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0, and, therefore, w(l) > w(l'). It follows from (25) that 1 w(1') > 2A w([jyp,y p]);?; 0.
We have proved that l E V implies that w(1) >
o.
72
THE GEOMETRY OF CLASSICAL DOMAINS
The cone V is homogeneous with respect to the entire group @o, so > 0 for all A E @o' We will now show that if w(AI) > 0 for all AE@o, then IE V. We apply a transformation of the form (23) with xp = (ljA)jyp to such an I. Then I = Arp + l' + yp is carried into w(AI)
1
[" = Ar p+ [' - 2A [jyp, yp]. Moreover, consider the vector I;' = (exp (ad L ijrp))l". It is not difficult to see that 1 [~ = [tArp+ [' - 2A [jyp, yp]. By hypothesis,w(l~) > 0 for all t, and, therefore, w(l') ~ (lj2A)w([jyp,yp])' The domain V is open, so w(!') > (lj2A)w([jyp,yp])' We have proved that 1= Arp + l' +Yp is such that w(AI) > 0 for alIA E@o, then w(k(l)) > 0, where k(1) = AI' - tUyP' yp]. It is not difficult to show that k(exp (adLx)l) = (exp (adLx))k(l)
for all x EjL'. As a result, if weAl) > ofor alIA E@o,then w((exp (adLx)) k(/)) > 0 for all x EjL'. By the induction hypothesis, this implies that k(l) E V' and, therefore, lEV. This completes the proof of the lemma. We will now prove that the domain Sl is a Siegel domain of genus 2. This requires us to verify statements 1, 2 and 3 of the first proof. Statement 1 follows from the fact that the set of linear forms of the form weAl), where A E@o, contains a complete system of linearly independent forms. Indeed, if the forms wCAI) were linearly dependent, this would also be true for the forms w([a, I]), where a EjL. As a result, in this case there exists an ao =f:. 0 such that w([a o, l]) = 0 for all l. This is clearly impossible, because w([ao,jaoD < 0 for ao =f:. O. Statement 2 is obvious. It remains to prove 3. We have F(Bu, Bu)
= AF(u, u), A = exp(adLjl)
B = exp(aduj[),
[EL.
As a result, w(AF(u, u)) = w(F(Bu, Bu))
= w([j(Bu), Bu]) > 0
for all A E@o' By our lemma, this implies that F(u, u) E V. This completes the proof that Sl is a Siegel domain of genus 2.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
73
In concluding this section, we will present a method permitting explicit representation of polynomial inequalities defining the cone V. We will prove that if V is a cone of rank p, there exist p polynomials Pl(l), ... ,Pi!) such that the cone V coincides with the set of points Pk(l) > 0,
k
= 1, ... ,p.
°
If A > 0, p = 1, the cone V consists of vectors of the form Ar l , A > We can take A for Pl. We will now present an inductive construction of the polynomials Pl(l), ... ,Pil). As in the proof of the preceding lemma, we will use relationships (25). Assume that Pl(l'), ... ,Pp_l(l') are polynomials that define a cone V' by means of the inequalities Pl(l') > 0, ... ,Pp-l(l') > 0.
We substitute AI'--![jyp,yp] for 1', and denote the resulting polynomials in lby P 2 (l), ... ,Pil). We also setPl(l) = A. It follows from considerations stated in the proof of the lemma that the inequalities P 1(l) > 0, ... ,Pi!) >
°
describe the cone V. It is not difficult to show that the degree of Pk(l) is no greater than 2k - l • The considerations we have given make it possible to explicitly write the polynomials corresponding to the given j-algebra. When the cone V is the set of all positive definite symmetric matrices, the polynomials P ll) are the successive principal minors of these matrices. The polynomials we have constructed have the following important property: This property is important in the theory of special functions (S. G. Gindikin [2] and [3]). Section 6. Universal j-algebras
Let B be a normalj-algebra. In this section we will describe use of the algebra B to construct a j-algebra with the following properties: (1) A admits the decomposition A=B+G
(1)
74
THE GEOMETRY OF CLASSICAL DOMAINS
where B is a j-ideal and G is a semisimple j-algebra. (2) Every normal j-algebra N with the ideal B and the factor algebra G' can be represented in the form
W=B+G',
(2)
and there exists aj-homomorphism ¢: G' ~ G such that [g:
bJ =
[¢(g'),
bJ.
(3)
It is natural to call such an algebra A a universalj-algebra. We will first describe the construction of the algebra G. Let B be a normalj-algebra, and let
B=L+jL+U
(4)
be its canonical decomposition. Consider the set G of all real linear transformations u ~ pu, such that
U
E U, (5)
(uadjL)p = p(adujl)
for all
lEL.
(6)
Here and in what follows P(Ul' uz) = CO([Ul' uz ]). It is clear that if Pb pz E G, then [pl,Pzl = PI pz - PZPl belongs to G, i.e., G is a Lie algebra. The transformations adujl commute withj, as a result of which it follows from axiom III thatjEG. We now define the endomorphismj in G by means of the following formula: j(p) = t[j, pJ.
(7)
Moreover, we denote the set of all pEG such thatj(p) = 0 by Go. It is easy to verify thatjz(p) +P E Go for allp E G. The form co is defined in the following manner. Any real linear transformation P of the space U can be written in the formpu = au + 13ft, where a and 13 are complex linear transformations of U and ft is the complex conjugate of u. It is not difficult to show that if pu satisfies (5), a can be described by a skew Hermitian matrix in any orthonormal basis, while 13 can be described by a symmetric matrix. We set co(P) = (lji) spa. It follows immediately that {G, Go,j, co} is aj-algebra. We now show that the algebra G is completely reducible. Indeed, let U0 be a subspace of the space U invariant with respect to G; we will
75
THE GEOMETRY OF HOMOGENEOUS DOMAINS
show that its orthogonal complement U 1 is also invariant with respect to G. Indeed, if Uo E U, U 1 E U 1 and pEG, then p(uo, PU 1) = - p(puo, Ul) = 0.
As a result,pul E U1. We now define the algebra A in the following manner: A =B+G.
We introduce a commutation operation so that: (1) on Band G it coincides with the commutation already defined, (2)
(8)
[G,L+jL] = 0,
(3)
[p,
u]
(9)
= pU.
It follows immediately, as the reader can easily show, that A is a j~algebra.
It remains for us to prove universality ofthej-algebra A, i.e., that any normal j~algebra N in which B is an ideal can be represented in the form (2), and that there exists aj~homomorphism ¢: G' -?- G for which (3) is true. Let N be a normalj-algebra in which the algebra B is an ideal. As we showed in Section 4, the orthogonal complement G' of B is aj-subalgebra of the algebra N. Using the theorem of Section 3 on the form of the roots of any normalj~algebra, we can show with no difficulty that
[G', L+ jLJ We set
Pg=adug,
0,
[G', u]
where
c
U.
gEG'.
(10) (11)
We now show that (11) defines a j-homomorphism of the algebra G' into the algebra G. First of all, we must verify that the operators Pg belong to G. We have P(piUl), u 2) + P(Ub piu2)) = w([g, [u 10 U2]]) = 0.
(12)
It follows from this and (10) that Pg E G. We now show that the mapping g -?- Pg is a j-homomorphism. It follows from the fact that [G', u] c Uthat P[91,92J
= P91 P92 -
P 92 P 91 •
(13)
It remains to verify that (14)
76
THE GEOMETRY OF CLASSICAL DOMAINS
It follows from axiom II of the definition of normalj-algebras that pg+jPjg+jpgj-Pjgj or
0,
[j, Pjg] = j{j, [j, pg]],
(15) (16)
which is equivalent to (14).
Section 7. Canonical Models of Bounded Homogeneous Domains 1. This section is devoted to describing realizations of bounded homogeneous domains in the form of Siegel domains. Recall that a Siegel domain is said to be homogeneous if its group of quasilinear transformations is transitive in it. Let ~ be some complex manifold. A fibering of the manifold ~ is said to be analytic if the base ~ 1 of this fibering is a complex manifold and the projection ¢ of the manifold ~ onto ~ 1 is a complex analytic mapping. All of the fiberings encountered below are analytic. Note that, as a rule, the fiberings discussed below are not locally trivial as complex analytic fiberings, and, simultaneously, as real analytic fiberings they are direct products. Henceforth, we agree to say that a fibering of a manifold ~ is homogeneous if the set of analytic automorphisms of ~ that preserve the fibering is transitive in ~. There is a natural homogeneous fibering for every homogeneous Siegel domain of genus 3. Some homogeneous fibering is therefore associated with each realization of a bounded domain in the form of a homogeneous Siegel domain. The fundamental result of the present section consists in the fact that the converse is also true, namely, that the following theorem is true. Theorem 3. Let ~ be a bounded domain in C". With each homogeneous analytic fibering of the domain ~ lve can associate a realization of the domain ~ in the form of a homogeneous Siegel domain of genus 3 whose base is the base of the given fibering of the domain ~. The plan of the proof for this theorem is as follows. First we describe the construction of the Siegel domain of genus 3 that corresponds to a given universalj-algebra. Then we prove that with each homogeneous fibering of the bounded domain ~ we can associate a j-ideal of the normal j-algebra associated with the domain ~. Theorem 3 then follows from the theorem of Section 6 on universality.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
77
2. Let B = L+ jL+ Ube a normalj-algebra. We denote the universal j-algebra corresponding to the algebra B by A. In this paragraph we will describe the Siegel domain S of genus 3 in which the transformations in the group whose Lie algebra is A are quasilinear transformations. Let V be a cone, and let F(u, v) be the vector function corresponding to the algebra B. A group whose Lie algebra is isomorphic to the algebra B is transitive in the domain H 5; C"+ m defined by the relationship
Imz-F(u, U)E V.
(1)
We denote by ® the group of affine transformations of the domain H of the form Z -1-
where
exz,
ex = exp(adLjl),
U -1-
f3u
(2)
f3 = exp(adujl).
As we showed in Section 5, transformations of the form z -1- exz are transitive in the cone Vand they satisfy the following relationship:
F(f3u, f3u) = exF(u, u),
U E
U.
(3)
Consider the set K of all antilinear transformations u -~ pu of the space U that possess the following properties:
F(pu, v) = F(pv, u)
(4)
(V is the closure of the cone V)
F(u, u)-F(pU,pU)E V F(u, u) =J. F(pu, pU),
f3p
pf3 for
f3
(5)
if u =J. 0
of the form
adujl.
(6)
(7)
We denote the set of all antilinear transformations u -1- pu for ¥00
~
II
L lim inf p2(Z~III), wim») k=lm->oo
~
Lp2 (a k , bk ).
(6)
k=l
Expression (4) clearly follows from (5) and (6). Our statement is proved. Again let ~ be an arbitrary classical domain. We denote some geodesic by z(s). We will show in Sections 10, 11, and 12 that the topology of an affine complex space contains a limit belonging to F for z(s) as s -1- + 00. Let ZE~ and aEF. We agree to say that points z and a may be connected by a geodesic if there is a geodesic z(s) such that z(o) = z and z( + (0) = lim Z(8) = a. s->
+ 00
Generally speaking, two arbitrary points z E ~ and a E F cannot be connected by a geodesic, as follows from Theorem 7. We should note that any two interior points of ~ may always be connected by a unique geodesic. Theorem 7. Let ZE~. Any component!F contains exactly one point a that may be connected to z by a geodesic. The set ~a of points in the domain ~ that may be connected by geodesics to a given point a E!F, is analytically equivalent to some Siegel domain of genus 2. A general proof applicable to arbitrary symmetric domains could be constructed for Theorems 4, 5, and 6 at the present time. The author, however, found it desirable to preserve the original computation, for it may prove useful upon a first reading and for construction of examples. Sections 10 and 11 give proofs of Theorems 4, 5, and 6 for classical domains of the first, second, and third types.
90
THE GEOMETRY OF CLASSICAL DOMAINS
In the past (I. I. Pyatetskii-Shapiro [11 D, this theorem replaced the more powerful Theorem 6 in applications to the theory of automorphic functions. Let ff be some boundary component. We denote the set of all points in domain ~ that are connected by geodesics to a point a E!F by ~ It follows from Theorem 6 that domain ~ "fibers" into sections ~ a' a E ff. This fibering is not an analytic fibering in the usual sense of the word, because it does not have the local structure of a direct product. As we will show in Sections 10 and 11, this "fibering" coincides with the natural fibering in the corresponding canonical realization of ~ in the form of a Siegel domain. From now on we will use the following terminology: A component consisting of one point is said to be zero-dimensional. A component ff of an irreducible classical domain is said to be a component of genus one if the corresponding fibers ~ a are Siegel domains of genus 1. A component ff of an irreducible classical domain is said to be a component of genus 2 if the corresponding fibers ~ a are Siegel domains of genus 2. In Sections 10 and 11 we wi11list all components and the fibers ~ a corresponding to them for classical domains. It will become clear from this enumeration that a component of genus 1 is always zerodimensional for irreducible domains. The converse is not true. For example, all components of the balllz l 12 + IZ212 < 1 are zero-dimensional, but they are components of genus 2. In Sections 10 and 11 we will show that ~ is analytically equivalent to some bounded homogeneous (generally speaking, non symmetric) domain. The existence of nonsymmetric bounded homogeneous domains was first discovered in this way. We will use the following terminology for reducible domains. Q'
Q
~
Let where the
~1' .. " ~P
=
~ 1 X ...
x f'2 P'
(7)
are irreducible domains. A component ff=ff 1 x, .. xff p
(8)
is said to be a component of genus 1 if each factor ffil ;;;; k ;;;; p) is either a component of genus 1 or an "ideal" component, i,e., coincides with ~k'
THE GEOMETRY OF HOMOGENEOUS DOMAINS
91
Components (8) of genus 2 for domains of type (7) are similarly defined. The remaining components of the form (8) are called components of genus 3. The following important subgroups of the group G of all analytic automorphisms of the domain ~ may be associated with every component ff: G 1 (ff)-the set of all transformations of G that map ff into itself; GzCff)-the set of all transformations of G that leave every point of !F fixed; G3 (ff)-the set G of all transformations g of G that leave every point of ff fixed in the sense determined by the interior Riemann geometry of the domain~. This means that for any geodesic z(s) such that lim z(s) = a EF, s .... +00
the limit relationship lim p(z(s), gz(s)) = s .... +00
o.
holds. We agree to denote the maximal commutative normal subgroup of the group G3 (ff) by Giff). We agree to denote the centralizer of the group Giff) in the group G 1 (ff) by Gs(ff). It is clear that GV+ 1 (!F)(l ~ v ~ 3) is' a normal subgroup of the group GvCff). In the following sections we will show that the group G3 (ff) coincides with the group A of "parallel translations" that correspond to the canonical realization of the domain ~ in the form of a Siegel domain. Section 10. Classical Domains of the First Type
Classical domains of the first type are described in the following manner (Siegel [1]). Let p "?:q > be an integer. We will consider p x q matrices C as points in a pq-dimensional complex space. The domain that interests us, ~, consists of the matrices Z, such that
°
Eq-Z*Z > 0, where Eq denotes the identity matrix of order q. The group G of affine transformations of an m = p+q-dimensional
92
THE GEOMETRY OF CLASSICAL DOMAINS
complex space that preserve a Hermitian form with p minuses and q pluses is a group of analytic automorphisms of the domain in question. More accurately, there is a correspondence between each square matrix M of order 111 = p+q such that M*HM=H,
H=(
-Ep
0 ),
o
(1)
Eq
and an analytic automorphism
M=(~ ~)
Z ..... (AZ+B)(CZ+D)-"
of the domain. t The boundary F of the domain
~
Eq-Z*Z ~ 0,
(2)
consists of all Z such that
det(Eq-Z*Z)
= O.
(3)
We will begin by enumerating all the boundary components of the domain ~. We prove the following lemma as a preliminary. Lemma 1. Let cP1(t), ... , cPlI(t) be a/unction analytic on the disk It I < 8: M = sup (lcP1(t)12 + ... + IcP,lt)12). It I <e
IcP1(0)12+ ... +lcP,lO)12 ~ M,
Then
(4)
where the equality occurs if and only if all a/the/unctions cPk(t)(1 ~ k ~ n) are constant. Proof The following equations are a consequence of the Cauchy integral formula: cPk(O) =
~J.2l! cPk(pe
2n
1
11
thus
iO
)
de,
k = 1, ... , 11; P < 8;
0
k~1 IcPk(0)12 ~ 2n
I2l! 0
11
k~1 1cPk(pe
2
iO )I
de < M.
If we have the sign of equality here, then
l¢k(OW
= 2~
f:'
l¢k(pe"W d8, k
= 1, """' n,
(5)
and, consequently, cPk(t) = const, k = I, ... , n. The lemma is proved.
-r It can be shown that all analytic automorphisms are of the form (2) if p =f- q. If, however, p = q, we also have the automorphism Z --+ ZI (Klingen [1]).
THE GEOMETRY OF HOMOGENEOUS DOMAINS
93
The following theorem contains a description of all components of the domain ~. Theorem 1. The set of points in F of the form
o )r Z r
,z*z < E q - n
1~ r
~ q,
(6)
p-r
q-r
forms a component $71' that is analytically equivalent to a classical domain of the first type with parameters p - rand q - r. Any boundary component may be mapped by some analytic automorphism of the domain ~ into the component $7", 1 ~ r ~ q. Every point F belongs to some component. Proof It is easy to verify directly that !FI' is a regular analytic subset of points of the boundary F of the domain ~. As a result, in order to prove that !F r is a component, it is sufficient to prove that any analytic curve contained in F and intersecting !FI' is entirely contained in !Fr. Let Z =Z(t), 1tl < e be some analytic curve, Z(O) E!F" Z(t) E F for all t, It I < e. We represent Z in the form
Z12 Z22 r
The inclusion Z
c
)r p-r.
q-r
Fimplies that Eq-Z*Z
~
0, whence, in particular,
Er-zt1 Z11 -Z~\ Z21
~ O.
(7)
It follows from (7) that if Z(t) E F, then p
r
L L \Zij(t)\2 ~ r, It I < e.
i= 1 j= 1
Furthermore, Z(O) E!FI' implies that P
I'
L L1 IZij(0)12 = i= 1
1'.
j=
As a result, by the lemma we proved above, the Z ij(t)(l ~ j ~ r, 1 ~ i ~ p) are constant, and, therefore, Z11(t) == En Z21(t) == O. In order to prove that Z 12(t) == 0, it is sufficient to note the equivalence of inequalities Eq-Z*Z ~ 0 and Ep-ZZ* ~ 0 (Siegel [1], p. 135). Using the same arguments as above, it is easy to show that Z12(t) == 0
94
THE GEOMETRY OF CLASSICAL DOMAINS
by means of the inequality Ep - ZZ* ~ O. It remains to show that ZIit)Z22(t) < Eq_r for all t, It I < 8. The inequality ZI2(t)Z22(t) ~ Eq_r follows from the fact that Z(t) EF. We must show that det(Eq_r -ZI2(t)Z22(t) =1= 0 for all t, Itf < 8. If this is not so, then there exists a to, Itol < 8, such that Eq-r-Z1'2(tO)Z22(tO) ~O,
Then there is a column vector
~o,
det(Eq_r-Zi2(to)Z2ito)) = O.
such that
~~(Eq-r - ZI2(tO)Z2ito»~0 = O.
We set ~(t) = Z(t)~o. It is clear that for all t, It/
Z'i2Z22' The lemma is completely proved. We will now turn to 'the proof of Theorem 5. We must construct canonical realizations in the form of Siegel domains for domains~.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
97
According to the theorem stated in Section 5, there must be as many realizations as there are typical components (in our case, q). We will first describe a compact complex symmetric manifold D that has the same relationship to ~ as the Riemann sphere has to the unit disk. The manifold D is called the dual manifold of the manifold ~. We denote the set of all complex m x q (m = p +q) matrices U of maximal rank by Q. If UEQ, then UREQ, where R is a nondegenerate square matrix of order q. We agree to say that the matrices U and UR are equivalent. The set of all equivalent matrices forms a class. The set of classes admits a natural complex structure. It is not difficult to verify that it is the compact symmetric manifold D. We will now prove that ~ may be realized in the form of a domain in D. Let H be the Hermitian matrix defined by relationship (1). We will consider the set Q H of matrices U such that U*HU > O.
(16)
It is easy to verify that if U E QH, then UR E QH' where R is any nondegenerate q x q matrix. Thus, every class of equivalent matrices is either entirely contained in QH' or its intersection with Q H is empty. The set of classes contained in Q H is clearly a domain in D. We set
u=(~:):.
(17)
q
Condition (16) may be written in the form U*HU= -U[U 1 +U1U 2 >0.
(18)
As a result, if U E QH, then the matrix U 2 is nondegenerate. Thus, it is possible to select a unique representative of the form
GJ for every class contained in QH' Condition (16) implies that Eq-Z*Z> O.
We have proved that ~ may be realized in the form of a domain in D. Note that the classes belonging to the boundary F of the domain D
98
THE GEOMETRY OF CLASSICAL DOMAINS
!!2 consist of the U for which the determinant of the matrix U* HU is equal to zero and U*HU ~ O. In other words, F consists of those Z satisfying the following conditions: (1) the determinant of the matrix Eq-Z*Z is equal to zero, and (2) Eq-Z*Z ~ O. We will now turn to describing the canonical realizations in the form of Siegel domains. A general method for finding the canonical realizations consists in the following. Let H be an arbitrary Hermitian matrix of order 111 with p positive and q negative characteristic roots. We denote the class of those 111 x q rectangular matrices U such that U* HU > 0 by QH' We agree to say that two matrices U and UR, where R is a nondegenerate q x q matrix, belong to the same class. We will now prove that the set DH of classes belonging to Q H is a domain in D (the dual manifold) that is analytically equivalent to our domain ~. Indeed, there is a non degenerate matrix M of order 111 such that
M*HM = Ho.
Here Ho indicates the Hermitian matrix defined in (1). The mapping maps Q H into Q Ho ' It induces an analytic automorphism of D that maps DH into DHo' In order to realize DII in an affine complex space, it is sufficient to find a method for setting up a correspondence between each class of matrices U E Q H and a point in the affine complex space. The analytic automorphisms of DH may be described in the following manner. Let G be the set of all 111 x m matrices A such that A *HA = H. Every matrix A corresponds to an analytic automorphism U -+ A U of the manifold Q. It is easy to see that an analytic automorphism of D that maps DH into itself corresponds to it. The boundary of DH in D clearly consists of those classes of matrices U such that (1) det(U* HU) = 0,
(2) U* HU ~ O.
The invariant r(Z1' Z2) introduced above for a pair of points (see Lemma 2), is the same as the rank of the matrix ut HU2 • First of all, in fact, the rank of the matrix ut HU2 is independent of the choice of
THE GEOMETRY OF HOMOGENEOUS DOMAINS
99
matrices U 1 and U 2 and depends only on the classes to which they belong; furthermore, it is clear that we have W 12 = V; HU 2
for appropriately chosen U l and U2 • The last conclusion may be stated in the form of the following lemma, which extends Lemma 2. Lemma 3. We denote the rank of the matrix ut HU2 by r(Ul , U2 ). The matrices U 1 and U2 are mapped onto a point in one boundary component of the domain DH by the mapping n -+ D if and only if (19)
We will now write the canonical realization Sq corresponding to a zero-dimensional component of the boundary of the domain~. Consider a matrix H of the form
o ,
PI
= p-q.
o As we can easily verify, p characteristic roots of this matrix are equal to - 1, and the remaining q are equal to 1. We partition U in the following manner:
V=
VI
q
V2
Pl.
U3
q
Condition (16) may be written in the form
W
= V*HV = i(V; v 3 -vj Vl)-V; V 2 > o.
(20)
We will show that if U E nH then the matrix U 3 is nondegenerate. Indeed, otherwise there would be a nonzero vector b such that U 3 b = O. Then b* uj = 0 and, therefore, b *W b = i( b *V t V 3 b - b *V 3 V 1 b) - b *V; V 2 b = - b *V; V 2 b ~ O. We have been led to a contradiction that proves that the matrix U 3
100
THE GEOMETRY OF CLASSICAL DOMAINS
is nondegenerate for U E QH' It follows from this that every class of equivalent matrices V E Q H contains a unique matrix of the form
Substituting a matrix V of this form into (20), we obtain the inequality 1:- ( U 1 -U *1 ) - U'~i U 2> 0 .
(21)
l
This inequality defines some unbounded domain in pq-dimensional complex space (whose coordinates are the entries in the matrices Vb k = 1,2). The domain constructed is the Siegel domain of genus 2 that was described in Chapter 1, Section 2. "V-,Te must now prove that the transformations preserving an "infinitely distant" zero-dimensional component are linear. It follows from Lemma 3 that membership of a matrix V in. some zero-dimensional component is a consequence of the equations U*HU
= i(Ui
u 3 -UiV 1 )
V;V2
= O.
U sing this fact, we can easily verify that the class of matrices V that contains a matrix of the form
(22)
is carried into a zero-dimensional component under the mapping Q -+ D. We naturally assume it to be "infinitely distant". We will now find the automorphisms of the domain that leave point (22) fixed. The matrices A corresponding to such automorphisms satisfy the condition
E
A 0
(23)
o where R is some nondegenerate q x q matrix that depends, generally speaking, on A.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
We partition A into blocks
A=
[AU
A12
A2l
A22
A13 A 23
A31
A32
A33
PI
q
q
101
r Pl'
q
and write (23) in the explicit form
Au
A12
A13
E
All
A21
A22
A 23
0
A21
A3l
A32
A33
0
A3l
nR.
(24)
whence A21 = 0, and A31 = O. Furthermore, we can easily show that A 32 = 0 by using the fact that A* HA = H. A transformation with such a matrix A is linear. Indeed,
A12
A13
Au U I +A12 U 2 +A13
A22
A 23
A22 U 2 +A 23
o
A 23
E
A33
(25)
E It is easy to verify (see Chapter 1, Section 2) that transformations of the form (25) form the full group of affine transformations of Siegel domain (21). We now turn to describing the remaining canonical realizations. Consider a matrix H of the form
o
0
o
o o
, Pl=p-r,
Ql=q-r.
(26)
102
THE GEOMETRY OF CLASSICAL DOMAINS
It is easy to see that p characteristic roots of the matrix H are equal to - 1, and that the remaining q are equal to + 1. We partition U into blocks in the following manner:
u 11 U12 u=
r
U 21
U 22
P1
U 31
U 32
q1
U41
U 42
r
r q1 We now write condition (16) in the form
"I I I where (27)
; (U12 U 12 - U;2 U 42) + U: 2 U 32 - Ut2 U 22' I
j
We will prove that if U E OH then the matrix U 31 U32) (28) ( U 41 U 42 is nondegenerate. Assume that this is not true for some U satisfying (27). We multiply U by a nondegenerate square matrix Q such that for = UQ the entries of the last column in the matrix
a
are all zero. Let e denote a ql-dimensional vector whose coordinates are all equal to zero, except for the last, which is nonzero. It is easy to see that a32 e = 0, a42 e = O. As a result, e* W 22 e = - e* Ut2 U 22 e ;£ O.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
103
We have obtained a contradiction. Thus, matrix (28) is always nondegenerate. As a result, any U satisfying (27) may be normalized so thatt (29) We write formulas (27) in the following manner:
W
(30)
* W 12 = W 21 W 22
*
1 12 - U 21 =-;-U I
U 22 ,
= E q1 -Ut2 U 22 ·
The relations (30) define an unbounded domain Sr, r = Q-Ql' in a pQdimensional complex space whose coordinates are the entries of the matrices Uu, 1 ~ i,j ~ 2. We will now prove that Sr is the Siegel domain of genus 3 for which the component fFr serves as a base. We first prove that (31)
if and only if (32) Indeed, it follows from (31) that Q*WQ > 0
t Normalization is the selection of a unique representative U for each class of equivalent matrices in QH.
104
THE GEOMETRY OF CLASSICAL DOMAINS
for any nondegenerate matrix Q. We assume, in particular, that
Q= (
E
- w2l
fV 21
r
o ) > o.
Then
(33)
W22
It is clear that inequalities (31) and (33) are equivalent and, therefore,
so are (32) and (33). We may therefore write relation (30) thus:
1
-:-(U 11 - U 11 ) - U 2l U 21 1 * * I
-CiU 12 +uf, U22 lWzlC:U 12 + Uf, U22l* >0, W 22 = E q1 - U 22 U 22 >
j
(34)
o.
After parentheses are removed, the first of these inequalities takes the form
-iU 12 Will U't2 U 21 +iU't1 U 22
w2l
UI2 > O.
(3,5)
In order to make it clear that the domain obtained is a Siegel domain of genus 3, we will introduce some new notation that is characteristic of Siegel domains. We set t = U 22 ,
Z
=
2U ll ,
U
=
(U 12 , U 2l ),
v = (V12' V2l ).
It follows from (30), as we should expect, that t may vary over a classical domain of the first type with parameters PI' q1. We define an operator
for separating the real part in the space of z: (36)
We take the cone of all Hermitian positive definite matrices of order r for the cone V.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
105
Finally, we set
w;l Vt2 + viI U 22 w;l ui2 U 21 +1-i(U 12 w;l V 21 + V 12 W;/ U 2*2 U 21 ).
LtCu, v) = viI U 21 + U 12
(37)
We can directly verify that our domain is the Siegel domain corresponding to the cone Vand the function Llu, v). It is easy to show that the following identity is valid: 1 (Epi - U 22 ui2)-1 = Eqi + U 22 U;2 Ui2.
This identity may be used to represent LtCu, v) in the form
L/u, v)
= V!l(E pi - U 22 U 2*2) -lU 21 + U 12 W;;l Vt2
+1-i(U 12
w;l ui2 V 21 + V 12 w;l ui2 U 21 )
The boundary F of this domain consists of those matrices U such that det U*HU = 0,
U*HU ~ 0.
(38)
Consider the matrices U E F of the following form:
z*z < E.
U=
(39)
It is easy to verify that a class of equivalent matrices may contain no more than one matrix of the form (39). Furthermore, if U and are defined by (39), then
a
° )
W=U*H(J= ( 0 *_ Eqi -Z Z
°
'
whence it is clear (see Lemma 3) that U is a component analytically equivalent to the component !Fr. This component is naturally treated as "infinitely distant". We will prove that the analytic automorphisms of our domain that map component (39) onto itself are quasilinear transformations
106
THE GEOMETRY OF CLASSICAL DOMAINS
(defined in Chapter 1, Section 3), while the analytic automorphisms leaving the point
l
Er
0
o
0 I
(40)
fixed are linear transformations. In fact, let A be a matrix corresponding to an analytic automorphism leaving component (39) fixed. Then for any U of the form (39) there is a of the same form and a nondegenerate q x q matrix B such that
a
AU=
aBo
(41)
We partition A and B into blocks in the following manner:
A=
All
A12
A13
Al4
r
A21
A22
A 23
A24
PI
A31
A32
A33
A34
ql
A41
A42
A43
A44
r
r
PI
ql
r
E12 )' B22 ql ql
B= (Ell B21
r
Formula (41) may be written in the form
[All A21 A31
l A41
A12Z+A13
I= A 32 Z+A" J A22Z+A23
A42Z+A43
whence
Since
Z is
arbitrary, we find that
B12
1
ZB 21
ZB22
I
E2l 0
B22
1 r Ell
l
0
r
THE GEOMETRY OF HOMOGENEOUS DOMAINS
Thus,
A-
I
All
A12
A13
A141
o
A22
A 23
A24
0
A32
A33
A34
.
I'
A*HA =H.
107
(42)
l
0 0 0 A44 J. It is easy to verify that transformation of the following type corresponds to each A of the form (42)
E 0
E 0
o o o
Z
0
Z
E
o
E
0
0
0
thus Z = (A22Z+A23)(A32Z+A33)-1. (43) It immediately follows from (43) that a matrix A of the form (42) corresponds to an analytic automorphism of the domain that leaves the point (40) fixed if and only if (44) A 23 = 0, A32 = O. We will now prove the converse, namely, that a matrix A of the form (42) and subject to supplementary conditions (44) corresponds to every automorphism of our domain that leaves the point (40) fixed. In order to do this, it is sufficient to prove that an analytic automorphism of our domain that leaves (40) fixed leaves every "infinitely distant" component fixed. This last immediately follows from the fact that (1) a component must be mapped into a component under an analytic automorphism, and (2) every boundary point, according to Theorem 4, is contained in a unique component. We will prove that the analytic automorphisms of our domain that leave point (40) fixed are linear transformations. The set of all matrices A of the form (42) and subject to supplementary conditions (44) is generated by two of its subgroups. The first of these consists of the matrices A of the form
o E o 0 o 0
(45)
o
E
lOS
THE GEOMETRY OF CLASSICAL DOMAINS
where (46)
i (At4 - A 14 ) = A!4A 24 - Ai4A 34' The second of these subgroups consists of the matrices A of the form
where
All
0
0
0
0
A22
0
0
0
0
A33
0
0
0
0
A44
AllA!4 = E,
A!2 A 22
E,
(47)
(4S)
Ai3A33 = E.
As we can easily show, a transformation of the form
~11 -+ U 11 +A12 U 21 +A 14 -
U 12 A 34 -A 12 U 22 A 34 -A13 A 34'
1
021-+ U2l+A24-U22A34, U 12
-+
U 12 +A 12 U 22 +A 13 ,
U 22
-+
U 22 ·
(49)
J
corresponds to each matrix A of the form (15). Expression (46) may be used to verify that the transformations obtained are "parallel translations" in the sense of Section 3 of Chapter 1. A transformation U 11
-+
All U 11 At1'
U 12 -+ All U 12 Ai3' U 21
-+
A22 U 21 A1~'
U 22
-+
A22 U 22 Ai3'
(50)
corresponds to each matrix of the form (47). It follows from (49) and (50) that a linear transformation of our domain corresponds to each matrix A of the form (42) that satisfies supplementary conditions (44). A quasilinear transformation of our domain corresponds to each matrix of the form (42). It is not difficult to see that in order to prove
THE GEOMETRY OF HOMOGENEOUS DOMAINS
109
this, it is sufficient to show that a quasilinear transformation of our domain corresponds to a matrix A of the form
E
0
0
0
0
A22
A 23
0
0
A32
A33
0
0
0
0
E
A=
(51)
Direct computation shows that the following transformation of our domain corresponds to a matrix A of the form (51) : V ll
---+
U ll -U 12 (A 32 U22+A33)-lA32 U 2U
U 12 ---+ U 12(A 32 U 22 + A 33 )-1, U 21
---+
(A 23 U 22 + A 22 )-1 U 21,
U22
---+
(A22 U 22 + A 23 )(A 32 U 22 + A 33 )-1.
(52)
Thus, every transformation of the domain S that maps an "infinitely distant" boundary component into itself is a quasilinear transfor~ mation. Proof of Theorem 5 for classical domains of the first type. Let D be a classical domain of the first type with parameters p and q, p ;;; q. As we have already seen, there are exactly q typical boundary components. We have constructed exactly the same number of Siegel domains having the typical boundary components as bases. We will prove that for any boundary point of domain ~ there is an analytic mapping of ~ that satisfies the requirements of the theorem and maps ~ onto one of the Siegel domains that we have constructed. Let a given point belong to a component analytically equivalent to the component ff r • It is clear that there is a mapping of ~ onto Sr under which the given point is mapped onto a point of the form (40). As we have already shown, the transformations that leave this point fixed are linear transformations of domain Sr and the transformations leaving the component of this point fixed are quasilinear transfor.. mations of the domain Sr. Theorem 5 is therefore completely proved for domains of the first type. In Chapters 3 and 4, which are devoted to the theory of automorphic
110
THE GEOMETRY OF CLASSICAL DOMAINS
functions, we will need a criterion for "convergence" of a sequence of points to a point in an infinitely distant component. We will now give a general definition of "convergence". Let S be some Siegel domain with base ff. As usual, we denote the points of S by w = (z, u, t), and we denote the points of ff by t. Let V and LtCu, v) have the same meaning as in Section 3 of Chapter 1. Definition 1. Let Q be some domain in ff, and let r be a vector in V. We will say that the set of all w (z, u, t), such that Imz-ReLtCu, U)-ZE V,
to E Q
(53)
is the cylindrical domain SeQ, r) in S. Definition 2. Let W, W2' ... be a sequence of points in S. We agree to say that lim Wy = to,
tEff,
(54)
v .... 0Cl
if for any cylindrical domain SeQ, r) (where to E Q) there is a that for v > Vo
WyES(Q,r).
Vo
such (55)
Let U 1 EQH' U 2 EQH; we set B(U 1,U2) = U~HU1(UiHU1)-1U:HU2' }
W(U 1, U 2) = U~ HU 2.
(56)
It is not difficult to see that (1) the matrix B(U1 , U2 ) depends not on U 1 but only on the class to which U 1 belongs; (2) when U2 is replaced by any other matrix in the same equivalence class, B(U1 , U2 ) and W(U1 , U2 ) are transformed in the same way: B
-+
R*BR,
W
-+
R*WR,
(57)
where R is a nondegenerate square matrix depending on U2 and on the given analytic automorphism; (3) the pair of matrices B(U1 , U2 ) and W( Uu U2 ) is the joint invariant (with respect to analytic automorphisms of the domain D H ) that corresponds to a pair of points in DH , i.e., they are transformed according to formula (57) under the analytic automorphism. The following lemma provides a criterion for convergence to some point in an "infinitely distant" component.
111
THE GEOMETRY OF HOMOGENEOUS DOMAINS
Lemma 4. Let Uo be some matrix mapped onto a point in an "infinitely distant" component under the mapping n -+ D. A sequence of points in our domain converges to the given point in the component if and only if
(58)
lim B(U n , U o) = W(U o, Uo),
n .... oo
where the Un En are arbitrary preimages of the given sequence of points. Proof It is clear that we may set
Vo
=
E
0
0
0
0
E
0
0
u(n) 11
0
0
U(II)
0
E
E
0
=
Vn
22
without loss of generality. Direct computation shows that
_(i(u~ni-U'!t»)-1 B(Um U o)-
o
W(U o• U o) =
0 (E-
)
utt) u~ni)-1
G~).
It is clear from the definition of "convergence" that the sequence UII converges if and only if (u~ni - U'!t)-1 -+ 0, u~ni -+ 0, which is equivalent to (58). Lemma 4 then follows. Without proof we will now give formulas for the Riemann distance and for geodesics. We will assume that domain ~ is realized as described at the beginning of this section. It can be shown that the Riemann metric, which is invariant with respect to analytic automorphisms, is given by the formula (Klingen [1]) ds 2 = a((Eq-Z*Z)-1 dZ*(E p -ZZ*)-1 dZ);
a(A) is the trace of the matrix A. The distance between two points Z1 and Z2 is given by the following expression: 2
p (Z1,Z2)
1 ~
= 4- k=f...;1 In
2
1 + rt
- 1t' -r k
rk
Al - 1
= --" Ak
where A1 , ... , Aq are the characteristic roots of the matrix R(Z1,Z2) = (Eq-Z,! Z1)-1(Eq-Z,! Z2)(Eq-Zt Z2)-1(Eq-Zt Z1)'
112
THE GEOMETRY OF CLASSICAL DOMAINS
Any geodesic in !!2 is analytically equivalent to a geodesic of the form (Hua Loo Keng [1]) th Ct 1 8
0
0
0
th Ct 2 8
0
0
0
th Ct q 8
0
0
0
(59)
Z(8) =
where Cti + '" + Ct: = 1, s is the arc length. Using (3), we can easily verify that a limit (that is a point in F) exists for any geodesic Z(s) when s-+
+00.
Let Zl E!!2 and Z2 E F. We agree to say that points Zl and Z2 may be connected by a geodesic if there is a geodesic Z(s) such that
Z(O) = Zl
lim Z(8) = Z.
and
s .....
+ CIJ
Generally speaking, two arbitrary points Zl and Z2 cannot be connected by a geodesic. Every point Zl E!!2 may, however, be connected by a geodesic with any component; this assertion is a consequence of the following lemma. Lenmw 5. Two points Z 1 E!!2 and Z 2 E F may be connected by a geodesic if and only if all of the characteristic roots of the matrix R12 =
w1l w 11 w2l
fV 22
(60)
[Wij is defined by (12)] are equal to zero or one. In particular, let Zl E!!2 and let!F be any component. There is a unique point Z2 E!F that lnay be connected by a geodesic to Zl' Proof. Assume that the points Zl E!!2 and Z2 E F may be connected by a geodesic Z(s), Z(O) = Zl' Without loss of generality,t we may assume that Zl = 0 and Z(s) has the form
(P~S)
~}
P(8)
=
th Ct 1 8
0
0
0
th Ct 2 8
0
0
th Ct r S
0 Ct 1 ~ Ct2 ~ ... ~ Ct r
(61)
> O.
t The characteristic roots of the matrix R, as (13) implies, are invariant under the analytic automorphisms of the domain PJ.
113
THE GEOMETRY OF HOMOGENEOUS DOMAINS
It is easy to verify that R(Z1) Z(s))
=
(
E- 0p2(S) 0E)'
and, therefore,
We have shown that if two points Z1 and Z2 may be connected by geodesics, the characteristic roots of the matrix R2 are equal to zero or one. We will now prove the converse, i.e., that a sufficient condition for points to be connected by geodesics is that the characteristic roots of the matrix R12 be equal to zero or one. Without loss in generality, we may assume that 0
0
°
X2
0
0
0
Xr
0
0
0
X1
Zl=(::
::}
Z2 =
Xk
~
o.
Direct computation shows that the characteristic roots of the matrix R are equal to 1-xi, ... , l-x;, i.e., Xk is equal to zero or one. It is not difficult to use (59) to show that the required geodesic exists. It remains to prove the second assertion of Lemma 5. Without loss of generality, we may assume that Z1 = 0 and ff has the form shown in (10). It is not difficult to show, if we use the criterion we have proved, that Z E ff may be connected by a geodesic to Z 1 = 0 if and only if
We denote the set of points in the domain ~ that may be connected with a given point Z E~ by ~z(Z Eff). It follows from Lemma 5 that ~ "fibers" into fibers ~z. It is not difficult to verify that this fibering is the same as the natural fibering of Siegel domains (34). In other words, a fiber consists of points in domain (34) with fixed U22 . It is clear from (52) that all fibers are analytically equivalent. As a
114
THE GEOMETRY OF CLASSICAL DOMAINS
result, it is sufficient to describe the fiber ~o corresponding to anyone value of U22 • For simplicity, we assume that U22 = 0. It follows from (34) that ~o is given by the inequalities
1 (U 11-:l
U·~i1 ) -U 21 * U 21 - U~· 12 U i'2>0,
whence it is clear that ~o is a Siegel domain of genus 2. The group G3 (.%) (see Section 5) is the same as the group A of "parallel translations" of the domain S. Let Z(s) ( - 00 < s < + (0) be a geodesic entering the component .%. As we know, Z(s) = g(s)Zo, where g(s) ( - 00 < S < + (0) is some oneparameter subgroup of analytic automorphism of the domain ~. Let a = Z( + (0) E.%. It is clear that g(s)a = a and, therefore, g(s) E G1(.%). We will prove that g(s) E G2 (.%). Consider the homomorphism of G1 (.%) onto G'(.%) = G1 (.%)JGzC.%). It maps g(s) into the compact group g'(s), since g(s) (-00 <s < +(0) leaves the point a fixed. The characteristic roots of the matrix of g(s) in representation (42) are real. As a result, g'(s) == F. Furthermore, let g E G3 (.%); then
limp(Z(s),g(Z(s))) = 0. for any geodesic Z(s), (Z( + (0) E .%). Assuming that Z(s) = g(s)(Zo), we find that lim p(g(s)Zo, g(g(s)Zo)) = 0, s-+
whence
+ ex:>
lim g-l(S)g(g(s)) = E. s-+
+ ex:>
It is not difficult to show by means of direct computations that the set of matrices defined by the last requirement coincides with the set of matrices A of the form (45). The assertion in question follows from (45) and (49).
Section 11. Classical Domains of the Second and Third Types In this section the results of Section 10 for domains of the first type are extended to classical domains of the second and third types. As we know, domains of the second type are described in the following manner.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
115
Let p > O. We will consider p xp skew symmetric matrices Z as points in a tp(p-l)-dimensional complex space. The set of Z such that (1)
forms a bounded domain {0. The analytic automorphisms of domain {0 are described in the following manner. Consider the set G of all square matrices M of order 111. = 2p that satisfy the conditions M*HM = H,
where
H
= (
M'KM = K,
-0E pO) Ep
K = (0 ,
Ep
EOp ).
(2)
(3)
An analytic automorphism of the domain {0 Z
-7
(AZ+B)(CZ+D)-1
corresponds to each square matrix l1f
M=G :} The boundary F of this domain clearly consists of all Z such that
det(E-Z*Z)
= 0,
E-Z*Z ~ O.
(4)
We will begin by enumerating all boundary components of the domain {0. Theorem 1. The set ofpoints oftheform
o ) 2r Z 2r
, Z*Z < E p -
2n
Z' = -Z,
p-2r
p-2r
in boundary F forms a component :F,. analytically equivalent to a classical domain of the second type with parameter P1 = P - 2,.. Any boundary component may be mapped into a component :Fr(1 ~ r ~ tp) by som.e analytic automorphism of the domain {0. Every point in F belongs to some component. The proof of this theorem proceeds in exactly the same manner as the proof of the corresponding theorem in Section 10. As in Section 6, we assume that (6)
116
THE GEOMETRY OF CLASSICAL DOMAINS
It is easy to verify that the rank r(Z1' Z2) of the matrix W 12 is an invariant of the pair of points (ZbZ2) under the analytic automor-
phisms of the domain !:0. We will now state a criterion for membership of two points in one component. Lemma 1. Two points Z 1, Z 2 E F belong to one component if and only if (7)
The proof of this lemma is exactly the same as the proof of the corresponding lemma, Lemma 2, in Section 6. We now turn to describing the canonical realizations of the upper halfplane type for the domain!:0. Let n denote the set of all complex rectangular 2p x p matrices U of maximum rank that are such that U' KU = 0, where K is a non degenerate symmetric matrix. If U En, then UR En, where R is any square non degenerate p x p matrix. We agree to say that the two matrices U and UR are equivalent. The set of all equivalent matrices forms a class. The set of classes is a compact complex symmetric manifold D. The dual manifolds D (defined in Section 6), of the domain!:0 which correspond to different K, are analytically equivalent. As a result, Kmay be chosen in any convenient manner. We will assume that K is of the form (3). Let H be the Hermitian matrix defined in (3). Consider the set nH of matrices U En, such that U*HU>
o.
(8)
It is easy to verify that every class of equivalent matrices is either
entirely contained in nH , or its intersection with nH is empty. The set of classes contained in nH is a domain in D. We set U=(U 1 )" p. U2 P
(9)
P It is easy to use the techniques we used in Section 10 to prove that !:0 may also be realized in the form of a domain in D. It is not difficult to see that the boundary F of domain !:0 consists of those U En, for which the determinant of the matrix U* HU is equal to zero and U* HU ~ O. In other words, F consists of those Z satisfying
THE GEOMETRY OF HOMOGENEOUS DOMAINS
117
the following conditions: (1) the determinant of the matrix Ep-Z*Z is equal to zero, (2) Ep-Z*Z ~ 0, (3) Z' = -Z. We will now give a general method for finding the canonical realizations. Let Hbe a Hermitian matrix of order 2p withp positive roots and p negative characteristic roots, and let K be a nondegenerate 2p x 2p symmetric matrix, where HK- 1 HK- 1 = -E.
(10)
Consider the set OR of 2p x p matrices U such that H*HU > 0,
U'KU = 0.
(11)
We agree to say that the matrices U and UR, where R is a p x p nondegenerate matrix, belong to the same class of OIl' The set DR of classes belonging to Q ii is a domain in D (see Klingen [1]). That is analytically equivalent to our domain £0. The analytic automorphisms of this domain may be described in the following manner. Let G be the set of all 2p x 2p matrices A such that A* HJ1 = Hand A'KA = K. With each A E G we associate the analytic automorphism U -7 A U in O. It is easy to see that an analytic automorphism in the set of classes corresponds to it. The boundary of the domain £0 consists of the classes such that
U'KU = 0,
U*HU ~ 0,
det(U*HU) = 0.
First we will describe the realization S[tp] corresponding to the zerodimensional components of the boundary of the domain £0. We will separately discuss the two cases depending on the parity of p. Case I. p even. We set
H=C~£p i:p). K=(~J J
= (_0£,
:'). where s =!P.
Jr
(12)
j
It is easy to see that (10) is satisfied when Hand J( are chosen in this way· We partition Uinto blocks in the following manner:
u=(U )p. 1
U2 P P
118
THE GEOMETRY OF CLASSICAL DOMAINS
Conditions (11) imply that
~(UI u1-ui U 2) > 0, U~JU2 = U~JU1'
(13)
1
As we can easily verify, U2 is nondegenerate, and, consequently, there is a unique representative of the form
(~) in each class. It follows from (13) that
~(Z-Z*) > .
0, JZ = Z'J ,
l
i.e., S[tP] is a Siegel domain of genus 1. Case II. p odd: p = 2s+ 1. We set
H=
o o
o
-1
o
0
K=
010
o
0
0
o o -J s
0
1
o .
1 0
0
0 0
0
(14)
We partition U into blocks:
u=
V 11
U 12
2s
U 21
U 22
1
U 31
U 32
U41
U 42
2s-
1
2s
It is easy to prove (see the analogous situation in Section 10) that the matrix (
U31 U41
U32) U42
is nondegenerate. As a result, we can normalize U with the conditions
119
THE GEOMETRY OF HOMOGENEOUS DOMAINS
Relationships (11) imply that
U'11J = JU 11 , W = (W11 W 21
u*21 U 21,
*) Jt 11 = -:-1 ( U 11 - .U 11 T
l
W12 = Wi\
= -iU 12 ,
W 22 = l.
We have been led to the following domain in a -!-p(p-I)-dimensional complex space (whose coordinates are the entries of the matrix U 12 and are independent of the entries of the matrix U 11 ):
~(U11-U~I)-U12Ui2-JUI2U'J>O,
U 11 J=JU 11 . }
(15)
Thus, in this case, the domain S[tP] is a Siegel domain of genus 2. We will now describe the remaining realizations. From now on we will not distinguish between even and odd p. We set 0
H=
0
iE 2s
0
0
0
Jr
0
0
Er
0
0
-Er
0
0
0
0
Er
0
0
Er
0
0
0
0
0
-Js
0
0
0
- i E 2s
where
0
J, = (
0 -Es
K=
:}
2s+r.
p=
We partition Uinto blocks in the same way we did in Section 10:
U=
U 11
U 12
2s
U 21
U 22
r
U 31
U 32
r
U 41
U 42
2s
2s
r
(16)
120
THE GEOMETRY OF CLASSICAL DOMAINS
and we prove as we did there, that if U satisfies conditions (11), then the matrix
is nondegenerate and, consequently, each class U E OH contains a unique representative such that U 31 = 0, U 32 = En U41 = E 2s ' U42 = 0. Relationships (11) imply that U'11 J s = J s U 11, W 11
U;2 = - U 22 '
U;l = JU 12 ,
1 11 - U 11 * )- U 21 * U 2b = :-(U I
W 12 = wil =
~I U12 -
uil U 22 '
(17)
W 22 = E- Ui2 U 22 ' fV12) > 0.
J
W22
The inequalities (17) define an unbounded domain S in a tp(p-I)dimensional complex space (whose coordinates are the independent entries of the matrices Ul1 , U 22 and U 12 ). , As in Section 6, we may write the inequalities defining the domain in the following form:
l(V 11 - U*) U~·21 W-221U 21 - UT 12 W-221 U~'12 11 -
-:1
- iU 12
}V;l ui2 u21 + iUil V 22 w;l Uj2 > 0,
)
(18)
E- vi2 U 22 > 0.
In order to make it clear that the domain obtained coincides with some Siegel domain of genus 3, we introduce some new notation characteristic of Siegel domains. We set
t = U 22 '
Z
= 2U 11 ,
U
= U 12 '
V
= V 12 .
It follows from (18) that t belongs to a classical domain of type 2 with parameter r = p-2s.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
121
Let the cone V consist of all 2s x 2s Hermitian positive definite matrices Y such that YJs = Js Y. We set
L/u, v) =
u12 Wi} vfz +J V12 W;l u 12 J'
It is not difficult to see that our domain is the Siegel domain corresponding to the cone Vand the form Lt(u, v). The boundary F of this domain consists of all matrices U such that
U*HU ~ 0
det U*HU = 0,
(20)
(compare this with formula (38) of Section 10). Consider the matrices U E F of the following form: E zs
U=
0 0 0
~1
~' j,
Z'= -Z,
Z*Z < Er •
(21)
It is easy to prove the following results, which are analogous to the results of Section 10: There is no more than one matrix of the form (21) in a class of equivalent matrices. The matrices U form a component analytically equivalent to the component!Fn which we naturally treat as "infinitely distant". The analytic automorphisms of our domain that map component (21) into itself are quasilinear transformations (defined in Section 3 of Chapter 1), while the analytic automorphisms leaving the point
rE 2s o o o
0
0
(22)
E 0
fixed are linear transformations. In order to prove the last assertion, it is first necessary to prove that the matrices A corresponding to the analytic automorphisms of the
122
THE GEOMETRY OF CLASSICAL DOMAINS
domain S that map an "infinitely distant" component into itself have the form
Au
A=
°
A12
A13
A141
A22
A 23
A24
o
A32
A33
A 34
° ° °
A*HA = H,
so that
j'
A44 A'KA = K;
(23)
(24)
Hand K are defined in (16). As we did in Section 10, we will prove that the transformations corresponding to these matrices are quasilinear transformations of our domain. The matrices A corresponding to the analytic automorphisms leaving point (22) fixed are separated by the supplementary conditions
A 23 = 0,
A32 = 0.
(25)
Linear transformations of our domain (see Section 10) correspond to such matrices. The proofs of the second fundamental theorem and Lemma 4 of Section 10 for classical domains of type 2 proceed in exactly the same way as they did above. We now give, without proof, formulas for the Riemann distance and geodesics (Klingen [1]). We will assume that the domain {0 is realized in the same way as at the beginning of the section. It is possible to prove that a Riemann metric invariant with respect to the analytic automorphisms is given by the formula. ds 2 = a{(Ep-Z*Z)-1 dZ*(E p-ZZ*)-1 dZ); (26) a(A) denotes the trace of the matrix A. The distance between two points Z1 and Z2 is given by the following formula: 2
1
p (Z1' Z2) = -
8
Ik p
==1
2 1 +..jrk In ----=.,
l-.Jr k
Ak-l rk = - - , Ak
(27)
where A1 , ••• , Ap are the characteristic roots of the matrix
R(Z1' Z2) = (Ep-Zi Z1)-1(Ep-Zi Z2)(Ep-Z! Z2)-1(Ep-Z! Z1).
THE GEOMETRY OF HOMOGENEOUS DOMAINS
123
Any geodesic in !:0 is analytically equivalent to a geodesic of the form
th (Xl s.j
0
0
0
ih (Xl s.j
0
0
0
th (Xr s.j
0
0
0
0
0
Z(s) =
r = [j-p], j
= (
0 -1
0
~}
where the last row is added if p is odd, (Xi + ... + (X; = 1, and s is the arc length. We will omit that statement and proof of Lemma 5, since they are completely analogous to those given in Section 10. The fiber !:0 o corresponding to U22 = 0 is given by the inequality
~(Ul1 l
Url)- U 12 Url -JU 12
U~l J' > O.
(28)
As we should expect, it is a Siegel domain of genus 2. We now turn to a discussion of classical domains of type 3. These domains are described in the following manner. Let p > O. We will treat p x p symmetric matrices Z as points in a j-p(p + 1)-dimensional complex space. The set of Z such that (29) forms a bounded domain!:0. The analytic automorphisms of the domain !:0 are described in the following manner. Let G be the set of all 2p x 2p matrices A such that A*HA = H,
A'JA = J,
where
(30) (31)
An analytic automorphism of the domain !:0 (32) corresponds to each square matrix A.
124
THE GEOMETRY OF CLASSICAL DOMAINS
The formulas for a linear element of a metric invariant with respect to the analytic automorphisms of the domain ~ and for the distance between two points Zl and Z2 are similar to the formulas given in Section 10 (p. 91). Any geodesic in ~ is analytically equivalent to a geodesic of the form
o
o o
Z(s) =
(33)
o where exi + ... + ex; = 1 and s is the arc length. Let a be the set of all complex rectangular
°
2p x p matrices U of maximal rank and such that U'JU = 0, where J is defined in (30). If U E a, then UR E a, where R is a p x p nondegenerate matrix. We agree to say that the matrices U and UR are equivalent. The set of all equivalent matrices forms a class. The set of classes is a compact complex symmetric manifold D. We will prove that ~ may be realized in the form of a domain in D. Let H be the Hermitian matrix defined in (30). Consider the set aH of matrices U E a, such that U*HU > 0.
(34)
It is easy to verify that every class of equivalent matrices is either completely contained in aH or does not intersect aH • The set of classes contained in aH is a domain in D. We set 1
U=(U )P. U2 P
(35)
As we did in Section 10, we can prove that ~ is realized in the form of a domain in D. The boundary F of domain ~ consists of the U E a, such that the determinant of the matrix U* HU is equal to zero and U* HU ~ o. In other words, F consists of the Z satisfying the following conditions:
(1) the determinant of matrix Ep-Z*Z is equal to zero, (2) Ep-Z*Z ~ 0, (3) Z' =Z.
THE GEOMETRY OF HOMOGENEOUS DOMAINS
125
The proofs of the Theorems 4, 5, and 6 for domains of type 3 proceed in exactly the same manner as the proofs for the cases of domains of types 1 and 2. As a result, we will state only the necessary changes in the statements and formulas.
Lemma 1. The set of points of the form
z=
z(p-" P-'J,
1 (36)
Z/=Z,
z*z < Ep-r
J
in the boundary F forms a component that we denote by ,~il ~ r ~ p). Any boundary component is analytically equivalent to a component :Fr. Every point in F is contained in some component. Lemma 2. Two points Zl' Z2 E F belong to one component if and only if (37)
where r(Zi'Z) denotes the rank of the matrix Wij = Ep-Zt Z). We now turn to describing the canonical realizations of domains of the third type. A general method for obtaining such realizations consists in the following. Let H be a Hermitian matrix of order 2p with p positive and p negative characteristic roots, and let J be a nondegenerate asymmetric matrix of order 2p, so that (38) Consider the set OH of 2p x p matrices U such that U*HU> 0,
U'JU = 0.
(39)
We agree to say that the matrices U and UR, where R is a p x p nondegenerate matrix belong to the same class. The set of classes contained in OH is a domain in D (see Klingen [1]). That is analytically equivalent to our domain !:0. The analytic automorphisms of this domain are described in the following manner. Let G be the set of a1l2p x 2p matrices A such that A*HA = H,
A'JA = J.
We associate an analytic automorphism U ~ A U in OH with each A E G. It is easy to see that there is a corresponding analytic automorphism in the set of classes.
126
THE GEOMETRY OF CLASSICAL DOMAINS
The boundary of the domain f» consists of the classes that satisfy the conditions
U'JU = 0,
U*HU ~ 0,
det U*HU = 0.
It is possible to prove that two boundary points U1 and U2 are mapped onto each other under an analytic automorphism of the domain f» if and only if the ranks of the matrices W l = ut HU1 and W 2 = ui HU2 are the same. We may deduce from this that the boundary decomposes into p transitive parts. Thus, there are exactly p canonical realizations. We will first describe the realization Sp corresponding to a zero-dimensional component of the boundary of domain f». We set
H=
(-iEp°
(40)
It is easy to verify that (38) is satisfied when Hand J are selected in this way. We partition Uinto blocks in the following manner:
Conditions (39) imply that l(u*2 -:1
°
(41)
U 1 - U*1 U) 2 > ,
It is easy to verify that U 2 is nondegenerate and, consequently, every
class contains a unique representative of the form
It then follows that from conditions (41)
~(Z-Z*) > 0, 1
i.e., S is a Siegel domain of genus 1.
Z' = Z,
(42)
127
THE GEOMETRY OF HOMOGENEOUS DOMAINS
This domain was introduced by Siegel, as a result of which it is frequently called Siegel's generalized upper halfplane of degree p. We now turn to descriptions of the remaining realizations. We set
H=
0
0
0
iEs
0
-Er
0
0
0
0
Er
0
0
0
0
-iEs
0
0
0
Es
0
0
Er
0
0
-Er
0
0
-Es
0
0
0
J=
(43)
It is easy to verify that condition (38) is satisfied. We partition U into blocks just as we did in Section 10, i.e.,
u=
u 11
V 12
S
U 21
V 22
r
U 31
V32
r
U 41
U 42
S
(44)
r
S
As we did in Section 10, we prove that if U satisfies (39), then the matrix
(
U U32) 31
U 41
U 42
is nondegenerate and, consequently, each class U E nH contains a unique representative for which U31 = 0, U 32 = E r , U l l = E s ' U42 = o. From (39) we obtain U~1
= U 11 ,
U~2
=
U 22,
U~1
=
U
12,
W12) >0, H/22
where
(45)
128
THE GEOMETRY OF CLASSICAL DOMAINS
Inequalities (45) define an unbounded domain S in a -!p(p+ 1)dimensional complex space (whose coordinates are the independent entries of the matrices U11 , U 12 , and U22 ). An "infinitely distant" component consists of matrices of the form E
0
0
Z
0
E
0
0
, Z'=Z, Z*Z<E.
(46)
We now turn to describing the groups Gig;), k = 1,2,3. As we did in Section 10, it is possible to show that group G 1 (ff) consists of all transformations with matrices A of the following form:
A=
Au
A12
A13
S
0
A22
A 23
2r,
0
0
A33
S
21'
S
A*HA = H, A22 =
(47)
S
A'iA = J,
(Bl B2} B3
B4
where Hand J are defined by (43). The following automorphism of a component corresponds to each transformation with such a matrix A:
E
0
E
0
o Z o E o 0
0
Z
0
E
0
0
whence it follows that matrix A in G 1 (g;) belongs to GzCg;) if and only if
THE GEOMETRY OF HOMOGENEOUS DOMAINS
129
In addition, it is possible to prove that the fiber f» 0 corresponding to 0 is the following domain:
U22
1 * *_ -:-(U 11 -U 11 )-U 12 U 12 - U 12 U 12 > 0, 1
i.e., some Siegel domain of genus 2.
E
CHAPTER 3
Discrete groups of analytic automorphisms of bounded domains Section 1. Introduction
Let r be a discrete group of analytic automorphisms of some bounded homogeneous domain {0. We will call the functions that are meromorphic in {0 and invariant with respect to the group r automorphic functions. It is not difficult to show that the degree of transcendence of a field of automorphic functions is no less than the complex dimension n of the domain {0 (see C. L. Siegel [7] and Lemma 2, Section 4). As we know, the degree of transcendence of a field of merom orphic functions on a compact analytic normal space is no greater than its dimensions (Remmert [1]). Thus, in order to prove that the degree of transcendence of a field of automorphic functions is no larger than n, it is sufficient to imbed {0/r in a compact analytic normal space M in the form of an everywhere dense set and show that any meromorphic function on {0/r extends to all of M. In order to do this, it is sufficient to show that the complex dimension of M' = M - {0/r is no greater than n - 2, where n is the dimension of {0. In this chapter, we will, for any arithmetic group in the sense of A. Borel, give an explicit construction of an extension of the factor space {0/r with all of the necessary properties.t t The unit disk I z I < 1, where
1, is an exception, for the dimension of 2. Here we must impose additional assumptions on the definitions of automorphic functions so that it will be possible to extend them analytically to all of M. 131
M'
=
n
M - flZjr is 0, i.e., larger than
II -
132
THE GEOMETRY OF CLASSICAL DOMAINS
For the case in which r is Siegel's modular group, a construction for such an extension was first given by Satake [2]. Satake's extension coincides with the one obtained from the general construction of the present chapter. A. Andreotti and H. Grauert [1] recently proposed a very elegant general method for proving the theorem on the degree of transcendence of a field of automorphic functions. As we will show in Section 7 of this chapter, the conditions under which this method is applicable hold (with trivial exceptions) for all arithmetic groups of analytic automorphisms of symmetric domains. The construction of the extension M of the factor space {0/r is applicable to discrete groups of analytic automorphisms of arbitrary bounded homogeneous domains. It is stated in this form in the present chapter. It is understandable that, in this case, the space M is, as a rule, not compact, but it is always an analytic normal space. This can be proved by means of a theorem due to H. Cartan. We should also note that our construction of the extension M is applicab'le not only to homogeneous domains, but, in general, to arbitrary Siegel domains. The examples of complex nqrmal spaces arising as a result of this construction will be of value to us. As a rule they are not complex manifold, even if they are compact. U. Christian [2] discussed this in detail for the spaces that arise in the study of Siegel-Hilbert modular groups. When {0 is a symmetric domain and r is an arithmetic group, the space M is always compact. We will use a recent result of A. Borel [3] to prove this proposition. We will now turn to a section-by-section survey of the contents of the present chapter. Section 2 presents the construction of the extension M for the factor space {0/r for the case in which {0 is a bounded homogeneous domain and r is a discrete group of its analytic automorphisms. In Section 3 we will use one paper ofR. Cartan as a basis forintroduction of an analytic structure into the space M. Satisfaction of Cartan's conditions is established by means oflemmas proved in Sections 4 and 5. In Section 6 we will show that if {0 is a symmetric domain and r is an arithmetic group of its automorphisms, then M is compact. In Section 7 we will show that, except for certain trivial exceptions, any arithmetic group is a pseudoconcave group in the sense of Andreotti and Grauert.
133
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
Section 2. Construction of the Extension of the Factor Space
~/r
Before we give this construction in the general form, we will consider it in a very simple, almost trivial case, the case in which {0 is the ordinary upper halfplane, i.e., the set of points of the form z = x+iy, 0< y, and r is an ordinary modular group. We let denote the space obtained by adding all of the rational points on the real axis and the point 00 to {0. The space M is the factor space 9)e/r. It is clearly sufficient to introduce a topology into the space in order to introduce one into the space M. We provide a topology in by means of a neighborhood base. A neighborhood of a point Zo E {0 is defined as usual. A neighborhood of the point 00 consists of those points of the domain {0 that have the form y = Imz> c > 0 and, naturally, the point 00 itself. A neighborhood of point z = r (r is a rational number) consists of the points of domain {0 that lie inside some circle tangent to the real axis at the point z = r and the point r itself. It is clear that the space M constructed in this manner is homomorphic to a two-dimensional sphere. It is very important to give the proper definition of the analogs of rational points in the example given above so that we can extend this scheme to the general case. Let {0 be a bounded homogeneous domain, and let r be a discrete group of analytic automorphisms of the domain {0. The most important point of the construction discussed below is the notion of a r-rational homogeneous fibering. Consider some homogeneous fibering of the domain {0. Let ® be the group of all analytic automorphisms of the domain {0 that preserve the given fibering. By ¢ we will denote the natural homomorphism of the group ® onto the group ®' of analytic automorphisms of the base, and by 3 we will denote the group of parallel translations of this fibering (see Chapter 2, Section 7). We will say that a homogeneous fibering is r-rational if: (1) the factor space 3/11 where 11 = r n 3, is compact, and (2) the group r' = ¢(r n ®) is a discrete subgroup of the group ®'. In what follows we will call r' the induced group. It is not difficult to see that there is a single family of rational fiberings for any two commensurable subgroups r 1 and r 2. Let us consider an example. Let G be an algebraic group defined over the rational numbers Q. Assume that the group GR is transitive in the
we
we
we
134
THE GEOMETRY OF CLASSICAL DOMAINS
domain £0, and let r = Gz • Now, a homogeneous fibering is rational if and only if the group of its parallel translations is an algebraic subgroup of the group G and defined over Q. Indeed, if the fibering is rational, the factor space 3//)., where /). = 3 n Gz is compact. This implies that 3 is a subgroup of the group GR. Moreover, we know (Chapter 2, Section 7) that 3 is an algebraic subgroup of the group of all analytic automorphisms of the domain £0. As a result, the subgroup 3 is an algebraic subgroup of the group GR. Compactness of the factor space 3//). implies that the subgroup 3 is defined over Q. We will now prove the converse, i.e., that if the subgroup 3 of some homogeneous fibering is an algebraic sub-group of a group G and defined over Q, this fibering is rational with respect to the group r = Gz • The group 3 is unipotent, and, consequently, the factor space 3/3z is compact (A. Borel, Harish-Chandra [1]). Consider its center 30' and let Go be the normalizer of the group 30. The group r in the definition of a rational fibering is, as we can easily show, commensurable with the group Gz, where G = G o/3. As a result, the group r' is automatically discrete in this case, i.e., the second condition of the definition of a r-rational fibering is a consequence of the first. This fact is clearly related to the fact that algebraic Lie algebras contain, along with any given element, all of its replicas. Let M denote the set-theoretic union of the domain £0 and the domains £0' that are bases of r-rational homogeneous fiberings of the domain £0. When £0 is the unit disk and r is an ordinary modular group, the space coincides with the space introduced in the example discussed at the beginning of the present section. In what follows, the domains £0' will sometimes be called r-rational components. The group r is naturally defined in the space IDe. As in the simple example considered at the beginning of this section, we define M to be a factor space IDe/r. Now our problem is to introduce a topology into the space M. Toward this end, we now construct a fundamental system T(rno) of open subsets M o =£0/r for each point rnoEM; henceforth, we will define topologies by means of these subsets so that the subsets of T(rno) will be the intersection of the neighborhoods of the point rno and Mo· Assume that the point rno is contained in Mo. Let Zo denote one of its preimages in £0. The system of sets that we seek is the image of a
me
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
135
fundamental system of neighborhoods of the point Zo under projection onto Mo. We will now consider the case in which m o E£0'jr', where £0' is the base of some r-rational homogeneous fibering and r' is the discrete group induced on £0'. Let Zo be some preimage (in £0'), of the point rno and let U denote some neighborhood of the point Zo in £0'. Moreover, let A(U) be some section in £0 over U, i.e., a subset of £0 for which the projection onto £0' induces a homomorphism with U, and J1(U) is the union of all orbits of the group 3 of parallel translations of this fibering that pass through points of the set A(U). The bounded holomorphic hull O(J1(U)) of the set J1(U) is an open subset in £0 (see Chapter 1, Section 4). T(mo) consists of the images of sets of the form O(J1(U)) under projection onto £0 jr, where U is an arbitrary neighborhood of the point Zo in £0' and A(U) is an arbitrary section over U. In what follows, we agree to call sets of the form 0(J1(U)) cylindrical sets. It is not difficult to show that our construction is independent of the selection of the point Zo in £0'. It is clear that if 0lET(mo) and OoET(mo), then there exists an 0 3 ET(mo) such that 0 3 E 0 1 n O2 , Now we can introduce a topology into M. Let mo EM, and associate the set W of all lnEM for which there exists an OET(m) such that Om E 0 with each set 0 E T(mo)' The family of sets W obtained in this manner is a base for the neighborhoods at the point mo. In addition, we will assume that the following assumptions are true: (A) for any point Zo E £0', where £0' is the base of some r-rational fibering, there exists a cylindrical set of the form O(J1(U)) (U is some neighborhood of the point zo) with the following property: if Zl E 0(J1(U)), yZl E 0(J1(U)), where y Er, then ypreservesthegivenfibering; (B) For any two differentpointsm 1 ,m2 EM, there exist sets 0 1 and O2 , OkET(mk), k= 1,2, whose intersection is empty. It follows from (B) that the topology in M is Hausdorff. Indeed, let ml and m 2 be two distinct points of M. Let 0 1 ET(md, O2 ET(m 2 ) be such that their intersection is empty. By W k , k = 1,2 we denote the set of all mEM for which there exists an OmET(m) such that Om C Ok for k = 1,2. It is clear that W 1 n W2 = 0. It is clear that conditions A and B are always satisfied. For spaces M obtained from arithmetic groups defined on symmetric domains, satisfaction of these conditions is a corollary of the results of A. Borel [3].
136
THE GEOMETRY OF CLASSICAL DOMAINS
Section 3. Analytic Normal Spaces
In this section we will introduce an analytic structure into space M. In the case of one complex variable, we can prove that M is a complex manifold. This is not so in the general case. As we will prove below, however, M is an analytic normal space. We now give the definition of analytic normal spaces. First we will present the notion of a ringed space (H. Cartan [3]) due to Serre. A Hausdorff space X such that a subring Ax of the ring of all germs of continuous complex-valued functions is defined at every point x EX, is called a ringed space. We denote the set of rings Ax by A. Homomorphisms and isomorphisms of such spaces are defined in the usual way. Let U be a domain in the complex space eN. An analytic subset in U is a closed subset V c U such that in a sufficiently small neighborhood of each of its points, it is the set of common zeros of some finite number of functions analytic in this neighborhood. We should note that every analytic set is a ringed space if we take the set of functions induced by functions analytic in some neighborhood of a point x for Ax. A ringed space is said to be an analytic space if for each of its points there is a neighborhood isomorphic as a ringed space to some analytic subset in eN' A ringed space is said to be normal if every local ring Axis an integrally closed integral domain. Recall that an integral domain is a commutative ring with no zero divisors. An integral domain 0 is said to be integrally closed if every solution of the equation yll+a i yll-i + ... + all
= 0,
(where ai' ... , all EO) that belongs to the quotient field of integral domain o belongs to O. For example, the ring of integers is an integrally closed integral domain. A point x E X is said to be regular if it has a neighborhood isomorphic to a domain in eN. We will prove that it is possible to introduce the structure of an analytic normal space into the space M defined in Section 2 in such a manner that on each M j = !?)r'($P), where $Pj is some r-rational
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
137
component, the structure will coincide with the natural analytic structure already there. We will use H. Cartan's theorem on extensions of analytic normal spaces (H. Cartan [1]) to prove this statement. Let X be some locally compact space. V is an open set that is everywhere dense in X and W = X-V. We will assume that the structure of an analytic normal space of dimension 111 is defined on V. H. Cartan posed the following question: Does X have the structure of an analytic normal space with the following properties?
(0:) it induces the indicated structure in V; (fJ) W is an analytic subspace of X with dimension less than m.
As H. Cartan remarked, if such a structure were possible, it lVould be unique.
Indeed, let x EX. The ring Ax of germs of continuous complexvalued functions at the point x is uniquely defined by the following condition: a function f belongs to Ax if and only if it is continuous in some neighborhood U of point x and at each point y E Un V it belongs to By (By is the ring of germs of analytic functions at the point y). As Cartan proved, satisfaction of the following three conditions is sufficient for existence of the required structure: (1) Any point Xo E W has a fundamental neighborhood system whose intersection with V is connected; (2) Every point Xo E W has a neighborhood U in which the functions continuous in U and analytic at every point of V n U separatet all points
ofVn U; (3) The structure A of the rings of germs of continuous functions that naturally appears in W induces the structure of an analytic normal space of dimension less than 111 in W.
We will prove the following proposition, which we will use later on, as an example of an application of Cartan's theorem: Let fifi be some complex manifold, and let r be a discrete group of its analytic automorphisl11s. It is always possible to introduce the structure of an analytic normal space into thefactor space fifi/r. Proof First of all, note that if the group r contains no nontrivial transformations with fixed points, then the natural mapping fifi -+ fifi /r
t We say that a function/(z) separates points Zl and Z2 if/(Zl)
=I=-
/(Z2).
138
THE GEOMETRY OF CLASSICAL DOMAINS
is locally one-to-one and, therefore, defines the structure of a complex manifold in £0 /r. Now for the proof of our assertion. We denote the set of all points in £0 that are not fixed points for elements of the group r by £0 0 , Now we set X = £0/r and V £0 o/r. It is clear that £0 0 , and, therefore, Vas well, are complex manifolds. We will show that the conditions of CaI'tan's theorem are satisfied for X and V. Let Xo E W = X-V. We denote any preimage of this point in £0 by Yo. lt is not difficult to see that any fundamental neighborhood system of the points Yo is mapped onto a fundamental neighborhood system of the point Xo by the natural mapping £0 -+ £0 /r. lt immediately follows that condition 1 of Cartan's theorem is satisfied. In order to prove that condition 2 is satisfied, it is sufficient to find for any point YoE£0-£0 o a neighborhood U in which analytic and r-invariantt functions separate all r-nonequivalentt points. We denote the set of all l' E r such that Y(Yo) = Yo by r o. We choose a neighborhood U of the point Yo so small that l' E r 0 is a consequence of nonemptiness of Un y(U). Furthermore, we may assume that the neighborhood U is r o-invariant. Let Yb Y2 E U. If Yl i= YY2 for any l' E r 0' then there exists a functionf(z) that is analytic in U and such that !(YYl) = 0
and !(YY2) = 1
for all l' Er o' The function
cp(z) = If(yz) )'ErO
is r-lnvariant and separates the points Yl and Y2' We have proved that condition 2 is also satisfied. We now note that in a sufficiently small neighborhood U of point Yo, it is possible to choose a coordinate system in which all transformations in r 0 are linear. The set of fixed points for r 0 in this coordinate system will be a linear subspace, and, therefore, a submanifold of the manifold £0. The mapping of £0 - £0 0 onto W is locally one-to-one and, consequently, the structure induced in W is that of a complex manifold. Thus, condition 3 is also satisfied. Our proposition is proved. t Functions such that the functional equation fez) = f(yz) holds for all y Erare said to be r-invariant. t Two points Zl and Z2 are said to be r-equivalent if Z2 = yZl for some yEr.
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
139
We will now prove that the space constructed in the preceding section has the structure of an analytic normal space. We will do this by applying Cartan's theorem in the case X = M, V = Mo. Verification of condition 1. Let Xo E ff' jo' and consider the sequence of cylindrical domains QIl with bases BII E JiP jo' Xo E BII" We assume that
Then, as we can easily see, there is a fundamental neighborhood system Un of the point Xo such that Un n M 0 is the image of QIl under the natural mapping flfi -+ flfi /r. Every cylindrical domain Qn is connected. As a result, Un n Mo, as a continuous image of Qn, is also connected. Verification of condition 2. Again let Xo E JiP jo • Lemma 2 of Section 9 implies the existence of a cylindrical domain Q with base Be JiP jo ' Xo E B, and the property that if Z E Q and yz E Q, where l' E r, then yXo = Xo. Let U denote the neighborhood of the point Xo such that Un Mo is the image of Q under the natural mapping flfi -+ flfi/r. Every function that is continuous in U and analytic in Mo n U induces some r o-invariant analytic function in Q (r 0 is the set of all l' E r such that y(xo) = xo). We map the domain flfi onto some Siegel domain S so that all transformations leaving the point Xo fixed become linear transformations, while all transformations mapping JiP jo onto itself become quasilinear. Note that the Jacobian of any transformation Yo E r 0 in the domain S is equal to 1. As a result, the concepts of r o-invariant functions and r o-automorphic forms coincide (see the definition in Section 4). Continuity of a function in U implies that the function induced by it in Q is "analytic at infinity" in the sense of Section 5. The converse will follow immediately from Lemma 1 of Section 5, namely that any r o-invariant function that is analytic in Q and "analytic at infinity" is induced by some function that is continuous in U and analytic in UnMo· It will follow from Lemma 4 of Section 5 that any two r o-nonequivalent points of the domain Q can be separated by functions that are analytic in Q, ro-invariant, and "analytic at infinity". Verification of condition 3. We will now prove that the structure of the rings of germs of continuous functions that naturally appears in W induces the structure of an analytic normal space of dimension less than
140
THE GEOMETRY OF CLASSICAL DOMAINS
111 in W. It follows from Lemma 1 of Section 5 that the ring structure induced on each Mk = flfiur£ coincides with the natural analytic structure already there. Let Wo denote the set of all Mk c M that are closed in M. In addition, let W 1 be the set of all Mk c M whose boundaries, i.e., Mk-Mk, belong to WOo We define W 2 , W 3 , etc., similarly. It is clear that W k = W for some k. We will now show that Wj is an analytic normal space whose dimension is equal to the maximal dimension of the Mk contained in it. This is clear for i = 0, because Wo is the union of no more than a countable number of closed disjoint sets M i , and the structure induced on each Mi coincides with the structure of an analytic normal space that is already there. Assume that we have already proved that Wj has the structure of an analytic normal space; we will now prove that Wj + 1 also has the structure of an analytic normal space. Wj + 1 is the union of no more than a countable number of closed disjoint sets. It is clear that it is sufficient to prove our assertion for any of these closed subsets. We again apply Cartan's theorem for the proof. For the same reasons as above, conditions I and 2 of this theorem are satisfied. Condition 3 is satisfied by virtue of the induction hypothesis.
Section 4. Poincare Series
In this section we will study Poincare series. The results of this section and the next will be used to prove Lemma 4, from which follows satisfaction of one of the conditions of Cartan's theorem. First we define automorphic forms. Let flfi be some domain, and let r be a discrete group of analytic automorphisms of this domain. Definition l.t A function fez) that is analytic in flfi is said to be a r -automorphic form of weight /11, if it satisfies the functional equation J(Y(Z))j~I(Z)
= fez) for all
"I E r,
(1)
where irCz) denotes the Jacobian of the transformation z -+ y(z) .. We should note that r-automorphic forms are transformed in the t Thi'J definition departs somewhat from the usual terminology. Properly speaking, automorphic forms are solutiqns of the functional equation (1), where additional hypotheses concerning the behavior close to certain boundary points are imposed on the solutions (see the example, Section 2). All the same, we will use this definition.
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
following manner under an analytic mapping z -+ Z1 domain £0 onto some domain £0 1 : 1(z) -+ 1(cp -1(Z 1»)j;-1(Z1)'
141
= cp(z) of the (2)
When the domain £0 is bounded, there is a very convenient method for constructing r-automorphic forms by means of Poincare series. Let r be an arbitrary discrete group of analytic automorphisms of a bounded domain £0. The series (3) is uniformly convergent in any compact subdomain of the domain £0 (Siegel [1], p. 103). As a result, the series
I
h(y(z»j~'(z),
111
~ 2,
(4)
YEr
where h(z) is a bounded function analytic in £0, is a function analytic in £0. It is easy to verify that equation (1) is satisfied by this function and, therefore, it is an automorphic form of weight m. Preliminary discussion of the construction of some special fundamental domain will be useful in the study of Poincare series. Definition 2. By afundamental domain, we will mean a closed domain bounded by a finite or countable number of real analytic manifolds, where every point in the domain £0 must have at least one point r-equivalent to it in the fundamental domain, and any two interior points of the fundamental domain must be r-nonequivalent. Let F denote the set of points in £0 for which Ii/z) I ;;; 1 for all y E r. We will prove that ifr 0 contains no nontrivial mappings with lacobians having modulus identically equal to 1, then F is a fundamental domain in the above sense. Indeed, it is clear that F is closed. Let Fo denote the set of points z E F such that Il/z)1 < 1 for all y except y = 8. It is clear that Fo is an open set. The points contained in Fbut not in Fo are clearly contained in one of a countable number of hypersurfaces Iliz) I = 1. Every point Zo E D has an equivalent point z' E F. In view of the convergence of series (3), in fact, among the numbersl/zo), YEr, there is one with maximum modulus. We denote it by lyo(zo). Set z' Yo z. Then (z 0) I ~ 1 IJ. ( ') I = IJ. (Yo (Zo) I = IIj jiz) I- , y z
and, therefore, z' E F.
y
)')'0
(5)
142
THE GEOMETRY OF CLASSICAL DOMAINS
Let Zl EFo and Z2 = Y(Zl) EFo; then
jy-1(Z2)j/Z1) = jy-1(Y(Zl»)j/Z1) = je(Zl) = 1. This is impossible (see the definition of Fo). Consequently, there are no r-equivalent points in Fo. We will now prove a property of the fundamental domain we have constructed that will be useful in what follows. We will prove that any closed set Do c D can be covered by a finite number of images of F. Indeed, if Zo E Do and Zl = 1'1 Zo E F, then (6)
Because series (3) converges uniformly, Do may contain only a finite number of different I' for which the last inequality is true. Now consider the space in which there are nontrivial mappings I' E r with lacobians identically equal to one. It is clear from (3) that there are only a finite number of such mappings. It is also clear that they form a group, which we will denote by r o' F is defined as above, and Fo is defined as the set of all Z such that Ii/Z) I < 1 for all I' Er except Y E r o. As we did above, we will prove that any point in f0 is equivalent to some point in F and that if two points Zl and Z2 in Fo are equivalent, then Z2 = YOZb where YoEro. The following lemma proves that it is possible to separate r-nonequivalent points in f0 by means of Poincare series. Lemma 1. For any two points Zl and Z2 that are not equivalent with respect to r, there exist Poincare series ¢l(Z) and ¢2(Z) of the same weight such that (7) Proof Without loss in generality, we may assume that Zl and Z2 belong to F. Note that when m ~ CIJ
(8) We choose hi(z), i = 1,2, so that the following conditions are satisfied:
h 1(Zl)hi z 2)-h 1(Z2)hi z 1) h/yzj)
= 0, if
Ij/z j)1 = 1,
0,
(9)
yZj =1= Zj'
(10)
=1=
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
143
Conditions (9) and (10) are consistent because the points Zl and Z2 are not equivalent. We denote the number ofy such that yZj = Zj by Sj. Itis clear that the y having a common fixed point form a group. By Lagrange's theorem, therefore, it follows from yZj = Zj that ySj = 8 and, therefore,
(11)
the lemma follows trivially from (9) and (11). We should note that a similar method may be used to prove a somewhat more general assertion, namely: let Zl' •.. , zp be some system of pairwise distinct points. Then there exist Poincare series CPl' ... , CPP of the same weight such that
CPl(Zl) ... CPi Z l) CPl(Z2) ... CPi Z2)
0
i= .
(12)
Expression (12) clearly implies the existence of a Poincare series with sufficiently large weight and any preassigned values atthe points Z 1, ••• , Z po Lemma 2. Let Zo E F be a point that is notfixed under any transformation in the group r except the identity transformation. Then there exist Poincare series CPo, ... , CPIl such that
CPo ... CPIl oCPo oCPn OZl ···OZl
OCPo OCPIl iJz 1 ••• iJzll
i=
o.
Z=Zo
Proof As above, we seek them in the form
cplz) =
L hlyz)j~(z). )'Er
(13)
144
THE GEOMETRY OF CLASSICAL DOMAINS
For sufficiently large m, the behavior of the functions CPi(Z) depends only on the terms in series (13) for which IJ/zo)! = 1. We set
/z) =
L
h/yz)j~J(z).
liy(zo) I = 1
It is clear that it is sufficient to select hi(z) so that
0 '" n 00
011
OZl '" OZl
00
:f 0.
8n
OZII '" oZn
z=zo
This is not difficult to do, if we use the fact that the points yZo are different for different y. We should note that without the restriction that Zo be a fixed point of none of the transformations, our assertion is false. Indeed, let the point Zo be a fixed point of transformations in r. These transformations form a finite group K. We expand the Poincare series cp(z) in a power series about z. For simplicity in the calculations we will assume that Zo is the origin. By a well-known theorem of Cartan (see Bochner S. and Martin U. T. [1]), we may assume that the l' E K are linear or even unitary, transformations. Thus,
cp(yz) = 8cp(Z), where 8 is a root of unity and l' E K. As a result, either cP or some of its partial derivatives are equal to 0. The following lemma may be proved by arguments similar to those given above. Lemma 3. Let Zo be a point that is fixed under none of the transformations in the group r except the identity transformation. Then for all sufficiently large m, for some a > 0, and for some any ak t ... k" (s = kl + ... +kll ~ am) of complex numbers, there exists a Poincare series cp(z) such that
(14)
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
145
This lemma generalizes to the case in which Zo is a fixed point. That is, it is possible to prove that the only restrictions on the possible selection of the ak( ... k" are consequences of the functional equations for the automorphic form for the 'Y that leave the point Zl fixed. Further on it will be important to be able to estimate Poincare series close to the boundary of a domain. We agree to adopt the following notation: Let £0 be some bounded domain; z -4 w = ¢(z) is some one-to-one analytic mapping of £0 onto some domain £0'; r is a discrete group of analytic automorphisms of the domain £0; and r' = ¢r¢ -1 is the corresponding group of automorphisms of the domain £0'. Note that the domain £0' may be unbounded. A subset T c £0' is said to be a proper subset if there exist 8 and N such that in any polycylinder C( wo, 8), t where Wo E T, has no more than N r'-equivalentt points. Lemma 4. Letfo(z) be some Poincare series. Thefunction few) =fO(¢-l(W»)j;-l(w) is bounded on any proper subset of the domain £0'. Proof It is sufficient to prove that the function
k(~) =
L: !j/¢-1(W))!2!jljJ_l(W)!2 )'Er
is bounded on any proper subset. Indeed, the inequality !f(w)! ~ c(!c(w))1Il/2, where c is the maximum modulus of the function h(z), follows immediately from (4). Let T be a proper subset of the domain £0'. We will prove that the function k(w) is bounded on T. The general properties of analytic functions can easily be used to derive the inequality
f
d(J,
C(wo, r)
where d(J is a Euclidean volume element of an affine complex space. t C(wo, e) is the set of points W (of an affine complex space) such that the modulus of any difference of coordinates W - Wo does not exceed e. t We assume that C(wo, e) C {J2' for any WOE T.
146
THE GEOMETRY OF CLASSICAL DOMAINS
We now apply this inequality, assuming that
few) =
( 00
iffor any cylindrical domain seQ, r) (to E Q) there is a Vo such that for v > Vo the point Wv belongs to SeQ, r). Definition 4. Let few) be some function in S. We agree to say that limf(w) = A, w->to
iffor any sequence
Wv E
S such that
the limitf(w v) exists for v --+ Lemma 1. The limit
00
and is equal to A.
limf(w) = ljJ(t) w->t
exists for any r-automorphic few) of weight J.1 that is "analytic at infinity". The limit function ljJ(t) is analytic in $7. Proof We denote by 30 the subgroup of 3 consisting of transformations of the form
(1)
where a is an arbitrary real vector.
148
THE GEOMETRY OF CLASSICAL DOMAINS
We set ~o r n 30. It is clear that ~o is a commutative group with the same number of generators as the dimension of group 30' i.e., ~o is a lattice for 30. Let the function fez, u, t) be a r-automorphic form of weight /1. The invariance of/with respect to the transformations of the group ~o implies that it is possible to expand/in a Fourier series of the following form:
f(z,u,t) = It/Jp(u,t)e 2ni (P,z),
11
(p,z) =
p
L
PkZk,
(2)
k=l
where P runs through the dual lattice of ~o, i.e., the lattice consisting of all vectors P such that (p, a) is an integer for any a E ~o. It is clear that each of the functions If p(u, t) is analytic in the domain
c m x $7. We denote the dual cone by V'. (The set of P such that (p,y) > 0 for all y E V, y =1= 0 is called the dual cone.) We will prove that if the r-automorphic form/(w) is bounded in any cylindrical domain, then t/J p(u, t) == 0 for all p E V'. (V'is the closure of V'.) We use Parseval's equality
4n (p,y), ~Lflf(x+iY, U, t)12 dx = L It/Jiu, t)12 eL p
mes
(3)
where L is the parallelepipedt of the lattice ~o. If t/J ilio, to) =1= 0, then for all sufficiently large y
e-41t(p,y)~
1 2f mes Llt/J p(uo, t o)/ L
If(x+iy,uo,to)12dx~C,
(4)
where C is some constant.t It immediately follows from inequality (4) that
(p, y) > C 1,
if y - rEV,
(5)
where C 1 is some constant.
t The fundamental parallelepiped is the parallelepiped constructed of the generators of the lattice Llo. :j: This means that there is a vector rE V such that if y - rE V, then inequality (4) is satisfied.
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
149
Substituting AY for y in (5), where A is a real number, and letting A go to 00, we obtain, from (5), the inequality which is true for any y E V;
(p, y)
~
0
implies that p E V'. We will now prove that the function I/Jo(u, t) is independent of u. Indeed, the factor space 3/(r n 3) is compact, and, therefore, the function I/J o(u, t) has 2m, periods with respect to u, where m is the complex dimension of u. Hence we conclude, by means of Liouville's theorem, that I/J o(u, t) is independent of u. Now let Wv = (zv' uv, tv) be a sequence of points in S that converges to the point to E D. We will now show that a limit, which is equal to I/Jo(to) , exists for few) as w ~ to. Without loss of generality, we may assume that U v is bounded and tvE Q, where Q is a fixed neighborhood of the point to' The Fourier series of an analytic function converges absolutely; Thus,
I
II/Jp(u, t)1 e- 2rr (p,y)
t
The lemma is proved. Later on we will need a criterion for "analyticity at infinity". Lemma 2. A r-automorphic formf(w) is "analytic at infinity" if and only ifin series (2) the Fourier coefficients I/J p(u, t) == 0 when p E V'. Proof Necessity of the conditions of the lemma follows from the proof of Lemma 1. Sufficiency is proved in the following manner.
150
THE GEOMETRY OF CLASSICAL DOMAINS
Let SeQ, r) be some cylindrical domain in S. Consider the subset in r) that consists of the points of the form
SeQ,
1m z - Re LtC u, u) - rEV, } lul 1 so that (Yo, uo, to) is not a fixed point for the principal part of any transformation. Hence, by the argument used in Lemma 1 of this section, it follows that the Fourier coefficients !/J /u, t) for the expansion of the function few) in a series of the form (2) are equal to zero for almost all U o and to if P E V'. It remains to note that the functions !/J p(u o, to) are analytic with respect to the set of u, t and, consequently, their being equal to zero on an everywhere dense set implies that they are equal to zero everywhere. By Lemma 2, we conclude that the function few) is bounded in any cylindrical domain. This completes the proof of the lemma.
Section 6. Arithmetic Groups in Symmetric Domains In this section we will show that the space M for an arithmetic group which acts on a symmetric domain is compact. In order to prove this we must use A. Borel's work on the fundamental domains of arithmetic groups (A. Borel [3]). We will now present his results: Let G be a semi simple linear algebraic group defined over the field of rational numbers Q. By a Satake subgroup we agree to mean an algebraic subgroup B £ G defined over Q with the following properties:
r
(1) B is solvable and split over Q; (2) a maximal solvable and split (over Q) normal subgroup of the normalizer S(B) of the group B coincides with B. Let !£ be the Lie algebra of some Satake subgroup, and represent it in the form W+ S)(:, where W is a subalgebra consisting of semisimple elements and S)(: is a nilpotent ideal. Since the sUbjective representation of Win S)(: is completely reducible, S)(: can be represented in the form of the sum of spaces 97:0; consisting of n such that
[a,nJ = a(a)n,
aEW,
nES)(:o;'
(1)
154
THE GEOMETRY OF CLASSICAL DOMAINS
Here o:(a) is some linear form on Sll. The linear forms o:(a) for which ~)(a is nontrivial are called roots of the algebra !l'. It can be shown that Sll contains a vector a o such that o:(a) > 0 for all roots 0:. We will denote the set of all a E Sll such that o:(a) > 0 for all roots 0: by V(Sll). We now introduce a partial ordering into Sll by setting a 1 > a2 if a1 -a2 E V(Sll). We denote the maximal compact subgroup of the group GR by K. Let X = GR/K be the corresponding symmetric space. Consider the set L of all orbits of the form {BRx}, where XE X and B is some Satake subgroup of the group G. Note that Xc L, because the identity subgroup is a Satake subgroup. The group GQ is naturally defined in the space L, namely, a transformation . BRX~gBRX=gBRg-1gx corresponds to each gEG Q • r,which will remain fixed to the end of the argument, is a group commensurable with Gz . We set (2)
It is clear that
r\xc s.
(3)
We now introduce a topology into S using the same method as that used to introduce a topology into the space M in Section 2 of the present chapter. Namely, with each point S E S we associate a fundamental system reS) of open sets r\x. A topology will be introduced into S in such a manner that the sets of the system reS) prove to be the intersections of the neighborhoods in a fundamental neighborhood system of the point y and r\x. We will now define the system of sets r(so)' Let So E r\X; then, for the system r(s) , we take a fundamental neighborhood system of the point s. Now let So = B R xo, where B is some nontrivial Satake subgroup of the group G. We denote the maximal unipotent normal subgroup of the group B by N. The orbit BR Xo fibers naturally into orbits of the group N R • There is a natural isomorphism between the orbits of the groups N into which the orbit B R x fibers and the Lie algebra SllR (recall that Sll is the maximal commutative subalgebra consisting of semisimple elements of the Lie algebra of the group B). We can use this isomorphism to carry this partial ordering in Sll over to the set of orbits. Let N Rx 1 be some orbit contained in BRx O, and let U(e, N Rx 1 ), where e > 0, be the set of all x E X such that (4)
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
155
where p is an invariant distance in X and Q X1 denotes the union of all orbits of the group NR that are larger than the orbit NR Xl in the sense of the partial ordering in the set of orbits. The projection of all sets of the form U(e, NR Xl) under the natural mapping of X onto r\X forms the system T(so). It is not difficult to verify that the family of sets obtained in this manner has the following properties: (1) if Ul E T(so) , U2 E T(so) , then there exists a U3 E T(so) such that
(2) if So E r\X, the intersection of aJI sets in the family T(so) coincides with the point So, (3) if So ¢ r\X, the intersection of all of the sets in the family T(so) is empty. We now define a topology in S in the following manner. Let So E S and Uo E T(so); denote by 0 0 the set of all s E S such that for each there exists a set Us E T(s) contained in Uo. It is not difficult to use properties 1, 2, and 3 to show that the family of sets 0 0 forms a fundamental neighborhood system for the point So. Borel's remarkable result consists of the fact that the space S is compact and Hausdorff. Even when Xhas a complex structure, however, the space S, as a rule; does not. We will now change the construction of the space S so that the space obtained does have a complex structure. Assume that Xis a complex symmetric space; then, as we know, Xis a bounded homogeneous domain in CII • Denote the set of bounded holomorphic hulls of the orbits {B R x} of the subgroups of the Satake group G by 1:. Moreover, set (5)
The natural mapping of L onto 1: induces a mapping of S onto S. Henceforth we will discuss S with the topology induced by the mapping S -+ S. In this topology S is compact. Below we will prove a lemma from which it follows that S coincides with the space M introduced in Section 2 of the present chapter and, therefore, is a complex compact sPFlce. Let ~ be a bounded homogeneous domain. A subalgebra R of the
156
THE GEOMETRY OF CLASSICAL DOMAINS
Lie algebra of all analytic automorphisms of the domain!» is called a Satake subalgebra if (1) R is solvable and split over R.
(2) a maximal solvable and split (over R) ideal of the normalizer of the algebra R coincides with R. We also agree to call a subgroup of the group of analytic automorphisms of the domain!» whose Lie algebra is a Satake sub algebra a Satake subgroup. We have the following lemma.
Lenuna. Let!» be a bounded homogeneous domain. A fibering into bounded holomorphic hulls of the orbits of the Satake subgroup B is a homogeneous analytic fibering of the domain!». We denote the normalizer of the group B in the group of all analytic automorphisms of the domain!» by sJC(B), while 'we denote a maximal commutative normal subgroup by A. The group ,3 of parallel translations of the given fibering is a maxim.al unipotent normal subgroup in the centralizer of the group A. We will first show how this lemma implies our statement about the space S and then we will prove it. lt is sufficient to show that the space we for the group r coincides with the space 1:. In order to do so, it is clearly necessary to show that: (1) a fibering into bounded holomorphic hulls of the orbits of a given Satake subgroup B of the group G is a r-rational fibering in the sense of Section 2 of the present chapter; (2) the maximal solvable and split over Q normal subgroup of the group of aut om orph isms of the domain !» that preserve the given r -rational fibering is a Satake subgroup B of the group G, and the fibering into bounded holomorphic hulls of the orbits of the group B coincides with the initially adopted fibering. We will first prove the first statement. Let B be some Satake subgroup of the group G. lt follows from the lemma that the fibering into bounded holomorphic hulls of the orbits of the group B R is a homogeneous analytic fibering. In order to show that this fibering is rrational in the sense of Section 2, it is sufficient to show that the group ,3 of parallel translations of this fibering is defined over the field of rational numbers Q. First note that the normalizer SJC(B) of the group B is an algebraic subgroup of the group G and is defined over the same field as B, i.e., over Q. Moreover, the maximal commutative normal subgroup A of the group SJC(B) and its centralizer ,3(A) is clearly an algebraic subgroup of the group G. As a result, the maximal unipotent normal subgroup of the group ,3(A) is an algebraic subgroup of the group G and
157
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
is defined over Q. It remains to note that this normal divisor, by the lemma, coincides with.8. We will now prove the second statement. Consider some r-rational fibering. As we noted in Section 2, the subgroup .8 of this fibering is an algebraic subgroup of the group G and is defined over Q. As a result, the normalizer 9((.8) of this subgroup in G is also defined over Q. The group ~(R(.8) coincides with the group of all analytic automorphisms of the domain !!fl that preserve the given r-rational fibering. Its maximal solvable and split (over R) normal subgroup B is generated by the group .8 and the directing subgroup of the given fibering (see Theorem 4, Section 8, Chapter 2). As a result, by what we proved in Section 4 of Chapter 1, the bounded holomorphic hull of the orbits of the group BR coincides with the fiber of the given fibering. Thus, we have proved, by means of the lemma, that 1: = and, therefore, S = M. It now remains to prove the lemma. Proof of the lemma. The normalizer m(B) contains a maximal solvable subgroup of the group of all analytic automorphisms of the domain !!fl and, consequently, is transitive in the domain !!fl. Let W denote the Lie algebra of the group ~(B). We can thus introduce the structure of a j-algebra into W in a natural manner (see Section 2, Chapter 2). As was proved in a paper contained in the appendix, any algebraic j-algebra can be repreEented in the form
we
(6)
vV=2'+j2'+K+U+W' ,
where 2' is a commutative ideal, J( is a compact algebra, j(K) = 0, W' is a semisimple algebra that is invariant with respect to j, and [u, U]
c
2',
[2'+j2'+K, W'] = 0,
[U, W']
c
U.
(7)
In addition, 2' contains an element 1o such that Ulo, I] = 1 for all = tu,
IE 2', Ulo, u)
(8) Consider the following subalgebra of the algebra W: L
2'+U+{jlo},
(9)
where {}lo} denotes the one-dimensional linear space generated by the vector }lo. It is not difficult to see that L is a solvable ideal of the algebra G and, therefore, is contained in the Lie algebra of the group B. With the
158
THE GEOMETRY OF CLASSICAL DOMAINS
representation of the Lie algebra W in the form (6) we can naturally associate a homogeneous fibering n of the domain ~ with the base ~', in which the Lie algebra W' is transitive, and fiber a Siegel domain of genus 2 in which a group whose Lie algebra is
is linear. It is not difficult to show (see Section 4, Chapter 1) that the bounded holomorphic hulls of the orbits of the groups expSP
and
exp(SP+jSP+K+U)
(10)
coincide with the fiber of the fibering n. The obvious relationship expSP c B c exp(SP+jSP+K+U)
(11)
implies that the bounded holomorphic hulls of the orbits of the group B are also a fiber of the given fibering. We will now show that the maximal commutative ideal of the algebra W coincides with SP. Let SP m be a maximal commutative ideal of the algebra W. It is clear that
SPlllcSP+jSP+K+U. It follows from (8) that SP m = SP n SP m+ U n SPm+(jSP+K) n SP m.
(12)
We will first show that (jSP+K)n SPill =0. Let gl =jll+klE(jSP+ K) n SP m • It follows from the definition of j-algebras (see Section 2, Chapter 2) that
[g 1,1 0] +j([jg l' 10J) +j([g 1,j 10]) - [jg 1,j 10] E K.
(13)
It follows from (8) that [jgl,loJ
= [gl,j IO] = 0,
[jlo,jgl] = [jlo, -1 1] = -1 1 , As a result, [gi' 10J = 11, Thus, if gl = jli +kl E SPill' we have 11 E SP m • Moreover,
(14)
DISCRETE GROUPS OF ANALYTIC AUTOMORPHISMS
159
But (14) contradicts the fact that the Lie algebra .fi'1II is commutative. The proof that .fi'1II n U = 0 is similar, but somewhat more complex. In order to prove this statement, we construct a differentiation ljJ for the j-algebra W. We set
(15)
W',
joEW'
where j is an endomorphism of the complex structure in Wand adujo
j
on
U o = [fli', UJ.
(16)
The existence of such an element jo in semisimple j-algebras is a well-known fact. It is not difficult to show that ljJ is a differentiation for thej-algebra W and, therefore, ljJ carries a maximal solvable ideal of the algebra W into itself. As a result, ljJ E W. We will now show that Un.fi'1II = 0. If U E Un .fi'IlP then ljJ(u) c Un .fi'1II. It follows from the definition of j-algebras that [ljJ(u), u] = [ju, u] :j:. 0, if u :j:. o. As a result, u = 0 and, therefore, Un .fi'1II = 0. We have proved that .fi'1II c .fi' and, therefore .fi'1II = .fi'. The centralizer of the ideal .fi' is clearly equal to .fi'+ U + W'.
(17)
A maximal nilpotent ideal of the algebra (17) is equal to and, therefore, coincident with, the Lie algebra of the group of parallel translations ,8 of our fibering. This completes the proof of the lemma. Section 7. The Andreotti-Grauert Method
Andreotti and Grauert [1] proposed a very elegant method for proving that the degree of transcendence of a field of automorphic functions is no greater than the complex dimension of their domain of existence. We will now present a brief outline of this method and show that it applies to our case. The ingenious notion of pseudoconcave boundary points plays a central role in this method. Let X be a strict subdomain of a domain ~, i.e., contained in X together with its closure. We denote the boundary of the domain ~ by X'. We agree to say that the domain X is pseudoconcave at a point
160
THE GEOMETRY OF CLASSICAL DOMAINS
X' if for any neighborhood U of the point Zo and any function ¢(z) that is regular in U there exists a point Zl E Un X such that
Zo E
(1)
Andreotti and Grauert proposed the following simple criterion for verification of pseudoconcavity. If there exists a two-dimensional complex plane E = E(tl' t2 ) passing through the point zo, an infinitely differentiable function q(z) defined on E, and a neighborhood of the point Zo in E such that LgaP~a~p > 0, q(z) O. It follows from (4) that we can select c so large that the set Fc = Fo +P 1 ( c) + ... + Pic) will have the following properties: (1) For any point Zo EFo there exists a Yo E r such that Yo Zo E Fc ' and (2) For each point Zo E!!fi contained in both boundary of P t and the boundary of 0 t there exists a Yo E r such that Yo Zo E Fc. We will now show that the set Fc selected in this manner has the properties of the set X that we discussed in the definition of pseudo·· concave groups. We split the boundary of the set Fc into two components !F 1 and !F 2' The component !F 1 consists of all points Z E Ot, 1 ~ t ~ S, such that Pt(z) = c. On the other hand,!F 2 consists of all remaining boundary points. It is clear that if Zo E!F 2' then the orbit F
162
THE GEOMETRY OF CLASSICAL DOMAINS
rZo of the point Zo is an interior point of Fc. Let Zo E!F l' As we know, the form
is positive definite at any point in the domain!». As a result, Zo is a pseudoconcave boundary point of the domain Fc. Thus, we have shown that any arithmetic group is a pseudoconcave group in the sense of Andreotti and Grauert.
CHAPTER 4
Automorphic forrl1s Introduction
In this chapter we will study automorphic forms for discrete groups of analytic automorphisms of symmetric domains. Certain results, e.g., the construction given in Section 1 for Fourier-Jacobi series, hold true for arbitrary Siegel domains of genus 3. In Section 3 we will use Fourier-Jacobi series to prove the theorem on algebraic relations for arbitrary arithmetic groups in symmetric domains. This proof is the third in this book. Two others, one based on a construction of a compactification, the other based on the Andreotti-Grauert method, were given in Chapter 3. In itself, the proof we will give for the theorem on algebraic relations is not, at the present time, particularly interesting. As far as we are concerned, however, its use of the apparatus provided by Fourier-Jacobi series is of value. In Section 2 we will study automorphic forms. At the end of Section 2 we will give an outline of Selberg's method for computing the dimension of spaces of automorphic forms of a given weight. Section 1. Fourier-Jacobi Series
Let S be a Siegel domain of genus 3 given, as usual, in the form (see Section 3, Chapter 1)
1m z- ReLtCu, u) E V,
tEf»
where V is some cone and f» is the base of the domain S. We will assume that the domain S was obtained by means of the construction given at the end of Section 3 of Chapter 1. Let 3 be the group of parallel translations of the domain S, and let 30 be the normal divisor of 3 consisting of all transformations of the form z-+z+a,
U-+U, 163
t-+t.
(1)
164
THE GEOMETRY OF CLASSICAL DOMAINS
Let r be a discrete group of analytic automorphisms of the domain S. We set r 0 = r n 20' r 1 = r n 2. In the present section we will assume that the factor space 2/r 1 is compact. As we showed in Section 3 of Chapter 3, in this case the numbers of generators of the groups r 0 and r dr0 are, respectively, n and 2m, where n is the dimension of 20 and 2m is the dimension of 2120' Since ji w) = 1 if Y E 2, each r -automorphic form is invariant under the transformations Y E r l' As a result, it can be expanded in a Fourier series (w) =
L l/Jp(u, t) e
21ti
(p.z)
(2)
p
where p runs through the set of linear functionals on the group 20 that are integer-valued on r o. Further study of automorphic forms in Siegel domains is based on a detailed investigation of the Fourier coefficients l/Jiu, t) in series (2). We will show that the functions (u, t) as functions of u are Jacobian functions. In connection with this, we will call series (2) a FourierJacobi series. Note the following relationship, which is necessary for what follows. Let (c l , a1) and (c 2, a2) belong to ~; then 2Q(e 1,c 2)Ero,
(3)
where Q(c 1 , C2) is defined in Section 3 of Chapter 1. Indeed,
The following relation is a consequence of the functional equation for an automorphic form:
fez + a +2iLtCu, e(t)) + iLtCe(t), e(t)), U + e(t), t)
fez, u, t)
(4)
= l/Jp(u, t)exp[ -2ni{p, a +2iLtCu, e(t))+iLtCe(t),e(t))}].
(5)
=
for all (c, a) E~. As a result, we find that
l/Jp(u + e(t), t) for the functions l/Jiu, t). The expression in the exponent is a linear function of u. We write it in the form (6)
165
AUTOMORPHIC FORMS
where bp(t) and [3p(t) are defined by the following relationship: (b p([3t),(U) = 4 n((p, Lt((U(, C(t)()),) ())}. p t) = 2 n p, L t c t), c t) - 2ni p, a
(7)
It is clear that bp(t) and [3it) depend on c. Let C(l), ... , c(2m) be a basis for the lattice of r' = fl.jrl. For the basis, relationship (5) may be written in the form
ljJ p( u + C(k) (t), t) = ljJ p( u, t) exp [{b~) (t), u} + [3~k)(t)],
(8)
where b~k)(t) and [3~k\t) are given by (7) upon substituting C(k)(t). Recall that Jacobian functions c(u) are functions that are analytic in an m-dimensional complex space and satisfy the relationships ljJ(U+Ck)
= ljJ(u)e(bk.u)+P\
k
= 1, ... ,2m,
(9)
where C l , ... , C2m is some set of vectors linearly independent over the reals. The matrix C whose columns are the vectors c 1 , ... , C2111 , is called the period matrix. It is known that functions satisfying relationships (9) need not exist for an arbitrary selection of vectors Ck and bk • Recall (see Siegel [1], pp. 50-4) that the following conditions are necessary for existence of functions satisfying (9) : (a) every element of the matrix R=
~(B'C-C'B)
2m
is an integer (here B is the matrix with columns b l , ... , b 2m ); ([3) the Hermitian matrix 1 _ H=-G'RG i
is negative definite (G is found from the conditions CG = E III , CG = 0). Furthermore, the dimension of the space of Jacobian functions with given Ck, bk, and [3k is always finite and no greater than 2111 ..jdR (dR is the greatest common divisor of the minors of a matrix R of maximal possible order for which the minors are not all equal to zero; in particular, if R is nondegenerate, then dR is the determinant of the matrix R). If the matrix H = i-1G'RG is positive definite, then the dimension of the space of Jacobian functions is equal to 2111 ..jdR (it is easy to verify that here the matrix R is nondegenerate).
166
THE GEOMETRY OF CLASSICAL DOMAINS
We will now prove that condition (0:) is satisfied. It follows from (7) that 2nirsk = (b s' ck) - (b k, cs) = 4n(p, L t ( C(k)(t), c(s)(t») - 4n(p, Lt(c(s)(t), C(k\t»)
= 4ni(p, Q(c s' ck» (rsk is the element of the matrix R that is located at the intersection of the sth row and the kth column). The fact that rsk is an integer follows from (3). We now turn to verification of condition ([3.) We write Llu, v) in the form L~l)(U, V)+L~2)(U, v), where L~l)(U, v) is the symmetric part of the form Llu, v) and L~2)(U, v) is the Hermitian part. As we know, such a representation is unique. Let wtCt E flJ) denote the set of all vectors p in an n-dimensional real space that are such that (p, LF)(u, u» ;;:;; 0 for all u.
(10)
We will prove that ([3) holds if and only if p E Wt. Note that (7) associates some vector function bpet) that is analytic on flJ with each vector function e(t) of our set so that if
c'(t) -+ then
b~(t),
c"(t) -+
b~(t),
(111 c' + 112 c")( t) -+ 111 b~( t) + 112b~( t)
for any real 111 and 112' As a result, the relationship between e(t) and bp(t) may be written in the form (11)
where the K~i)(t), t = 1,2 are square complex matrix functions of t that are linearly dependent on p. Generally speaking, the matrix functions K~i)(t) are not analytic. Substituting the expression for bpet) into (7), we can easily see that the following relations are valid: (K~l)(t)C(t), u) = 4n(p, L~l)(U, c(t»), } (K~2)(t)C(t), u) =
4n(p, L~2)(u, c(t»).
It is easy to verify that the matrix K~2)(t) is Hermitian.
(12)
AUTOMORPHIC FORMS
167
It follows from (12) that K~2)(t) is non-negative if and only if p E Wt • It remains for us to show that the matrix H given in ([3) is equal to K~2)(t)
up to a positive real factor. Substituting the expression for R in terms of Band C into the formula defining H, we find that . H = -(2n)-lG'(B'C-C'B)G = (2n)-lBG = (2n)-1(K~1\t)C + K~2)(t)C)G
= (2n)-1 K~2)(t).
Here we have used the relationships CG
= CG = E,m CG = O.
The following lemma follows directly from our discussion: Lemma 1. Let r be a discrete group of quasilinear transformations of the Siegel domain S, where the factor space B/(r n B) is compact. If the convex hull of the vectors LtCu, u) coincides with V' for all t in some open subset of the domain!», then any r-automorphicformf(w) will be bounded in any cylindrical domain. Proof We expand few) in a Fourier-Jacobi series. It follows from
our discussion that when the hypothesis of the lemma is satisfied, the Fourier coefficients are ljJp(u, t) == 0, if pE V'. It follows from this and Lemma 2 of Section 5 of Chapter 3 that the function few) is bounded in any cylindrical domain. The first such type of "effect" was, in another context, discovered, in fact, by the German mathematician Kocher while he was proving the following important theorem (Kocher [2]): Kocher's Theorem. Let H be Siegel's upper halfplane, i.e., the set of all complex symmetric matrices Z = X + i Y, where Y is positive definite. We denote by r the group of transformations of the domain H that have the form
Z-+A'ZA+S, where A is any unimodular integer matrix and S is any symmetric integer matrix. Then a r-invariantfunction analytic in H will be bounded in any domain consisting of points of the form Z + iT, where Z E Hand T is an arbitrary positive definite matrix.
The lemma proved above shows that the Kocher "effect" occurs, as a rule, for Siegel domains of genuses 2 and 3.
168
THE GEOMETRY OF CLASSICAL DOMAINS
We will now prove a lemma that includes Kocher's theorem as a special case. This lemma contains the conditions that must be imposed on a discrete group of affine transformations of a Siegel domain S of genus 1 in order for the Kocher "effect" to occur. As we know, the affine transformations of a Siegel domain of genus 1 S have the form z
-+
Az+a,
(13)
where A is the matrix of an affine transformation of cone V into itself and a is an arbitrary real vector. As a result, the group of all affine transformations of S into itself may be treated as a group of pairs (A, a) with law of composition (14)
We denote the subgroup of group G consisting of elements of the form (E, a) by d. Let G' denote the set of all affine transformations of cone V and let G~ denote the subgroup of G' that consists of the unimodular affine transformations of the cone V. Consider the natural homomorphism G -+ G'. If r A = r n d has as many generators as the dimension of the group d, then r' = r /r A is a subgroup of the group G~. Indeed, an automorphism (A, a)(O, b)(A, a)-l = (0, Ab)
of the lattice of r A corresponds to each (A, a). It is clear that the determinant of the matrix of an automorphism of the lattice equals ± l.
Lemma 2. Let r be a discrete group of linear transformations of the domain S such that the factor space G' /r', where r' = r /r A has finite volume. Then any r-invariant function is bounded on every cylindrical domain. Proof Letf(z) be some analytic r-invariant function. We expand it in a Fourier series: n
fez) = LAp e21ti (p,z), p
(p, z) = L
PkZk'
(15)
k=l
Note that invariance of fez) with respect to the group r entails a relationship between the coefficients Ap. Let (A, a) E r, then f(Az+ a) =fez) and, consequently, AA'p = Ap e21ti (p,a) , (16)
AUTOMORPHIC FORMS
169
where A' is defined by the condition (Ay, p) = (y, A'p). From now on, we agree to call the vectors P and A'p associate vectors. The lemma will be proved if we show that Apo = 0 for any PoE V'. We will prove this fact by contradiction. Let Apo :j:. 0 for some PoE V'. We denote an arbitrary point in V by Yo and fix it. By Parseval's equality,
L IApl2 e- 4n
(p,yo)
0 there exists an n. nee) such that for all
n1 > nee), n2 > nee), (fill -1,'2,1,'1 -1,,) < e. It is easy to use the functional equation of an automorphic form to show that
where ~1 is any fixed subdomain of the domain ~ such that ~1 c ~. Using the general properties of analytic functions, we can easily use this fact to show that the sequence of functionsf,(z) uniformly converges in any subdomain ~1 of the domain ~ to some function lo(z). The functional equation of an automorphic form of weight 111 will clearly be satisfied by the limit function/o(z). We denote the upper bound of the numbers (f" Ill) by c. It is clear that
f
I
lJo(z) 2plll(Z) dv
~
Bn501
lim
r f,,(z) I2plll(Z) dv ~ C,
II->OOJB
where ~1 is an arbitrary sub domain of the domain result,
(fo'/o) Thus,
=
fo(z)
~, ~1 c~.
As a
fB IfoCzWp"'Cz)dv "" C. = limf,,(z) E A(r, 111). /1-> 00
We will now show that A(r, m) c
t Recall
mer, m).
that a cylindrical domain associated with a given fibering is a subset with the following properties: (1) the projection of P onto the base !!)' of the fibering is strictly contained in !!)' (i.e., its closure belongs to !!)'), (2) the intersection of P with any fiber is contained in the bounded holomorphic hull of some orbit of the group of parallel translations.
pc!!)
175
AUTOMORPHIC FORMS
Let T be some sufficiently small cylindrical domain with base Q in some r-rational component fF. We will assume that T is chosen so that if two points Zl and Zz are equivalent with respect to r (i.e., Zz = '}'Zl' where ,},Er), then ,},Er1(fF). When Tis selected this way, we have the inequality (3) Map the domain f» onto the corresponding canonical Siegel domain S. After substitution of variables, integral (3) takes the form (4) where A(W) is a solution of equation (1) in the domain S, E>y(w) = J(¢(W))jlll(W) , and dv is an invariant volume. The function E> yew) is invariant under the transformations in the group r 3(fF) so it may be expanded in a Fourier-Jacobi series E>y(W) = t/J /u, t) eZ1ti (p,z), (5)
I
p
It is easy to verify that A(W) = A1(y-ReLi(u, U))Az(t) and that an invariant volume has the form
Let A(yo, uo, to, e) be the set of points W = (z, U, t) E S such that
u-uol < e,
t-tol < e,
IY-Ayol < e,
A ~ 1,
xEL,
where L is the fundamental domain of the group r 4(fF). Finiteness of the integral
f
A(yo, 110, to; B) i
It/Jp(U, t)eZ1ti(p,Z)lzAIIl-l(W)dXdYdUlduzdtl dt z
(6)
P
follows from (4) when Yo, uo, to, and e are suitably chosen, and, therefore (after integrating with respect to x), the sum
ff
A'''-'(w)i'''p(u, tW e- 4 ,,(p,y) dy du, dU 2 dt, dt2
(7)
176
THE GEOMETRY OF CLASSICAL DOMAINS
is also finite. It is easy to verify that the individual terms in this sum may be finite only when P E V'. It follows from this and Lemma 2 of Chapter 3, Section 5 that the function 03""(z) is finite in any cylindrical domain with base in fF. Thus, we have proved that (j, f) < CfJ implies that condition b is satisfied. It is not difficult to interpret the space mer, m) as the set of all crosssections of some analytic fibering into complex lines over M, where M is the space introduced into Chapter 3. As a result, if M is compact, the dimension ofm(r, m) is finite. The dimension of A(r, m) is afOl·tiori finite. The technique ofFourier-l acobi series (see Section 3 ofthe present chapter) can be used to reduce computation of the dimension of mer, m) to computation of the dimension of A(r, 111). A. Selberg [1] gave a method making it possible to do this for the case in which f» is a symmetric domain. We should note that, as a rule, dimA(r,m) = CfJ for nonsymmetric domains. We will now briefly summarize his method. Let km(z, u) denote a function of z E f» and u E f» that has the following two properties:
(1) the integral operator
f..
km(z, u)f(u) dv
(8)
commutes with the operators Tgf(z) = f(gZ)j;'(Z); (2) the function k(z, u) is analytic with respect to z. It is easy to show that property (1) is equivalent to the following functional equation: knlgz, gu)j;l(z)j";m(u) = k lll (z, u)
for all g E G (G is the full group of analytic automorphisms of the domain f»). Apparently, this proposition can be proved for all arithmetic groups. As far as the author knows, however, the argument has never been completed. We should note that, for arithmetic groups, asymptotic formula (14) is a corollary of general theorems on the dimension of the zero-dimensional cohomology group of a coherent sheaf. R. P. Langlands [1] gave an exact formula for the dimension of a space of automorphic forms of weight m that is applicable to the case in which the fundamental domain of the group r is compact.
AUTOMORPHIC FORMS
177
Section 3. The Theorem. on Algebraic Relations It is well known Siegel [7] that the theorem on algebraic relations is a simple corollary of the following estimate of the dimension N m of the space mer, m) (1)
where Nis the complex dimension of the domain f». In the present section we will prove estimate (1) for a certain class of discrete groups (quasinormal discrete groups) in symmetric domains f». It can be shown that all arithmetic groups are quasinormal discrete groups. We will not present the proof of this proposition, for it is very similar to the arguments that we used to prove the Andreotti-Grauert conditions for arbitrary arithmetic groups. Definition of quasinormal discrete groups: Let f» be a symmetric domain, and let r be a discrete group of analytic automorphisms of the domain f». The group r is said to be quasinormal if there exists a finite number of homogeneous r-rational fiberings with bases g; 1, ... , g; p that satisfy the following conditions: (l) there exist, corresponding to the given fibe_ring, cylindrical sets = f»jr is entirely contained in
T 1 , ... , Tp with the property that the fundamental domain B
(2) (2) the fundamental domains g;dr~ are compact, where r~ is the group induced on g;k (see Chapter 3, Section 2); (3) there exist, corresponding to the given fibering, cylindrical sets T~, K = 1, ... ,j}, with the properties that Tk C T~ and for any'}' Er and any k and I (3)
where C1 is constant and Ct:lz) is the Jacobian of the mapping of the domain f» onto the corresponding Siegel domain. Let jEm(r, m), and set 0 k(z) = f(z)Ct:"m(z), 1 ~ k ~ p. It follows . from the definition of mer, m) that 0 k (z) is bounded in any cylindrical domain corresponding to the given fibering with base g;k' We now turn to the proof of estimate (1).
178
THE GEOMETRY OF CLASSICAL DOMAINS
Theorem 1. Let r be a quasinormal discrete group of analytic automorphisms of some classical domain f». The dimension of the space EJ! of aut0l11OJ1Jhic forms of weight p, is finite and does not exceed (4) where C2 is a constant depending onlv on rand N is the complex dimension of the domain f». Proof Let M' be the maximum modulus lE>k(Z)1 in the domains Tk and let Mbe the maximum of Iek(z) I in the domains Tk. It follows from (3) that for any functionf(z) E E that is not identically zero we have the inequality (5)
where C 3 = Inle!l. The function E>k(Z) may be expanded in a Fourier-Jacobi series. It follows from Lemma 3 of Section 1 that if all the coefficients t/J /u, t) of the functions E>iz) vanish when Ipi ~ 'r, then sup \E>k(Z)\ < zeT k
C4
e- CST sup \E>k(z)l,
(6)
zeT'k
where C4 and c 5 are constants. It is clear from (5) and (6) that there exists a C6 such that if all the Fourier coefficients t/J p(u, t) of the functions ek(z) are equal to zero when Ipi < C 6 p, for all k(1 ~ k ~ p), thenf(z) == O. It remains for us to note that in view of Lemma 5 of Section 1, it is sufficient to impose no more than C 7 p,"+lII+k = C 7 p,N linear homogeneous conditions on fez) for these coefficients to become zero. The theorem follows from this.
CHAPTER 5
Abelian modular functions Section 1. Statement of Fundamental Results
One of the most interesting and important classes of arithmetic groups is the class of groups to which the theory of abelian functions can be applied.t This chapter is devoted to such groups. First of all, we will recall the classical connection between elliptic functions and ordinary modular functions; in essence, this connection led Gauss to discover the latter. As we know, elliptic functions are doubly period meromorphic functions of one complex variable. The set of elliptic functions with a given period lattice forms a field. A pair of periods COl' CO2 is said to be fundamental if any vector in the period lattice is an integral combination of these periods. Modular functions and modular groups appear naturally in the study of the manifold F of all nonisomorphic fields of elliptic functions. Modular functions are functions on the manifold F, or, in other words, functions on a pair offundamental periods with identical values on pairs of periods corresponding to isomorphic fields of elliptic functions. As we know, two fields of elliptic functions are isomorphic if and only if the period lattices corresponding to them can be obtained from each other by a linear transformation (z -+ o::z) of the complex plane. As a result, we can construct the manifold F in the following manner. Consider the set Q of pairs of complex numbers (COl' CO 2 ) such that the ratio C02/C01 is not real. Fields with fundamental periods (COl' CO2) and (co~, CO2) are isomorphic if and only if there exists a complex 0:: and an integer matrix A with determinant ± 1 such that (1)
t Recall that an abelian function is a merom orphic function in CP that has 2p periods that are linearly independent over the field of real numbers. 179
180
THE GEOMETRY OF CLASSICAL DOMAINS
The manifold F can clearly be obtained by identification of all pairs (COl' CO 2 ) that correspond to isomorphic fields of elliptic functions. This identification can be carried out in two ways. First we identify the pairs (COl' CO2) and (aco l , aco 2 ), where a is an arbitrary nonzero complex number. This leads to a set consisting of the complex plane with the real axis removed. Let K denote the connected component of the set we have obtained. For example, we can assume that K is the upper halfplane: Im7: >0. It is easy to verify that two points 7:1 and 7:2 correspond to isomorphic fields of elliptic functions if and only if there exists an integer matrix A with determinant = ± 1 such that a7: l +b
7:2=--C7: l +d'
Thus, F can be treated as a factor space Kjr, where K is the upper halfplane and r is a discrete group of analytic automorphisms of the domain K. The group r obtained with this method is called a modular group. Unfortunately, the technique we have given above does not generalize immediately to the case of p > 1 variables. The difficulty lies in the fact that the set F of nonisomorphic fields of abelian functions of a given number p of variables is not a manifold when p ~ 2. The reason for this is that not any 2p vectors that are linearly independent over the reals in a p-dimensional complex space can be periods of a nondegeneratet abelian function. It is well known that systems of periods of nondegenerate abelian functions form an everywhere dense set in the manifold of all systems of 2p vectors independent over the reals in a p-dimensional complex space. We must introduce the following change into our method. In the set of systems of periods we select analytic manifolds n and apply the technique described above for constructing an ordinary modular group to each such manifold. In this case, we do not only identify systems of periods that correspond to isomorphic fields of abelian functions, but systems of periods in which the isomorphism of fields of abelian functions extends to some neighborhood. (The neighborhood is selected in the given manifold n.)
t An abelian function of p variables is said to be nondegenerate if there are p analytically independent shifts in its set of shifts, i.e., functions of the form/(z + r), where rECp.
ABELIAN MODULAR FUNCTIONS
181
The manifold F obtained with this method from Q can be treated, as we will prove in this chapter, as a factor space Kjr, where K is some classical domain and r is a discrete group of analytic automorphisms of the domain K. This group r is called the modular group corresponding to the manifold Q. Thus, there is an infinite number of different modular groups associated with the abelian functions of a given number of variables. Some of these modular groups, in contrast to the ordinary modular groups, have compact fundamental domains. Sections 2 and 3 of this chapter contain a classification of all modular groups associated with abelian varieties. We will now present a technique for constructing modular groups associated with abelian functions more concretely. Our description of the manifold Q uses certain notions drawn from the theory of "complex multiplication", which notions we will now restate. Let OJ denote a matrix whose columns are the fundamental periods of some abelia,n function of p variables. As we know, the Riemann-Frobenius conditions for the periods of a non degenerate abelian function of p variables consists in the following: There exists a rational 2p x 2p skew symmetric matrix R such that OJROJ'
= 0,
iwROJ'
>
o.
(2)
The matrix R is called the principal matrix for matrix OJ. Generally speaking, there is no single principal matrix for a matrix OJ. The set of all principal matrices for a given matrix OJ forms a cone [l}l. Let A be an integer square matrix for which there exists a p x p complex matrix a such that OJA
=
aOJ.
(3)
The set D of all such matrices A forms a ring. Such matrices A are sometimes called "multipliers". This is connected with the following circumstances. Let P denote the set of all meromorphic functionsJ(z) in CP whose periods are the columns in a matrix OJ. The mapping (z)f -+ J(az), where a is defined by (3), is clearly an endomorphism of the field P. Let Wdenote the set of all matrices of the form
IrkAk k where the rk are rational numbers.
182
THE GEOMETRY OF CLASSICAL DOMAINS
Henceforth we will call the algebra mthe algebra of endomorphisms of a field of abelian functions. It is well known that the algebra has a positivet involution A ---7 A a = RA'R - 1
m
where R is an arbitrary principal matrix. These properties are characteristic, i.e., we have the following theorem. Theorem. 1. Let mbe an algebra of rationalm.atrices of order 2p, and let R be a skew symmetric matrix. If the lnapping (4) is a positive involution of the algebra m, then there exists a matrix w for which R is a principal matrix and mis the algebra of all endomorphisms (A. A. Albert [1]). Let Q(m, R) denote the set of all matrices w for which (1)
wRw'
m
0,
iwRw' > 0,
Em
is the algebra of endomorphisms for w, i.e., for any A (2) there exists a p x p complex matrix a such that wA = aw. It is clear that if WE Q(m, R), then {3w E Q(m, R), where {3 is any nondegenerate complex matrix. We agree to say that the matrices w and {3w are equivalent. We denote the set of classes of equivalent matrices wEQ(m,R) by K(m,R). We have the following theorem. (Section 2 of this chapter.) Theorem 2. K(9.(, R) is the product of classical domains of the first three types (see Chapter 2): The dOl1iain K(m, R) depends only on the real span m of the algebra m. It is clear that the fields of abelian functions corresponding to the matrices wand {3w, where {3 is an arbitrary nondegenerate matrix, are isomorphic. However, this is not the only case in which they are isomorphic. It is well known that two fields of abelian functions with period matrices w 1 and W 2 are isomorphic if and only if there exist matrices U and {3 such that (5) where {3 is a complex nondegenerate matrix and U is a unimodular integer matrix.
t An involution A ---7 Acr of the algebra Sll is said to be positive if S(AAcr) > 0, where S(B) is the trace of the matrix B.
ABELIAN MODULAR FUNCTIONS
183
It follows from (5) that vm 2 V- 1 =
m1 , Vf!lt2 V' =
f!lt 1 ,
(6)
where m1 and m2 are the corresponding algebras of endomorphisms, while f!lt 1 and f!lt 2 are the corresponding sets of principal matrices. It can be shown that the algebras of endomorphisms for all co EQ(m, R), except for the union of a countable number of submanifolds of smaller dimension, coincide with m. If one principal matrix for a matrix co is Ro and the algebra of all endomorphisms is m, then, as we can show with little difficulty, any principal matrix is of the form AR o, where A Em, AD' = A, and all characteristic roots of A are positive. The cone of principal matrices is therefore uniquely defined by one principal matrix Ro and the algebra mof endomorphisms. As a result, for all co E (m, R) with the same position (i.e., for which the algebra of endomorphisms coincides with m), the cone f!lt of principal matrices is the same. Let L(m, R) denote the set of all unimodular integer matrices V such that V2i.v- 1
=
m,
Vf!ltV'
= f!lt.
(7)
With each matrix V E L(m, R) we naturally associate the following transformation in Q EL(m, R): co
---7
coV.
(8)
Some analytic automorphism in K(m, R) clearly corresponds to each transformation of the form (8). We should note that some of the transformations of the form (8) induce identity transformations in K(m, R). We denote the group of transformations obtained in K(m, R) by rCA, R). We have the following assertion. Theorem 3. The group R) is an arithmetic group (Section 2). Note that, as a rule, R) does not coincide with the set of all integer matrices of some linear algebraic group; instead, it is some extension of this set. In what follows we will call the group R) a modular group, and we will call the meromorphic functions that are invariant under R) modular functions. The following theorem is a consequence of Theorem 3.
rem, crm,
rem,
rem,
184
THE GEOMETRY OF CLASSICAL DOMAINS
Theorem. 4. Afield of modular functions is afinite algebraic extension of a field of rational functions of n variables, where n is the complex dimension of the domain K(m, R). So-called modular abelian functions are interesting in certain cases. We will now define them. Consider the space C(m, R) of pairs (co, z), where co EQ(m, R) and z is a p-dimensional complex vector. A function f(co, z) that is meromorphic on C(m, R) is said to be a modular abelian function if it is invariant with respect to the following transformations:
(co, z) -7 (J3co, J3z)
(1)
(9)
where J3 is any nondegenerate matrix of order p, (10)
(2) where
CO k
is any column vector in the matrix co,
(co, z) -7 (coU, z)
(3)
(11)
where UEL(m, R). We have the following proposition. Theorem 5. A field of modular abelian functions is a finite algebraic extension of a field of rational functions of n + p variables, where n is the complex dimension of K(m, R). Theorem 5 can be used with an ordinary method to prove that each modular abelian function can be represented in the form of the ratio of modular abelian forms. These are functions that are holomorphic in C(m, R) and multiplied, under transformations (9), (10), and (11), by some factor. An example of such functions is the theta-function. The general theory of modular abelian forms is very interesting and, as far as we are concerned, may have important applications to the theory of numbers.
Section 2. The Domains K(9J, R) In this section we will present an explicit description of the domains K(m, R) and, in particular, we will prove Theorem 2 of Section 1. Let 1 be an algebra over Q of rational matrices of order 2p, and let Rl be some skew symmetric matrix defining a positive involution in 1 ; assume, moreover, that 2 is some other algebra and that R2 is a skew symmetric matrix defining a positive involution in m2 •
m
m
m
ABELIAN MODULAR FUNCTIONS
185
We agree to say that two pairs (m!, R!) and (m 2 , R 2 ) are equivalent if there exists a unimodular integer matrix U such that (1)
where R k , k = 1,2, denotes the set of all skew symmetric matrices R defining positive involutions in mb k = 1,2. If, however, there exists a rational and not necessarily unimodular matrix U for which (1) is valid, then we agree to say that the pairs (m!, R!) and (m 2 , R 2 ) are isomorphic. Let Si( denote the real span of the algebra m, i.e., the set of all matrices of the form
where the rk are real numbers. R is similarly defined. We agree to say that two pairs (m!, R!) and (m 2 , R 2 ) are equivalent over the reals if there exists a real matrix U such that (2) It is not difficult to prove that:
(1) If two pairs (m!, R 1 ) and (m 2 , R 2 ) are equivalent over the reals, the domains K(m 1 , Rd and K(m 2 , R 2 ) are analytically equivalent. (2) If two pairs (m 1 , R 1 ) and (m 2 , R 2 ) are isogenous, the modular groups r(~!, R 1 ) and r(m 2 , R 2 ) are commensurable. The algebra 21 is always semisimple and, therefore, is the sum of simple algebras m1 , .•. , As a result, there exists a basis in which all of the matrices in mhave the form
mm.
(3)
It is easy to verify that, in this basis, the matrix R is always of the form (3). We will show that
(4)
186
THE GEOMETRY OF CLASSICAL DOMAINS
In order to do so, it is sufficient to show that each class of equivalent matrices OJ contains a matrix of the form
OJ
=(:'
~).
0
o
(5)
OJ m
Let OJ be some matrix in Q(m, R), and consider the algebra of all complex matrices (X such that (XOJ = OJA, where A E m. This algebra is isomorphic to the algebra mand, consequently, is also the sum of simple algebras isomorphic to the algebras m1 , ... , mm. In this proper basis, all matrices (X of this algebra are of quasidiagonal form
(6)
A change of basis is equivalent to substitution of f30J for OJ, where f3 is some p x p complex matrix. We have thus showed that every class of equivalent matrices OJ in Q(m, R) contains a class in which all matrices (X are of quasidiagonal form (6). It immediately follows from the relationship (XOJ = OJA that such an OJ must be of the form (5). We have thus proved (4). It follows from our result that when we are trying to find K(m, R), it is sufficient to restrict the discussion to the case in which mis a simple algebra. We will now show how to reduce all cases to the case in which ~( is a division algebra. As we know, every simple algebra is a matrix algebra over a division algebra. Let mbe a simple algebra consisting of order 2p matrices. By 1 we denote its maximal division subalgebra. Then there exists an integer m such that all of the elements A E mare of the form
m
_(a~l
1m
a
)
A-. a m1
, •••
amm i
(7)
ABELIAN MODULAR FUNCTIONS
Consider the following involution in
A
-+
AO' = (b KS )'
187
m: bKS = a~K'
(8)
mi'
where a -+ aO' is a positive involution in lt is known that any positive involution is of the form
A-+A' = H-iAO'H where HO' = Hand His positive definite (this means that any representation maps H into a matrix with positive characteristic roots). We select a basis in the space of representations so that the matrices A E are of the form
m
(9) where au -+ Au is a fixed (the same for all i, j) representation of the algebra lt is easy to verify that there exists a matrix R Ef!ll that, in this basis, is quasidiagonal:
mi'
R=
(~1 o
~) ..
(10)
Rl
We will now show that each equivalence class of matrices contains a matrix of the form
WE Q(m, R)
(11)
Let W be an arbitrary matrix in Q(m, R). The algebra of all matrices ex such that exw = wA, where A E m, is isomorphic to the algebra and, consequently, is a matrix algebra over some algebra isomorphic to the algebra
a
mi'
188
THE GEOMETRY OF CLASSICAL DOMAINS
In the proper basis, all matrices (X of this algebra are of the form
(X
(
(x~i
•••
(Xim)
=. (XIII 1
, ..•
(12)
(XIII1/!
mi'
where the (Xij generate an algebra isomorphic to the algebra As we have already noted above, a change of basis is equivalent to substitution of f30J for OJ, where f3 is some p x p complex matrix. It remains to note that if (XOJ = OJA, where A is of the form (9) and (X is of the form (12), then OJ must be of the form (11). We have proved that K(m, R) = K(m i , R1)' Consequently, to find K(m, R), we can limit discussion to the case in which is a division algebra. Let be some division algebra over the field of rational numbers Q, and assume that it has a positive involution a -+ aO'. Note that all remaining positive involutions a -+ at are of the form at = baO'b -1, where bO' = band S(b k ) > 0 for all k, where S(b) denotes the trace of the element b. Our problem consists in describing all representations a -+ A(a) of the algebra mfor which there is a skew symmetric matrix R such that
m
m
(13)
where a -+ aO' is the given involution in m. Henceforth we will call the matrix R a principal matrix. It is clear that if there exists a matrix R such that (13) holds for one positive involution, then there is also a matrix R with analogous properties for any other positive involution. Let R denote the set of all skew symmetric matrices R defining positive involutions in the given space of representations of the algebra m. We agree to say that two representations a -+ A1(a) and a -+ A2(a) are isogenous if there exists a rational (not necessarily unimodular) matrix U such that (14)
VR i V' = R 2 , where R1 and R2 are cones of principal matrices.
(15)
189
ABELIAN MODULAR FUNCTIONS
Since, for any two representations of the algebra mby matrices of the same order, there exists a matrix U for which (14) holds, classification of nonisogenous representations of the algebra reduces to the following problem. Given some representation of the algebra m describe all possible cones R for this representation, where the cones Rl and R2 are clearly not assumed to be different if there exists a rational matrix U such that
m
UR 1 U ' = R 2 ,
UA = AU
for all
A Em.
(16)
Note, moreover, that each cone R has a skew symmetric matrix Ro such that for all A E m (17) where a ---7 aD' is some fixed (to the end of the proof of the theorem) positive involution in the algebra m. This matrix is naturally defined nonuniquely. The general form of such matrices is (18) where aD' = a and a belong to the center of the algebra m. In the space in which a representation of the algebra is defined, consider the scalar product (x, y) = x'Ro 1 y. It clearly has the following properties:
m
(x, y)
= -(y, x),
(Ax, y) = (x, AD'y)
for all
(19) A Em.
(20)
The converse is also clear, i.e., it is clear that a scalar product with properties (19) and (20) can easily be used to restore the matrix Ro. As we know, any representation of the algebra mcan be described in the following manner. Consider the vectors
(21)
where XiEm. The set of vectors of the form (21) forms a linear space X of dimension kl' over the field of rational numbers, where l' is the rank of the algebra mover the field Q.
190
THE GEOMETRY OF CLASSICAL DOMAINS
With each element a E
mwe associate the following transformation:
x= (Xl) xa = (X~ a). -+
Xk
(22)
Xk a
As we know, the algebra !l' of transformations that commute with the transformations of mconsists of matrices of the form
(b~lXl+"'+blkXk)
X.l') X
=
.
-+
Bx =
( . XIe
.
(23)
,
.
bk1 Xl + ... + bkk Xk
where B = (bij) is a square matrix with entries from m. We introduce an involution into the algebra !l' by means of the following formula: (24) Our problem consists in describing all scalar products (x,y) with the following properties: (x, y)
- (y, x),
(25)
(xa, y) = (x, yaG).
(26)
Let e i denote a vector whose elements are all equal to zero, except for the element in the ith place, which is equal to e (e is the identity of the algebra 520. We have, by (26), (27)
m;
where x, yE (e i , ejY) is a linear function of the algebra exists an hij E msuch that (eb ej y)
S(h ij y),
where Sea) denotes the trace of the element a in the algebra It follows from (27) and (28) that (e x, e y) = S(hij yx G) = S(xGhij y), i
j
m, so there (28)
m. (29)
ABELIAN MODULAR FUNCTIONS
191
whence it follows that (X,y)
= L(eiXi,ejy) = LS(xf hijY) = S(x 0, we can associate an analytic automorphism w -+ wU of the manifold Q(Q2P,1). This transformation induces some analytic automorphism of the domain K(Q2P, 1), which automorphism we will denote by u(z). It is easy to show that the set r 2p U r 2p is the union of a finite number of sets of the form r 2pUv, v = 1, ... , r. Let I(z) EP. Then any elementary symmetric function s of I(Ul(z)), ... ,/(Ur(z)) is one of Siegel's modular functions. We will show that the function s is representable in the form of a ratio of modular forms whose Fourier coefficients belong to the field k. In order to do this, it is sufficient to show that all of the functions l(uvCz)) are representable in the form of ratios of forms that are modular with respect to certain congruences of a subgroup of the group r 2p and have Fourier coefficients in the field k. As we know, every rational matrix U satisfying the relationship UIU' = AI, A > 0, can be reduced by means of multiplication by matrices in r 2p to the form (2) As a result, we can assume that the matrices uv , v = 1, ... , r are of the
ABELIAN MODULAR FUNCTIONS
197
form (2). We wiil use the well-known interpretation of K(Q2p, J) as the upper halfplane:
Z = X + iY,
Y > O.
(3)
An analytic automorphism of the domain K(Q2p, I) of the form Z --* AZA' + T,
T = T'
B£i2-l,
(4)
corresponds to each U of the form (2), where A and T are rational matrices. It is clear that if ¢(Z) has rational Fourier coefficients, then the Fourier coefficients of ¢(U(Z)), where U is of the form (4), belong to the field k. What we have said implies that an elementary symmetric function s of /(U1(Z)), ... ,/(uz(Z)) is representable in the form of a rational function of /0'/1' ""/N with coefficients in the field K. We will now turn to proving that the field of definition of modular functions is the field k. We will first consider the case in which the center of the algebra 2r is either the field of rational numbers or its imaginary quadratic extension. The natural imbedding of O(2r, J) in O(Q2p, J) induces an analytic imbedding of K(2r, J) in K(Q2p, J). Consider the subgroup r' consisting of the transformations in the group r 2p that map K(2r, I) into itself. As we showed at the beginning of this section, r' is a subgroup of finite index in r(2r, I). As a result, it is sufficient to show that a field of functions automorphic with respect to the group r' is defined over k. The functions /EP separate all points in K(Q, J) that are not equivalent under the group r 2p' This implies that the restriction of Siegel's modular functions to K(2r, J) generates the field of all r'-automorphic functions of K(2r, J). Consider the field p' of all functions representable by the restriction of functions / EP to K(2r, J). It is sufficient for us to show that the additional (in comparison to those already in P) relations between the functionsfEP' are defined over k. Let F denote the subgroup of the group of analytic automorphisms of the domain K(Q2P' J) that consists of the transformations that leave every pointZ E K(2r, J) fixed. The rational automorphisms are dense in F. Indeed, with each A E 2r such that AA = AE, where AE Q, we can associate a rational automorphism of the domain K(Q2P' J) and, as we can see with no difficulty, such A form an everywhere dense subset in F. Let/o'/1' "',/N be a system of functions in P such that all of Siegel's (f
198
THE GEOMETRY OF CLASSICAL DOMAINS
modular functions are rational functions of them. Let U be a rational automorphism of the domain K(Q2p, J) that is contained in F. Then
1,/Z) = 1,/U(Z)), 0
~ 11 ~
N,
ZEK(91,J).
(5)
We will show that relationships (5) correspond to certain algebraic relations between the restrictions of the functions/o, ""/N to K(91,I). As we noted above, the set r 2pur 2p consists of a finite number of cosets r 2p Uv , v = 1, ... , l', and the coefficients of the polynomial r
wet,!, U) =
IT (t-f(Uv(z))),
f(Z)EP,
v= 1
can be rationally expressed in terms of /0, ... , /" with coefficients in k. It follows from (5) that (6) Relations (5) and, therefore, (6) form a basis for additional relations in P'. Ordinary methods can be used to show that all of relations (6) follow from a finite number of such relations. A somewhat more complicated form of this argument can be used to show that a field of functions automorphic with respect to a congruence of a subgroup of the group r' is also defined over k. This implies that a field of functions automorphic with respect to r(91, R), where R is any skew symmetric matrix, is defined over k. The same proof can be generalized in the case of an arbitrary algebra 91. In this argument an imbedding of K(91, R) in K(K, R), where K denotes a maximal absolutely real subfield of the center of the algebra 91, is used instead of the imbedding of K(91, R) in K(Q, R).
CHAPTER 6
Classification of bounded homogeneous domains Section 1. Introduction
As we noted above, and as we will show in the appendix, there is a one-to-one correspondence between normal j-algebras and bounded homogeneous domains. Thus, classification of bounded homogeneous domains reduces to classification of all normal j-algebras. In this chapter we will construct algebraic apparatus convenient for construction of examples and classification of normalj-algebras. We will use this same algebraic apparatus to describe homogeneous imbeddings of bounded homogeneous domains ~ in the Siegel disk Kw In particular, we will show that there is always a finite number of different homogeneous imbeddings of the Siegel disk Kn in the Siegel disk Km. We will also note that there is a continuum of homogeneous imbeddings of the n-dimensional ball Iz 112 + ... + IZnl2 < 1 in the Siegel disk K,'Z' and, because of this, there is a continuum of different homogeneous bounded domains. Let us now turn to a section-to-section survey of the contents of this chapter. Section 2 contains certain auxiliary propositions from linear algebra. In Section 3 we will introduce the notion of a complex, show that a complex corresponds to each normal j-algebra, and that each such complex is uniquely determined. Section 4 presents the construction ofthej-algebra corresponding to a given complex and contains a discussion of several examples. Section 5 presents a description of all homogeneous imbeddings of a given bounded homogeneous domain in the Siegel disk Kw Section 6 considers the problem of the characteristics that distinguish j-algebras that are transitive in a given domain~. We will show that the number of essentially differentj-algebras of this type is finite. 199
200
THE GEOMETRY OF CLASSICAL DOMAINS
Section 2. Isometric Mappings
Let X and Y be Euclidean spaces. A nonnegative scalar product defined on their tensor product X x Y is called an isometric scalar product or isometry if (x x y, x x y) = (x, x)(y, y).
(1)
We should note that isometric scalar products may be degenerate. The simplest example of an isometry is the following scalar product: (2) where X k and Ys form orthonormal bases in X and Yand (jkm is the Kronecker delta. Another example of an isometric scalar product can be constructed in the following manner. Let X be an (not necessarily associative) algebra over the field of real numbers in which there exists a scalar product with the following property: (3) All such algebras are well known, i.e., such an algebra is either the algebra of complex numbers, or the algebra of quaternions, or the algebra of Cayley numbers. Then the following scalar product can be defined on the tensor product Xx X: (4) We should note that the isometric scalar product obtained in this manner is always degenerate. Note that with each isometric scalar product defined on the tensor product of two spaces X and Y we can associate a bilinear mapping of X and Y into some Euclidean space Z. This mapping is defined in the following manner. Let Zo denote the subspace of the space X x Y consisting of all Z E X X Y such that (x x y, z) = 0
for any
x, y.
(5)
CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS
201
Let Z denote the factor space X x YjZo. The scalar product inherited by Z from Xx Y is no longer degenerate. Note that the bilinear mapping (x, y) --* (x y) E Z, which bilinear mapping is the composition of the tensor product X x Y and a homomorphism of the space X x Y onto Z, has the following property: 0
(x 0 y, x 0 y) = (x, x)(y, y).
(6)
We will call such mappings isometric mappings. It is also clear that an isometric scalar product in the tensor product of X and Y corresponds to each of their isometric mappings into some Euclidean space Z. So-called continuable transformations will playa very important role in what follows. Let X, Y, and Z be Euclidean spaces, and let (x,y) --* (x o y) EZ be an isometric mapping. Definition 1. A linear transformation x --* ax of the space X is said to be continuable if there exists a transformation [3 of the space Z such that a(x) 0 y = [3(x 0 y), a*(x) 0 y
(7)
[3*(x 0 y).
Here a* and [3* denote the adjoints of a and [3. We can obtain a complete description ofa continuable transformation in the following manner. Let ny,y" y, y' E Y denote the linear transformation of the space X that is induced by an isometric mapping, i.e., such that (ny,y'(x), x') = (x 0 y, x' 0 y')
for all x, x' E X. We have the following lemma. Lemma 1. A transformation x --* ax is continuable
(8)
if and only if (9)
for all y, y' E Y. Proof Let a be a continuable transformation; then
(ny,y,(a(x)), x') = (a(x) 0 y, x' 0 y') = ([3(x 0 y), x' 0 y')
= (x y, [3*(x' y')) = (x y, a*(x') = (nV,y'(x), a*(x')) = (any,y'(x), x' 0
0
0
0
y')
(10)
202
THE GEOMETRY OF CLASSICAL DOMAINS
It remains to show that if (9) holds, then the transformation a is continuable. Let fi denote the transformation defined on the space Xx Y in the following manner: fi(x x y) = a(x) x y. (11) We will now show that
jj maps
the space Zo into itself. Note that
if and only if (12)
for any y E Y. Indeed, if
then (z, x x y) = 0 for any x E X, Y E Y. As a result,
0= (z,xoy) = I(XkoYk,Xoy) k
whence follows (12). _ It is clear from (12) that if a satisfies (9), then f3 maps Zo into itself. and, consequently, induces some linear transformation f3 onto Z = Xx YjZo. Lemma 1 is proved. We also have the following elementary proposition. Lemma 2. The bilinear function ny,y' defines an isometric scalar product if and only if (13) ny,y = (y, y)E, for any y, y' E Yand (14)
where Yb ... 'YIII are arbitrary vectors in Y and Xl, ... 'X IIl are arbitrary vectors in X. We leave the almost obvious proof of this proposition to the reader. Let Xl' X 2 , ••• , Xp be Euclidean spaces. We assume that an isometric
CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS
scalar product is defined on X k x X k + 1 (1 these scalar products are compatible if k-l k+l n a , a' n b, b'
=
~
203
k ~ p) and agree to say that
k+l k-l n b, b' n a , a'
(15)
where n~,~,l and n~,t,l are induced linear transformations in X k, i.e., such that for all x, x' E X k (n~~:(x), x')
= (a x x, a ' x x'),
(n~,t,l(x), x')
a, a ' E X k- 1,
(16)
(x x b, x' x b'), b, b ' E X k+ l'
We will show that if the scalar products defined on X k x X k + 1 are compatible, then we can uniquely define scalar products on all tensor products of the form X k XX k+ 1 X
... X k+ s'
1 ~ k < k+s ~ p,
(17)
in such a manner that (Xk
X ... x k+ s, x~ X •.. x~+s)
= (n~k.X'JXk+ 1) x ... x k+ s, x~+ 1 X ... x~+s),
(18)
(19) where nk and nk + s respectively denote the linear transformations induced on X k + 1 and X" +s -1 by the isometric scalar product in X k x X k + 1 and X k +s 1 X X k +S' First of all, we should note that relationships (18) uniquely define the scalar products. We need only verify that these scalar products are nonnegative and that relationships (19) follow from (18). We will carry out the proof by induction on p. For p = 2 our assertions are direct consequences of the definitions. Assume that p > 2 and that our assertions have been proved for p 1. It is clear that it is sufficient for us to prove that the scalar product defined by (18) on Xl x X 2 X .•• X Xp is nonnegative, and that (19) is valid for this scalar product. Successively applying (19), we can easily obtain the equation (20)
where band b ' are vectors in X 3 •
204
THE GEOMETRY OF CLASSICAL DOMAINS
Let z = Ix~ k
x... x;x ... x;; then
Here we have used (20) and (15). (n~, I) composed of the matrices n~, 1 is nonnegative definite. The matrix pi = (nt, I) has the analogous property. It follows from (15) that
It follows from (20) that the matrix p3 =
(21) Using (21), we can easily show that the matrix P = (nt, 1 n~,I) is nonnegative definite. We have thus proved that the scalar product defined on Xl x ... x Xp is positive definite. Successively applying (18) to both parts of (19), we can easily obtain a relationship that follows from (15). We leave the details to the reader. Note that in certain cases there is a complex structure in one of the spaces X or Y. It clearly carries over to their tensor product, and the definition of the isometric scalar product requires introduction of the additional requirement of invariance with respect to this complex structure. Section 3. Complexes
By a complex of rank p we mean a set consisting of: (1) Euclidean spaces A km , 1 ~ k < m ~ p, (2) Hermitian spaces Ck , 1 ~ k ~ p, and (3) isometric scalar products defined on the tensor products of the spaces (1) where the scalar products defined on Atk X A km
and
A km X Ams
(2)
are compatible. We should note that the dimension of each of the spaces A km and C k may be equal to zero. In this section we will describe the construction of the complex K( G) corresponding to a given normalj-algebra G, and we will show that the algebra G is uniquely determined by its complex.
205
CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS
We should also note that the homogeneous domain corresponding to the complex K is a Siegel domain of genus 1 if and only if dim em = 0, 1 ~ 111 ~p. Let us briefly recall certain results obtained in Section 3 of Chapter 2. Let G be a normalj-algebra, K its commutator, and H the orthogonal complement of K. The algebra H is commutative and its representation onto K is completely reducible, so K can be represented in the form of the sum of root spaces Ka. Each of the spaces Ka consists of all x E K such that [h,x]
CI.(h)x
foraIl
hER.
(3)
The linear forms CI.(H) are called roots. Let CI.!, ... , Cl. p denote all roots such thatjKalll c H. When the are appropriately labeled, every root has the form t(Cl.k±Cl.IIl) ,
1 ~ k < 111 ~ p,
Cl. 1 , ... , Cl. p
tCl.m , Cl. lm 1 ~ m ~ p.
(4)
Notation:
Moreover, we set (6) (7) In addition, let
xgn
denote the orthogonal complement of I,X ktm in t
Xkl1Z' and, similarly, let ZI~ denote the orthogonal complement of I,Zms in Z11l' Moreover,
ZO = IZ~. k
XO = I x2s, k<s
(8)
In addition, We set Xl = [XO, x°], x 2 = [XO, Xl], etc. Similarly, let Z! = [XO,ZO], Z2 = [XO,Zl], etc. We will now show that the union X of the spaces xk" k = 0, 1, ... , coincides with I, X klll , and that the unionZ of the spaces zk, k 0,1, ... , coincides with I,Zk' It is not difficult to k
use induction to show that X is a Lie algebra. Note that X kk + 1 = x2k+ 1 and, therefore, belongs to X. We will now show that X ks C X by induction on s-/c Thus, assume that we have proved that X ks C X under the assumption that s- k < d; we now show that this is true when s-k = d. We have X ks = x2s+ IX kts . t
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THE GEOMETRY OF CLASSICAL DOMAINS
By definition, X~s c X and, moreover, X kts = [XkP Xts] c X, because X is a Lie algebra and, by the induction hypothesis, so is X kt and Xts C S. We will now show that Z = I,Zk' First we will show that [X,Z] c Z. By the definition of Z, it follows that [Xo,Z] c Z. We use induction on n to show that [X", Z] c Z. Assume that we have proved that [X"-l,Z] c Zfor n < no. We now show that [x,z] c Zfor XEX", ZEZ. It is clear that it is sufficient to verify this for [XO,X Il - i ], wherexoEXo, x"- 1 E X II - 1 ; we have
[[XO, X
ll
-
i
],
z] = [XO, [x -l, z]] + [x -l, [XO, z]]. ll
ll
By the induction hypothesis, [xIJ - l ,Z]EZand, therefore, [XO,[xIJ - l , Zl]EZ; similarly, [xo, z] EZ and, therefore, [xll -l, [xo, zl] EZ. We will now show that Zk c Z for all k. We prove this by induction on k, beginning with k = n. It is clear that ZII c Z, because ZII = Z~. Assume that we have already proved that Zk C Z for all k > s. We will show that Zs c Z. We have (9)
By definition Z~ c Z. Moreover, Zst = [Xst' Zt] c Z, because Zt c Z, by the induction hypothesis. In addition [Xk1m rill] = Yk1ll • The following lemma summarizes what we have proved. Lemma 1. A normal i-algebra G is generated by its subspaces iRk, R k, X~m' andZ~. As we noted in Section 2 of Chapter 2, the form w on G is not uniquely defined. In what follows, it will be convenient for us to fix it so that w(rk) = .h k = 1, ... ,p, where rk denotes an element of Rk such that [irk' tk ] = rk· We have the following relationships (see Section 3 of Chapter 2): [x,jz]
j[x,z]
([x,z], [x,z]) = (x,x)(z,z)
where
X
EX k1ll ,
ZEZm +
I
(10) (11)
(Xms+ Y IlIS )'
m<s~p
Consider the set of Euclidean spaces X~1II and Hermitian spaces Z~.
It follows from (11) that the commutation operator in G induces isometric scalar products in the spaces XE1II x ZI~ and X~m x X~s' We will
207
CLASSIFICATION OF BOUNDED HOMOGENEOUS DOMAINS
show that these spaces form a complex, and that the isometric scalar product of this complex can be used for unique restoration of any algebra G. Notation: p
Um
Zm+
I (X mt + Y mt ), t=m+l
Gm = jRm+ Um+R,w
(12)
As we know, Gm is an elementary i-algebra and
[Gm,U k] c Uk
for
k < m.
(13)
Consider the i-algebra G = Gt+Gk+G,w We set
L = Rt+Rk+ Ytk , U = U;+ Uk'
(14)
where U; is the orthogonal complement of X tk + Ytk in Ut. Then
Gt+Gk = L+jL+U.
(15)
As we showed in Section 6 of Chapter 2, the representation ofiL onto U is complex linear and
[jL, Gill] = 0,
[Gill' U]
C
U.
(16)
The operators adg on U are symplectic operators, so their commutability with if implies that they commute with (jf)*. Let x, x' E X tk ; it is then clear that the operators nx,;x' = (adx')*ad x carries Uk into itself and commutes on Uk with any operator adg, g EGm • The operator n x , Xl is clearly uniquely defined by the relation
(nx,x{u), U') = ([x, u], [x', U/]),
x, x' EX tk , U, U' E Uk.
(17)
The following lemma states the proposition we have obtained. Lemma 2. The operators adg, g E Gm and nx,x' commute. It immediately follows from this lemma that the spaces X km , Yklll , and Zk are invariant under nx,x'. It also follows from Lemma 2 that
nx,x'([x kr , X rm ]) = [nx,x,(Xkr), x rm ]'
x, X E X tk ,
(18)
where Xkr E X kn Xrlll E X nll , and, therefore, the spaces X krm are invariant under the operators nx,x' . The space X~/1I is the orthogonal complement of L X krm and, therefore, is also invariant under the operators nx Xl. I"
'
We will now show that the set of spaces X~m and Z~ forms a complex.
208
THE GEOMETRY OF CLASSICAL DOMAINS
In order to do this, it is sufficient to show that the isometric scalar products in the spaces X?k x x2m
and
x2m x X~IS'
(19)
X?k
and
x2m x Z,~
(20)
X
x2m
are compatible. Expression (19) immediately follows from Lemma 1 of Section 2 and (18). Expression (20) can be proved analogously. We will now show that an algebra G is uniquely determined by its complex. We set where kl < k2 < ... < k r • It is clear from (18) that the spaces X~l ... kr are invariant under the operators nx,x' and are mutually orthogonal. Using (18), we can easily use induction to show that the scalar products in X21 ... k r are uniquely determined. This also shows that the commutation operation in X is uniquely defined. Similarly, it can be shown that the commutation operation is uniquely defined if x E X km , ZEZm. Moreover, using the fact that the representation of Gm onto Uk is symplectic, we can easily show that the commutation operation in G is uniquely defined by the commutators [x km , xms] and [Xkm, ZIIJ, where XkmE X km , zmEZIII. We leave the details to the reader. Section 4. Construction ofj-algebras
In this section we will present a method for constructingj-algebras by means of a complex, and we will also consider several examples. First of all, note that a complex of rank 1 consists of one Hermitian space C 1 . An elementary j-algebra G = jR+ C1 +R corresponds to such a complex. We will now introduce the notion of an ideal of a complex. As we will show below, the ideals of the complex of the algebra G are complexes of the j-ideals of this algebra. Henceforth, we agree to denote the complex span of a space A by [A]; note that if A is a Euclidean space, then [A] is a Hermitian space. An ideal of a complex of rank p is any complex of rank 'p, 1 ~ P < p, that is obtained from the initial complex in the following manner: Akm=Akm , l~k<m~p, Ck=C k+ I [AkIllJ, (1) p