbothP and i are equal to
one, and by its one-dimensional
center. It is thereforealso generatedby its
derivedgroup and the groupof elementsconsideredin (5), whichproves(4).
Anyelementg e Lb commuteswith the Cayley transformcb; therefore,
we have by 1.9, appliedto Lb operatingon Sb and on Db:
(6 )
Jb(X
g) = T(g) ,
(7)
Jb(o,
g) = T(g0) ,
g) = A(g) ,
ib(Y
ib(Ob,
Given x C Sb, there exists 1 e
Lb
g) = *(g0)
such that
(x
Sb; y e
Db;
ge
Kb)
(g e Lb)
,
=
ab(X)
Ob 1
l.
The points x and
of Ub, henceJb(x, g)= Jb(ob l, g) by (i), and the
l belong to the same fibre
latterfunctionaldeterminant
has to be comparedwithib(Ob 1, g). The desired
relationshipthenfollowsfromthe cocycleformulaand what has alreadybeen
proved.
Ob
1.12. PROPOSITION. Let a
Then
a) =I1i
Jb(o,
Write a
=
- umv (u
=
exp (X1Xl+ *
liblilt
b e
+ XX,) be an element of A.
(coshXi)-m?o
exp (X1Xl+ ***+XbXb); v = exp (Xb+1Xb+l.+ **
tXt)).
We have thenu e Zb, V C Lb, and therefore,by 1.11,
Jb(o, a)
Since Cb Th'Cb
givenby
=
= Jb(o, u) Jb(o, v) .
u' = exp (X1Hl+ *
+ XbHb) e Kc, the action of u on Sb is
s *u = Ad u'(s),
(s e Sb)
hence
L(
1
)
Jb(X,
U) =
T(Cb-*'*Cb)
exi mo
l1=1<i!b
On the otherhand, a standardcomputationon the three-dimensional
simple
v0of v is
group(see e.g. [24, p. 316]) showsthat the Kc-component
v0= lij-t exp log coshXi-Hi.
and our assertionnow followsfrom1.11 (5), (6).
1.13. COROLLARY. Let hJ(X)= exp X(X1+
Then Jc(o,he(X))=
hc(X)) if c _ b, and
monotonically to zero as X ->if c > b.
Jb(o,
* + XJ) (1 < c
Jb(o,
By 1.11, we have
Jc(o,
h,(X)) = exp -X Xmo0c,
_t
X e R).
hJ(X)) Jc(o h,(X))-l tends
BAILY AND BOREL
458
and
Jb(o, h,(X)) =
(c _ b),
exp -X.m,.c ,
Jb(o, h,(X)) = exp -X *m *b(cosh X)-mo(C-b)
(c > b)
whenceour assertion.
1.14. Remark on the Bergman kernel function. Let Kb(z, w) be the
Bergmankernelfunctionin Sb. We have therefore
(z, We Sb; ge G)
(z, We Sb; k e K) .
Kb(Z, w) I Jb(Z, g) HJb(W, g) ,
Kb(z.k, w*k) = Kb(z, w)
Kb(z g, w g)
=
Since G = K. A. K and o is fixedunderK, thisshowsthatKb(z, z) is completely
determined
by Kb(o-a, o-a), (a e A), whichis givenby
Kb(o a, o a) = Kb(o, o)
*
I Jb(o,
a)
I2 .
We maytheninsertthe expressionof Jb(o, a) givenby 1.12; the formulathus
obtainedin the two extremecases b = 0, Sb = D, and b = t have been given,
in a slightlydifferentform,and with the value of the constantKb(o, o), by
Bott-Koranyi[27,5.7] and Koranyi[27, 5.5] respectively.
of Ad g in bb (cf. 1.3)
Our next aim is to relateJb(X, g) to the determinant
wheng e Zb. For this, we need the followinglemma:
1.15. LEMMA. Let X be irreducible. Let u (resp. v) be themultiplicity
be the
the
of
R-roots(Yi + yj)/2(i j) (resp. yi/2in the case BCJ). Let
restriction of ab to ab (1 ? b ? t). Then the weights of ab in g, for the
adjoint representation are 0, Vb, and possibly +2*.Vb. Let Pb (resp. qb) be
themultiplicity of Vb (resp. 2.Vb):
t
= o.
u ()qb
(i ) if RT? iS of typeC, and b = t, thenPb =
(ii) if R is of typeCt and b#t,thenPb= 2*u*b*(t- b),qb= b + u (2).
(iii) ifR(i is of typeBCt, thenPb =v.b + 2.u.b.(t-b),
qb= b +
The R-rootsare linearcombinationsof the simpleones with coefficients
0, +1, ?2. Since ab annihilatesall the simpleR-rootsexceptab, this proves
the firstassertion.
We have 2b = jt in the case (i), and 2b = Yb/2 in the othercases. In the
of the R-roots-y(i < t), whichare
case (i), Pb is the sum of the multiplicities
all equal to one, and of the R-roots(yi + yj)/2(1 < i < j ? t), while qb = 0.
of theR-roots-y(i ? b)
In cases (ii) and (iii), qb is the sum of the multiplicities
and (yi + yj)/2(1 < i < j _ b). In case (ii), Pb iS the sum of the multiplicities
of the roots(yi ? yj)/2(1 < i ? b < j ? t), and in case (iii), we have to add
also the multiplicities
of the rootsyi/2(1 < i _ b), whencethe lemma.
In the next proposition,
the importantpointis not the explicit value of
Vb
459
COMPACTIFICATION
but rather the fact that it is >0 and completelydeterminedby Pb
and qb. This will play an importantrole in our discussionof Eisenstein
nbS
series.
1.16. PROPOSITION.Let 72b be the restrictionto Zb of thefunctional determinant Jb, and let Xb(g) = det Ad'b g (g e Pb). Then 7)b(g) = X,(g)-nb if
g e Abe Vb, and ] b(g) I = I Xb(g) -lnb if g e Zb, wherenb =1 in case (i) of 1.15,
and nb
(p, + 4qb).(2pb + 4qb)-1in the cases (ii), (iii) of 1.15.
By 1.7 (1'), we have
Jb(X,
9)
= lb (9)
det (Ads+Cb g *Cb) Tb(Cb *g*Cb)
(X C Sb; g C Zb)
Both Xb and 72bare rationalcharactersof Zb. Theyare thereforeequal to
one on Vb and on the derivedgroupof Zb. On the compactsubgroupK n Zb
they are both of modulusone. Since Zb is generatedby its intersectionwith
K, a semi-simplesubgroupLb, its unipotentradical Vb, and Ab (see 1.3), it
remainsto check1.16 on Ab.
The groupAb belongsto the center of the maximal reductivesubgroup
Z(Ab) of the parabolic subgroupPb; hence,the weights of Ab in bb are the
restrictionsof the positive R-roots which are not equal to one on Ab and
therefore
=
(2)
Xb(g) =
det Ad'b g
=
Vb(g)Pb+2qb
(g e
Ab)
We may write a e Ab in the form a = exp \(X1 + *** + Xb). Therefore (1.6)
we have c-1a cb = expX(H1 + *
+ Hb).
Let v4 be the image of Vb under
Intcb1. Its value on cb- .a*Acb is again equal to the restrictionof Yb in case (i),
of Yb/2in cases (ii), (iii), where vyare now coordinatesin it with respect to
the base (Hji2). By definition
?b(a-1) is equal to the productof the values on
- 24(a)rb where the exponent
of
a
the
Cb
roots
,
Ce
7+.
Therefore
cb-1.
7)b(a-1)
rb is in case (i),
the numberof elementsof 7w+,
whichis equal to Pb;
in case (ii),
the number of elements of 7w+restrictingto one of (yi + yj)/2,
(1 ? i ? b < j _ t), plus twice the numberof elementsin 7w+
restrictingto
one of vy(1 _ i _ b), or of (yi + yj)/2(1 _ i < j < c), whichgives
(3)
rb--
Tub(t-b)
+ 2b + ub(b-1)
= pb/2 + 2qb;
in case (iii)Y
it is the same as in case (ii) plus the numberof , e 7w+restrictingto one
of vy/2(1 < i _ b). Accordingto Lemma 14 in [22], this last numberis half
the multiplicity
of the R-rootvi/2,whichgives
460
BAILY AND BOREL
( 4)
rb = u b(t-b)
+ 2b + u b(b-l)
= pb/2+ 2qb,
+ vb/2
and our assertionfollowsfrom(2), (3) and (4).
1.17. PROPOSITION. We keep the notation of 1.11, 1.16, and let q,
mO/mb. Then
IJb(X, g)
i
g) |
|ib(b(X),
iXb(g)
-'
(x e Sb; g e N(Fb))
,
In view of 1.3, we may write g = lIz (1 e L(Fb), z e Z(Fb)). By the cocycle
formula
Jb(X,
g) =
Jb(x,
l, z)
l)*Jb(X
Since z acts triviallyon Fb, we have
Mb~Y,
=
l)
Mb~Y,
l
z)
,
(ty
e Fb).
The proposition
followsthenfrom1.11, 1.16.
1.18. PROPOSITION. Let 1? b < d ?
X
Vb:
Sb is as in 1.6. Then
(i)
Jb(X ,g
(ii) the
=
functional
(g
g),
Jd(Vb,d(X),
e
t
=
Lbd
Vd
? 1: Sb
-Sd,
where
x e Sb).
Z(Fb)0;
determinant
and
A(x,
bd)
of
V2b,d is
constant along the
fibres of the canonical projection ab. b ) Fb.
The groupZ(Fb)? is the semi-direct
productof a reductivegroup Rb with
Lie algebra mb + ab + I', in the notationof 1.3, by the unipotentradical Vb of
Pb= N(Fb), and it is containedin Z(Fd). Both Jb(x, ) and Jd(x, ) are equal
to one if z e Vb by 1.11. Therefore,it sufficesto prove(i) wheng e Rb. By
the definition
of the Cayleytransform,
therefore,
C_1*cdC LbC;
cb 1cd centralizes Rb, and (i) followsfrom1.7 (1').
rule forfunctionaldeterminants,
we have
By the composition
(x,
Vzb,d) * Jd(Vb, d(x),
g)
=
Jb(X,
g)
* j(x
g9,
(x
Vb d)
g e
G Sb,
GI);
of Z(Fb)? on the fibres
therefore,
(ii) followsfrom(i) and fromthe transitivity
of ab, (1.7).
We end this section with a result which will be used in discussing
rationalboundarycomponents.The followinglemmawill be needed.
1.19. LEMMA. Let X be irreducible.
(i)
(ii)
[X, g('yi.Y)/2]
# {O} (X e
-(Y{i}Y)/2
BCE,
[X, gli/21
if R(D is of type
{0};
+
Then
1 <
{O} (x C
i <
j
-1{/2O{0};
t)
1-i
<
t).
on tx
PROOF OF (i). By Lemmas13, 15 of [22], the roots, e 4) restricting
to (- yj + 'j)/2 (i # j) are compact,those restrictingto ('yj+ 'j)/2 are in wu,
and , v , + 'yjis a bijectivemap of the firstset Cij ontothe secondone P3j.
Let ci (resp. pij) be the C-subspaceof gc spannedby the vectors E,, (,a Cij,
461
COMPACTIFICATION
resp. , C Pij). With ctbeing as in 1.6, we have
(1)
Ad ct(cij) n g=,(Yj-Yi)12
=
Ad ct(pii)ng
('Y+Yj)/2
,
and spanned by Ad ct(Ey.). From the
Furthermore,giwcis one-dimensional
result just quoted,and the standardfact [E,, EJ # 0 if " + v is a root,we
get then
[get, 9(-YiYj)12]
(2)
in particular,we may write
x=
= 9('Y
Yj)/2
[y, u]
(Y C g( yYi?,Y2,ufCG-y.)
It is well-knownthat,given v C g, (a C 1go),thereexists v' C g-, such that
[v, v'] is a non-zeromultipleof the elementh, e a such that 8(ha) = (S,.a)
of v undera suit(a C R?).
(One maytake, forinstance,forv' the transform
such that [z, y]
able Cartaninvolution.) Thereexists thereforez C g(y+y,)/2
c-h(-y,,)I2(c # 0). We have then
#
.
[[z, y], u] = c- 0(h(yt=y?)/2)'U
Since [z, u] = 0, because 3?/2 + -j/2 is not an R-root,the Jacobi identity
showsthat [x, z] -[[y, u], z] # 0, whichproves(i).
PROOFOF (ii). Let C' (resp.Pi) be the set of compact(resp. non-compact)
rootswhichrestricton at to 'yi/2and c' (resp. pi) the space spanned by the
vectorsE,, (e C C', resp. be C Pi). We have
Ad ct(c' + pi) ng= -Y/2,
therefore(ii) is equivalentto
(3)
i(e'
+ pi) n (c,+
pi)
By Lemma 14 of [22],the map av-- a + yt is a bijectionof Ci =-C' onto
C Pi), thereexistsv C Pi (resp.v C C') suchthat
Pi. Hence,givenbe C' (resp.bce
v
Then
#
0. Since the left hand side of (3), beingstable
b + =ye.
[E,, EJ
undertc, is spannedby rootvectors,this proves(3).
be the center of Vb
in Pb. Then c is the
direct sum of 1b+ fb, whichis an ideal of c,and of its intersectionwith m=
i(a) n f. In particular, C/(Lb.Vb) is compact.
1.20. PROPOSITION. Let X be irreducible. Let
?
(1 b ? t), and C be the connectedcentralizer of
Wb
Wb
The ideal bb is the sum of the rootspaces gewherea runsthroughthe Rrootsof the form
(t + -j)12 (1 _ri < b < j
b(yi
) + j)/2 (1
<
7i (1 _ i
b) I
t), i
i < j < b)
462
BAILY AND BOREL
togetherwith7t/2(1 < i < b) in the case BC,. We want to prove that
the sum of the rootspaces g,, wherea runsthroughthe roots
(1
+ Jj)/2,
(-mi
t'b
is
j < b).
i
The relation[ga, go]c &+O, and the structureof Rap, show that Vob containsthe
rootspaces just listed. Of course Bo is stable under a, hence is the sum of
its intersectionswith the g,. In orderto proveour assertion,it is therefore
enoughto show that
1b
(
ng(',+',)/2=
ib
< j
t),
and that
I~b
ngsYi/2
=
{0}
<
5(1i
(1
i
i b),
if R4) is of typeBCE, but this followsfrom1.19.
The Lie algebra c is also stable undera. It is obviousthat it containsthe
0. Lemma1.19 showsmoreover
ga
C tb andthatc fl a fl fb
that,if ga. Qb
C
c 0. The propositionfollowsthen fromthe facts about Pb and fb
then g, n C
recalledin 1.3.
REMARK.Proposition1.20 was suggested by a statementin [31, ? 3.3]
whichbecomesessentiallyequivalentto 1.20, if the wordnormalizerthereis
replacedby centralizer.
2. Relative root systems
For mostof the facts recalled below, we referto [14]. As was already
pointedout in 0.3, the ground fieldsmay be assumed to be containedin C,
whichis thenour universal field, althoughthe results of 2.1, 2.2 are valid
in greatergenerality.
2.1. Relative roots. Let G be a connectedreductivek-group.Its maximal
k-splittoriare conjugate over k and their commondimensionis the k-rank
rkk(G) of G. Let S be a maximal k-splittorus of G. The k-roots,or roots
relative to k, or restrictedroots are the non-trivialcharactersof S in the
adjoint representationof G, and the relativeWeylgroupkW = kW(G) is the
quotientN(S)/Z(S). We denoteby kA or kP(G) the set <>(S, G) of k-roots.It
is a root systemin X*(T) 0 R
V. This means in particularthat, with
respectto a scalar product( , ) on V invariantunder k W, the group k W is
generatedby the reflectionsin the hyperplanesorthogonalto the k-roots,
leaves kA stable, and that 2(a, 8) (p3,6)I3 C Z for all a, ,JGkG . For every
.
a C kA we put
ga=
{x C g I Ad s(x) =
Sa.X(S
C S)}
COMPACTIFICATION
463
Then g is the directsum of the gal(a eke ) and of the Lie algebra s(S) of the
centralizerof S.
Givenan orderingin X*(S), we denoteby kA the set of simple k-roots.
A subset of kA is connectedif it is not the unionof two non-empty
disjoint
subsetswhichare mutuallyorthogonal.
2.2. Parabolic k-subgroups.An algebraicsubgroupP of G is parabolic
if the quotientspace GIP is a projectivevariety. P is thenconnected,equal to
its normalizer,and is the normalizerof its unipotentradical.
Let U be the subgroupnormalizedby S whose Lie algebra u is the sum
of the ga,wherea runsthroughthe positivek-roots(forsomefixedordering).
Then U is a unipotentk-subgroup,normalizedby Z(S).
For every subset 8 of kA, let So = (naeo ker a)'. We let kPo be the subgroup generatedby Z(S6) and U; it is the semi-direct
productof Z(S6) and of
its unipotentradical U6c U. The splitradical (0.5) of kPO is the semi-direct
product So -U6. Every parabolic k-subgroupof G is conjugateover k to one
and onlyone kPO. Moreover,two parabolic k-subgroupsare conjugate in G
if and only if they are conjugate over k. The groupskP6 are the standard
parabolic k-subgroups(fora given choiceof S and U). If 8 = 0, then kPO =
kP = Z(S) * U is the minimalstandard parabolic k-subgroup.We can write
uniquely
(M normalk-subgroup,M n S finite),
Z(S) = M. S
and M is anisotropicover k, i.e., rkk (M) = 0.
For 8 c kA, we let [8] be the set of k-rootswhichare linear combinations
of elementsin 0, and let kLo be the smallestconnectedk-subgroupnormalized
by Z(S) whoseLie algebra ktOcontainsthe subspaces g, (a e [1]). It is easily
seen that [81 = kD(kLG), that S n kLo is a maximalk-splittorus of kLo, and
that kLo is semi-simple
withLie algebra
JO =
EaE[6]
ga
+
[gal
ga]
Moreover,we have kPO = M6OkLo * So * U6,whereMOis theidentitycomponentof
M n Z(kLo). If 8 is connected,then kLo is almost k-simple,since otherwise,
by [14, 5.11, 8.5] kLo wouldbe the almostdirectproductof a k-groupwithout
k-rationalunipotentelements# e, and of a k-groupcontainingall unipotent
elementsrational over k of kLo, and kLo could not be generatedby unipotent
k-subgroups.
We shall sometimesdenoteby k T a maximalk-splittorus and by k U the
unipotentradicalof a minimalparabolick-subgroup.
We recall finallythe Bruhat decompositionGk = Pk-N(S)k-Pk [14, 5.15].
BAILY AND BOREL
464
Moreprecisely,let nw,be a representativein N(S), of w c , W. Then G, is the
disjointunionof the doubleclasses
Pk-n.wPk
-
Uk -
Ukn.Z(S)k-
If G is connected,but not reductive,it is the semi-directproductof its unipotentradicalR.(G) by a reductivek-subgroup(we are in characteristiczero),
and R.(G) is containedin every parabolic subgroup. Since N(S) nfR(G)is still valid, with P a
z(s) n R.(G), it followsthat the above decomposition
minimalparabolick-subgroup,and N(S) being the normalizerof S either in
G or in a maximalreductivek-subgroupcontainingS.
2.3. Fundamental highest weights relative to k. There is a basis of
X(S) 0&Q over Q consistingof elementsd, e X(P)(a e k4) such that (da, 3) =
3 ki) where Ca are positive integers. The restrictionto So of the
Caeaa3(a , G
forma basis of X(S6) 0 Q. Let d e X(S) be a
elementsda(a e 0' = QA0-)
linearcombinationof elementsda(a e 0') with strictlypositive integralcoefficients. Then there exists an absolutelyirreduciblerepresentationp: G
subspace V' c V stable
GL( V) definedover k, and a unique one-dimensional
by d(g) [14, 12.2, 12.13].
underP6 and on whichg C Po acts via multiplication
The charactersda(a C 0') will be calledfundamentalhighestweightsforPo.
Let
We have then
(p C Po)
Xo(p)= det(Ad.,p)
Xo=
(ea C Q; ea > 0)
eEO eada
In fact, by definition,
X0is the sum of the weightsof S in uO. These are
the positive roots which involveat least one of the elementsof 0', each root
beingof coursecountedwith its multiplicity.X0is stable under N(S) n Fo,
hence under the fundamentalsymmetries
s,,( C 0); thereforeit is orthogonal
ea
to 0, and is a linearcombinationof the elementsda(a e 0'). The coefficient
of d, is equal to (Xo,a) *c'1. Let X1be the sum of the positiveelementsin [1]0,
and X = X0+ X1. Then (X1,a) ? 0 fora e O'
countedwiththeirmultiplicities,
and, by a standardargument(X,a) > 0, (in fact it is equal to (c + 2d) (a, a),
of a and 2a), hence(x0,
a) > 0 and ea > 0.
wherec and d are the multiplicities
2.4. Restrictionof relative roots. Let K be an overfieldof k, T a maxihomomal K-splittorusof G containingS and r: X(T)
X(S) therestriction
morphism.Two orderingsof X(T) and X(S) are compatibleif a > 0, r(a) # 0
implyr(a) > 0 (a e X(T)).
The existenceof an orderingon X(T) compatiblewith a given ordering
on X(S) is immediate[14, 3.1]. Let ,A and kA be the sets of simplerootsfor
-
COMPACTIFICATION
465
compatibleorderings. Then we have
( i ) kA C r(KA)
C
kA U{O}.
C KA and a eHK n r-'(0). If / is connected,then
(ii) Let 0 c 4,
A*
r(Q) n kA is connected. If 0 is connected,thereexists a connectedsubset0'
of KA containinga such that
0 c r(0') c 0 U {O} .
For the proofs,see [14, 6.8, 6.15, 6.16]. A simple K-root will be said to
be critical if it restrictsontoa simplek-root. Thus a simpleK-rooteitheris
criticalor restrictsto zero.
(iii) Let 0 be a connectedsubsetof A, and assume thereexists a unique
greatest connected subset *r of KA such that r() n kA- 8. Then kLo= KL+.
In particular, KL' is defined over k.
PROOF. Let f8e k4'. Then the space gois the directsumof theeigenspaces
of T, wherea runsthroughthe K-rootswhoserestrictionto S is equal to
,8. It is a standardfact about root systemsthat, if a roota is expressedas
linearcombinationof simpleroots,then the set of simple roots which occur
is connected. Thereforeif ,3e [0] and a e K?D
in a with a non-zerocoefficient
restrictsto 8, then a e [*]. This impliesthat kLoc- KLI. Moreover,Z(S)
that KLYP C Z(S) *kLO. ConnormalizeskLo, and it is clear fromthe definitions
sequently,kLo is a normalsubgroupof KL/. However the latter is almost
K-simple(2.2), whenceour assertion.
ga
2.5. PROPOSITION. Let k be an algebraic numberfield,kvits completion
with respectto an archimedeanvaluation v, and G a connectedreductivekgroup. Then every maximal torus definedoverkvof G is conjugateoverkv
to a maximal torus definedoverk.
(i) We show firstthat, if L is a connectedk-group,then Lk is dense in
(Lkv)0 in the usual topology. By [32, p. 41], there exists a genericallysurjective rational map of an affinespace into L which is definedover k. In
other words, we may finda Zariski k-opensubset U of an affinespace, and
a k-morphism
f: U ? L whose image contains a non-emptyZariski k-open
subset V of L. Since we are in characteristiczero,f is separable,and there
Zariski k-opensubset U' of U such that f: Ukv Lkv is
exists a non-empty
open. Of course f(Uk) c Lk and Uk is dense in Ukv. Thus, Lk is densein a
open subsetof Lkv, hence in (Lkv)0.
non-empty
(ii) If kv= C, thenall maximaltoriof GkVare conjugate,and 2.5 amounts
to theexistenceof a maximaltorusdefinedover k, whichis known[32,p. 45].
Assumenow that k = R. Let T be a maximaltorusof G definedover R and
H' the set of regularelementsof (TR)'. It is well known that the set C of
466
BAILY AND BOREL
conjugates of elementsof H' by elementsof (GR)Y is an opensubsetof (GR)0.
By (i), it containsan elementx rational over k. Then Z(x)0 is the desired
maximaltorus.
2.6. COROLLARY.
Let k be a subfieldof R. Then G has a maximal torus
definedoverk whichcontainsa maximal R-split torusof G.
2.7. REMARKS.(1) Proposition2.5 is also valid foran arbitraryconnected k-groupG. In fact,let U be the unipotentradical of G and iu: G = G'
G/U the canonicalprojection. Let T be a maximaltorusdefinedover kVof G.
Then wz(T)is a maximaltorusofG' [9, ? 22], obviouslydefinedoverkv. By 2.5,
it is conjugateover kvto a maximaltorus T' definedover k of G'. Since U is
andk is perfect,themapGk G'-G is surjectivebyRosenlicht'scrossunipotent,
sectiontheorem,whencetheexistenceofx e GkVsuchthatxTc 7r-'(T') = Q. The
groupQ is a connectedsolvablek-group,henceits maximaltoridefinedoverkv,
are conjugateover kVand one of themis definedover k (see e.g. [14,11.4]; or,
in characteristiczero, Borel-Mostow,Annalsof Math. 61 (1955), 389-405).
Proposition2.5 is then of course also true if maximaltori are replaced
by Cartansubgroups,sincethe latterare the centralizersof the former.
( 2 ) Althoughthis will not be needed in this paper,we pointout that,
if G is a connectedk-group,Gk is densein Gkq,,
notonlyin (Gkv)0,as was shown
in (i). If kV= C, thenGkVis connected,and thereis nothingnew to prove. If
kV= R, thereremainsto show that Gkmeets every connectedcomponentof
of G, (or
GR. By 2.6, applied to a maximal connectedreductivek-subgroup
by remark(1) above), thereexists a maximaltorus T of G definedover k and
containinga maximalR-splittorusof G. By [14, 14.4], each connectedcomponentof GR containsone of TR, so that we are reducedto the case of a
torus,whereour assertionfollowsfroma resultof Serre quoted in [25, 5.1].
of [25], this means essentiallythat G has the weak
In the terminology
approximationpropertyfor archimedeanvaluations. As a matterof fact,it
has been checkedhereonlyforone such valuation,but the case of several is
easily reducedto that of one by consideringthe groupRkQG.
2.8. Let k be a subfieldofR and G a connectedsemi-simple
and absolutely
simplek-group.Let RT be a maximalR-splittorusof G containinga maximal
k-splittorus kT. We let r: X(R T)
X(k T) be the restrictionmap, endow
X(RT) and X(kT) with compatibleorderings,and denoteby A, RA the correspondingsets of simplerelativeroots.
We assume furtherthat the riemanniansymmetricspace K\GR,where
K is a maximalcompactsubgroupof GR, is a boundedsymmetricdomain,
necessarilyirreduciblesince GR is simple. Then R(D is eitherof typeC, or of
467
COMPACTIFICATION
type BC, (1.2). In fact, the followingproposition,and its proofare valid
{a1, . ., aj, we use the canonicalnumunderthat last assumption.On R'=
beringof 1.2. For each /8e 4, let m(hS)be the greatest value of the index
i such that r(ai) = /. We number the elements /89, , /9. of A in such a
way that i < j if and onlyif m(,Si)< m(/9j),and then write m(j) for m(/9j).
2.9. PROPOSITION. We keep the notation and assumptions of 2.8. We
assume that dimkT > 0. Then
(a) kq) is of typeBC8 if eitherR(D is of type BCt or R1 s of type Ct
and r(at)- 0, and is of typeC8 otherwise. The numberingof kA definedin
2.8 is thecanonical one.
ofoneand onlyonesimpleR-root.
(b) Each /9CkA is therestriction
By our choice of the numberings, any final segment (/9t,
**i,,/9)
in
kA
(possiblywith zero added), is the restrictionof a finalsegmentof RA, hence
is connected(2.4). Conversely,any connectedsubset 0 of kA containing9a, /9b
(a < b) contains/i foreveryi betweena and b. In fact, there exists by 2.4
(ii) a connectedsubset 0' of RA containingarm(b)such that 0 c r(O') c 0 U {O}.
The set 0' containsthenat least one simpleR-roota, with c < m(a). In view
of the structureof RA, the set 0' must then contain all simple R-roots with
index i betweenc and m(b), hencein particularall R-rootsaCm(j)(a < j < b),
whenceour contention.
This showsthat the graph of kA is a chain (no branchpoint). kA is thereforeof one of the types A8,B8, C8,G2,F4, BC8, where the firstfive symbols
referto the standardCartan-Killingclassification.We now distinguishsome
cases.
( i ) R" is of typeBCt. In this case, the set of a C RA whose double is a
root spans X(RT) 0 R, hence contains at least one elementa whose restriction is not zero. Thus r(-) and 2 r(-y)are k-roots,and kA is of type BC8.
Moreover, the highest root in RA is -y1 2(a1 +
+ at); therefore, if /9C kA
*
> 4 in
is the restrictionof at least two simpleR-roots,then,8has a coefficient
the highestk-root,whichis impossiblein typeBC,. Also, am(s) + * + at =
Ym(s)/2and its double are roots, hence /9sand 2/89are k-roots; and, by the
above, the connectedsubsets of kA containing/9,are the finalsegments.This
showsthat our numberingis the canonicalone,and endsthe proofofthe propositionin this case.
Fromnow on, RA is of typeCt. Its highestrootis then
2(al +
* a + at-,) + at
a
Let ctdenotethe numberof simpleR-rootrestrictingonto,8/.
(ii) Assumethat r(at) # 0. Then the highestk-rootis
BAILY AND BOREL
468
a = 2c1.Sl1+
*.
+ 2 c-.1S-1
+ (2(c.-1)
+ 1),s8
> 4 in 6, and k(IDis of
If c >_2 forsomei < s, thenthereis a coefficient
3, which must thenbe the
type F4. There is in this case in a a coefficient
3 in the
of f,3. However, in F4 the simple root with coefficient
coefficient
highest root is not an end point of the graph. Thereforect = 1 for i < s.
Assume now that /,5= r(a) = r(aj) for some j < t. (There is at most
> 5 in a in the systemsunder
one such j since no simpleroothas a coefficient
>
considerationhere). Then /,3has a coefficient 3 and all othersimplek-roots
2. This occursonlyin G2. If m(1) < j, then
have coefficient
3q92 = r(2(a3j +
...
+ at-,) + at)
is a k-root,whichis absurd. If j < m(1), then
r(i(Ym(l)
+ 1j)
+
2(/91
I2)
is a k-root,whichis absurd because 81 + I2 is a k-root,and G2 has no root
whosedoubleis a root. This provesthat ct= 1 (1 < i _ s), hencethat
8=-2(X81 + * -+ +SS-l)
+ 's
ThereforekAI) is of typeCt, and the numberingis the canonicalone.
(iii) Assumethat r(at) = 0. In this case the highestk-rootis
a = 2.cl,91 +
..
+ 2.c./S, .
By the classificationof rootsystems,this impliesthat k'I is of type BC, and
that ct = 1 (1 ? i ? s). Furthermore, 2,/9= r(Ym(s)) is a k-root, so that again
the numberingis the canonical one. This completesthe proofof the proposition.
2.10. COROLLARY. (a) The proper maximal parabolic k-subgroupsof G
are also proper maximal among parabolic R-subgroups.
(b)
Let * be an initial (resp. a final) segmentof RA consisting of all
n 4.
roots which come before (resp. after) a critical root and 0 = r(*)
Then RL3Z= kLo and is defined over k.
PROOF OF (a). Let P be a propermaximalparabolick-subgroupof G. It
followsfrom2.2 that thereexists a simplek-root,8 such that P is conjugate
over k to the group kPo (O = kA- {/E?}). Let a be the unique critical simple
R-rootwhichrestrictsonto,8 and * = RA {-a}. Then To = So by 2.9. Furthermore,since the given orderingsare compatible,we have RU C Z(k T) *kU,
and thereforeRP* c kPO. Since RP* is a propermaximal parabolic R-group,
we have RP* = kPo, whichproves(a).
PROOF OF (b). By 2.9, 0 is an initialor finalsegmentof k4, and * is the
469
COMPACTIFICATION
greatestconnectedsubset of KA such that 0
lows from2.4 (iii).
4A n r(*). Therefore(b) fol-
3. Rational boundary components
3.1. Let G be an algebraic group definedover Q. A subgroupF of GQ
is arithmeticif forone (and hence for every [13, 6.3]) faithfulQ-morphism
with p(G)z.
p: G GLm,the groupp(F) is commensurable
We recall that, if f: G G' is a surjective Q-morphismof G onto a Qgroup G', and F is an arithmeticsubgroupof G, thenf(F) is also arithmetic.
(See [13, 6.11] for isogenies,[11, Th. 6] for the generalizationto surjective
morphisms.)Since we are interestedin automorphism
groupsof symmetric
spaces, we may,withoutrestrictinggenerality,limit ourselves to centerless
groups wheneverconvenient. Moreover,it followsfrom[13, 6.11] that, if G
is an almostdirect(or a semi-direct)productof two Q-subgroupsG1,G2and F
is an arithmeticsubgroupof G, then (F n Gi) is an arithmeticsubgroupof
G. (i = 1, 2) and (F f G1)*(F fl G2)is commensurable
withF.
Let G be simpleover Q. Then there exists an algebraic numberfieldk
and an absolutelysimplek-groupG' such that G = RkIQG' [14, 6.21 (ii)], where
RkIQ is the functorof restriction
of the groundfield[38, Ch. I], fromk to Q.
-
3.2. LEMMA. Let k be an algebraic numberfield. G' a connectedsemisimple and absolutelysimple k-group,and G = RkIQG'. Let K be a maximal
compactsubgroupof GR, X= K\GR,and r be an arithmeticsubgroupof G.
( a) If K has the same rank as G, in particular if X is a bounded
domain, thenk is totallyreal.
(b) If X/P is not compact,then G' has no compactfactor f{e},and
rkk(G') # 0.
Let V be the set of normalizedarchimedeanvaluationsof k, and kVthe
completionof k withrespectto v e V. Then
GR
lvev GV
[38, 1.3.2]
hence
X = Hve
Z Xv
and K(v)
~(XV= (K nG
v)\G' V)
K n G' is a maximal compact subgroup of G'V. If kV= C, then G'
is a complexLie group, viewed as real Lie group,and its rank as such is
twice the rank of K(v), whence (a).
The groupsG' are the simplefactorsof GR. If one of them is compact,
then G' - GQ consists of semi-simpleelements,hence GR/F is compact [13,
11.6], whichproves(b), and G has no properQ-parabolicsubgroup.
470
BAILY AND BOREL
domain,H(X) its groupof holo3.3. (i) Let X be a boundedsymmetric
and Is (X) its groupof isometrieswithrespectto the
morphicautomorphisms
underlyingriemannianstructure. Let f be the Lie algebra of H(X). It is
knownthat Ad tj H(X)0 c Is X = Aut t. Thus H(X) is identifiedwith a
group of finiteindex in the group of real points of an algebraic R-group,
namelyAut t5c. Assume that we have put on Aut 15ca structureof Q-group
subordinatedto its natural R-structure. This is equivalent to puttinga Qstructureon f; i.e., fixinga Lie subalgebra JQover Q of f such that k =
dQ0 QR. An arithmeticsubgroupF of H(X) is thenan arithmeticsubgroup
of Aut 1c, viewed as a Q-group. More correctly,one should say that F is
arithmeticallydefinable,since this definitionpresupposesthe determination
of a Q-structure,forwhichthereis usuallya wide choice. However,we shall
that Ad fjchas
just say arithmeticforthesake ofbrevity. It is thenunderstood
witha semi-simpleQ-groupG which has no center;i.e., with
been identified
Ad gc,and H(X) witha subgroupof Aut R. The space X is thenthe quotient
of AutgR,or H(X), or AdgR, by a maximalcompactsubgroup.
The group G is the direct product of its normal simple Q-groups Gi
spaces (K n GiR)\GiR, which
(1 ? i ? m), and X the productof the symmetric
domains. Let Fi= r n GR (1 < i ? m) and
are thenalso boundedsymmetric
F' be the subgroupgeneratedby the Fi. It is arithmetic,normal,of finite
index,in F.
of X/P. It turns out that the
Our problem is the compactification
(cf. 8.9). Since X/F' is the
X/F
X/F'
no
difficulty
from
to
offers
passage
productof the Xi/Fi,the essential case to consideris when G is simpleover
Q, and FcGOR
(ii) We introducesomenotationpertainingto our main case of interest.
We keep the assumptionof (i), and assume moreoverG to be simpleover Q,
and I c GO. Then G -Rk,QG', whereG' is an absolutelysimplek-group,and
k a totallyreal numberfield. Let E be the set of distinctisomorphismsof k
betweenelementsof X and normalized
into R. There is a 1-1 correspondence
a
k
archimedeanvaluationsof given by I [ = a(a) I (a e k), and we have GkV
(-G')R, [38, Ch. I]. We maythenalso write
x L=I~oe Xq,
(XI =
K(o)\GGR
Ka\ GR)
where X, is an irreduciblesymmetricboundeddomain. For simplicity,we
shall also writeGOfor 9GOR
We assume furtherthat if F is an arithmeticsubgroupof G, then X/F is
not compact. This implies(3.2) that no X, is reducedto a point and that G'
has a non-trivialmaximal k-splittorus,say S'. Then 9S' is a maximala(k)-
471
COMPACTIFICATION
splittorusof 6G'; thereis a canonicalisomorphism
(pa: S'
an isomorphismof
onto a(k4)(G')
kid
=kga.
I~S
whichinduces
Furthermore the maximal Q-split
subtorusS of RkQS' is a maximalQ-splittorusof G. It is canonicallyisomorphic to S' and is diagonallyembeddedin RkIQS'. This means more precisely
that the projectionprq of S into 6G' is the compositionof the canonicalisomorphismsup:S o S' and
(Pa:
S'
-
6S'. The isomorphismalso induces an isoWe shall identifyk(D(G), a(k)(D(6G'),and Q@b(G)
morphism
of QiP(G)ontok4D(G').
by meansof these isomorphisms.
In each group 6G', we choose a maximal R-split torus To DISf , contained
in a maximaltorusdefinedover a(k) (apply2.6 to Z(7S')). We fixan ordering
on X(S'), hence,using upand qi, also an orderingon X(IS') and X(S). For
each a, choose an ordering on X( To) compatiblewith the given one on
X(IS'), and let r: X( To) X(aS')
X(S) be the restrictionhomomorphism.
By 2.9, the canonicalnumberingon the set RAo of simple R-roots of G with
respect to To is compatibleby restrictionwith the canonicalnumberingof
QA.
Let kZA {iJ51
For i between1 and s, we let c(i, a) be the index
**1 *38}.
of the criticalsimpleR-rootof IG restrictingon Si. Then, the remarkjust
made showsthat i < j impliesc(i, a) < c(j, a) forall a e `.
A sequenceof elementsindexedby X will often be denotedin boldface
and used as a multi-indexor a multi-exponent.In particular,let b be between 1 and s. Then
,
Fb
T1e
Il
Fc(ba)
is the product of the standard boundarycomponentsFC(b,O) of XI, where
standard refersto the choice of To and RAI. It is also understoodthat the
Lie algebra of K(a) is orthogonalto that of To. Since c(j, a) is an increasing
functionof j, foreach a, we have Fj c F1 (1 ?i < j _ s).
Let F = II, Fi(,) be a productof standardboundarycomponents.We let
SF= II Si(,)be the productof the unboundedrealizationsassociated to the
Fi(a), (1.6), JF be the functionaldeterminantin SF, and iF be the functional
in the Harish-Chandraboundedrealizationof F. If F= Fb we
determinant
also write Sb, JA, Ah forSF, JF and jF In the notationof 1.8, we have therefore
JF(X,
g) =
flo Ji(0)(XI,
go)
(X = (x,), g = (go); XI e Xa, g, e G,),
Fi(,), go s L(F0)) .
By 1.11, applied to each irreduciblefactor of X, the functionaldeterminantJF(x, g), (g e N(F)), is constantalong the fibresof the canonicalprojectionaF of X ontoF, (definedin 1.7, remark).
jF(X,
g) =
oii(0)(XI,
go)
(X = (x,), g = (g,); XI e
472
BAILY AND BOREL
The naturalcompactification
X of X is the productof the natural compactificationsX, of the X,. We shall also write Ob forthe pointwithcomponents0c(b,o) E X,, in the notationof 1.5. Thus
Fb
=
Ob*N(Fb)
=
Ob *L(Fb)
=
ObhG(Fb)
(1 ?
,
b
_ s)
We recall that G(Fb) = N(Fb)/Z(Fb). We shall denoteby tUb the naturalprojection of N(Fb) onto G(Fb). Applying1.3 and 1.5 to each irreduciblefactor
of X, we see that G(Fb) is connected, that N(Fb) = L(Fb) Z(Fb), and that
.
Z(Fb) is the greatest normal subgroup of N(Fb) with identitycomponent
Z(Fb) .
3.4. It will be sometimesconvenientto use the followingvariationon
the notionof arithmeticgroup.
Let H be a connectedreal Lie group. A subgroup P is of arithmetic
type, or arithmeticallydefinable,if there exists a connectedQ-groupG, a
continuoussurjectivehomomorphism
H with compactkernel N and
f: G'
an arithmeticsubgroupiF of G such that f(P') = P. Since N is compact,the
group P is thenobviouslydiscrete.
Let H be semi-simple.It is easily seen that, withoutrestrictinggenerality, G maybe assumedto be semi-simple,
and to be almostsimpleover Q if
H is simple. If G is simpleover Q, and dimN > 0, thenH/r is compact. In
fact, we have in this case G = Rk/QG',
wherek is an algebraic numberfield,
and G' an absolutelysimple k-group. The group GR is the productof the
groups G' where k, runs throughthe archimedeancompletionsof k, and
these are the simplenormalsubgroupsof GR. ThereforeN containsat least
one of them,G' consistsof semi-simpleelements,and GR/P' is compact [13,
11.6]. This impliesof course the compactnessof H/P and of K\H/P, where
K is a compactsubgroupof H.
3.5. Rational boundarycomponents.Let G be a connectedsemi-simple
Q-group,whose symmetricspace of non-compact
type,X= K\GR,whereK
is a maximalcompactsubgroupof GR, is a boundedsymmetric
domain. For a
discretesubgroup P of GR and a boundarycomponentF of X, (1.2), we let
P(F) be the image of rPn N(F) in G(F) = N(F)/Z(F) by the natural projection. The componentF is said to be P-rationalif
(i) the quotient U(F)/(U(F) n r) is compact,
(ii) the group1(F) is discrete.
Clearlya P-rationalcomponentis P'-rationalforany groupF' commensurable with F. In particular,if F is P-rationalforone arithmeticgroup P, it
is so for all arithmeticgroups; in that case, we shall dropthe prefixP- and
speak of rational boundarycomponents.
-
COMPACTIFICATION
473
Let now F be arithmetic.We remarkfirstthat (i) is equivalentto
(i)' N(F)c is definedover Q.
The implication(i)' -(i) follows fromthe standardfact that, if U is a
unipotentQ-groupand P an arithmeticsubgroupof U, then UR/F is compact.
Assumenow (i) to hold. Let V be the smallestalgebraic subgroupof U(F)c
containingU(F) n F. It is invariantunder all automorphismsof C, since
F cmGQ,hence it is definedover Q. Since U(F)/F is compact,the quotient
to euclidean
UR/VR is compact, too. But UR and VR are homeomorphic
spaces, hence U(F)c= V, which shows that U(F)c is definedover Q. The
group N(F)c, being equal to the normalizerof its unipotentradical,is then
also definedover Q.
It will turnout that (i)' (ii) in our case; howeverwe have preferredto
whichmakes sense for any symmetricspace and any
startfroma definition
Satake compactification.
3.6. Clearly,(ii) is impliedby
(ii)' the groupF(F) is of arithmetictype.
Assuming(i), we now prove that (ii)' is impliedby either of the two
followingequivalentconditions:
(iii) Thereexists a normalconnectedQ-subgroupC of N(F)c containing
U(F)c and L(F)c and such that CR/L(F). U(F) is compact.
(iv) Thereexists a connectednormalQ-subgroupB of N(F)c, contained
in Z(F) , containingU(F)c, such that Z(F)/(BR n G') is compact.2
We show,to begin with,that (iii) and (iv) are equivalent. First assume
(iii). Let H be a maximal connectedreductiveQ-subgroupof C, and L a
maximalconnectedreductiveQ-subgroupof N(F)c containingH. We may
writeL = H.H', withH' normal,definedover Q, and H n H' finite. Then
D = H'. U(F)c is a normalQ-subgroupsuch that Z(F)/DI is compact. Moreover, DRnG' cmZ(F) by (1) of 1.3, whence (iv). The other implicationis
provedin the same way.
of
Assume(iv) holds. The projectionN(F)
G(F) is thenthe composition
the restrictionto N(F) of the Q-morphism
N(F)c >N(F)c/B withthe projectionof N(F)/(BR n G?) ontoG(F), whichhas a compactkernel,whence(ii)'.
We note finallythat if G = G1 x ... x Gmis a directproductof normal
Q-subgroups,thena boundarycomponentF = F1 x ... x Fmof X is rational
if and onlyif Fi is a rational boundarycomponentof Xi = (K n Gi)\GiRfor
all i. This follows immediatelyfromthe two followingfacts: the parabolic
subgroupsof G are the productsof the parabolic subgroupsof the Gi; the
2 These conditionsare of course fulfilledif
Z(F)c is definedover Q. In fact, this is
the requirementmade in [10]; however, it has turned out to be too restrictive.
474
BAILY AND BOREL
withthe productof the groups r n Gi, whichare
group F is commensurable
arithmetic(3.1).
3.7. THEOREM. We keep the assumption of 3.3 (i). A boundarycomponent F of X is rational if and onlyif N(F)c is definedoverQ. If F is
rational, 17(F) is of arithmetictype. The map F v--N(F)c defines a bijection of theset of proper rational boundarycomponentsontotheset ofproper
maximal parabolic Q-subgroupsof Gc.
By the last remarkof 3.6, we mayassume G to be simpleover Q. If X/P
is compact,where F is an arithmeticgroup,then rkQ(G)= 0 [13, 11.4, 11.6],
G has no properparabolicQ-subgroup[14, 8.3-5],and there is no properrationalboundarycomponent.
Assumenow X/F to be non-compact.In the notationof 3.3 (ii), we have
G = RkIQG'with G' absolutelysimpleand k totallyreal. Let F be a boundary
componentof X. If F is rational,then N(F)c is a Q-subgroupby 3.5 (i)'.
Assume converselythat N(F)c is definedover Q. We have then N(F)c
RkIQP, whereP is a parabolick-subgroupof G', [14, 6.19], hence
N(F)= fN(F0) ,
Ff= lz F0, Y
(N(F0) = (7P)R n G0,a YE)
whereFJ is a boundarycomponentof X,. Let V0' be the center of the unipotent radical V' = Ru(P) of P, and C' the connectedcentralizerof V"'in P.
The groups V', V0 and C' are clearlydefinedover k. It follows immediately
fromthe propertiesof the functorRkIQ that C = RkIQC' is the centralizerin
N(F)c of the center V0= Rk/QVJof the unipotentradical of N(F)c, and that
Co =- II, CO,whereCOis the connectedcentralizerin N(Fo) of the center of
U(F0). By 1.20, COcontainsL(Fo). U(Fo) and the quotientCo/L(Fo).U(Fo) is
compact. Thereforecondition(iii) of 3.6 is fulfilled;since, togetherwith (i),
it implies(ii), (ii)', by 3.6, our firsttwo assertionsare proved.
Let F be rational. Then N(F)c = II, N(F0)c and N(Fo)c is a proper
maximalparabolicR-subgroupof ?G', hence N(F)c is a propermaximalparabolic Q-subgroupof G.
Conversely,let P be a propermaximal parabolic Q-subgroupof G. We
have P= RkIQP', where P' is a propermaximalparabolick-subgroupof G',
and thereforeP= II, "P, and "P' is a propermaximalparabolic a(k)-subgroup of ?G'. By 2.10, "P' is also a propermaximalparabolicR-subgroupof
?G'; consequently(1.5), ?PR n GO = N(Fo) whereFo is a boundarycomponent
fIFo). The boundarycomponentF is then
of X, and PR n G = N(F), (F
rationalby the firstpart of the theorem. Since two boundarycomponents
with the same normalizerare identical,the proofof the theoremis complete.
475
COMPACTIFICATION
3.8. THEOREM. We keep the notationand conventionsof 3.3 (ii). Let3
= kLo(b) (cf. 2.2) and Lb = RkIQLf,(1 _j _ s). If b #s,
*
0(b) =
1}, Lb
{1,b+l,
then(Lb,R)=
L(Fb). In particular L(Fb)c is definedoverQ, almostQ-simple,
and its Q-rank is equal to s - b. For any arithmeticsubgroupF of G, the
quotientFs/F(Fs) is compact. Given a rational boundarycomponentF, there
exist one and only one index b (1 < b ? s) and an elementx e GQsuch that
F= Fb x.
for all a6e . Since Lb = HLb,
By 2.10 (b), we have Lb =
L(F,(b,O))c
this proves the firstassertion. The set 0(b) being connected,Lb is almostksimple,(2.2), henceLb is almostQ-simple[14, 6.21 (ii)].
For each a, the indexc(s, a) is the last criticalindex,therefore(1.3, 1.5)
Z(FS) containsS'. Let C be the connectedcentralizerin N(FS)c of the center
of the unipotentradical of N(FS)c. By 1.20, the intersectionof G' lnc with
0S' is finiteforeverya e 1. In particular,S normalizesC and S c is finite.
This impliesthat the Q-rank of C is zero, forotherwisetherewouldexist a
maximalQ-splitsubtorusT of N(F)c withdim(Tn c) # 0, and it could not
the conjugacytheorem
be conjugate to S (since C is normal),contradicting
formaximalsplittori. It followsthen from[14, 8.5] that the unipotentelements of CQ all belong to the unipotentradical of C, and that X(C)Q= 0.
The quotient CR/(fnc) is then compact [13, 11.8]. Since the projection
N(FS) G(FS) maps CR onto G(FS) and F f CR into F(FS), we see that
G(Fs)/F(Fs) is compact,whenceour secondassertion.
Let F be a rationalboundarycomponent.Then N(F)c is a propermaximal parabolic Q-subgroupof G. On the otherhand,the groupN(Fb)c is, in
(+(b) = QA-{f b};
the notationof 2.2, the standardparabolic group QP34(b),
.
=
.
=
all
the
are
1,
*..
1,
properstandard
, s)
b
(j
., s). The groups QP34(j)
maximalparabolicQ-subgroups(2.2); thereexists thereforeone and onlyone
b forwhichwe may findx e GQ such that x N(F)c x- = N(Fb)c. We have
thenF= Fb.x, whichends the proof.
c
of X such
3.9. COROLLARY. Let F, F' be rational boundarycomponents
thatF' c F. Let b, c be the integerssuch that F c FbhGQ and F' c FC.GQ.
Then thereexists g e GQsuch that F. g = Fb, F'.g = F,.
There is nothingto prove unless b # c; in particular,we may assume
b # s. Therefore(3.8), L(Fb)c is definedover Q, almost Q-simple,and we
may apply 3.8 to X= Fb. The proof of 3.9 is thenthe same as that of the
3We hope the reader will not be unduly confused by the occasionally similar (or, by
chance, even coinciding)notation used for real Lie groups in ? 1, and for their complex
formsin ?? 2, 3. Also, Lb has a differentmeaning here than in ? 1.3.
BAILY AND BOREL
476
similarremarkmade in 1.5, using 3.8 to insurethat u, v maybe taken in GQ
and L(Fb)Q respectively.
3.10. REMARKS.(1) The group G(Fb) is the quotient of L(Fb) by its
with the adjoint groupof L(Fb).
center,which is finite;it may be identified
Thus, if b # s, the groupG(Fb)c may be viewedin a canonicalway as a group
of N(Fb)c onto
definedover Q, in such a way that Cb inducesa Q-morphism
G(Fb)c,and that J(Fb) is arithmetic.The imagesof P n L(Fb)c and S n L(Fb)c
under Cb are a minimalparabolic Q-subgroupand a maximal Q-split torus
respectively.
of 3.7, that F is rationalif N(F)c
( 2 ) Theorem3.8 shows,independently
is definedover Q and conjugate to one of the groupsN(Fb)c with b # s. Our
originalproofof 3.7 consistedof 3.8 and of a separate discussionof the case
b = s. The proofof 3.7 given here,which is based on 1.20, was suggested
by [31].
3.11. PROPOSITION. We keep the notation and assumptions of 3.3 (ii).
Let Xb(g) =det Ad,,g (g e N(Fb)), whereu = u(Fb) is theLie algebra of U(Fb),
and let 7b bethe restrictionto Z(Fb) of thefunctional determinantJb. Then
thereexists a rational numbernb > 0 such that
(g e Ab* U(Fb))
(ge Z(Fh))
)7b(g) = Xb(g)-""
1)7b(g)
I = I Xb(g) j-'b
Using the notationof 1.16, we may write
Xb(g)
(
=laeoedet
)-1e
Adv, g,
(g
Xc(b,a)(ga)
= (g0), Ha = U(FC(b, )))
and similarly
T ?7c(b,0)(g0)
(2)
)2b(g)
By 1.16, thereexists a rationalnumbern,(bO) > 0 such that
(3)
1)c(b,a)(g)
I = I Xc(ba)(g)
I|nc(ba)
Z(Fe
(g G
(boa)))
(g G Ac(bp,)) ( 4)
)c(ba)(g) = Xc(b0a)(g)-nc(ba)
Let g = (g,) e QA. We have alreadyremarkedthat g, = qpo,(g), in the notationof 3.3; by the propertiesof the functorRk/q, this implies
(5)
det Ad,,g, =
X,(bp,)(g0)
(a e
; ge QA) .
The propositionwill thenbe a consequenceof (1) to (5) once it is shown that
of a. By 2.9, k(J is of typeC8 if and onlyif R4(DG') is of
nc(bo) is independent
Thus if kD is of type C, and b = ,
typeC,, and c(s, a) = t, forevery a (.e
e
If either
we are in case (i) of 1.15 for every a, hence n(b, a) = 1 (a G).
477
COMPACTIFICATION
b # s or ken is of type BC8, then,foreverya e E, we are in one of the cases
(ii), (iii) of 1.15. Then nc(b,a) is given by an expressionwhichdependsonlyon
of the restrictionsto AC(b,o) of a,,c(b,a) and 2Ca%,c(ba). In view
the multiplicities
made in 3.3, these are also the multiplicities
of
of the different
identifications
the restrictionsof 18b and 2.*1b to Ab,and theyare consequentlyindependent
of a. This provesour assertion;and, in view of 1.16, showsthat
(6)
nb
=
1
(7)
nb
=
(Pb
(keF
+
4qb)
*(2Pb
+
of type CQ and b = s)
otherwise,
4qb)1
of the restrictionsof lab and 2. Ib to the
wherePb and qb are the multiplicities
intersection
QAbof the kernelsof the simpleQ-rootsSi (i + b).
3.12. The functional determinant on Fb. Let qb (qc(b,U)0XE where
is the rational number> 1 attached by 1.17 to X2 and the boundary
qc(b,C
componentFC(b,O). Put
JF(9b(X),
g)
b
-=
tI
[
g)c(bc)(1bc(ba)
(Xo)
ga)
JqC(ba)
(g
= (g,)eAN(Fb))
If we apply1.17 to each componentFc(b,,) and use 3.11, we see that
(1)
5 Jb(X,
g)
I
I jF(Ub(X),
g)
b
I
Xb(g)
I-nb
,
(X G Sb;
g G N(Fb))
3.13. Let B be a normalconnectedQ-subgroupof N(Fb)c satisfying(iv)
of 3.6. We may write B as a semi-direct
productover Q of Sb. U(Fb)c by a
reductiveQ-groupH. WritefurtherH = H' *JDHas an almostdirectproduct
of its connectedcenterH' by its derivedgroup DH. We knowthatZ(Fb)/ U(Fb)
is the almostdirectproductof BR! U(Fb) by a compactgroup. Furthermore,
Z(Fb)/QAb U(Fb) moduloits derivedgroupis compact. This followsfrom1.3,
1.5, appliedto each factorN(FC(b,O)). It followsthen that BR/U(Fb) modulo
its derivedgroup is compact,thereforeHR' is compact. Since Jb and Xb are
bothcharacterson Z(Fb), theyare equal to one on ?DH, whence the equality
*QAb*U(Fb)) Let now F. be an arithmeticsubgroupof N(Fb)c. The group J' n H is
commensurablewith (r7 D H') . (rF nD?H), whereboth factorsare arithmetic
[13; 6.4, 6.11]. Since H' is compact,the group J' n H' is finite. Of course,
Xb takes the values ? 1 on Po. It followsthenthat the image of Fo n B under
Jb is a finite
group(of rootsof unity). This provesthereforethe
( 1)
-nb
~~~Xb(g)
-
Jb(Xg
(g G (9)H)
g)
3.14. PROPOSITION. We keep thepreviousnotation. Let F0 be an arith-
meticsubgroupof N(Fb)c. Then thereexists a positiveinteger d such that
Jb(X, g.y)d
=
Jb(X, g)d
(x E
X, g E N(Fb), 7 E ro nB)
3.15. PROPOSITION. Let G be as in 3.3 (i). Assume thatG has no normal
BAILY AND BOREL
478
Q-subgroupofdimension3. Theneveryproperrational boundarycomponent
has complexcodimension> 2.
It sufficesto prove this when G is simple over Q. If it is absolutely
simple,thenour assertionfollowsfrom1.5 (iii). Let now G be not absolutely
simple. Thenthe set E of 3.3 (ii) has at least two elements. By 3.7, a rational
boundarycomponentF is a product]l, Fo, where F, is a proper boundary
componentof XO. We have then dimcX, - dimcF, > 1 for each a; since
cardX _ 2, we are done.
4. Fundamental sets and compactification
Q-group,and P a minimalparabolic
4.1. Let G be a connectedsemi-simple
Q-subgroup. We writeP = M. S. U as in 2.2 and let QAbe the set of simple
Q-rootsforthe orderingassociatedto U. Since M centralizesS, we also have
P = S. V where V = M. U is the semi-direct
productof M and U.
We shall oftenwriteQA forS1. For t > 0, let
(1)
QA,= {aeQAI al < t(S eQA)} A
A fundamentalpropertyof this subset is given by the:
LEMMA. Let oi be a compactsubsetof (Ma U)R. Then the union of the
sets a *Ada-', (a E QAt)is relativelycompact.
Since S centralizesM and normalizesU, it is enoughto prove this when
v C UR. But then QA, is a bounded set of operatorson UR by a F->Ad a, in
of QAt. Since the exponentialis a homeomorphism
view of the verydefinition
of UR onto UR,our assertionfollows(see [13, Prop. 4.2]).
4.2. Let K be a maximalcompactsubgroupof GR whose Lie algebra is
orthogonalto that of SR, with respect to the Killing form,11the natural
projectionof GRonto X= K\GR,and o = z(K). A Siegel domain @5"= @"t
in GR (withrespectto K, S, and U) is a set of the form
=
e tff
(f
=
(oI compactin VR).
K. QA,*oi
In this paper,we shall in fact be more concernedwith the intersection
Af = (K n P) QA, *oi of (2" with PR, to be called a Siegel domain in P, and
.
with
e
=
et,@=
0O25
= 0o25Y
to be called a Siegel domain in X. If we replacethe sign ? by < in 4.1 (1),
and cvby an open relativelycompactsubset of VR,then we get open Siegel
domains in GR,PR and X. We note thatQAcentralizesM and that K n Pci M,
so that we may also write @2' = QA,*a' with c' = (K n P) *Ai.
479
COMPACTIFICATION
The Q-rootsare, in a naturalway, positive-valuedfunctionson PR which
are equal to one on VR. Thus we also have
(s G (2t; a
Spa<t
QA).
4.3. Let F be an arithmeticsubgroupof G. It is knownthatthereexists
a Siegel domain@2and a finitesubset C c GQsuch that a7 = @2*C is a fundamentalset forF in X; i.e., verifies
(Fl) 2.F = X, and
(F2) given x E GQ,the set of yE F forwhich&2xn go $ 0 is finite(the
Siegel property).
in GR,but this is clearly
7u-'(&2)
(See [20]. The resultis stated therefor&2' w
equivalent.)
If S = {e}, then X/F is compact,and conversely[13, 11.6, 11.4]. Of course
any compactset has the Siegel property(forany discontinuousgroup F, and
any x E GRin fact),so that in this case, any compactset verifying(Fl) is a
fundamentalset. Also, in this case, a(n open) Siegel domainis just a(n open
relatively)compactset.
REMARK.In consideringlater arithmeticsubgroupsof H(X), we shall
semi-simple
implicitlyuse a slightextensionof these resultsto non-connected
groups,whichis not stated in the literature,but reducesreadily to the connectedcase. Namely,if G' is a Q-groupwhose connectedcomponentis G, we
replace P and K by theirnormalizersin G' and G' respectively.Let r be an
arithmeticsubgroupof G'. The onlynon-obviouspointis to see that a Siegel
domainhas the Siegel propertyin G'. To see this, one uses the corresponding
resultin G and the followingfacts:
(i) G'Qis generatedby GQand N(P)Q, whichfollowsfrom[14, ? 4.13];
(ii) if h E N(P)R, then 2 h is containedin a Siegel domain(with respect
to K, SY P).
.
4.4. We now assume that G is simpleover Q, X is a boundedsymmetric
domain,F c G' and X/I is not compact. We take the notationand conventionsof 3.3 (ii). In each groupGO,we shall use the notationof ? 1. Putting
we maywrite,
(2)
(3)
=
(y
y,$)/2,
i+-7G
(1 < i _ to)
a,,,t, = 'to 9
(R"u
of typeCt,),
a,, tg = t'g /2 .
(ReDo
of type BCt,) a
In the productof the groups TSR, the torus QA is containedin the identity
BAILY AND BOREL
480
componentof the intersectionsof the kernels of the roots cay,(a, GRAE,
1 < i < t_ i not critical). An elementa
(a,) E QA satisfiesthereforethe
relations
(a e , 1 < i < to,i not critical)
(4)
ya ,(log a,) = y.,+1(loga,)
with the conventionthat y7t = 0 if i > to. Moreover,SR beingdiagonally
imbeddedin GR (see 3.3), we have
(5
=
)ate
aaa^i
a
1, . . Y ;ff5
c Uj)y i=
(i=
Therefore, a E QAt if and only if it verifies (4), (5), and
y,,o(loga,) - y,,?1(log a,)
(6)
?
(i
2 log t
c(j, a))
if c(j, a) # to, and
7o t,(loga,) _ log t (resp.7, t,(loga,) < 2 log t),
if tois critical;i.e., if to= c(s, a), and if R'D, is of typeC (resp. BC).
The subset QAt0c' = @2'n P is containedin everystandardparabolicsubgroup,in particularin the groupsN(Fb). From 3.8 and the constructionof
the followingassertion: For b # X,
the groupsL(Fb), we deduce immediately
et n L(Fb) is a Siegel domainof L(Fb) with respectto K n L(Fb), S n L(Fb),
and U n L(Fb); its image under-tr'bis a Siegel domainof G(Fb); and
n N(Fb))
Ob * Dtb
is a Siegel domainof Fb.
If b = s, then Fb/F(Fb)
LEMMA.
4.5.
@ of a Siegel
in
the union
(O
?
b
domain
?
s);
is compact,and Ob
We keep the assumptions
domain
'2, in the natural
of the standard
The intersection
Fbn
moreover,
obtained
= Ob
rational
e is equal
any
Siegel
t nbQ"
D N(Fb)) is compact.
of 4.4.
and notation
components
boundary
and
domain
in
@5 itself
Fb
The closure
of X, is contained
compactification
to Ob -bQ')
in this way by taking
b
Fb
is a Siegel
domain
in
in
a Siegel
is contained
sufficiently
(O _ b _ s).
Fb
large.
1, 2, ...) be sequences of elementsin QAtand o reLet a, and vp(2
assume
that vp- v and that limad - dj exists forevery
spectively. We may
j _ s. If all the dj are > 0, then ox a, v, tends to a point of 25itself. Otherwise let b be the greatestindexsuch that db = 0. It followsfrom4.4 that
(1 )
y,,o(log
a,,,
)-
(i < c(b, a), a El,)
DOc
and that 'y/,,(loga,,,) has finitelimits if i > c(b, a).
limosa,s
We have then
obka,
where a is the element of L(Fb) n e such that
is the identity if b = s. This implies
at
- dj
(j
> b) if b :z s, and
481
COMPACTIFICATION
limpc.o-a, -v, =
ob~a-v
E
?b'
TLb(QA,
-))*
Conversely,given x G 0b* b(QA*a), it is clear that we can finda sequencea,*vp
r o *a,*v,. Taking intoaccountthe next to last paraas above suchthat x lim
graphof 4.4, this provesthe lemma.
The followinglemmais the analogue,in the presentcontext,of Lemma1
in [33].
4.6. LEMMA. We keep the assumptionsof 4.4. Let @2bea Siegel domain
in X, C a finite subset of GQ, and &2= SAC. Then &2 is contained in the
union offinitelymany rational boundarycomponents. There exist finitely
many elementsA e r (1 < i < q) having the followingproperty:for every
- E F such that f2yn Li # 0, thereexists yj verifyingthe condition a waya y for everyacC 7-' n Q.
Fb. c(c E C)
By 4.5, &2is containedin the unionofthe boundarycomponents
whichare all rational. Moreover,if we put 2b = e n Fb, then we may write
Q=
UX ZA\ CA where X runs through a finiteset A, CAE C, and
ZA =
Zb(X)
for a
suitable b(X). Thus ZA is a Siegel domainof Fb(x) and has the Siegel property
(4.4, 4.5).
For each pair X, 4aE A forwhichthereexists yE 1 verifying
(1)
n
zD.c CIL
tCX7
0,
C-z'.
chooseone element7YA{E F fulfillingthat conditionand let dx(. = C .7
e N(Fb(X))Q. Let y be an element
We have b(X)= b(4a), A, = . C F (A), and dx(b
Co'. We have ex (y) G N(Fb(x))Q.
of F satisfying(1). We defineex,(y) = C,.7'*
7A
Condition(1) implies
.
(2)
-Z
*t3'b(d
I) .fb(eX(e))I
n. z
0
o
If b s, then Ox is relativelycompact,and has triviallythe Siegel property
for any discontinuousgroup. If b # s, then t-b is canonicallya Q-morphism
(3.10 (1)). Since c,*F c i' is arithmetic[13, ? 6.4] and Z, is a Siegel domain
(4.4), the Siegel propertyobtains. In both cases, it shows that there are
when y varies throughthe eleonlyfinitelymanypossibilitiesfor tb(eA(7))
mentsof F verifying(1). We have thus shownthe existenceof a finitesubset
D. of c-'(N(Fb(x))Q)c,, such that (1) impliesthe existenceof z(y) C DA verifying
x.zT(7)
-
Xy,
(X C SACA).
.y n a
0, there are only finitelymany possibilities
This meansthat, if &i2
Dforthe actionof on Do
n a, whencethe lemma.
'
4.7. LEMMA. We keep the assumptionsof 4.4. Let X* be the union of
the rational boundarycomponentsof X. Then thereexists a fundamental
482
BAILY AND BOREL
set & = SAC for P such that X* = E2P =f2-GQ.
By 4.6, we have f2c X*, and hence Q2.GQc X*, for every fundamental
set of the typeconsideredin 4.2.
By 4.2, 4.3, and 4.5, there exists a fundamentalset fl such that
Qfn Fb is a fundamentalset forF(Fb) (O _ b _ s). For this set we have then
FbcL( fl n N(Fb)). Let now F be a rational boundarycomponent. There
exists CeGQ and b such that F= Fb c (3.8). Let ye N(Fb) nI. By the
Siegel property,there exists a finitesubset D of 1 (dependingon y), such
that
cczLD
f 2.y-.
.
This implies
i2 C.c C2 D cz 2.J';
therefore,
F=
Fb.c cz&
f2
and X* c war.1,whichcompletesthe proof.
4.8. We now turnto the general case and let F be an arithmeticsubgroup of H(X) as in 3.3 (i). We take the notationof 3.3 (i). The unionX*
of the rational boundarycomponentsof X is the productof the similarly
definedspaces Xi* correspondingto the differentsimple Q-factorsGMof G.
The group H(X) operates continuouslyon the naturalcompactification
of X
(11.2), and henceH(X)Q operateson X*. Furthermore,(11.2), the restriction
of each elementh e H(X) to each boundarycomponentis holomorphic.By
4.7, appliedto X* and Pi = 1 D GqR,and 4.3, thereexists thena fundamental
setfnfor inXsuchthat F2.rP=X*. ForxCX*, we letP=I7={yc,x
7z=x}.
The space X* will be endowedwith the topology5(fl, F) in which a fundamental set of neighborhoodsof x C X* is the set of all subsets U of X* containingx and havingthe followingproperties:Usy= U if yC Fx and Usy Dn
f
is a neighborhood
of x .y in Q, in the naturaltopologyof &2wheneverx .y E .
This topologywas introduced,in a similarsetting,in [33], and will be called
the Satake topology for X*.
4.9. THEOREM. We keep the assumptionsof 4.8. The Satake topology
is the unique topologyon X* having thefollowingproperties:
( i ) It induces the natural topologyon X and on the closure of any
fundamentalset U for any arithmeticgroup 1 c H(X).
(ii) The elementsof the group H(X)Q operatecontinuouslyon X*.
(iii) If x and x' are not equivalent with respect to F, then there exist
neighborhoodsU of x and U' of x' such that u.r uf
0.
nU
483
COMPACTIFICATION
(iv) For each x E X*, thereexists a fundamental set of neighborhoods
E
and USAn U = 0 if 7 X~P.
{U} ofx suchthatUSA= U ifyGx,
PROOF. We wish to appeal to Theorem1' of [33]. We firstconsidera
fundamentalset as in 4.8, used to definethe topology. As was remarkedin
4.8, every elementof H(X)Q operatescontinuouslyon X*, viewedas a subspace of X in the naturaltopology,and this impliescondition(2) of Theorem
1', loc. cit. As to the condition(3), it is just Lemma 4.6. Thus, Theorem1'
applies. Also, by [33, Remark2, p. 563], the topologyinduced by s(aq, F) on
X is the naturalone. This proves4.9, withH(X)Q replacedby P in (ii).
Let now c C H(X)Q, and let &2be a fundamental set for c P. c-l n P. It is
also one forP and c P c-' in view of the Siegel property.It followsdirectly
and fromthe fact that c acts continuouslyon X, that c
fromthe definitions,
carriesS(c *P c-1,&2)ontoS(I, &2.c). However,as pointedout in [33, Remark
3, p. 563], these two topologiesof X* are identical,whence(ii).
In the sequel we shall also writeS forS(f2,P).
4.10. PROPOSITION. Let F be a rational boundarycomponentof X, and
x C F. Then x has a neighborhoodU in X* verifying4.9 (iv), and such that
a rational boundarycomponentF' intersectsU if and only if F c F'. The
closure of F in X* is the union of the rational boundarycomponentsconX of X.
tained in theclosure of F in thenatural compactification
We let S and 2 denote respectivelythe Satake topologyof X* and the
X. Let Li be as in 4.8. Then S and
topologyof the naturalcompactification
Tfcoincideon Q2. We may (and shall) assume that x -P containsan interior
pointof f2n F. It followsfrom4.6 that we mayfindfinitelymany elements
PP such that Q n rP= xo.n,... ,xcym}. Let Ui (1 ? i _ m) be
,. . .,m
a subset of X * such that Us.7n is a neighborhood of x yt in Q2. Then U =
ui UsurI7 is a neighborhood of x, which satisfies 4.9 (iv) if the Ui are suf.
small,and we get in this way a fundamentalsystemof neighborficiently
hoods of x in S [33, p. 562]. By 4.6, f2is the unionof its intersectionswith
finitelymanyrational boundarycomponents.We may thereforeassume UJ
to be such that if the rationalboundarycomponentF' intersectsUs.7r, then
x 'ytbelongsto the closureof F' n Qi.
Let now F' n u $ 0. There exist then y E FXand an index i such that
F' n ui
t
0; by the conditionjust imposed
0 , whence F' .7-1 *i n Ut y7
U7 y
eitherin
onUi, this impliesthat x*ytbelongsto the closureof F'.y'1 .* rt K?Q
3; or T sincebothcoincideon Qi.
Consequentlyx belongsto the closureof F' .ye, and henceto that of F',
in bothS and 2. This provesour firstassertionand shows that if x belongs
484
BAILY AND BOREL
to the closure of F' in 5, then it does so in if,too. To provethe converse,
we mayassume G to be Q-simple,and F = Fb to be standard. ThenF, (c ? b)
belongsto the closureof Fb in boththe 5 and the iFtopologies.
Let F' be a rational boundarycomponentcontainedin the 1f-closure
of
F. There exist c ? b and g c N(F)Q such that F'.g = F, (3.8, 3.9). Since
in the 3-topology(4.9 (ii)), it followsthat F' is
g definesa homeomorphism
also in the s-closureof F.
4.11. COROLLARY. The quotientV*
X
X*/F,
endowedwith the quotient
topology,is a compact Hausdorffspace. V= X/V is an open everywhere
dense subset. V* is the finite union of subspaces Vi = Fi/F(Fi), where Fi
runs througha set of representativesof equivalence classes modulo F of
rational boundarycomponents. The closure of Vi is the union of Vi and of
subspaces Vj of strictlysmaller dimension.
The firstthreeassertionsare obviousconsequencesof 4.9 and 4.10, once
it is shownthat, whenG is Q-simple,Fb.H(X)Q/F is covered by the images
of finitelymanyrationalboundarycomponents.Let 2 = . C be the fundamentalset of 4.7, where(25is a Siegel domainand C a finitesubset of GQ.
Let F be a rational boundarycomponentof typeFb. There exist c e C and
/e F such that B c .k n F # 0. But the intersectionof e with the orbit
of Fb under G' is containedin Fb by 4.5; therefore,Fbhc.7 n F # 0, and
Fbhc = F, so that Fbh H(X)Q/F is a quotientof Fbh C.
The last assertionof the corollaryfollowsfrom4.10, because in X, the
closureof a boundarycomponentF is the unionof F and of boundarycomponentsof strictlysmallerdimension(1.5).
.
REMARK.The properrational boundarycomponentscorrespondto the
propermaximalparabolicQ-subgroups(3.7). Since parabolicQ-subgroupsare
conjugate if and only if they are conjugate by an elementof GQ(2.2), the
withthe orbitsof F in
Vi's of the corollaryare in one-to-onecorrespondence
the
P
runs
maximal
standard
parabolicQ-subgroups.
GQ/PQ,where
through
Let G be the quotient of the symplecticgroup Sp(2n, C) by its center,
and F = Sp(2n, Z) be Siegel's modulargroup. The propermaximalparabolic
Q-subgroupsare the stabilitygroupsof the rationalisotropicsubspaces. It is
elementarythat two rationalisotropicsubspacesof the same dimensionq are
of each otherby an elementof F, so that we have GQ= F *PQ for
transforms
everymaximalparabolicQ-subgroup. The boundarycomponentcorresponding to such a subspace is isomorphicto Siegel's upperhalf-planeof degreeq,
and thus V* =flo~q
Vq,where Vq is the quotientof the Siegel upper half
plane of degree q by the correspondingmodulargroup. We get therefore
485
COMPACTIFICATION
again the compactification
introducedby Satake and later consideredin [35].
4.12. TruncatedSiegel domains. In order to give a more precise descriptionof a fundamentalset of neighborhoods
of a pointin V*, we need to
introducecertain subsets of a Siegel domain. To definethem, we assume
again, forconvenience,that G is simpleover Q.
Let Y' = QAt*GObe a Siegel domainin PR and e = on Y'the corresponding
Siegel domainin X (4.2). Fix an index b ! s. For a positivenumberu and a
subset E C Fb, we let
(1)
{ 6E
eb(u,
E)
(2b(u,
E) = ofe(25(, E) .
5
gSb
=< , Ob'Se
El
and
( 2)
The sets describedby (1), (2) will be called Fb-adapted truncatedSiegel
domains. The subscriptb will sometimesbe omitted,if it is clear fromthe
context,or replacedby F.
An element$ e (' can be writtenuniquelyin the form
(3 )
al*hb a2.w
(a,
e 9Z(Fb) n QA,a2, L(Fb) n QA,hb XQAb, W Xo)
where QAbis the kernel of all simpleQ-rootsSi (i # b). Of course,all elementson the righthand side depend(continuously)on s. However,for simplicity,we shall not make this explicitin the notationas long as no confusion
arises. We collecta few remarksabout this decomposition.
( i ) The element hb annihilates all simple Q-roots except Aib, and a,
(resp. a2) annihilateshi fori > b + 1 (resp. i < b); hence,
84)
85)~~~~~~~~-
sjki- alsi
(i <b-1
i
(i >b +l)
a'i,
If the roots fi (i > b + 1) are multiplicativelybounded on a2, then a2 is
bounded;hence,thereexists t' such that the set of productsh al is contained
in QA,'. In view of 4.5, it is clear that the fi's (i > b + 1) are multiplicatively
boundedif E is relativelycompact,as will be the case unless the contraryis
stated. If so, the discussionof 4.5 implies
((6)
fl>05'(u,
E)
o=
h.b
(uo,
E)
(some us > 0)
(ii) The groupQAbcentralizesK' =KK PR, and is containedin the isotropygroupof 0b. Since 25' K'* QAt, (o ci VR),we have then
( 7)
and hencealso
hb.2(u, E)
Ci ((v,
E),
(v = uhIb, he QAb)
BAILY AND BOREL
486
Ob 0
Ob.Y'(u,
E)=
= Obh5(u,
Ob.2'(V,
Fh)
E)
(u >
0)
(u, v > 0).
(iii) Let (Xi) be the basis of QQ with respect to which the simple Q-roots
are written as in 1.2. Then the Lie algebra of QA n gZ(Fb) (resp. QA n L(Fb),
,b-1) (resp. Xi (i = b+ 1,* s),
resp. QAb) is generated by Xi - Xi,, (i = 1,
+ Xb).
resp. Xi +
...
4.13. LEMMA. We keep the assumptionsof 4.4. Let x e Fb, and let x*
be the image of x in V*. Let 2 = 5.C be a fundamentalset verifying4.7,
wheree is an open Siegel domain whose closure contains x. Then there
exist finitelymany elementsei e N(Fb)Q with the following property:the
image U in V* of U' = Ui c2(u,Ei) ei, where Et is a relatively compact
neighborhoodof x-e-1 in Fb, is a neighborhoodof x*; the set U (resp.
U'. *F) describesa fundamental set of neighborhoodsof x* in V* (resp.
x e X*) if u 0 and, independently,Ei runs througha fundamental set
of xe, 1 in Fb. Furthermore,we get an equivalent set of
of neighborhoods
of x* if we replace @Nbyany open Siegel domain @YD ( 5.
neighborhoods
-
It is clear from4.5 that b(u, E) containsan open neighborhoodof x in
set ofneighborhoods
(b, and 4.12 showsthat we get in thisway a fundamental
of x in 5. We let D be the finiteset of elementsc e C such that x ?e 25 c.
of x in
If d e D, then we get similarlya fundamentalset of neighborhoods
of xd-' in Fb.
@5d by takingsubsets 2b(u, E)-d, whereE is a neighborhood
Since Qi is the finiteunionof the closed subsets C, (c e C), it followsthat x
has a fundamentalset of neighborhoodsin Q2of the formUd (b(u, Ed) d,
of x *d-1.
whered runsover D and Ed is a neighborhood
F Q2is finite. Let yj (1 < j ? q) be elementsof r such that
By 4.6, x r n
x.1 n Q2 {x*Yi, . . *, Xq}, and put x 1j xj. Let J' be the isotropygroup
of xj in Q,
of x in r. From 4.10, it followsthat if Uj3.j is a neighborhood
of x in X *, and that we get in this way a
then Uj Uj31,,is a neighborhood
fundamentalset of neighborhoods.For Uj, we may bythe above take a finite
unionof sets 5b(u, E) *c, wherec e C is such that xj e *C, and E is a neighborhoodof xj *c-1in Fb. Fromthis,our firstassertionfollows,withei running
. Such elementsare rac
over the set of productsc * j-1forwhichx .j c-1
tional over Q, and theybelongto N(Fb), sincetheytransformx intoa point
of Fb.
As forthe secondassertion,the imagesof the sets Uj Uj, whichare constructedsimplyby replacing(25by (0' whileallowingc and yj to run over the
of Uj, are relativelycompactneighsame sets of objects as in the definition
borhoodsof x*, and foru - 0 and Ejs, decreasingthrougha basis of compact
487
COMPACTIFICATION
of xj c-1 for each j and c, we obtain a decreasingfamilyof
neighborhoods
compactneighborhoodsof x*; it followsfrom4.12 (6) that the intersection
of all these is just x*. Since V* is compactand Hausdorff,and since all the
spaces we considerare secondcountable,it followsthat these neighborhoods
of x* in V*, whichprovesthe last assertion.
also give a basis of neighborhoods
REMARK. Let En (n = 1, ***) be a decreasing sequence of neighborhoods
of x in Fb, whose intersectionis x, and let us be a sequence of real numbers
tendingto 0. Then the images in V* of the sets Us - Ui 2(u, En ej ) et
of x*.
forma fundamentalset of neighborhoods
4.14. LEMMA. Let b (1 ? b ? s) and a Siegel domain @5'in PR be given.
' D@5
in PR such that (b(u, E) =
Then there exists a Siegel domain
o.2'(u, E) is connectedfor every u > 0 and everyconnectedopen subsetE
of Obh*
PR= K'. A. N of PR where
We choosean Iwasawa decomposition
K'
A=(MfnA)xQA,
KnfP,
N-(MnN).UR.
We mayassume as beforethat QT C RT c T, and choosecompatibleorderings
on the root systems. The groups Z(Fb) n A.N and L(Fb) n A. N are connected (for otherwisethey would have infinitelymany components,which
is impossiblesince Z(Fb)c, (A.N)c, and L(Fb)c are algebraic); theyare also
of
contractible,and the productmappingyieldsa homeomorphism
(L(Fb) n A.N) x (Z(Fb) n A.N)
ontoA.N. Furthermore(L(Fb) n A) and (L(Fb) n N) are the A and N part
of an Iwasawa decompositionof L(Fb). It followsthat (a, n) - oval n and
of A x N ontoX and of
(a, n) -+ 0b* a *n yieldhomeomorphisms
(L(Fb) n A) x (L(Fb) n N)
ontoFb. We also have (a meaninghomeomorphism)
L(Fb) n A.N
Z(Fb) n A.N
(QA n L(Fb)) x (L(Fb)
(QA n Z(Fb)) x (Z(Fb)
n VR n A.N),
n VRn A.N),
are given by the productmapping.
wherethe homeomorphisms
Let us write 8' = K'*QAto*WO, (booc VR). We take t > to,and choose
open, relativelycompact, and connected in L(Fb) n VRn A-N and
(02,
ol
Z(Fb) n VR n A-N such that K'*01w(2 D K'-*0o, (note that K' meets every
connectedcomponentof PR). We claimthat c2'= e',., ((w= K' *O
01 (02), verifies
our conditions.Since K' is containedin the isotropygroup of Ob in PR, and
is the isotropygroup of o in PR, an elementaryargumentshows that it
sufficesto prove that K'\S'(u, x) is connectedfor every u > 0 and every
BAILY AND BOREL
488
X G Ob 25%.
We may write
x =
E
(a e L(Fb) n QA, Uw?o)
ob-a*aUw
An arbitraryelements E 2' can be writtenas a product
s
k'.al*hb*a2,*w-w2,
where a,, hb, a2 are as in 4.12, and k' e K', wi e oi (i = 1, 2). The remarks
made above implythat Ob*S x if and onlyif a2 = a', w2 w'. Thus w1runs
throughcoN,whichis connectedby assumption,k' throughK', and a,-hbruns
throughthe elementsy e Z(Fb) n QA satisfyingthe conditions
Yei< t (i < b),
y1 <
U. t,
whichdefineclearlya connectedset; whenceour contention.
We keep the notation of 4.11. Every point v* G V*
has a fundamental set of open neighborhoods U such that U v iis connected.
4.15.
PROPOSITION.
Let F' be a normalsubgroupof finiteindex of F. Then V* -V*(F) =
X*/F is the quotientof X*/F' by F/F'. Let Gi (1 < i < m) be the simpleQfactorsof G and, as in 3.3 (i), let F' be thegroupgeneratedby the F - GR n r.
Then X*/F' fi.X*/Fi. It sufficesthereforeto considerthe case whereG is
assume x E Fb. We use the
simpleover Q and F c G'. We mayfurthermore
notationof 4.13, 4.14, take 80 with C5 verifying4.7, and 2, containingx.
We choose the elementsej E N(Fb)Q as in the proofof 4.13; in particular,
x eT'lE o 25' Moreover, we may so arrange things that each xei' is contained in the (relative)interiorof Fbnfl ; this is an easy consequenceof the
two followingfacts: the fixedset C of 4.13 is finite,and the numberof elements in the intersectionof any orbit of I in X* with the closureof any
Siegel domain(constructedwithrespectto the given torus,ordering,etc.) is
finite.
Let E be a connectedopen neighborhoodof x in F, small enough so that
Ee7-1 cQOb ( for all i. Given u > 0, there exists g06 Ofb(U, E) such that
hence (loc. cit.),
x = obg0 (see 4.12 (ii)). By construction,ob-g90e7i E ?b(o;
in
the
subgroup
isotropy
therefore
there
exists
EeT-1);
ui
obey
E
b(u,
Ob
of Ob in N(Fb) such that Ui g90ei-' EOb(U,
EneiT).
are
maximalcompactin L(Fb) and
K
and
Z(Fb)
K
L(Fb)
n
The groups n
Z(Fb), and the firstone is the isotropygroupof Ob in L(Fb). Moreover,
Z(Fb)
-
(K
n Z(Fb)) *(P n z(Fb)) .
Since N(Fb) = L(Fb) . Z(Fb) by 3.3, we may write(not uniquely):
-i =
kf.kf'si
(kfe K n L(Fb); k1'e K n Z(Fb); si E Z(Fb)
n P)
489
COMPACTIFICATION
Let Y' be an open Siegel domainin PR satisfying4.14 whichcontains 8' and
the elementssi.go. The imagesin V* of the sets U U- gb(u, E'.e- 1).e,, as
u runs over all strictlypositive numbers,and E' over a basis of connected
open neighborhoods
of x containedin E, forma fundamentalset of neighborhoods of v*, and their interiorsforma fundamentalset of open neighborhoods. It is then enough to show that the interiorof U n x is connected.
This latterset is the unionover i, of the open sets o-.2(u, Eves1)*ei. We are
thereforereduced(4.14) to provingthat
(1)
0 ,
?o.gtb(u, E' . et1). et fl ?o. (u, E')
forall u > 0, all i, and all E' subject to our previousconditions.
We let h be an elementof QAbsuch that hVb ? t'1. U.
0 x and si*goe 2'; hence,si goe 25(t, E'), and
By construction,Ob* Si * go
therefore(4.12 (ii)):
h si go e 2(u,
( E')
(2
By 4.12 (ii), we have
u i -go-e,1E o * b(2(u, E'* et1).
But QAbcentralizes L(Fb) and K n Z(Fb); therefore,
o ~h~ui~g,.e-1
whence
(3
-
o-h-k'-*k"'si.gey1
o
oh Si goe
(25'(u, E'
-
o.h.si~.g0.e
*e- )-es,
which,togetherwith (2), proves(1).
4.16. PROPOSITION. We keep the previous notation. Let J(x,g) be the
functional determinantfunctionin the unboundedrealization associated to
the rational boundarycomponentF = Fb (3.3) and g X N(F)Q. Then J(x,g)
is multiplicativelyboundedon (F, where (F = 2b(U, E) (E compactin F)
is an F-adapted truncatedSiegel domain.
We have
( 1)
(2F =
o - )F
and
J(x,g) = J(o*S, g) =J(o, S) *J(o,S *g)
(s
F)
We want to estimate both factors on the right hand side. We write as in
4.12,
(2)
S
= a1.h.a2.q
)Z(F) nQA, a2e L(F) nQA, h eQAb,qua)
(ale
As pointedout in 4.12 (i), since s
thereexists t' ? t such that
(3)
eF(,
the elementa2 is boundedon
h-a, e QA,
(F,
and
BAILY AND BOREL
490
From (2), we get
(c = a2.h-a,.q((h-aj)-')
= ceheal
=
ahahqa(h-a1)lha,
S
hence,by 1.11,
J(o,s) = J(o,c)J(o, h).
(4)
By 4.1, and the above remarkon a2, the elementc varies in a compactset
when s
(5)
e (;
hence,
(S G (F)
J(o, s) J(o, h)
gives
The Bruhatdecomposition
whence
(U V E UQ,t E Z(S)Q,W E N(F)(S)Q)
g=u*t*w*V,
(d
s.g = d-f-v
v
-
c-h-a,.u.(h-a4)-1t-w;f
= w-1h-aj-w).
Since c varies in a compactset, 4.1 and (2) showthat so does d. We maywrite
(V' E L(F)
V- V V
n U, v"fE Z(F) n U)
f8}).
The elementv' commuteswiththe subtorusZ(F) n S = SO(0 - {ib+l9
Since 0 is a connectedcomponentof the set QAZ- {fb} of simple Q-rootsof
N(F), the Weylgrouprelativeto Q of N(F) also leaves S0 stable. Therefore
v' commuteswithf, we have s-g = dv'-f-v", and, by (1.11),
J(o,s g) -J(o, d -v') -J(o,f) J(o,v") .
(6)
.
But v' and v" are fixedand d varies in a compactset. Consequently
J(0,S g) AO,f ) (
( 7)
.
.
F)
The restrictionof J(o,g) to Z(F) is a character(1.8, 1.11), whichis of course
the relativeWeyl group of
equal to one on the derivedgroup. Furthermore,
withrespectto the hyperplanesannihiN(F) is generatedby the symmetries
latingthe simpleQ-rootsSi # sub,and hence it acts triviallyon QAb. We have
therefore
J(o,f) = J(o,h),
( 8)
and the propositionfollowsnow from(1), (5), (7), and (8).
II. AUTOMORPHIC FORMS
5. Poincare series
5.1. In this section,G is a complexconnectedreductivegroup defined
over R, whose group of real pointshas a compactcenter,H, a subgroupof
finiteindexof GR,K, a maximal compactsubgroupof H, V, a finitedimensional complexHilbertspace, and p: K
GL( V) a unitaryrepresentation.
-
491
COMPACTIFICATION
A functionf: H
V is of type p on the left (resp. right) if f(k -g)p(k).f(g) (resp.f(g k) = p(k-1).f(g)) (g e H, k e K). It is of finitetype,with
respectto K, or K-finite,on the right(resp.left) if the set of righttranslates
rkf (resp. left translates 1kf) of f by elements of K spans a finitedimensional
vectorspace over C of V-valuedfunctions. This is in particularthe case if f
is of type p, and the generalcase can be subsumedintothat one, at the cost
of changingV, as is easily seen.
5.2. The universalenvelopingalgebra G1t(g)
of g is identifiedin the customarymannerwith the algebra of rightinvariantdifferential
operatorson
H. In particular,if X e g and f is a differentiable
V-valuedfunction,then
Xf (g)
-
(dfeetfx.g))
The center2(g) of 4t(g) is thenidentifiedwith the algebra of left and right
invariantdifferential
operatorson H. A smoothfunction(or a distribution)
is called Z(g)-finite
if it is annihilatedby an ideal Gj{ of finitecodimensionof
Z(g). If Gj{ has codimensionone, such a functionis an eigenfunction
of Z(g),
whichis our maincase of interest. It is knownthat if f is Z(g)-finite,
and is
K-finiteon the right (resp. left), then it is annihilatedby an ellipticright
(resp. left)invariant,henceanalytic,differential
operator,and is consequently necessarilyanalytic. However,since analyticityis obvious for the functions to be consideredlater, we omitthe proof. The convergenceproofsfor
the Poincareseries could be based on that fact (and indeed are in [35, Exp.
10]). Here, we shall use instead the followingresult(in which,in fact, the
assumptionson H maybe slightlyrelaxed) of Harish-Chandra[23, Th. 1].
5.3. LEMMA. Let U be a neighborhoodof the identity in H. Let
f: H a V be a C-'-functionwhich is of finitetypeon the right (resp. left)
with respectto K, and is Z(g)-finite. Then thereexists a X Cc(U) such that
a(k. g. k-1')= a(g) (g E H, k E K) and thatf = f *a (resp. f = a *f).
As is usual, Cc refersto C--functionswithcompactsupport,and * stands
forthe convolution.Thus in particular
f *a(g) = |f(g *h-') a(h)dh
*
=
f(h) *a(h- g)dh ,
wheredh is a Haar measureon H, chosenonce and forall.
5.4. THEOREM. Let F be a discrete subgroup of H. Let f: Ha V be a
function which belongs to L'(H) (0 V, is Z(g)-finite, and is of finite type on
the right,with respectto K. Then the series
pf(g) =
Syerf
(g *7) ,
PAlf
1(g) =
:yer
f (g
If
I)
BAILY AND BOREL
492
are absolutelyand uniformlyconvergenton compactsubsets,and are bounded
on H.
It is enough to prove this for Plifll. Since F is discrete, we may find a
symmetriccompact neighborhoodU of e such that U2 n Fp {e}. By 5.3,
thereexists a E Cc( U) such that
f(g y)
h-').a(h)dh;
\f(g
H
(g E H, y E F)
this can also be written
f(g
)
f(g h-')a(h * )dh.
=
Let M be the maximumof Ia I on H. Since a has its supportin U = U-', we
have then
AM.5
IIf(g.y)
(1)
But Us D
n uri
1f(g h-1) *Idh.
0 if Y t a, hence
Adze f(g a) I11< M|f(g
dh < Me |f IIL1
h-1)11
which proves that PlIfl,is bounded on H.
Let now C be a compactsubset of H. Givens > 0, thereexistsa compact
subset D of H such that
(2)
X
H1-D
1f(h)1dh
e.
The inequality(1) can also be written
< M|
1f(g Y) IMI
(3)
f(h) 11dh
.
Let * be the set of elementsye F for which C.Y. uD D # 0; it is finite.
Given g E C, the translates g -Y U(y E F, YX *) are pairwisedisjointsubsets
of H - D, hence
.
Eyer-+
If(g
xY) || <
M.5
1f(h) || dh < Mat
(gC C),
fromwhichthe uniformconvergenceof Plifl,on C follows.
The above proof is due to Harish-Chandra. Our
originalargumentwas longer,and was a variation on one of Godement's
[35, Exp. 10].
( 2 ) If f is of finitetypeon the left,and satisfiesthe otherassumptions
of the theorem,then a similar argument,or the one of Godement,shows
5.5. REMARKS. (1)
493
COMPACTIFICATION
readilythat pf is absolutelyand uniformly
convergenton compactsets. However, it does not seem necessarilytrue thenthat pf is bounded.
5.6. The seriespf,wheref satisfiesthe assumptionsof 5.4, will be called
a Poincare series. Our next aim is to show that the usual Poincareseries on
boundedsymmetric
domainsare associatedin a simpleway to Poincare series
in the above sense.
Up to the end of this section,we assume that X
K\H is a bounded
symmetric
domain,let H = Go, and use the notationof ?1. Let, further,
,cc(x,g) = (elx *g)oE KcI
(g EH. x ED)
be the canonicalautomorphyfactorof the boundedrealizationD of X (1.8),
and let
(1)
ep(x,g) = p(P(x, g))I
where p also denotes the natural extensionto Kc of p. Since 1a(x,k) = k
(k E K), we have in particular,
fpp(o,
k) = p(k)
( 2)
To a function F: D
V, we associate f: H
(k E K) .
V by
(3)
f(g) = ,cj(o, g).F((g))
From (2), and the cocycleidentity(1.8), it followsthat
f(k-g) = p(k)*f(g)
( 4)
(g E H) .
(k EK, geGH)
It is easily seen that F e f is in fact a bijectionof the set of V-valuedfunctionson D ontothe set of V-valuedfunctionson H whichsatisfy(4).
5.7. LEMMA. We keep the notationof 5.6. Then
(a) thefunctionF on D is holomorphicif and onlyif Yef = 0 for all
YE Ph-;
(b)
(YeE-).
Let f: H
V be a function which satisfies 5.6 (4) and Y-f = 0,
Thenf is Z(g)-finite.
The secondassertionis provedin [35, Exp. 10, pp. 6-8]. As a matterof
of Z(g) if p is
fact,it is onlyexplicitlystated therethat f is an eigenfunction
irreducible,but the proofalso yields(b).
Part (a) is also known,and mentionedin [35, Exp. 10, p. 6]. For the sake
of completeness,we indicatea proof. If we view g as a Lie algebra of differentialoperatorson D, via the action of Go, then the elementsof pr are
the linear combinationswith constantcoefficients
of the partial derivatives
in
it is enoughto show
the
are
coordinates
Therefore
where
zi
a/aii,
p+.
(1
Yf(g)A= "jo
-e YF7{(Czg)
g)
(Ye2 p-; gye2
BAILY AND BOREL
494
Let firstYe gR. We may write
(Y+e?+; Yoefc)
Y= Y_+ YO+ Y+
(2)
where,obviously
Y
(3)
(+d(ety)o).
We have
Yf(e) = d (",(o, OF')-F(o eOF))|
dtt=
hence,using (3), and denotingby dp: gc
-?
of p,
gl(V) the differential
Yf(e) = dp(Yo)*F(o) + Y-F(o).
(4)
By linearitythis formulaextendsto all Ye g. If Ye p-, then YO= 0, and (4)
yields(1) forg = e. Now the correspondence5.6 (3) associates to the right
translatef ' = r, -.f(ge H) the functionF' given by F'(x) = ,p(x,g) -F(x -g).
in x, we have YF'(o) = "e(o,g)*YF(o g); on the other
Since ," is holomorphic
hand Y(r,f )(e) = Yf(g), hence (4), appliedto f ' and F', yields(1).
.
5.8. LEMMA. Let J(x,g) be thefunctionaldeterminantfunctionin the
boundedrealization of X. Then g v- J(o,g)a is in L'(H) for every integer
a > 2.
We know that J(x, k) = det Adp+k-' is a scalar independentof x, of
modulusone (k e K). By the cocycleformula,g I J(o,g) la is thereforeleft
and rightinvariantunderK, and in particularmay be viewed as a function
on D. We have
-
|I
GR
J(o,g) la dg =
D
I J(o, x) 1adx
,
wheredx is a suitable invariantvolumeelement. Up to a positivefactor,
dx =J(o,
x)
K ),
whereoj is the euclidean volume element in P+. The domain of integration
beingbounded,it is thenenoughto showthat IJ(o,x) I is boundedon D. Since
to checkthis on o-A, whereit followsfrom1.12.
GR = K-A-K, it suffices
5.9. Let X = X1 x
...
x Xq be the decomposition of X into irreducible
domains. The Xi's correspondcanonicallyto the almost
boundedsymmetric
simple,non-compact,almost direct factorsof the derivedgroupof GR,and
are thereforestableunderGR. Each g e GOinducesa complexanalytichomeomorphismgi of Xi such that
(x1, * * *, xq)*g
= (x1*g1, * * *,
xqgq)
(xi G Xi)
495
COMPACTIFICATION
LettingJi be the functionaldeterminantfunctionin the canonical bounded
realizationDi of Xi, we have
J(X'g) = Hi Ji(xi,gi).
If a = (a1, *.,
aq) is a sequence of integers, defineJa by
J(x, g)a = Ii Ji(xi,gi)ai
Let F be a discretesubgroupof H. Let Ap:D
V be a polynomialmapping.
Put
Pq,(x) = P(x)
(1)
=
yer J(Xy
)a-9(Xy).
Up to the fact that (forlater use) we allow a multi-exponent
a, this is just a
Poincare series in the usual sense. If it converges,it representsan automorphic formof weighta, i.e., it verifies
-p(Xwr)xEX;
Er
p(X) = J(X,_Y)a
as followsfromthe cocycleformula.To p we associate,as in 5.6, the function
V definedby
f: H
( 2)
f(g) -
J(o,g)a.9q)((g))
(g
c H).
Then by the cocycleformula,we have formally
P
(3)f(g)
-Ee
J(o,g)a.(p(g))
f(g')=
As before,let
(4)
Plif
i(g)
IIf(g Y) II
=yer
5.10. THEOREM. Assume ai > 2 (i
=
1, - **, q).
Then the series pf is a
Poincare series in the sense of 5.6. ConsequentlyP.,,pf and Pllfl,
converge
absolutelyand uniformlyon compactsets, and Pllfliis boundedon H.
By 5.6 (4), f is of finitetypeon the left. Together with 5.7, this shows
that f is Z(g)-finite.Since p is a polynomialmappingon D, it is bounded;
hence,f is in L'(H) 0 V by 5.8. We have
J(o,g-k) = J(o,g) l-
(det Ad k-l)ai
(k
= (k1,** kq),kiE Ki)
On the otherhand,since k acts on D by means of a linear transformation,
namely Ad?+k-1, it transformsp into a polynomialmappingof the same
degree, hence the set of transformsof p under K is containedin a finite
dimensionalvectorspace. It followsthenthat f is K-finiteon the right,too.
It satisfiesthereforeall the assumptionsof 5.4.
5.11. We concludethisparagraphwithsomeremarksto be used in ? 10, in
the applicationof our mainembeddingtheorem.Let v be the one-dimensional
BAILY AND BOREL
496
k I det Ad,+k-' of Kc. Then,in the notationof 5.6, the autorepresentation
morphyfactor," is just the functionaldeterminantJ in the boundedrealizationof X. Lettingp(m) (m E Z) stand forthe tensorproductp 0 v-, we have
(x EX, g EH) .
g) = J(X,g)- .,",e(X, g) ,
A of a finitedimensionalHilbert space we let
For a linear transformation
I A I2= Tr (A* .A), whereA* is the adjoint of A. We claim that, given p,
there exists m. such that the functionS: g
g) belongsto L1(H)
fep(m)(0,
p
( 1 )
",e(.)(X,
for all m > mi.
PROOF. We have
(g E H. kgk' E K nH) ,
p(o, k~gk') = k-p(o, g).k'
( 2)
so that 8 is rightand leftinvariantunderK. Clearly,fp(n)
=
Jn-ffip(m),
(n, m E Z); therefore,as in 5.8, it is enoughto show that S is boundedon A
form big enough. For a E A, we have
(3
()=
"(o, a)
=ao
IIexp
(log cosh yi*Hi),
(a = exp (y1X1+ *** + ytXt)),
as was recalledin 1.12. The transformations
p(m)(ao) are simultaneouslydiagonalizable,and theireigenvaluesare of the forma', wherea runsthrough
the weightsof p(m). These are the sumsm.op + X, whereX runsthroughthe
weightsof p. From 1.10, appliedto each irreduciblefactorof X, we see that
>(Hi) < 0 forall i. Consequently,thereexists mosuch that &(Hi) < 0 for all
i and all weightsa of p(m) if m > mo By (3), S is thenboundedon A.
V is a polynomialmapping,the series
As in 5.10, it followsthat if p: X
-
ETer
q(x .a)
J(X, y)m UP(x, g) .9
,
is a Poincareseries form > mo, and any discretesubgroupP of H, to which
the conclusionof 5.10 applies.
6. Poincare-Eisensteinseries
Q-group,H a subgroup
6.1. In this section,G is a connectedsemi-simple
of finiteindexof GR, K a maximalcompactsubgroupof H, V a finitedimenpF
sional complex Hilbert space, P a parabolic Q-subgroupof G, and X0:
det Ad,,p (p E P) the determinantof Ad p in the Lie algebra of the unipotent
radical U = R.(P) of P.
Furthermorewe assume P to be in the standardform(2.2). Thus P= QPo
where 0 c QA. Let 0' = QA - 0. We have then (2.3):
eada
x0=ZJae,9
where the
da
(eaonQ, ea>0),
are fundamentalhighest weights for P. For every set s,
497
COMPACTIFICATION
(5)aeo' of complexnumbers,we let A(p, s) be the complexvalued function
on PR definedby
(1)
Let f: H
(2)
-
Id.(p) I-s
A(p, s) ll
V be a continuousfunctionwhichsatisfies
f(g p) = f(g).A(p, s)
(g e H p EP nH).
Let F be an arithmeticsubgroupof G, containedin H, and rho
a subgroupof
finiteindexofr n P. The series
(3 )
Ef(g)=elro-g7
(g eH)
is called an Eisenstein series. It follows from a theorem of Godement
(unpublished;for a sketch of the proof,see [12]) that this series converges
absolutelyand uniformly
on compactsets if
(4)
(aced').
Rsa>ea
We note that since a rational character,definedover Q, takes only the
values + 1 on an arithmeticsubgroup,(2) impliesthat f is rightinvariant
underFO, so that the summationin (3) makessense,and Ef is rightinvariant
underF. We shall need the followinggeneralizationof this result.
6.2. THEOREM. We keep the notation of 6.1. Let f': H
V be a continuousfunctionwhichis rightinvariant under Ji,,.
and such that
-
m(g) = supp I1f'(g p) II. A(p, s) 1-'
(P E P nH)
is finitefor everyg E H, and is boundedon compactsets. If s verifies6.1(4),
then
(1)
EfZ
=
ye rv fr'(xf
y)I
convergesabsolutelyand uniformlyon compactsets.
We onlyshow how this reducesto the Godementtheorem. Replacingf'
by Iif ' f
I we may assume the s, to be real, and f ' to be a real-valuedpositive
function. Let f " be a strictlypositivecontinuousfunctionon H such that
f"(g p)
f"(g)-A(p, s)
(g e H, p e PRnH)
Such functionsobviouslyexist; we may for instance simplyput f"(k -p) =
A(p, s). Since A is trivial on P n K, this is legitimate,and definesthe required functionon H = K- (P nH). Writingg = kep (k X K, pE Pn H), we
have
y'(g)
- f
( "(g))-' = f '(k . p) . A(p, s)-1.i 'k-
f " has a strictlypositiveminimum
on K, and f '(k-p)A(p, s)-1remainsbounded
when k runsthroughK and p throughP n H by assumption. There exists
BAILY AND BOREL
498
thereforea strictlypositiveconstantc such that
(g E H)
f'(g) _3I
c*f"(g)
whencethe reductionto the case of 6.1.
We now introducea generalizationof Eisenstein series and Poincare
series.
series,to be called Poincare-Eisenstein
6.3. Let B be a normalconnectedQ-subgroupof P which containsthe
split radical S- U of P, and let C = P/B. This is a reductiveconnectedQgroupwhichhas no non-trivialrationalcharacterdefinedoverQ. The natural
projectionmapsH n P and PR onto subgroupsof finiteindexof CR. FurtherV verifies
more, if f: H
IIf(h-c) II = IIf(h)-A(c, s) II
(h E H. c E BRnH) ,
( 1)
then the restriction of
f A( , s)-1 I1 to H
I1
n
P is right invariant under B
witha functionon theopensubgroup(H
and maybe identified
n
P)/(H
n
n
H,
B) =
C1 of CR.
Let now P and Pm. be as in 6.1, and P, =Pr. n B. We definethe PoincareEisensteinseriesEf by:
Ef(h) =
(2)
(h E H)
eYerlrof(h-)
6.4. THEOREM. We keep the notationof 6.3, and assume that CR has a
V and f': C1-> V be continuousfunctionswhich
compactcenter. Let f: H
verify
II f(h) II A(bs) (k E K, h E H, bE B n H), where s = (sa)
(i) II f(k h b) II
is real, and
(ii) the function f' belongsto L'(C1) 0 V, is Z(c1)-finite(Cf.5.1), is of
finite type on the right with respect to some maximal compactsubgroup,
and is equal in normtof.A(, s)-, .
Then,if s satisfies 6.1 (4), the series 6.3 (2) convergesabsolutelyand
uniformlyon compactsets.
PROOF. Note firstthat
=
(1)
11f(h.b.p)
11= Ilf(h.p) II*A(b,s)
In fact, we have bep =p -b' (b'
lf(h.b.p)
II -
(h E H. be BnB H, pEP
nH)
.
p-'.b -p), hence
lf(h.p.b')
II-
Ilf(hip) I.A(b', s) .
But A(b',s) = A(b,s) since P acts triviallyby inner automorphismson its
character group, whence (1). The functionA, beingequal to one on P7,and
P,,being normalin ],,, this impliesin particular
(2e)f(htzf)
Let
f(h~u)
=
(h E H. z-e
Ea
499
COMPACTIFICATION
pf(h) =
(3
plif11(h)=
,oerS1Orof(h6a)
Efer/FO
(he H)
IIe (h *a)
Writeh = k, P,, wherekhE K and p, e P are determinedup to an elementof
K
P
P. In view of (i), (ii), we have, wr
denoting the projection H n Pa
c1,
p).A(p, s)'1 = A(Ph,
p1!f11(h
(4)
,
5).Pu1f'II(W(PhUP))
where
(5)
P1!f'11(wf(q))
=
EG~r"'/ro
(q e
flf'(wr(qa)) 1l
P n H)
By (ii) and 5.4, the series in (5) is uniformly
bounded;therefore,the lefthand
side of (4) is boundedwhenp varies in P n H and Ph runsover a compactset.
Moreover,(2) and (3) show that pf is rightinvariantunderP.. We may write
Ef - 1ye:er pf(h.Y) ,
(6)
(7)
E
f
= 57er/rF.p
yefl
p1fII(h 7)
so that the theoremnow followsfrom6.2.
7. Poincare-Eisenstein series on bounded symmetric domains
In this section,G is a simple,connectedQ-groupsatisfyingtheassumptions of 3.3 (ii); the notation of 3.3 is used.
7.1. Let F= Fb (1 ? b ? s) be a standardrationalboundarycomponent,
Ub:X-> F the canonicalprojection. The groupN(F)c is definedover Q, and
has a connectednormalQ-subgroupB, containingthe split radical Sb. Ub of
N(F)c, such that BR c Z(F) and Z(F)/B' is compact(3.6, 3.7).
Let JF or Jb, or simplyJ, be the functionaldeterminant
in the unbounded
p
realizationSb associated to F, and a polynomialon F, in the coordinatesof
the canonicalboundedrealizationof F. Let P be an arithmeticsubgroupof
G, contained in GI, P. - N(F)
n1,
integer. We shall considerthe series
( t)
and
FO
=
Bo nfi;
let 1 be a positive
E (x) = E,,n1JxZ)= E.Yerro )(Ub(X")) -J,(X, a)
(X xE).
For this to make good sense, each termof the righthand side should be
rightinvariantunderFo. Moregenerally,we wish to knowthat
( 2)
q(b(X g-x))
*
Jp(Xgu
g) (g
q
q(Ub(x
g))*JF(x
EGJXX
EX, XE)
Since BR c Z(F) acts triviallyon F, the invarianceof the firstfactor is
clear. For the second one, it is enoughthat 1 be a multipleof the integerd
of 3.14, as we shall assume.
BAILY AND BOREL
500
In what follows,E may be suppliedwith a subscriptconsistingof any
to characterizeit in a givencontext. It will be
subset of {p, 1,P} sufficient
series (P-E series for short) adapted to F. More
called a Poincare-Eisenstein
Eo g of E by
generally,for any g e GQ, we shall also considerthe transform
g, definedby
(X E X)
Eog(x) - J,(x, g--').E(x.g-')
(3)
By the cocycleidentity:
E o g(x)
(4)
=
5yer/ro
p(b(x
* g*))
JF(x, g .7)1
We shall see shortlythatthisseriesconvergesabsolutelyforsuitable1. It
representsthenan automorphicformof weight1 forthe group17 = g-17Fg.
7.2. THEOREM. There exists a positive integer lo such that, if 1 is a
positive multiple of lO,then the series Esf o g convergesabsolutelyand
uniformlyon compactsets.
to provethis forE.
It suffices
The groupN(F)c is the standardparabolicgroupQP,where0 = QA -{8bb
in the notationof 2.2. Thereforethe set 0' QA - 0 reduces to {Jb}, and
highestweight
Xb(P) = detAd,,p(p e N(F)c) maybe taken as the fundamental
relativeto Q (2.3). By 3.12 (1), we have then
(g E N(F))
|b-A(g, nb) ,
|J.F(X,9) |=
( t)
F(Ub(X))
whereA(g,nb) = Zb(g)-7b.
We claimthat 7.2 holdsif we take for t, the smallestpositive integerI
verifying7.1 (2) and:
I1.nbC Z.
( 2)
I
I qc(ba) >: 2
nb > 1;
(
Let f be the complexvalued functionon GRdefinedby
1
f(g) =q 9(Ub (C())) .JF (0, 9)l
( 3)
Since the maximalcompactsubgroupK leaves o fixed,and IJF(o,k)
k e K, we have
(k KnGGR ge GO) .
If(k~g) I = If(g) I
The cocycleformulaand 3.11 imply
(4)
( 5)
|If (g
gb)I
I|f (g) I* A(by 1
*
1, if
nb)
g
GRS BRn GR).
From (1), we get
(6)
If(g)
I?A(g,
lflb)
=p(Ub
6hC(g))
I jF(oh,
g)
K'
(g e
N(F)
n GR)
Taking (4), (5) and 5.7, 5.8 intoaccount,we see that all conditionsof 6.4
are fulfilledby If 1, if 1 is a positivemultipleof l; therefore
501
COMPACTIFICATION
(7)
Ef(g)
=yepYr0f(g
(g
7) '
e GO)
convergesabsolutelyand uniformly
on compactsubsets. Since
EP,I(o?g)= J(o,g)- Ef(g),
(8)
(g e GO)
by the cocycleidentity,the theoremis proved.
As in 6.4, we may writeE in the form
( 9)
E(o *g)
-
J(o,g)-' *Eyevrv pf(g *a)
where
(10)
pf
f/
(gX
9
)
Tp/p09(6b o (g X)) Jp(o
g. ).
The argumentused in proving6.4 shows that
|f(g b p) I = If(g p) IA(b,IAnb)
(ge GO,be Rn GR P E N(F))
therefore,if we put again
(11)
Plifii = ExerO/rOIPg -) I
we have
(12)
g, b) = pif11(g)
pllf11(k*
*A(b,I Anb)
(g e Gl, be G, n B, k e K)
As was provedin 6.4, pilf
1(gp)A(p, Inb)'1 is boundedwheng varies in a compact set of GOand p in P n GRI
We henceforthassume 1 to be divisibleby the integerlodefinedabove.
7.3. Fromthe geometricpointof view, an automorphicformco on X of
cross section,invariantunderP, of the comweight 1 for 17is a holomorphic
plex line bundle(Anv)-1 (n = dimcX), whereAMvis the nthexteriorpower of
the tangent bundle z to X. If X is realized as a domain in C7, then z is
canonicallytrivialized,and co is representedby a holomorphic
functionc! on
X whichverifies
(1)
co(x) = j(x, y)1(bo'(x 7) ,
(X E X, 7 E 1)
wherej is the functionaldeterminantin the coordinatesof the ambientvector
space.
By a slightabuse of language, we shall say that c) is a P-E seriesadapted
to Fb, if it is representedby such a series in the unboundedrealizationSb associated to F= Fb. Let F*
Fb* be another standard rational boundary
0
v
=
=
?,
c <
where
is the
component.We let ipF *
S, (
_s) v,: D
canonical isomorphism(cf. ? 1). The P-E series E adapted to F is then representedon Sb* by the functionE * given by
(2)
E*(X(x)) = j(x, i))1.E(x)
(x E Sb)
BAILY AND BOREL
502
wherej(x, v) is the functionaldeterminantof v at x. In studyingE *, and
more generallyE * o g (g e GQ), we shall use the followingnotation,where
J. J* stand forJF, JF*:
( 3)
a{(s) =7 J(O, s)1
(4)
a*(S) = J*(O, S)1
po
xe 9(/
/3(s)
(5)
(C
bb(X))a(sX)
(se GI).
,
Thus, /8is the functionpf of 7.2 (10), and we have, by 7.1 (4) and 7.2 (9):
(6)
(Eog)(o.s)
Moregenerally
(7)
-
a(sY)1.e
(E* o g)(o,s)
=d-1 o*S-1.
(s, g e GO)
(s.g 1y)
(8sg e GI; d = j(o, v)')
eS g-l1~
In fact,the obviousequality
( 8)
showsthat
j(x, +) J*(V(x), S) = J(x, s).j(x.s, V) ,
d~a*(s) =: a(s)j(o-s, v)l
(9)
On the otherhand
(E* og)(o.s)
(s e GI, Xe Sb)
-
-
J*(o.s, g-)l*.E*(o.s.g*-)
J*(o.s, g-')1*j(o*s.g-1,z)-1.E(o.s.g-1)
4)-1.E(o*s*g-1).
J*(o,s)-1.J*(o, s*g-1)1*j(o*s*g-1,
Togetherwith (9), this gives
(E* og)(o s) = a*(s)-1 d-1 a(s. g-1) E(o s g-1)
7.1 (3) of Eo g.
so that (7) followsfrom(6), and fromthe definition
7.4. LEMMA4. We keepthe notationof 7.3, fix b,b*,1, put A =flbZXb if
b > 1, AN*
=Inb*Xb* if b* > 1, and A= 0, A* = 0 otherwise.Let (b* = 2* be
an Fb*-adaptedtruncatedSiegel domain (4.12) in PR. For s e 2*, we write
s = a(s).v(s) (a(s) e QA,v(s) e VR). Then
a
( )c*(s)
_ac*(a(s)) _a(S)-A*, (S e 25*)
(b* ~< b; s e 25*)
( ii) o*(a(s))
a(s)-liti (b < b*; 1,e Z. 1,> 0. (b < ib*)
a*ii o(a(s))-a
lA-Ib<ilb*
By 3.12 (1), we have
*a(8At
Ia*(s)
=I
IJF*(b*(O*),
t'b*(S))
|qb
Xb*(S)Iflb*l
a* (a(s)) I =I jF*(Ob*(O*), 1Ob*(a(s))| qb | Xb*(a(s))1nb** l
4In this proof, s occurs in two capacities: as the Q-rank of G and as an elementof
a truncated Siegel domain. We trust this will not cause any confusion.
503
COMPACTIFICATION
By the definition
of a truncatedSiegel domain, ub*Q(*) is a set of elementsin
G(F*) whichbringthe originintoa compactset, and so is relativelycompact.
Since v(s) varies in a compactset, the set of elements-'b*(v(s)) is then also
relativelycompact. Thus the factorsj,* in the two above equalities are
multiplicativelyboundedon 25. Since Xb*(S) = Xb*(a(s))for any se N(F
this provesthe firstassertion.
If b* 0, then 2* is compact,and the remainingassertionsare obvious.
So we assume b* > 1.
As in 4.12, we may write, with referenceto the index b, an element
a E QA as a
a1,ha2; the set of all a1 tresp.h, resp. a2) whichoccurin this
way forma subgroupA1(resp. H, resp. A2)of QA, we have
H= QAb,
A2=L(Fb) nQA,
A.H=
Z(Fb) nQA,
Al, -z(Fb) nQA,
- ij/29
and the rationalcharacterA is trivialon A1.A2. We have i5(i
t8 (resp. f,8 - y8/2)in case CQ (resp. BC8). It is clear
(1 i < s), and 8
that any characterX of QA trivialon A1,A2is of the formX m
n(Y1+ ***+ Yb)
with somem e Q. In particularA= Mb(Yl + * + 'Yb), and ibn> 0, if b > 1,
in det Ad,, XbZ
because the simpleQ-rootsappear with positive coefficients
where u is the Lie algebra of U(F). We want to prove that Mb is independentof b forb ? 1. We have
-
(2)
A = 2Mb(l
+b~b2RR2
+
*** +
+
+ A-0
+ v -b
S
wherev = 1/2(resp. 1) in case C8 (resp. BC8). There is an analogousformula
forthe restriction
of A to each irreduciblefactor0G'of G (notationof 3.3 (ii)).
Moreover(3.3 (ii)) the firstsimplea(k)-rootof 0G'is the restrictionof onlyone
simpleR-root,with indexc(1, a). Using the fact that QA is diagonallyembedded in (Rk/QS')R and applying1.12 to Go = (aG')Oforeach a, we see that
mb
=
*L
mO,,.c(1
v)
)
wheremr,0is the positiveintegerassociatedto Goby 1.10, and denotedby mO
there. This expressionis indeed independentof b > 1. From now on, we
writem formb. From (1), we get
(3 )
aA
-
a2mPb)s.ll .I?<iba2migi
llb<.<S
a2mbfi
(b > 1, m > O,a e QA)
The rootsA (i > b*) are multiplicatively
boundedon a(s) (s e @5*). Thereforeif X = cl,51+ * + cj38 (ci e Q) is a characteron QA, we have
(4)
(ng (i
(e i
a(s)rticulartak a(i)ni
In particular,taking(i) intoaccount:
504
(
BAILY AND BOREL
5)
(6)
a*(a(s))
(1 f- b*
(8-mt
I8)-'l~;b
at*
a -2msvfs.
a-2mii
< s),
(s e *; b* = s)
These relationsand (3) yield(ii) and
(7)
a* (a(s))a()A
(8)
a*(a(s)).a(S)A
Ib<i<b*
,(s
b)i
a()2m(i
a(s)-2m(i-b)fi
a-;a2m(-8b)s.1Ib<is
e (*;
b < b* < s)
(b < b* = s)
whichproves(iii).
7.5. Our mainaim in this sectionis to study the behavior of P-E series
near boundarycomponentsin X*. For this purpose,we need to constructa
functionthat will help us to majorize these series in a certain way and to
studythemtermwise.
definedover Q, with
Let p: G GL( V) be an irreduciblerepresentation
highestweightA = n
bXb, wherenb is as in 3.11, such that VQ contains an
elementeo 7 0 which spans a line stable under N(Fb)c, and on which the
A. This always exists
lattergroupacts via its one dimensionalrepresentation
(2.3). We endow V witha hermitianstructuresuch that K and S are represented by unitaryand self-adjointoperatorsrespectively,and such that e,
has normone. The functionwe shall use is definedby
-
(l1)
c(g)=
p(g)e e I
(g cGR) .
l
Obviously
(2)
c(k.g.p)
C(g)*
pAl
(p e N(F))
-
It is also clear that if h varies in a compact set C and g e GR, then
c(g) c(h g). In particular,if A' is a Siegel domainin the minimalstandard
parabolicQ-groupP, then,using 4.1, we see that
.
c(a (s) g) I--1c(s g) y
( 3)
(s e
g e GR)
(2g Y
where,as in 7.4, a(s) denotesthe componentin QA of s. The main properties
of interestto us of the functionc are given by the followinglemma:
7.6. LEMMA. We keep the notationof 7.4, 7.5, choose t > 0, and let ao
be the elementof QA on whichall simple Q-roots/3itake thevalue t. Then,
(i)
aA *c(a.h)
(ii)
limalb*.o
j
a
A
*c(ao
*
h) (he GR; a C QAt).
aA ce(a. g) = 0 ifb* > band g X N(Fb*)*N(Fb),(g e GQ,a C QAt).
We referto the situationin 7.5. Since QA is representedby self-adjoint
operatorsin V, the space V is the direct sum of the mutuallyorthogonal
subspaces
V, = {v C V, p(a) Tv= agv, (a C QA)},
505
COMPACTIFICATION
correspondingto the differentQ-weights4ceof p. The space VA being onedimensional,spannedby the unit vectore,, we may write
p(h).eo = d(h).e, + 7,Pf( ,a
(f.(h) e V,; d(h) eR; h e GR)
whence
d(h)2+
c(h)-2 =
IIfp(h)112
and
p(a*h)*e,
,
Jr i
a+*f,(h)
aA*d(h)*eO
=
(a e QA, h e
GR).
It is known[14, ?12.14] that everyQ-weightis of the form
A
1A-ElS
-
mi(p) .,Si
(M()
e Z, mi(4) _ 0)
(
a-mi(i)ii).fk(h)
(a C QAh e GR)
hence
(1)
a-A.p(a h).eo
=
d(h)
-
eo +
Using (1) and the definition
(7.5) of c, we have
(2 )
Ic(a.h).aA
(I
-2 = d(h)2+ I
a-2mi(',i)
Since the mi(ji) are >_0, we have
a-2m~~
>
aO
f(h) 112
If
if
m(ii=t
forall 4ce
and all i, whence(i).
Let P - M. S. U be the minimalstandardparabolicQ-group(2.2) which
of the standardboundarycomponents.As recalled in
underliesthe definition
2.2, the elementg e GQmaybe writtenas
(3)
g = u-ng.u
(u, u'f UQ;nge N(S)Q),
whereng is uniquelydeterminedby g. Let Wg be the image of ngin the relative Weyl groupQW(G) = N(S )/Z(S).
We have a *g aua-* ng.n1*a*ng* , and consequently
c(a-g) = c(auma-lXng).(n-la-ng
)-A
But (4.1), a *u a-1 remainsin a relativelycompactset, since a E QA,, and so
the firstfactor is multiplicativelybounded. Therefore,we are solely concernedwiththe behaviorof
aA. (nll . a .n g)-A
The transformv =
Wg(A)
aA. a-w(A)
of A by Wg is a weightof p, and thereforehas the
form
= A -m(2-)I,
Thus we are reducedto studyingthe product
(mi(V) > O. mi(2) e Z)
506
BAILY AND BOREL
Since a C QAt,each factor is boundedabove. Let 0(2) = I,8ji mi(s) > 0}. It
A is
followsfrom[14, ? 12.16] that 0(2))U A is connected;by construction,
orthogonalto all simpleQ-rootsexcept 8b. Now, if g X N(Fb), then v ? A;
therefore8(2)) is non-empty,
and we musthave mb(2))# 0. It followsthat
(a X QAt)
ami<l)hi =0
limagb~oJnJi
whichproves(ii) if b = b. Let now b*> b, and assume that the limitis not
zero. Then mb*(2)) = O, and 0(2)) is contained in the set QA - {/3b*} of Q-simple
rootsof N(Fb*)c/U(Fb*)c.Then, by [14, 12.17], thereexist
ni
C N(S)Qn
such that ng = n1*2.
g = grg2,
n2CN(S)Q nN(Fb)
(Fb*) ,
Consequently
g1= usnd C N(Fb*)Q,
g2 =2tUo
C N(Fb)Q,
whichcompletesthe proofof (ii).
7.7. LEMMA. We keep the notationof 7.4, 7.5. Let g e GQ.
( i ) There exists a convergentseries of positive constantterms which
majorizes termwise,in the truncated Siegel domain (5*, the series (see
7.3 (7)):
(E* og)(o s) =
(ii)
if b* < b and
ay*F/F1 a*(s)-l.3(Sgl
g X N(Fb*).N(Fb),
*y)g
then
lim8,b*,0
g)
0, (sC
a*(s)l.
/(s
g)
- 0,
*);
(s
(iii) if b < b*, then
limsib*-o
a*(s)-lfJ(s
*)
The function/3is equal to pf, in the notationof 7.2(10), and is majorized
by Pilf
I. We have alreadyremarked(7.2(12)) that
pllfj1(h p) .A(p, nb X ) 1 = pjlfj(h .p)
. I p-A I
is bounded when h varies in a compact set of GR and p e P n GR. Since
c(h p) = c(h)Ip-A I by 7.5 (2), it follows then, as in 6.2, that pllf11c-1is
boundedon GI. There exists thena constanta > 0 such that
( 1)
I F~~~~~(h)
aI-a
c(h),
(h e GOR)
and thereforesuch that
(2)
6a5e7,s
a**(s)-1c(s h. y),
is a normalmajorantof (E* o h)(s) forall s, h e GO,which convergesin view
of 6.1.
We have c(s h) c(a(s) .h) fors in a Siegel domainand h C GI by 7.5 (3)
and a*(a(s))
ca*(s)for s C A*, (s = a(s).v, a(s) C QA,v C VR) by 7.4. Together
with (1), this yields
507
COMPACTIFICATION
a *(s) 1.,C(s .h) a*<.*(a(s))1.c(a(s).h)
(3)
(s G A*, he GI)R
Using 7.4, we get
a*o(s) *,S(s*h) |
ab+1 a(8) iA)*(c(a(s) *h) *a(8) A),
( 4)
wherethe firstfactoron the righthand side stands for the constantone if
b* < b and the li are strictlypositiveintegersif b* > b. Let, as in 7.6, a, be
the elementof QA on which the simple Q-rootstake the value t. Since the
firstfactoron the righthand side of (4) is boundedon A*, Lemma 7.6 (i) and
(1) implythe existenceof a constant8' > 0 such that
C'(h) - a'. c(a. h) *aA
(X*(s)-1.* (s *h)
As a consequence, E* o g is majorized in 5* by the series
>I
E
(s e 25*; he GI)
C'(g-1.ry)
whichconverges,as was notedabove (6.1 and 7.5 (2)), whence(i).
In view of (1), (2) it sufficesto prove the statements(ii), (iii) with
a*(s)-'.,8(s
g) replaced by
(ll-b?1
* *
(a C QAt)
aliii) * c(a g) aA
Let b* ? b and g X N(Fb*) N(Fb). Then the firstfactor is one, and the
second one tends to zero as aib* 0 by 7.6 (ii), which proves (ii).
Let b* > b. Then lb* > 0 by 7.4; hence,the firstfactor tends to zero.
The secondone remainsboundedby 7.6 (i), whence(iii).
7.8. THEOREM. Let E and E* be as in 7.1, 7.4. Let (F* be a truncated
Siegel domain adapted to F*; let s > 0 and g e GQbe given.
(i) Assume b < b*. Then thereexists uo > 0 such that
(s G @f* a(S)
| E og(o. s) I < S
(ii) Assume b > b*. Then thereexists uo> 0 such that
I E* og(o.s)
for all s e
-
j(o,
25F* satisfying
0)1
1yeg.N(F*)N(F)
a(s)$b*
<
nlr/f a*(s).
*(s -gly)
|
b* <
uo)
< e
Us.
By 7.7 (i) the series E* o g has a constant majorant series in A*, so that
we may investigate its behavior termwise; 7.7 (ii) and 7.7 (iii) allow us to do
this, withthe theoremas an immediateconsequence.
7.9. PROPOSITION. We keep the previous notation and assume b* < b.
Then the series
(1)
E* og(o.s) =
a*(s)1.
IEY(N(F*s(F)n
r) roo 8(
g .)
is an automorphicform on F*, for the group
Fr(F*) = (Z(F*) n g1rg)\(N(F*) n g-rg),
(ge GQ)
BAILY AND BOREL
508
lifted to X by means of the canonical projection aUb: X
J*(x,
F*, of type
)1.
Since b* < b we have a canonical factorizationa = Ub = TU* where
F is a holomorphicmap. In fact r may be identifiedwith the car: F*
nonical projectionof F* onto F, the latter being viewed as a standard
rationalboundarycomponentof F*.
constantalong the fibresof 0*.
We show firstthat E * o g is holomorphic,
By definition
-
E* og(o s)
-
a*(s)-l
Eyer/rF,
T6xer./ro
p(6(o * s * gyl
)) .J(o, s *g-1*o*)1
the range of y being as in (1). For fixeda, the productof the series on the
is a Poincareseries for p,,/rOon F, liftedto
right hand side by a(s g-.)1
functionon F*.
X by a. It is thereforea fortiori liftedfroma holomorphic
In orderto finishthe proofof our assertion,it sufficesto show that
(u c N(F *), v e N(F))
ax*(s)-l *a(s *u *v) ,
as a functionof o * s e X, and constantalong fibresof v*. The
is holomorphic,
equality7.3 (8) yields
a* (s)-1 *a(s*u*v) = J*(o, s)- .J(o, s.u.v)l
= j(o, VY)j(o -s', 22- J(O, s)-' -J(O, s-u -v)l
= d 8j(o *s, v)-' *J(o *s, u 8v)l .
in oks; by 3.3 (ii),
The factorson the righthandside are of courseholomorphic
is
transitivealong
of
1.7,
Z(F*)o
is
By
the
fibres
j(o.s, v)Y constantalong
v*.
the fibersof v*. In view of the cocycleidentity,it sufficesto prove
(2)
(3)
We have zu
J(xu ) = J(xz, u)
(z
CZ(F*), u e N(F*)),
J(xu, v) = J(xzu, v) .
= uz' (z' e Z(F*)).
Since b* <
b,
Fc
F, whence Z(F*) c Z(F),
so that (2) and (3) followfrom3.3 (ii).
' * *g. Thus y0 g- *. g (Y' c F), and it is clear
Let now y0e N(F*) n g-1
that the series on the righthand side of (1) remainsunalteredif s is replaced
by s *y0. Moreover
-
a*(s.'yo)-l= J*(o,s09 )- = J*(o, s)-.J*(o.s, go)-i
hence,
E* og(o.s)
= J*(o.s,
yo) *E* og(o*s*.y)
whichends the proofof the proposition.
,
509
COMPACTIFICATION
8. The operator 1D
8.1. Up to 8.7, we keep the notationand assumptionsof 3.3 (ii). As in
4.8, X* denotesthe unionof the rationalboundarycomponentsof X, endowed
of x C X* is one whichverifies
withthe Satake topology.A goodneighborhood
and tifF be
4.9 (iv) and 4.10. We let F denotea rationalboundarycomponent,
of N(F) ontoG(F).
the canonicalepimorphism
An automorphicformwoof weight1, forthe arithmeticgroup F, will be
said to be a P-E series adapted to F if its transformw o g undersomeelement
g C GQ which carries F onto the rational boundarycomponentFb is a P-E
series adapted to Fb for PI, (7.6).
8.2. The functionaldeterminantJb(x,g) (g e N(Fb), x C X) is by 3.3 (ii)
Fb. It definestherefore
constantalong the fibresof the projectionUb:X
an automorphyfactoron Fb forN(Fb), hencean actionof N(Fb) on the trivial
line bundleFb x C given by
-
(x e Fb, c e C, n N(Fb))X
(x, c).n = (x.n, Jb(x,n). c)
The N(Fb)-bundlethus definedwill be denoted by db. If F = Fb g-1 is
as above, the translationby g-l carriesdbover ontoan N(F)-bundle denoted
class of eF, viewed as an N(F)-bundle, dependsonly
by eF. The isomorphism
on F.
Let A be a discretesubgroupof N(F) whose image A' under tZ?F is discrete, and I an integer. An automorphicformwoforA', of typeit, is a A'invariantholomorphiccross section of d. The transformo o g is then an
automorphicformof type ib for Cb(A"). It is thereforerepresentedin the
canonicalboundedrealizationof Fb by a functionf whichsatisfies
(1)
f(x.X) = Jb(x,X)If(x)
(x C FbY,x
,
Ag)
where,by abuse of notation,we writeJb(x,g) forJb(x',g) if x e Fb, x'Cae'(x),
ge N(Fb).
Let c < b and Vbc: Sc, Sb be the canonicalisomorphism.We have
(2)
)bC(x),
g)
Jc(x, g).j(x
g, pb c)
(x e Sc, g e GI)
Let g e N(Fc). Then, using 1.11, we see that Jb(x,g) is constantalong the
fibresof the projectionoc: Sc Fc, and hencedefinesan automorphyfactor
on Fc, and an actionof N(Fc) on the trivialline bundle. However, (2) shows
that the automorphy
factorsgiven by Jband Jcare equivalent,and hence the
N(Fc)-bundlejust definedis isomorphic,as an N(Fc)-bundle,to ic.
-
8.3. Let U be an open subset of X* whichintersectsF, and A a discrete
subgroupof N(F) leaving U invariant. Let co be an automorphicformof
510
BAILY AND BOREL
weight 1 for A in U. Let ge GQ be such that Fag - Fb forsomeb. We say
og is represented,in the unboundedrealization
that wt)
extendsto F n U if wt)
Sb, by a functionf whichextendsby continuityto a holomorphic
functionf '
on (U n F) *g. The extensionf ' clearlyrepresentsan automorphicforma on
(u nF) g of type fi, forAl, or ratherforthe image 'Vtb(A") of Al in G (Fb).
Its transformco' underg-1is thenan automorphicformof type it for -trtF(A),
to be called sometimesthe extensionof co. This formdependsonlyon co and
F. In fact if Fa g' -Fb(g' C Ga), theng' = g *n(n e N(Fb)Q); hence,GOo g' is representedbyf *(x) = f(x,rn-1)
.J(x,n-1). The function
f * extendsbycontinuity
to f*' = x - f'(x, n-) Jb(x,
whichrepresentsa on. Its transformunder
g'-1is thenagain co' (cf. Remarkof 1.7).
Let d _ b, and fd be the functionwhichrepresentsGoo g in the unbounded
realisationSd associatedto Fd. Then co extends to F n U (wherestill F.gg
Fb) if and only if fd extends by continuityto a holomorphic
functionfd on
(U nF) g. This follows fromthe equality fa= f J( ,Y d,b)l and the constancy of J( , Ld,b) along the fibresof Ub. Thus, in order to check that o
extendsto U n F, we may use any unboundedrealizationSd (d _ b). Furthermore,the last remarkof 8.2 implieseasily that )' o g is representedby
fdjin the trivializationof V whichis given by Jd.
.
8.4. Local integral automorphicforms. Let F be an arithmeticsubgroup of G, and x C X*. We suppose x e F. An automorphic
formGoof weight
I for Fx on X n N(x), whereN(x) is a good neighborhood
of x, is integral if
it extendsto an automorphicformfor z:F(N(F') n ir)on F' n N(x) for every
rationalboundarycomponentF' whichmeetsN(x).
If yeN(x), then P',czI7x by 4.9; therefore,the restrictionto a good
neighborhoodN(y) ci N(x) of y of an integralautomorphicformon N(x) is
an integralautomorphicform,whose extensionto N(y) n F' is the restriction
of the extensionof coto F' n N(x).
The formcois integralon N(x) if and onlyif forevery g e GQ such that
Fag = Fb, the transformo o g is representedin Sb by a functionwhichexfunctionon F, n N(x) for all c ? b. In
tendsby continuityto a holomorphic
fact, as before, this condition is insensitive to a change of g; moreover, by
3.9, if F' nN(x) 7 0, there exists ge GQ such that F'.g - Fc, Fag = Fb
(c= b).
From3.9 we also deduce that o is integralif and only if for one g C GQ
which maps F onto Fb, the transformw og extendsby continuityto N(x) g
and is holomorphic
on (F' n N(x)) g for every rational boundarycomponent
F'.
.
511
COMPACTIFICATION
8.5. Integral automorphicforms. The operator 4. An automorphic
form o on X of weight 1, for F, is integral if its restrictionto X n N(x) is
integralforevery x X X* and every good neighborhoodN(x) of x. This is
the case if and onlyif foreveryF and g G GQsuchthatF. g - Fb is standard,
the transformGoo g is representedon Sb by a functionwhich extendsby continuityto a holomorphic
functionon Fb. Then this functionalso extends by
continuityto a holomorphic
functionon F, foreveryc _ b.
Let so be integral. It then has an extensionto any rational boundary
componentF, which is an automorphicformfor1(F) of type it, and which
will oftenbe denotedby 1'FO. The operator1 is, by definition,
the operator
which associates to co the collectionof automorphicforms JFO). The definition of the extension of 0o given in 8.3 implies that for any rational
boundarycomponentF and g e GQ:
(1)
(
0
-'o))og
=
I(o)
? g)
8.6. THEOREM. Let E be a P-E series adapted to the rational boundary
componentF, for the arithmetic group F, of weight 1. Then E is an integral automorphicform. Let F* be a rational boundarycomponent.Then
bDE - 0 if dimF* < dimF and F* C F.J. The operator 1'F maps the
module of P-E series adapted to F, of weight1, ontothemoduleof Poincare
series for r(F), of typed.
If the statementis true for E, F, 1, then it is also true for Eog, Fag,
and 1P (g e GO). We may therefore,withoutloss of generality,assume that
F= Fb is a standardrationalboundarycomponent.
In orderto provethe firstassertion,it is enough to show that, for any
x e X* and g e GQ such that x g e F, (1 < c < s), the transformEog is represented on S, by a functionwhich extends by continuityto a holomorphic
function around x g on Fc.
.
Let E* be the functionwhichrepresentsE on Sc. Then Eog is represented by the functiondefinedby (E* o g)(x) = Jc(x,g-l)l*E*(x *g-l). Let (0'
be a Siegel domainin P such that Fc n a2, where e = od.Y, contains x g in
its interior. This is possibleby 4.5. There exists a finitesubset C of N(FC)Q,
containing the identity, such that iFx.g,
*
where &2 - Uae 2(mu,Va) *a, runs
througha fundamentalset of neighborhoods
of x in X* when um 0 and Va
runs througha fundamentalset of neighborhoodsof x g. a-' (see 4.13). We
have
.
(E* og)(s -h) = J(s, h)-1. (E* ogh-')(s)
(s e (mu,Va), h e arPx.ga e C).
The function J( , h) is constant along the fibres of ac: Sc
Fc (3.3 (ii)), and
BAILY AND BOREL
512
boundedon c((u, Va) by 4.16. Consequently,we are reduced
is multiplicatively
to showingthat E * og. h-1extendsby continuityto a holomorphicfunction
on the interiorof c((u, Va) n F,. Since this series has a normalmajorantin
2(u, Va) by 7.7, this followsfrom7.8, 7.9.
In orderto studythe limitof E* og around x g on F,, it is enough to
x g. If c > b,
of f
considerits behavioron (mu,V), where V is a neighborhood
i.e., if dimF* < dimF, this limitis zero by 7.8 (i). Let now c = b. Theorem
7.8 (ii) impliesthat the limit is zero unless g.N(Fb) n 1 # 0, i.e., unless
Fbg-1 c FbP. Since
~Pb
(Eo?g) =
(<
Fb 9-1E )o g
this ends the proofof the secondassertion.
Let now g - e. Then 7.8 shows that the limitof E on Fb is the limit of
the "constant" terma(s)-'.b(s), whichis by 7.9 the Poincareseries of typeeb:
Exeroo/ro
9(O(b(X
* X))Jb(X,
XY1
(see 8.2). Furthermore,it is clear that every such Poincare series can be
of E.
obtainedin this way by suitablechoiceof p in the definition
factorfor N(Fb) definedon Fb by Jb is equal in
8.7. The automorphy
absolute value to iJb(Ub(X), g) jqb (cf. 3.12). Therefore,(5.10), it satisfiesthe
conditionimposedby H. Cartan [35 Exp. 10 bis; 19 p. 170] so that we may
apply Theorems2, 3 of loc. cit. to the Poincareseries formedby meansof Jb,
on Fb. Since everyrationalboundarycomponentis the transformof some Fb
by an elementof GQ,this implies:
(1) Let F be a rational boundary component of X, al,
a q points of
*t*,,
F, no two of whichare equivalentunderF(F), and t a positiveinteger. Then
thereis a positiveintegerlowith the propertythat, forany multipleI of lo,
there exist a Poincare series P of type d (see 8.2) whichhas pre-assigned
admissible(i.e., locallyinvariant,cf. [19, pp. 170-171]) Taylor developments
of order t (in suitable local coordinates), at ac, * , a,.
This meansin particularthat, forsuitable 1, we may findP whichis not
zero at a, and zero at a, (i _ 2).
8.8. PROPOSITION. Let F, F' be two rational boundary components such
1
that dimF > dimF', and that eitherF =F' or Fc4 F'.]?. Let x e F, y F
be not equivalentunder P. There is an integerI,, such thatif I is a multiple
of10,thereexistsan integralautomorphicformE whichverifies<>FE(x) # 0,
1'F,E(y)
=
0.
This followsfrom8.6 and the resultof Cartan mentionedin 8.7 (1).
8.9. We now want to extend8.8 to the case where G is not necessarily
513
COMPACTIFICATION
Q-simple,and whereP is an arithmeticgroup of holomorphicautomorphisms
of X. The notationof 3.3 (i) is used. In particular,X is the productof the
spaces Xi = (K n GiR)\GiR, the space X* is the productof the Xi*, where G?
runs throughthe Q-simplefactorsof G, and 17 is the productof the groups
pi>= 7 f Gq. We do not exclude the possibilitythat Xi/J7is compactfor
some i, in whichcase Xi Xi*. Let pro:X*
Xi* be the naturalprojection.
The notionof an integralautomorphicformon X for F is definedas in 8.5.
If Et is an integral automorphicformon Xi for Pi, of weight 1, thenthe
productof the formsEI opri is an integralautomorphicformof weight I on
X forP'. This shows firstthat 8.6, 8.8 are valid for F'.
Let F, F' be rational boundarycomponentsof X, such that F' ? F.]?,
or F = Ft dimF? < dimF and let x e F, ye Ft,y ixfr. LetJ(F)x be the
isotropygroupof x in F(F) = (N(F) n r)/(Z(F) n F), where N(F) and Z(F)
are respectivelythe normalizerand the centralizerof F in H(X). The orbit
of x underN(F) n F is the union of finitelymany orbits of N(F) nf1. By
8.6, 8.8 for I?, there exists losuch that forany multipleI of l we mayfind
a P-E series E' adapted to F of weight I for F' verifying
'(1FE(x) =
bF Ef(yf)
0,
(x' e xOP n F; x' i x.F' n F),
0=
IE'(x')
-_0 .
(vGfrn
Ff)
Assumenow that I is also a multipleof the orderof J(F)x, and put
Ef
\FEoy
y
E=
Each summandon the right-handside is an integral automorphicformof
weight I for P', hence E is an integralautomorphicformof weightI forP.
We claimthat
(2)
PsE(x)
#
,
?F IE(y)
= O,
whichwill proveour contention,if we take for lo some multipleof lo and of
the orderof J(F)x. We have, forany rationalboundarycomponentF*
(3)
(PF*-.-YE')
o
-
=P)F*(E
oyr) ,
(see 8.5). Let F' X Far and F* = F'. Then F* Y-1X F.oP', the left-hand
side of (3) is zero by 8.6, whencethe secondpart of (2) if F # F'. Let now
F * = F = Ff. If y
/ (N(F) n 11)r.P'then F. --' X2F.]P', the left-handside of
(3) is zero by 8.6, whence
4IFE(z)
whichimpliesPFE(y)
-F'\(flF).F'
((HFE')oy)(z)
(z e F)
0 and, because of the conditionsimposedin (1).
514
BAILY AND BOREL
x *J7',
If X
thenthe corresponding
summandis zero by construction
of E'.
There remainsto considerthose terms for which x y e x .17'. In that case
e (N(F) nf ) J(F)X. Since we sum modulo N(F) n ir, we may assume that
e r(F)X. Then jF(x, -/)f
= 1, since I is a multipleof the orderof P(F)x, and
the correspondingsummandis equal to 4PFE'(x). As a result,PFE(x) is a
non-zeromultipleof PsE'(x), whichends the proof.
.
.
8.10. Assumeagain forconvenience,G to be Q-simple. An automorphic
formof weightI is a cusp form if it belongsto the kernelof P.
It is known[35; Exp. 10, ? 4] that every cusp formwhose weight is a
multipleof some suitable fixedintegerlo is a linearcombinationof Poincare
series; in particular,if X/I is compact,every automorphicformof weight
ml, is a linearcombinationof Poincareseries. It followsthereforefrom8.6,
by an obviousinductionprocedureon dimF, that there exists an integer lo
with the followingproperty:every automorphicformfor I, of weight 1
underelementsof GQof
divisible by k,is a linearcombinationof transforms
P-E series forconjugatesof P underGQ.
III.
THE COMPACTIFICATION AS AN ANALYTIC SPACE
9. An analyticitycriterion
9.1. In this section,V is a locallycompactHausdorffspace, satisfyingthe
secondaxiom of countability,whichis the unionof a locally finitecountable
family of disjoint subspaces V0,V1, ...,
each of which is provided with the
structureof an irreduciblenormalanalyticspace.
An a-functionon an open subset U of V is a complexvalued continuous
fv is analytic (0 ?< i a< m). If we
functionon U whose restrictionto Un
associate to U the C-moduleof a-functionsdefinedon U, we get a presheaf
The
whichis easily seen to be a sheaf,the sheaf & of germsof CT-functions.
V
the
an
U
of
are
over
subset
continuoussections of (i
open
d-functions
definedon U. We let &, be the stalk of a at v e V.
9.2.
THEOREM. We keep the notation of 9.1 and make the following
assumptions:
( i ) For each positive integer d, the union V(d) of the Vi's whose dimension is ?d is closed. dimV0= dimV, dimVi < dimV0 if i # 0 and V,
is dense in V.
(ii) Each point v e V has a fundamental set of open neighborhoods
(Ua) such that U.,nlV is connectedfor everya.
(iii) The restrictionsto Vi of local (G-functions
definethe structural
sheaf of Vi.
515
COMPACTIFICATION
(iv) Each point v e V has a neighborhoodU, whosepointsare separated
by the CC-functions
definedon U.
Then (V, (G) is an irreducible normal analytic space and for each
d < dimV0,V(d) is an analytic subspace of (V, (G)with dimensionequal to
maxdimv.<d(dim Va).
The proofof 9.2, will be brokenup into several lemmas,and will be concluded at the end of 9.7. We note firstthat,in view of (i), the subspace Vi is
locallyclosed in V(d) (d -dim Vi), hence is locallyclosed in V.
We shall use the followingremarkon normalanalyticspaces.
9.3. LEMMA. Let Y be a normal analytic space. Then the ring of
analytic functions on Y is integrallyclosedin the ring of complex-valuled
continuousfunctionson Y.
which
Being normal,Y is the disjointunionof its irreduciblecomponents,
are open in Y. We may thereforeassume Y to be irreducible.
Let h be a continuous,complex-valuedfunctionon Y which satisfiesa
relation
(Y c Y)
( 1)
+ aO<n ct(y).hn-?(y) ?0
hhn(y)
wherethe a, are analyticfunctionson Y.
Let a e Y, /9a be the local ringof Y at a, and Ka be the fieldof quotients
T + at1Tn-1 + . . . + a. G (Da[T], where a, also deof Oa. Let P = P(T)
notesthe germdefinedat a by a,. We assume a to be a regularpoint. Using
the Gauss factorizationlemma,and (1), we can finda factor
Q =Tm + bi. Tm' +***++b
e OaT]
of P whichis irreduciblein Ka[ T], and such that
(2)
Qy(h(y)) = h-(y) + b1(y)*hm-1(y) + *** + bm(y)= 0
(Y G U)
small neighborhoodof a. Here Q, e CQT] denotes
where U is a sufficiently
Q by replacing bi by bi(y),(y e U). By confrom
obtained
the polynomial
siderationof the resultantof Q and dQ/dT, we see that the set of points
y e U, for which Qy and dQy/dThave a commonroot,is a properanalytic
subset Z of U. If y X Z, thenthe implicitfunctiontheoremand (2) show that
h is analyticaroundy. Hence, h is analyticat a set of points definedlocally
as the complementof a proper,local analyticsubset of U. It is then analytic
in U by Riemann's extensiontheorem[1, 44.42, p. 420]. Since the set of
singular points of Y is a properanalyticsubset,a furtherapplicationof the
Riemannextensiontheoremshows that h is analyticon Y.
9.4. LEMMA. We keep the notation of 9.1 and assume (i), (ii) of 9.2.
Then (TVis integrallyclosedfor everyv e V.
516
BAILY AND BOREL
Except forthe use of 9.3, the proof is the same as that of the correspondingassertionin [35, Exp. 11, p. 7], and we describeit briefly.
By (ii), v has a fundamentalsystemof neighborhoods
U such that U V0
is an irreducibleanalyticspace. Therefore,if f, g are i-functions
on U whose
product is identicallyzero, then one of them must be identicallyzero on
u nfV, henceon U by continuity.This showsthat CC,is integral.
Let now f, g e Gf,with g notidenticallyzeroandf/gin the integralclosure
of atv. There exists thena relationof the form
ai *(f/g)-
, n - 1)
*0
0,
If U is a sufficiently
smallneighborhood
of v, thenf, g may be viewed as
H-functionson U, and g is not identicallyzero on un v0. Since Un v is
normal,thereexists thenan analyticfunctionh on U n VOsuch that
(f/g)f +
(1)
O<jn
-0
(ai e (Gv;i
h(x)*g(x) = f(x)
(x E u n vo).
As in [35, loc. cit.] it followsfrom(1) and (i) that h extendsby continuityto
a continuousfunctionon U, whichwill thenverify
(2)
hn(x) +
EO<jgn
ai(x) ' hn-i(x) 0-
forall x E U. By 9.3, the restrictionof h to Vi n u is then analytic;hence,
h is an a-functionon U, and f/gE (TV.
9.5. LEMMA. We keep the notationof 9.1 and the assumptions(i), (iv)
of 9.2. Let v E V, U' be an open relativelycompactneighborhood
of v whose
points are separated by aC-functions,
and U be a neighborhoodof v whose
is
closure contained in U'. Then thereexistfinitelymany aC-functions
on
U which separate thepoints of U.
Let fl, **., fs be a finiteset of d-functionson U'. Define a holomorphic
map f: U'
Cs by f(u)
= (f1(u),
...
, f8(u)), and let
) Cs X C.
qT= f x f: U x
Let A and D be the diagonalsof U' x U' and Cs x Cs respectively.Clearly
c-'(D) D A, and we have qr'(D)- A if and onlyif f is infective.
U' x U' is the disjoint union of the locally closed analytic spaces
(u' n vi) x (u' n v7). Similarly (U' x U') - A is a disjoint union of locally
closed subspaces, each endowed with the structureof a separable normal
analytic space, namely the complementsof the diagonal in the subspaces
(U' n vi) x (u' n vj). Therefore U' x U' - A may be written as disjoint
union of countably many subspaces Mj, each of which is an irreducible
analytic space, locally analyticallyembeddedin some Va. The restrictionof
T to Mj is analytic,hence U n -'(D) n Mj is an analytic subspace. Let Mj,
517
COMPACTIFICATION
be its irreduciblecomponents,
and let
a,( fl -**,f*s)
-
maxj,,
dimMjk
Put au(fi, *..., f)
-1 ifall theMjkare empty,i.e., iff is infective
on
U. It is clearlyenoughto showthatif,au(f1,*.*, f3) - 0, thenthereexists
an openneighborhood
U" of U, and finitely
manyai-functions
f,.. *** f/ on
U" such that au(fl,
ft') < au(f,
...
*
fs)
Let us enumeratethe Mjk as Y1,Y2, *.., and let y -(ui, vi) E Yi (i
1, 2, *..). Then ui # vi, so there exists an i-functiongi on U' such that
Definego*on U' x U' by g*(x,y) = g(x) - gi(y).
gi(ui) # gi(vi) (i = 1 ...).
We may, and shall, assume that Igi I ! 1/2 on U', hence that g* < 1 on
U' x U'. Let U" be an open neighborhoodof U whose closureis contained
in U'. We claim that we may choose constantsci such that the sequence
Y(rm)=
cige convergesuniformlyon U" x U" to a functiong* such
that g*(ui,vi) # 0 (i = 1, 2, *..). In fact, supposingc1,*.., c1 chosen in
* , r-,
we select emverifying
such a way that g (ui,vi) #0fori1,
the followingconditions
4-;
I cr. g*(ui,vi)
cI
C
g*)(u
< 4m
0.
Minl<j<mI g(*u>(i,vi)
,VM) #
(1 ? i < m)
Then the constantsci are easily provento satisfyour condition.In this case,
c g convergesuniformlyon U" and is an a-functionon U" such
g E gci
that g*(x,y) = g(x) - g(y), (x, y E U"). This implies that g(ui) - g(v) =
1, *..), hence that
g*(ui, v,) # 0 (i
au(f1,***,f, g) < au(fi,..,f s)
9.6. LEMMA. We keep the notationof 9.1 and the assumptions9.2 (i),
9.2 (ii). Let U be a relativelycompactopen neighborhood
of v e V, f19,*.* , s
a finite set of a-functions on U which separate the points of U, and
f: u ~ (f1(u), .*,f.(u)) the associated mapping of U into C3. Then there
exists a relatively compact neighborhoodU' of v in U such thatf induces
0 1,... ) ontoan analytic (resp.
a homeomorphism
of U' (resp. U' nvi,i o,
locally analytic) set in someopen domain N of C8,and that f(U') is locally
analytically irreducibleat each of its points.
Let U1 be an open neighborhoodof v such that U1 is containedin U.
N of f(v) such that
Since f is infectiveon U, thereis an open neighborhood
f(U1- u1)n N is empty. Put U' = f-l(N) n U1. Let C be compact in N,
and C' -f-(C)
n U. The set C' is containedin f-(C) nUl, which is
compact. Let b belong to the closure of C' in U1. Then f(b)e Cc:cN, so
b E u1nl f'(C) c U'; thus be fl(C) n U' - c', so C' is compact. Conse-
BAILY AND BOREL
518
of U' onto
quently. f is properon U', and thereforeis a homeomorphism
f(U') c N. Now, let Vi1,* , Vir be those Vi of smallestdimensiondo which
meet U'. By 9.2(i), the intersectionof each with U' is closedin U' and since
f is properon U', it followsthat f((Vi n ... n vi r) n U') is a closedanalytic
subset of dimensiondoof N. Assume now that for some integerd ? do we
have proved that S -f((V(d)) n U') is a closed analytic set in N of dimension < d. Let Vj be of dimensiond + 1. By [21,Ch. V, C5, p. 162]f(Vj n U')
is analytic of dimensiond + 1 in N - S. Then, by a theoremof RemmertStein [21, Ch. V, D5, p. 169] the closureoff(Vi n U) in N is an analytic set
in N. The fact that f(U') is an analyticset now followsby inductionon d.
it
Sincef is bijectiveon U' and since each of its coordinatesis an (i-function,
followsthat foreach x E U', f inducesan injectionof the local ring of f(U')
at f(x) into(ix; the latterbeingan integraldomain,we see that f(U') is irreducibleat everypoint.
9.7. LEMMA. We keepthe notationof 9.1 and the assumptions of 9.2.
Let U' be as in 9.6, and put Y - f(U'). Let Y be thenormalizationof Y.
Thenf induces an isomorphismof ringed spaces of (U', ae J) onto Yk.
Since Y is analyticallyirreducible(of dimensiond = dimV.) at each
and we may
point,the canonicalprojectionof Y onto Y is a homeomorphism,
identifyY with Y, endowedwiththe structuralsheaf C whose stalk at y is
the integralclosure(D of the local ring(, of Y at y. We have to provethat
of C, ontoCf(u) foreveryu e U'.
f inducesan isomorphism
Let firstg e
(Df(u,).
There is a neighborhood of f(u) in which g defines a
continuousfunctionwhichsatisfiesan integraldependencerelation
( 1 )
gn(X) +
lO<jin
bi(x).g"-'(x)
= 0 ,
wherethe bi are analyticon Y aroundf(u). The functiongof is then continuousaroundu, and satisfiestherea relationsimilarto (1), withbi replaced
by ai = biof. The ai's are continuousaroundu. By 9.6, theyare a-functions;
hence (9.3), the restrictionof g o f to Vi aroundu is analytic. Thereforeg of
is an a-function.
of one comIt is well-known[8, p. 179] that an analytichomeomorphism
plex manifoldonto anotheris an isomorphism.Let a be an a-functionat
off(u). If N is chosensmall
u e U', i.e., a e (C., and let N be a neighborhood
n N (viewed as
enough,thena of-' is continuousin N and analyticon f(VO)
at
a subset of Y), except possibly the image points of the singularitiesof
V0. Hence [1, 44.42, p. 420], aof-' is analytic on all of f(Vo)n N. By 9.67
(f(U')- f(U' n vo)) n N is a properanalyticsubsetof Y n N. Hence, a of-'
of Of(u)ontoau.
is analyticon Y n N, so that, finally,f * is an isomorphism
519
COMPACTIFICATION
By 9.5, each point v E V has a neighborhoodU' as in 9.6. Therefore,
the firstassertionof the theoremfollowsfrom9.7. The assertionabout V(d,)
followsat once fromthe inductionprocedureindicatedin 9.6.
9.8. COROLLARY.Let U be an opensubsetof V, and f a continuousfunction on U whichis analytic on V0f U. Thenf is an aC-function.
This followsfromthe theoremand the Riemannextensiontheorem.
10. Analytic structure and projective embeddings
of the compactification
10.1. We now revert to the set up of 3.3 (i). In particular,G is a connectedsemi-simple
Q-group,withcenterreducedto {e}, whosesymmetric
space
is
a
X -K\GR ofnon-compact
boundeddomainand H(X) is the groupof
type
of X, in whichGI is offiniteindex. Moreover,
all holomorphic
automorphisms
X* is the unionof the rationalboundarycomponentsof X, endowedwiththe
Satake topology(4.8), IPan arithmeticgroup of automorphismsof X, V* =
of V X/F introducedin ? 4, and zc:X*
V* the
X*/IFthe compactification
canonicalprojection. There are finitelymany rational boundarycomponents
F. (O < i < m, F0 = X) such that V * is the disjointunionof the quotients
Vi
= Fi/I7(Fi). Since Vi
# Vi if i # j, we have Fi X Fj .I(i
# j).
10.2. The group F(F) acts in a properlydiscontinuousfashionon Fi;
hence, Vi is canonicallyendowedwiththe structureof an irreduciblenormal
analytic space [17]. We are thus in the situationof 9.1 and introducethe
sheaf a of germsof a-functionson V*. An a-functionon an open subset U
of V* is a continuouscomplex-valuedfunctionwhose restrictionto Vi n u is
analytic (0 : i _ m).
Let x e X* and v -w(x). Let U be a good neighborhoodof x in X*
(8.1). Then U' -(U)
may be identifiedwith U/Px,hence Vin u' with
of the analyticstructureon Vi and of
(F. n u)/(Ix n N(F,)). The definitions
if and only if f o wZis a continuous
U'
an
that
is
C
(d imply
(T-function
f:
functionon U, which is invariantunderIX, and whoserestrictionto F n u
is analytic for every rational boundarycomponentF. In particular, the
quotient a/w' of two integralautomorphicformsse,oG'for Px on U, of the
same weight,whereA' does not take the value zero in U, may be identified
withan a-functionof U'.
-
10.3. Let i be the index such that v E Vi. The canonical projection
as: X
F. (1.7, remark)inducesan analytic
mapa' of (X n u)l/, = v n u'
of v inVi n
ontoa neighborhood
ufU (U fnFi)/x. Let j be such that v E
and let w E Vi n U'. There exists y E F suchthatFj-3 n u # 0 and Fas y3 x.
520
BAILY AND BOREL
The canonicalprojectionuF,F?.Y
7w(F3.yn U)
inducesa holomorphic
map of
(Fry lnU)/(N(F3y) n rIx)
ontoa neighborhood
of v in Vi. We have the factorization(1.7, remark)
Let now f be a holomorphicfunctionaround v on Vi. The above remarks
implythat fo ai. extendsby continuityto an a-functionnear v in V, whose
restrictionto Vi aroundv is equal to f, and whose restrictionto Vj near w,
liftedto Fj3., is equal to
fOwo?F.,F-.Y
10.4. THEOREM. We keep the assumptions and notationof 10.1, 10.2.
Then (V*, CC)is an irreduciblenormal analytic space, in which each Vi is
embeddedas a locally closedanalytic space.
To provethe theorem,it is enoughto checkthat the conditions(i) to (iv)
of 9.2, with V and VT7
replaced by V* and V respectively,hold true in the
presentsituation.
Conditions(i), (ii) and (iii) are consequences of 4.11, 4.15 and 10.3,
respectively.
It remainsto checkthe separationof points by a-functions. Let v E Vi
= v. By 8.8, 8.9 thereexists an integralautoand x E F. be such that wZ(x)
morphicformE, of some weight1, such that bFAE(x) # 0. Let U be a good
neighborhoodof x in X *, on which the extensionof E does not take the
value zero, and let U' = w(U). Let p', q' E U' and j, k be the indicessuch
that p' E Vj, q' E Vk. Let p E F3 n '-(p'), q e Fk nr-'(q'). AssumedimFj >
dimFk. Since, by constructionwe have either j = k, or Fj, g Fk- *, there
exists (8.8, 8.9) a multipleV'- 1 m of 1, and an integralautomorphicform
E' of weightV'for17,such that
sJ?'FE
(p) # 0 ,
CFkE (q)
- 0.
The quotientE'/Em is thenan (i-functionon U' which separates p' fromq'.
Thus 9.2 (iv) also holdstrue in V*.
10.5. COROLLARY. Assume that G has no normal Q-subgroup of di-
mension 3. Let U be open in V*. Then every meromorphicfunction on
un v is the restrictionof a meromorphicfunctionon U. In particular,
the restrictionto V yields an isomorphismof the field of meromorphic
functionson V* ontothefield of meromorphicfunctionson V.
The assumption on G implies, by 3.15, that dim (V*
-
V)
?
dim V* -- 2.
COMPACTIFICATION
521
Therefore10.5 follows from 10.4 and a well-knownextensiontheoremon
normalanalyticspaces.
We recall the map f wr
of identifiesthe meromorphicfunctionson V
with the F-invariantmeromorphic
functionson X, i.e., withthe automorphic
functionsforP.
10.6. LEMMA. We keep the notationof 10.1, 10.2. There exist a weight
I and finitely many integral automorphic forms E, ***, E, of weight 1
such that the forms (iDx.Ej are nowheresimultaneouslyzero (O ? i !< m).
Givenx e Fi, thereexists a weightl, and an integralautomorphicform
of
Ex weight1 such that 14)DEx(x) / 0, (8.8, 8.9). There is thena good neighborhoodN(x) of x such that the extensionof E to N(x) is nowherezero. By
manysuch neighborhoods
compactness,V is coveredby the images of finitely
lemma
The
follows
then
the
1
for
l.c.m. of the lxJand for
by
taking
N(xj).
Ei's suitable powersof the Exi.
10.7. Let Ej (O < j ? N) be as in 10.6. If we trivializethe bundle Fli,
withholomorphic
the formsIDF.Ej are identified
functionswhichare nowhere
simultaneouslyzero; theirvalues at x e Fj are the coordinatesof a point in
C-+1 - 0. If we change the trivialization, these coordinates are all multiplied
by the same non-zeroconstant,hencedefinethe same point in the associated
projective space P(N, C). Thus, to x C F there is associated a well-defined
pointin P(N, C), whose homogeneouscoordinateswill be denotedby'1F.Ej(x).
Since the Ej are automorphicformsof the same weight, two points x and
x. ry(y
e P) will have the same imagein P(N, C), whencea map f: V*
P(N, C)
definedby
(X C Fi, i = O.*, m) .
f(zl(x)) = ((DFiE.(x), **
, iE.,(x))
outside
Since the quotientof two integralautomorphicformsis an (a-function
the set of zeros of the denominator,
mapping.
f is a holomorphic
10.8. LEMMA. We keepthe notationof 10.1, 10.2. There exist a weight
and
I
finitely many integral automorphic forms E0, ***, E, of weight l for
P(N, C) associated to the Ei's
r, satisfying10.5, such that themap f: V*
is a homeomorphism
of V* ontof(V*).
The proofis essentiallythe same as in the symplecticcase [2], and will
be describedbriefly.It is enoughto show that forsuitableEi's the mapf is
injective.
Let D and A be the diagonalsin V* x V* and P(N, C) x P(N, C) respectively, and S
(f x f)%(). Then S is an analytic subset of V* x V*
containingD, which is equal to D if and only if f is injective. Since, in
522
BAILY AND BOREL
a compactanalytic space, a decreasingsequence of analytic subsets is stationary,it is enough to show that if S
D, thenthereexists a similarmap
D
*
C)
Ve
associated
to
f':
integralautomorphicformsof some weight
P(N',
1' forwhichS' = (f' x f')-'(A) z S.
Let x e Fi, y e Fj (dimFi > dimFj), x' = r(x), y' w
r(y), be such that
(x', y') e S - D. Then x X y *P; by 8.8, 8.9 there exists a multiple 1' of 1and an
integral automorphic formE of weight 1 such that FE(x) # 0, 'I(FE(y) = 0.
in the E/'s, and E. We
We then take as Ej's all the monomialsof degree 1'/1
have S' ci S, and (x, y) XS', hence S' # S.
10.9. Let (F(]) be the graded ring of automorphicformsof positive
weight for P on X. It may be identifiedwith the set of invariantsof F in
a ringB = Ei2t Bi of holomorphic
functions,on whichH(X) operatesby
(f O g-)(x) -J(x, g)i f(x *g)
(f e Bi; x e X),
whereJ is the functionaldeterminantin some realizationof X as a domain
in euclidean space. Since X is connected,it follows that d(F) is integrally
closed [35, Exp. 17, No. 5]. We claimthat the subringdi'(r) of integralautomorphicformsis also integrallyclosed. Since d(F) is, this amountsto showing that, if h is an automorphicformof weight1 whichverifiesan integral
dependencerelation
hV +
(1)
L<o,,,
ai(x).hn-
= 0 ,
whereai is an integralautomorphicformof weight 1o i, then h is integral.
Let x e X* and U be a good neighborhoodof x. We may identifyh and the
functionson X r U. Moreoverthe ai's extend by conai's withholomorphic
tinuityto continuousfunctionswhose restrictionsto F n U are holomorphic
forany rationalboundarycomponentF. The relation (1) and the condition
9.2 (ii) implyagain, as in 9.4, that h extendsby continuityto a continuous
functionon U. If follows then from 9.3 that h is analytic on Fn U for
everyF. Thus h is integral.
10.10. Let
(Ei)0s<i be a set of integral automorphicformsverifying
10.8, A be the subring of A'(r) generated by the Ei's, and A its integral
closure. The latteris a finitelygeneratedalgebraoverC [15,Ch. 5, ? 3, No. 2]
and is containedin A'(P) by 10.9. It is elementarythat thereexists an integer
d such that the subringA(d) of elementsin A whose degree is a multipleof d
is generatedby Ad [15, Ch. 3, ? 1, No. 3, Prop. 3]. Moreover,A(d) is also integrallyclosed [15, Ch. 5, ? 1, No. 8, Cor. 3]. ThereforeA(d) is a normally
projectivealgebra overC, in the sense of [35, Exp. 17]. Let Ei (0 < i < M) be
a basis of Ad. Then
A(d)
-C[TO,
* , TM]/I, where I is the ideal of the rela-
COMPACTIFICATION
523
tionsbetweenthe Ei. The projectivevariety V(A(d)) c P(M, C) definedby I
is then normallyprojective. The map f: V* P(M, C) associatedto the Ei's
is well-defined,
injective; its image is an analytic, hence algebraic, variety,
containedin V(A(d)). It is in fact equal to V(A(d)) since otherwisetherewould
exist a polynomialPc CQTo,- - *, TM],not contained inI, such that P(EO, *** EM)
ofI. Thus
with the definition
would be identicallyzero on X, in contradiction
map of V* onto V(A(d)). Since both V* and V(A(d)
f is a bijectiveholomorphic
of analytic spaces. Thus we
are normalanalyticspaces,f is an isomorphism
have provedthe following:
-
10.11. THEOREM. We keep the notation of 10.1, 10.2. There exist a
weight 1 and finitely many integral automorphicforms E, of weight 1
whose extensionsto X* are nowhere simultaneouslyzero, such that the
associated map f: V* P(N, C) is an isomorphismof V* ontoa normally
projectivesubvarietyof P(N, C).
10.12. COROLLARY. Assume that G has no normal Q-subgroupof dimension 3. Then the field of automorphicfunctionsfor P is canonically
isomorphicwith thefield of rational functionson f( V*). In particular, it
is an algebraic function field of transcendencedegree equal to dimcX.
Every automorphicfunction is the quotient of two integral automorphic
forms of the same weight.
function
This followsfrom10.5, and fromthe fact that a meromorphic
on a projectivevarietyis rationalby Chow's theorem.
-
10.13. Let p: K0
GL(E) be a finitedimensional unitary representation
of K0. It defineson X a complexvectorbundle Up,the bundle associated by
p to G', viewed as principalK0-bundleby lefttranslations. The total space
is thereforethe quotient GOx KoE of GRx E by the equivalence relation
(g, v)
-
(keg,p(k).v)
It can also be written as P-Kc.
G' x P-.KE,
(k e KO, geGO , v e E).
where p is extended in the
obvious fashionto a representationof P- K whichis trivialon P-; hence,
it is a holomorphicvector bundle. An automorphicformof type d, is a Finvariant,holomorphiccross-sectionof d,. These formscorrespondin a canonicalfashionto theholomorphicV-valuedfunctionson X whichsatisfythe
relation
(x e X, ye r) ,
f(x. Y) = 4P(x,y)-'.f(x)
factorintroducedin 5.6. We let ip be the sheaf
whereftpis the automorphy
of germs of automorphicfunctionsof type p,for F on X/P. It is reflexive,
torsionless,and is knownto be an analyticcoherentsheaf [34, Exp. XX].
524
BAILY AND BOREL
REMARK. We have tacitly assumed that P operates on Up. This is cer-
tainlythe case if P CiG'. Otherwisewe assume that p extendsto a subgroup
K' of finiteindexof K n H(X) such that P ci K' . G'. Replacing K0 and KC
by K' and KC respectivelyin the above construction,
we see easily that the
actionof Go extendsto one of K'G1.
10.14. THEOREM.Assume that G has no normal Q-subgroupof dimension 3, and let CC,be as in 10.13. Then thedirect image i*KiU
in V* of the
sheaf of germs of automorphicforms of type d, is an algebraic coherent
sheaf. In particular, if U is an open subset of V*, every holomorphic
sectionof piP
over U n v extendsto a holomorphicsectionover U. The space
of automorphicforms of typedpis canonically isomorphicto the space of
holomorphiccross-sectionsof i*Upover V*, and is finitedimensional. The
ring of automorphicforms of positiveweightis finitelygenerated.
We identifyV* withits image underthe map of 10.11. Then the restrictionto V of the line bundle(9 of P(N, C) attachedto the divisor of a hyperplane is the sheaf ad of germsof automorphicformsof weightd. We know
(5.11) that if m is large enough,the productJdM.m p is an automorphyfactor
whichsatisfiesthe conditionallowingone to constructPoincareseries. Therefore, Theorem3 of [19] applies. It shows that given x e X/F, there exist
finitelymanyanalyticcross-sections
of the sheaf Up0 0n whichgeneratethe
fibreof ipat x. Since e - V has codimension? 2 (3.15), Serre's extension
theorem[36] applies,and yields the theorem,except for the last assertion.
We now knowthat A(F) = A'(P). Let 1 and Ei be as in 10.11. The automorphic formsof weightm 1 (m a positive integer) may be identifiedwith the
holomorphiccross-sectionsof 0-. They are thereforethe polynomialsof
degree m in the Ei's. This means that the algebra A(I7)('1is generatedby
the Ei's. Each space A(U)i is finite dimensional. Therefore,in order to
to show the existenceof an integer
establish the second assertion,it suffices
nosuch that
(n > no,p > no)
( 1)
A(F) A(P)p+.
A(I), (%+l).l
.
Since the sheaves involvedextendto algebraic coherentsheaves on a projective variety,the proofof (1) given by Serre [34, XX, nos. 9, 10] whenX/P is
compactapplies withoutchangeto our case.
APPENDIX
11. Connected
components
of automorphism
groups
In this section, we collect some partly known remarks on connected
components,whose use in the precedingsectionshas allowed forsomeslight
COMPACTIFICATION
525
technicalsimplification.X is a boundedsymmetricdomain,H(X) the group
of X, and Is(X)
(i.e., of automorphisms)
of complexanalytichomeomorphisms
the groupof isometriesof X withrespectto the riemannianstructuredefined
by the Bergmanmetric.
11.1. As is well-known,Is (X) and H(X) are semi-simpleLie groups,
X is the quotient of Is (X) by a
with finitelymany connectedcomponents,
maximal compact subgroupK, and Is (X) -H(X)0 is a non-compactsemisimple Lie group with centerreducedto {e}. Thus Is (X)0 = Ad g whereg is
it is knownthat Is (X) -Aut g.
the Lie algebra of Is (X). Furthermore,
Assumenow X to be irreducible. Then its isometrics are either holomorphic or anti-holomorphic,and H(X) has index twoin Is (X). In fact,
of g associatedto the Lie algebra f
let g = f + p be the Cartandecomposition
of K. Then Is (X) = KenP (P = exp p) and H(X) = (KfnlH(X)) -P. The
identitycomponentS of the center of the identitycomponentof K is oneInt k (k C K) is either the identityor the
dimensional,and an automorphism
by V -1 in the
inversions H- s51 on S. On the otherhand, the multiplication
tangentspace X0 of X at K is inducedby Ad s, where s0 is an elementof
order4 in S. ThereforeIntk(s0) is equal either to s0 or to s-1. The transin the second
formationk is holomorphicin the firstcase, anti-holomorphic
one. In particular
(1)
H(X)
nK
=Z(S),
and H(X) has index ? 2 in Is (X). On the otherhand,thereis clearlya linear
A on X0 whichcarriesthe given complexstructure
orthogonaltransformation
ontoits conjugate. By standardfacts on simplyconnectedriemanniansymmetricspaces, A extendsto an isometryof X, whichis thenanti-holomorphic,
henceIs (X) # H(X).
If X is the productof r irreduciblecomponents,it is clear that Is (X)
of
(resp. H(X)) is generatedby productsof isometries(resp. automorphisms)
the different
factors,and permutationsof isomorphicfactors.
These remarkshave alreadybeen made by E. Cartan, who has also given
the structureof H(X) in all irreduciblecases; it is connected,except in the
cases mentionedin 11.4, whereH(X)0 has indextwo in H(X), [16, p. 152].
11.2. PROPOSITION.Let D be the natural compactificationof X (1.4).
The action of H(X) on X extendsbycontinuityto a continuousaction on D,
and the restrictionof h e H(X) to any boundarycomponentis holomorphic.
to provethis for K n H(X) since H(X) is generatedby this
It suffices
group and by H(X)0. But, if k e K n H(X), then k commuteswiththe ele-
526
BAILY AND BOREL
ments0consideredin 11.1, and the extensionto Pc of Ad k leaves the two
subspacesp+,p- stable; hence,the actionof k on X extendsto a lineartransformationof p+. Thereforek operatescontinuouslyon D. Furthermore,
the
boundarycomponentsare opensubsetsofcomplexaffinesubspacesofP+; hence,
the restrictionof k to such a componentis holomorphic.
11.3. PROPOSITION.Let G be a connectedsimple algebraic group defined
over R such that the symmetricspace of non-compacttype X of GR is a
bounded symmetricdomain. Then H(X) n GR - G'. The group GR has
eitherone or two connectedcomponents.
Since G is simple,X is irreducible,and we may identifyGR with a subgroupof Aut gR, namelyAd gc n Autg9. We keep the notationof 11.1; in particular,K n GRis a maximalcompactsubgroupof GR, and GR = (K n GR)PThe second assertion follows fromthe firstone and 11.1; in view of
11.1 (1), the firstassertion is equivalent to: GR n z(s) is connected,which
we now prove.
Being the centralizerof a torusin Gc, thegroupZ(S)c is connected,since
G is (cf. [9, ? 18]). It is definedover R, and its Lie algebra is tc. Thereforef
is a compactreal formof fc,and K0 is the identitycomponentof a maximal
compactsubgroupL of Z(S)c. Since Z(S)c is connected,so mustbe L, whence
z(s) nK= K .
11.4. REMARK. In the typeIV, (IV refersto Siegel's notation; it is III
in [16]) of boundedsymmetric
domains,G - PSO(n + 2, C) is the quotientof
the special orthogonalgroupin n + 2 variablesby its center,GR = PSO(n, 2),
and K n GR is the group of elements of determinantone in 0(n) x 0(2)
(dividedby {+1} if n is even). Fromthis we see readilythat
(a) if n is odd, GR= Is (X), GO = H(X), and
(b) if n is even, Is (X)/Is (X)=
Z, + Z2,
The situation(b) also occursfortypeInn (n
>
Z2,
GR/G= H(X)/H(X)R
Z2
2).
11.5. LEMMA. Let G be a connectedsemi-simplegroup definedover R.
Assume that Go has a center reduced to {e}, and has the same rank as its
maximal compactsubgroups. Then the centerof Gc is reducedto {e}.
Let t be a Cartan subalgebraof a maximal compactsubgroupK of GR.
The assumptionimpliesthat tc is a Cartan subalgebra of g. It is the Lie
algebra of a maximaltorus Tc of Gc,whichis definedover R, and whose subgroup of real points is compact. The latter is then necessarilya maximal
compactsubgroupof TO, and is connectedand equal to exp t. Let now z be
in the centerof Gc. It belongsto Tc and is of finiteorder;hence,z and exp t
generatea compactsubgroupof Tc. Thus z e exp t n GR,and z = e.
527
COMPACTIFICATION
REMARK. If we dropthe assumptionon the rank of K, the lemma becomesfalse as is shownby the case whereGR - SO(p, q), (p, q odd).
11.6. We now revertto the notationof 1.3, 1.5, and prove that G(F) is
to
connected,as asserted in 1.5. An obviousreductionshowsthat it suffices
do this when X is irreducible.In view of 1.5 (1), (2) this amountsto proving
that the group Pb/Zb of 1.3 is connected.
The groupQC= PC/Zb,C is almostsimple,connected,definedover R, and
QO= Ad lb. On the otherhand,the symmetricspace Fb of non-compacttype
of Lb is a boundedsymmetricdomain; hence, Lb has the same rank as its
maximal compactsubgroups. By 11.5, we have thenQC= Ad qc, whichimplies that QR is a subgroupof Is (Fb). By 11.4, all elementsof Pb inducecomof Fb; therefore(11.3), the image of Pb in QR
plex analytichomeomorphisms
is connected,equal to Ad fb. The kernel of the homomorphism
Pb
QR is
Pb n Zb,C. This is a normalsubgroupof Pb, withLie algebra Sb, whichcontains
Zb. It is thereforeequal to Zb (see 1.3), whencethe result.
-
UNIVERSITY
INSTITUTE
OF CHICAGO
FOR ADVANCED
STUDY
REFERENCES
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functionsand theproblemof moduli,Bull. Amer.
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Math. Soc., 69 (1963), 727-732.
, "On compactification
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, Fourier-Jacobiseries, ibid.
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(Received February 21, 1965)