The symplectic geometry of a new kind of Siegel upper half space of order 2 (I)

0.3 MB·

by  Tianqin Wang

THE SYMPLECTIC GEOMETRY OF A NEW KIND
OF SIEGEL UPPER HALF SPACE OF ORDER 2 (I)
6
1
Tianqin Wang, Tianze Wang, and Hongwen Lu
0
2

n Abstract. In this paper, we introduce a new kind of Siegel upper half space and consider
a the symplectic geometry on it explicitly under the action of the group of all holomorphic
J
transformations of it. The results and methods will form a basis for our number theoretic

7 applications later.
1

]
G
Keywords: Siegel upper half space; symplectic geometry; group action.
S
.
h
Mathematics Subject Classification 2010: 11F55, 11F46, 11F50.
t
a
m
[ 1. Introduction and notations

1 Let R denote the field of real numbers, and C the field of complex numbers by conven-
v tion. For any positive integers m and n, let C(m,n) denote the set of all m n matrices Z
68 with entries in C. For any Z C(m,n) we use tZ to denote its transpose, w×hich is a n m
2 matrix in C(n,m). For any po∈sitive integer n, the well known so-called Siegel upper×half
4 space H of order n is, by definition,
0 n
.
1
0 Hn := Z C(n,n) tZ = Z, ImZ > 0 ,
∈ |
6
n o
1
: where ImZ (resp. ReZ) denotes the imaginary (resp. real) part of Z, i.e., if Z = (zij)
v
then ImZ = (Imz ), ReZ = (Rez ), and ImZ > 0 means that ImZ is positive definite.
i ij ij
X Here, and throughoutthis paper, the symbol ”:= ” is used to indicate that the right hand
r side of an equality is the definition of the left. Clearly, if n = 1, then H is reduced to the
a n
classical Poinca´re[P] upper half plane H = H .
1
The symplectic geometry on H by the action of the group of linear fractional trans-
formations, or, by the order 2 symplectic group Sp(2,R) = SL (R), possesses an almost
2
full satisfactory understanding nowadays. And this understanding has led to many great
applications both in mathematics itself and the spread area of other branches of subject.
One of the most fascinating examples is the basic role it plays to the theory of classical
automorphic forms and the applications in the field of number theory.
For n > 1, the symplectic geometry on H , acted by the order 2n symplectic group
n
Sp(2n,R), were studied systematically by Siegel[S2] in 1936 for the first time, and lots of
very important essential results were established. Besides many other important applica-
tions it leads to, it then becomes the concrete basis for the theory of Siegel modular forms
and Jacobi forms, see, e.g. [AZ], [DI], [EZ], [M], [S3], [Sk3], [Sk4], [SZ1] and [SZ2]. And
Project Supported by the NationalNatural Science Foundation of China, GrantNumber 11471112.
1
these subjects nowadays are becoming more and more important, active and fertile fields
of mathematics.
In this series of papers, we will introduce a new kind of Siegel upper half space Hˆ
2
of order 2, see (3) below; and give a relatively systematic argument for the symplectic
geometry on it, in the light of Siegel’s classical work[S2]. This first paper will focus
only on the relatively pure geometric part of our work. Although one might notice that
our new object is not irreducible by Cartan[C] and the geometry might be viewed as a
topological productoftwoPoinca´reupperhalfplanesHinessenceasweproceed,itshould
bepointedoutinadvancethattheredoexistmanyimportantinterestingandusefulresults
valuable for researching at least from the explicit point of view. Furthermore, besides the
independent importance just from the geometric point of view, one will find that the
geometry in this paper will naturally become the essential basis for our number theoretic
applications that follow in this series later. In other words, the principal novelties of this
paper can be realized mainly from the following two aspects of view: one is a systematic
and explicit formulation of the principal geometry of the new kind of Siegel upper half
space Hˆ in (3) below, and the other is the presentation of the ideas and explicit methods
2
of transforming the non-irreducible object Hˆ to the irreducible one H; and the latter will
2
have further important applications in the forthcoming number theoretic researches. By
theway, wewanttopointoutfurtherthatwhentheactionofadiscretegroupisconsidered
there would arise some new difficulties to be overcome.
Throughoutthis paper, we will use the following specific notations. The letter ε is used
to stand for 1 or 1, that is ε = 1. The letters p and q are always used to denote the
− ±
following matrices of order 2,
1 1 1 0 1
p := − , q := .
√2 1 1 1 0
(cid:18) (cid:19) (cid:18) (cid:19)
The capital letters P and Q are always used to denote the block matrices
p 0 q 0
P := , Q := .
0 p 0 q
(cid:18) (cid:19) (cid:18) (cid:19)
Then simple computations show that
1 1 1 p 1 0
p 1 = , P 1 = − ,
− √2 1 1 − 0 p−1
(cid:18)− (cid:19) (cid:18) (cid:19)
and
q2 = I, Q2 = I.
Here and throughout this paper, I is used to denote the identity matrix of proper order
n 1, which is not necessarily the same at different occurrences. We always use the
≥
capital letter J to denote the block matrix
0 I
J := .
I 0
(cid:18)− (cid:19)
In this paper, J is also assumed to be of order 4, or equivalently, the above blocks I and
0 are assumed to be of order 2. Then the well known symplectic group Sp(4,R) of order
4, which will be denoted by Ω throughout this paper, is as follows
2
A B
Ω := Sp(4,R) = M = A,B,C,D R(2,2), tMJM = J .
2 C D | ∈
(cid:26) (cid:18) (cid:19) (cid:27)
2
Notice that we clearly have
P, Q Ω .
2
∈
Recall that the Siegel upper half space H of order 2 is defined as
2
τ z
H = Z C(2,2) tZ = Z, ImZ > 0 = Z = 1 τ ,τ ,z C,ImZ > 0 .
2 ∈ | z τ2 | 1 2 ∈
(cid:26) (cid:18) (cid:19) (cid:27)
n o
And the action of Ω on H is defined by
2 2
f : Ω H H
2 2 2
× → (1)
(M, Z) W = f(M, Z)= M < Z > .
7→
Here, and throughout this paper, we will always use the definition
M < Z >:= (AZ +B)(CZ +D) 1, (2)
−
A B
for any Z H and M = Ω .
∈ 2 C D ∈ 2
(cid:18) (cid:19)
The materials of this paper are arranged as follows. In 2 we will first give the exact
§
definition of our object we will work with throughout our series, i.e., the definition of
the new kind of Siegel upper half space Hˆ of order 2. Then we will give an initial
2
formulation of the action group. 3 is arranged to give an alternative formulation of
§
Hˆ and the corresponding action group. 4 is devoted to the investigation of the bi-
2
§
holomorphic mappings of Hˆ . The result together with the arguments in 2 will give a
2
§
complete formulation of the action group. This then becomes the basis for our further
arguments. In 5weconsiderthereducedformofapairofpointsinHˆ ,whichwillbeused
2
§
to simplify largely the formulation of our results in the following sections. In 6 the most
§
important symplectic metric is built. This is a cornerstone for the materials that follow.
7 is devoted to consider the geodesic line and the distance connecting two points. This
§
of course presents one of the most important intrinsic feature of the so called geometry.
In the last 8 the corresponding symplectic volume element is considered explicitly.
§
2. The new kind of Siegel upper half space and the action group
BasedonthewellknownSiegelupperhalfspaceH oforder2,wenowgivethedefinition
2
of the most important object Hˆ in this paper:
2
Hˆ := Z H Q <Z >= Z . (3)
2 2
{ ∈ | }
From now on, this Hˆ will always be called the new kind Siegel upper half space of order
2
z z
2, as expressed in the title of this paper. Note that, for any Z = 1 2 C(2,2),
z3 z4 ∈
Q < Z >= Z if and only if qZ = Zq, i.e., z = z , z = z . Thus by t(cid:18)he defini(cid:19)tion of H ,
1 4 2 3 2
τ z
Hˆ = Z = τ,z C, Imτ > Imz . (4)
2 z τ | ∈ | |
(cid:26) (cid:18) (cid:19) (cid:27)
Remark 1. From (4) one can see easily that the freedom of the elements Z in Hˆ over C
2
is 2. Further, in view of the form of the matrices Z, we call them bi-symmetric. Recall
3
that the well known Siegel upper half spaces H and H have freedoms 3 and 1 over C
2 1
respectively. Thusfromthisfreedompointofview, ournewkindofSiegelupperhalfspace
Hˆ can be viewed as an intermediate case between the cases of H and H . Therefore, by
2 2 1
comparing the classical outstanding work of Siegel [S2] in 1936, it might be interesting to
establish a basis for the symplectic geometry of the new object Hˆ .
2
The first main result in this paper is an explicit formulation of the maximal subgroup
of Ω , which can act on Hˆ by group action.
2 2
Theorem 1. Let Ωˆ be defined as
2
Ωˆ := M Ω M < Z > Hˆ for all Z Hˆ .
2 2 2 2
{ ∈ | ∈ ∈ }
Then we have
Ωˆ = M Ω MQ = εQM .
2 2
{ ∈ | }
And so Ωˆ is the maximal subgroup of Ω , which can act on Hˆ under the action given by
2 2 2
(1).
To prove Theorem 1, we first give a preliminary lemma.
A B
Lemma 1. Let M = Ω . Then
C D ∈ 2
(cid:18) (cid:19)
M < Z >=Z for all Z Hˆ if and only if M = εI or M = εQ.
2
∈
Proof. The sufficiency is obvious by definition. So we only need to prove the necessity,
i.e., we need to prove that M = εI or = εQ if M < Z >= Z for all Z Hˆ . From
2
∈
Z = M < Z >= (AZ +B)(CZ +D) 1 we get
−
AZ +B = Z(CZ +D)= ZCZ +ZD. (5)
Taking Z = τI with τ C and Imτ > 0, which is clearly in Hˆ , then the above equality
2
∈
becomes τ2C +τ(D A) B = 0, and this leads to
− −
B = C = 0, D = A (6)
by considering the limits as τ 0. Substituting these into (5), we see that the matrix A
→
must satisfy
AZ = ZA (7)
a a z τ
for any Z Hˆ . Now, if one puts A = 11 12 , Z = , then by (7) and
∈ 2 a21 a22 τ z
(cid:18) (cid:19) (cid:18) (cid:19)
direct computation we have a τ +a z = a τ +a z and a z +a τ = a τ +a z,
11 12 11 21 11 12 12 22
which clearly imply that a = a and a = a respectively. This then enables us to
12 21 11 22
assume that A is of the form
a b
A =
b a
(cid:18) (cid:19)
with a, b R. Again, in view of M Ω , there holds A tD B tC = I. So we also have
2
∈ ∈ −
A tA = I. By this and direct computation, we easily obtain a2+b2 = 1, ab = 0. So now
there exist exactly two possibilities: one is b = 0, a = 1 and we derive A = εI, and the
±
4
other is a = 0, b = 1 and we derive A = εq. This together with (6) implies that M = εI
±
or = εQ as what we need. The proof of lemma 1 is thus complete.
Now, we turn to the proof of Theorem 1. By the definition of Ωˆ in Theorem 1, for any
2
M Ω ,itisinΩˆ ifandonlyifM < Z > Hˆ for all Z Hˆ .ButbythedefinitionofHˆ ,
2 2 2 2 2
∈ ∈ ∈
M < Z > Hˆ ifandonlyifqM < Z >= M < Z > q,i.e., qM < Z > q 1 = M < Z >,or
2 −
∈
Q < M < Z >>= M < Z > by definition 1. Using the simple property of group action,
one can see easily that this last equality is also equivalent to M 1QM < Z >= Z.
−
Thus by Lemma 1 we can derive M 1QM = εI or = εQ. However, the first case of
−
(cid:0) (cid:1)
M 1QM = εI is impossible since this would imply Q = 1 which is clearly impossible.
−
In other words, the set of M Ω satisfying M 1QM = εI is void. So there is no
2 −
contribution to Ωˆ from this ki∈nd of case. And thus we can only have the latter case
2
of M 1QM = εQ, i.e., QM = εMQ, or in other words, the contribution to Ωˆ of the
− 2
M Ω comes exactly from the latter case of QM = εMQ. So this proves that M Ωˆ
2 2
∈ ∈
iff QM = εMQ, as desired by the first part of Theorem 1. As for the other parts of the
theorem, the maximal property of Ωˆ is obvious from its definition, and the remaining
2
things can be derived easily from the relative definitions by using the conclusion of the
first part. The proof of Theorem 1 is thus complete.
3. An alternative formulation of Hˆ and the action group
2
In this section, we first come to give an alternative formulation of the new Siegel upper
halfspaceHˆ ,whichisisomorphictoHˆ undersome”conformaltransformation”, andcon-
2 2
sider the corresponding action group, see Eˆ and Ωˆ below in (10) and (12) respectively.
2 Eˆ2
Then present another main result in this paper: The action of Ω on Hˆ is transitive. To
2 2
this end, we will first give some further conventions for notational convenience. For any
m m matrix A and any m n matrix X, we useX¯ to denote the conjugate of X, i.e., the
× ×
matrix with all of its elements being the complex conjugates of that of X, and we denote
A X := tXAX¯, A[X]:= tXAX.
{ }
In this way we can write
τ z
Hˆ = Z = τ,z C, Imτ > Imz
2 z τ | ∈ | |
(cid:26) (cid:18) (cid:19) (cid:27)
1
= Z C(2,2) qZ = Zq, ImZ = (Z Z¯)> 0 .
∈ | 2i −
(cid:26) (cid:27)
0 I
And for any Z C(2,2), in view of J = , we have by simple computation,
∈ I 0
(cid:18)− (cid:19)
Z Z Z
J := J = ( tZ I)J = tZ Z
I I I −
(cid:20) (cid:21) (cid:20)(cid:18) (cid:19)(cid:21) (cid:18) (cid:19)
and
Z Z Z¯
J := J = ( tZ I)J = tZ Z¯.
I I I −
(cid:26) (cid:27) (cid:26)(cid:18) (cid:19)(cid:27) (cid:18) (cid:19)
These show that
Z
tZ = Z if and only if J = 0,
I
(cid:20) (cid:21)
5
Z
and when J =0, we have
I
(cid:20) (cid:21)
1 Z
ImZ > 0 if and only if J > 0.
2i I
(cid:26) (cid:27)
Thus we can write
Z 1 Z
H = Z C(2,2) J = 0, J > 0 ,
2 ∈ | I 2i I
(cid:26) (cid:20) (cid:21) (cid:26) (cid:27) (cid:27)
and
Z 1 Z
Hˆ = Z C(2,2) J = 0, J > 0, qZ = Zq
2 ∈ | I 2i I
(cid:26) (cid:20) (cid:21) (cid:26) (cid:27) (cid:27)
1 Z
= Z C(2,2) qZ = Zq, J > 0 .
∈ | 2i I
(cid:26) (cid:26) (cid:27) (cid:27)
Z
Here the last equality comes from the fact that qZ = Zq implies J = 0. As for the
I
(cid:20) (cid:21)
action group Ωˆ (sometimes also called motion group) of Hˆ , we have
2 2
Ωˆ = M Ω QM = MQ = εMQ
2 2
{ ∈ | ± }
= M R(4,4) J[M] = J, J M = J, QM = εMQ
∈ | { }
n o
= M R(4,4) J[M] = J, QM = εMQ . (8)
∈ |
n o
A B
Further, for M = Ωˆ , Z Hˆ , by putting U := AZ+B, V := CZ+D, then
C D ∈ 2 ∈ 2
(cid:18) (cid:19)
Z A B Z U
M = = . So
I C D I V
(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
U A B Z Z Z
J = J = (Z I) tMJM = J = 0,
V C D I I I
(cid:20) (cid:21) (cid:20)(cid:18) (cid:19)(cid:18) (cid:19)(cid:21) (cid:18) (cid:19) (cid:20) (cid:21)
and
1 U 1 A B Z 1 Z¯ 1 Z
J = J = (Z I) tMJM = J > 0.
2i V 2i C D I 2i I 2i I
(cid:26) (cid:27) (cid:26)(cid:18) (cid:19)(cid:18) (cid:19)(cid:27) (cid:18) (cid:19) (cid:26) (cid:27)
Thus
1
tUV tVU = 0, (tUV¯ tVU¯)> 0.
− 2i −
Now we come to establish a transformation which is similar to the so-called ”conformal
transformation”oftheplaneofcomplexnumbers. WeclaimatfirstthatZ+iI isinvertible
for any Z Hˆ . In fact, for any v = v(2,1) C(2,1) such that (Z + iI)v = 0, one has
2
iv = Zv,∈iv¯= Z¯v¯, i tv = tv tZ = tvZ. S∈o
− −
1 1 1 1
(Z Z¯) v = tv(Z Z¯)v¯= ( tvZv¯ tvZ¯v¯)= ( i tvv¯ i tvv¯) = tvv¯ 0.
2i − { } 2i − 2i − 2i − − − ≤
6
On the other hand, since ImZ is positive definite, we have
1
(Z Z¯) v = tv(ImZ)v¯ 0.
2i − { } ≥
The combination of the above yields 1(Z Z¯) v = 0, so gives rise to v = 0. This then
2i − { }
proves the invertibility of the matrix Z+iI as being stated above. Now we can define the
above mentioned ”conformal transformation” of Hˆ into C(2,2) as follows
2
ψ : Hˆ C(2,2)
2
−→
Z Z := ψ(Z) := (Z iI)(Z +iI) 1. (9)
0 −
7−→ −
From this definition it is not hard to see that Z is bisymmetric and I Z Z¯ is positive
0 0 0
−
definite and Hermitian. The first thing is because qZ = Z q which follows easily from
0 0
qZ = Zq. The second is a consequence of the relevant arguments of Siegel []. Thus we are
naturally led to define a domain Eˆ in C(2,2) as follows
2
Eˆ := Z C(2,2) qZ = Z q, I Z Z¯ > 0 . (10)
2 0 0 0 0 0
∈ | −
n o
And (9) maps Hˆ into Eˆ . Conversely, if Z Eˆ , then Siegel [] has proved that I Z is
2 2 0 2 0
∈ −
invertible. Thus we can also define the following map
φ : Eˆ C(2,2)
2
−→
Z Z := φ(Z ) := i(I +Z )(I Z ) 1. (11)
0 0 0 0 −
7−→ −
By this definition it is also easy to prove that Z Hˆ for any Z Eˆ , thus (11) maps
2 0 2
∈ ∈
Eˆ into Hˆ . Further, direct computations show that the composition of (9) with (11) is
2 2
the identity mapping of Hˆ and the composition of (11) with (9) is the identity mapping
2
of Eˆ . Hence the mappings (9) and (11) are all invertible and they are inverse mappings
2
of each other, and whence both of them are one-to-one correspondence. Therefore the
domain Eˆ defined by (10) can be served as another formulation of our Siegel upper half
2
space Hˆ . In particular, one has
2
φ(0) = ψ 1(0) = iI, ψ(iI) = φ 1(iI) = 0.
− −
Next, we come to consider the action group on Eˆ corresponding to Ωˆ . First of all, we
2 2
define
iI iI I 0
L := , R := − .
I I 0 I
(cid:18)− (cid:19) (cid:18) (cid:19)
Note that there holds
1
J[L]=t LJL =2iJ, J[L 1]= J, QL = LQ,
−
2i
1 1 I 0
J L = tLJL¯ = − = R.
2i { } 2i 0 I
(cid:18) (cid:19)
Then for any M Ωˆ , we define a corresponding matrix M C(4,4) by
2 0
∈ ∈
A B
M := 0 0 := L 1ML,
0 C D −
0 0
(cid:18) (cid:19)
7
where A , B , C , D C(2,2), and we put
0 0 0 0
∈
Ωˆ := L 1Ωˆ L = M = L 1ML M Ωˆ . (12)
Eˆ2 − 2 0 − | ∈ 2
n o
Notice also that (11) is indeed the one to one correspondence
L : Eˆ Hˆ
2 2
−→
Z Z = i(I +Z )(I Z ) 1 = L < Z > . (13)
0 0 0 − 0
7−→ −
This in combination with (1) implies that for any Z Eˆ there holds
0 2
∈
M < Z >= (L 1ML) < Z >= L 1M < Z >= L 1 < W > .
0 0 − 0 − −
This shows that W := M < Z > is an element of Eˆ , by noting that W = M < Z > is
0 0 0 2
in Hˆ since Z is. Thus if we note also that Ωˆ is a group with matrices multiplication,
2 Eˆ2
then it can be checked easily that we have defined a group action of Ωˆ on Eˆ as follows
Eˆ2 2
Ωˆ Eˆ Eˆ
Eˆ2 × 2 −→ 2
(M , Z ) W = M < Z > . (14)
0 0 0 0 0
7−→
To have a better understanding of the group Ωˆ defined by (12), we need to give a more
Eˆ2
explicit expression of it. To this end, we first note that by (8) one can derive easily that
Ωˆ = M C(4,4) J[M] = J, J M = J, QM = εMQ .
2
∈ | { }
n o
Thus for our purpose we only need to transform the constrains on M in this expression
to that on M in (12). This can be done directly from M = LM L 1. In deed, it is not
0 0 −
difficult to find by direct computations that J[M] = J, J M = J and QM = εMQ are
{ }
equivalent to J[M ] = J, R M = R and QM = εM Q respectively. Thus we have
0 0 0 0
{ }
Ωˆ = L 1Ωˆ L = M C(4,4) J[M ]= J, R M = R, QM = εM Q . (15)
Eˆ2 − 2 { 0 ∈ | 0 { 0} 0 0 }
Further, if we let
0 I
F := JR = ,
I 0
(cid:18) (cid:19)
A B
then for any invertible matrix M = 0 0 C(4,4) with A , B , C , D C(2,2)
0 C0 D0 ∈ 0 0 0 0 ∈
(cid:18) (cid:19)
we can verify easily that
M 1FM¯ = F iff FM¯ = M F iff C = B¯ , D = A¯ .
0− 0 0 0 0 0 0 0
And, for this kind of M , we can also verify that R M = R together with J[M ] = J
0 0 0
implies FM¯ = M F. This proves that { }
0 0
A B
Ωˆ M = 0 0 A , B C(2,2), J[M ] = J, QM = εM Q .
Eˆ2 ⊆ 0 B¯0 A¯0 0 0 ∈ 0 0 0
(cid:26) (cid:18) (cid:19)(cid:12) (cid:27)
(cid:12) 8
(cid:12)
(cid:12)
A B
Conversely, for any invertible matrix M = 0 0 C(4,4) with J[M ] = J, one can
0 B¯0 A¯0 ∈ 0
(cid:18) (cid:19)
verify easily from FM¯ = M F that R M = R. This proves that
0 0 0
{ }
A B
M = 0 0 A , B C(2,2), J[M ] = J, QM = εM Q Ωˆ .
(cid:26) 0 (cid:18)B¯0 A¯0(cid:19)(cid:12) 0 0 ∈ 0 0 0 (cid:27)⊆ Eˆ2
(cid:12)
Gathering together the abov(cid:12)e we therefore obtain
(cid:12)
A B
Ωˆ = M = 0 0 A , B C(2,2), J[M ] = J, QM = εM Q
Eˆ2 (cid:26) 0 (cid:18)B¯0 A¯0(cid:19)(cid:12) 0 0 ∈ 0 0 0 (cid:27)
(cid:12)
= M0 = BA¯00 BA¯00 (cid:12)(cid:12)A0, B0 ∈ C(2,2), A0 tA¯0−B0 tB¯0 = I, A0 tB0 = B0 tA0, QM0 = εM0Q .
(cid:26) (cid:18) (cid:19)(cid:12) (cid:27)
(cid:12) (16)
(cid:12)
Next, we come to give a(cid:12)purely algebraic lemma which will be useful for our further
arguments.
k k
Lemma 2. Suppose that K = 1 2 is a positive definite real matrix of order 2,
k k
2 1
(cid:18) (cid:19)
x x
then there exists a invertible real matrix K = 1 2 of order 2 such that
0 x x
3 4
(cid:18) (cid:19)
K K tK = I, qK = εK q
0 0 0 0
with ε = 1.
±
Proof. First of all, from the positive definiteness of K we see that there holds k > k .
1 2
| |
So we separate the proof into two cases according to k = 0 or not. If k = 0, the result
2 2
1 0 0 1
1/2 1/2
is obvious by taking K0 = ±k1− 0 ε or K0 = ±k1− ε 0 . As for the case of
(cid:18) (cid:19) (cid:18) (cid:19)
k = 0, we note at first that the condition qK = εK q is equivalent to K being of the
2 0 0 0
6
x x
form K = 1 2 by direct computation. Thus to prove the lemma, we are led to
0 εx εx
2 1
(cid:18) (cid:19)
consider the solvability of the matrix equation
x x k k x εx 1 0
1 2 1 2 1 2 = ,
εx εx k k x εx 0 1
2 1 2 1 2 1
(cid:18) (cid:19)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
and again by direct computations, this can be shown to be equivalent to the solvability of
the system of the algebraic equations
k x2+2k x x +k x2 = 1,
1 1 2 1 2 1 2
k x2+2k x x +k x2 = 0.
(cid:26) 2 1 1 1 2 2 2
However, this is clearly true since on noting k > k and k = 0 one can easily give the
1 2 2
| | 6
solutions of the system of the equations as follows
x1 = 12 ε1(k1 +k2)−1/2+ε2(k1−k2)−1/2
 x2 = 12 (cid:16)ε1(k1 +k2)−1/2−ε2(k1−k2)−1/2(cid:17)
(cid:16) (cid:17)
where ε1 = 1, ε2 =1. The proof of Lemma 2 is complete.
± ±
Now we can state the main theorem in this section.
9
Theorem 2. The action of Ωˆ on Eˆ is transitive, so is the action of Ωˆ on Hˆ .
Eˆ2 2 2 2
Proof. For the first assertion we only need to prove that for any Z Eˆ it is in the same
0 2
∈
orbitof0 Eˆ . PutK = I Z Z¯ ,whichisclearlyreal,positivedefinite,andbisymmetric.
2 0 0
So by Lem∈ma 2 there exist−s an invertible real matrix A such that A I Z Z¯ tA =
0 0 0 0 0
−
A K tA = I and qA = εA q. Thus if we let B = A Z , then by (16) it is easy
0 0 0 0 0 − 0 0 (cid:0) (cid:1)
A B
to verify that the matrix M = 0 0 is in Ωˆ . Also, by the definition of B we
0 B¯ A¯ Eˆ2 0
0 0
(cid:18) (cid:19)
clearly have M < Z >= (A Z +B )(B¯ Z +A¯ ) 1 = 0. Thatis, for any given Z Eˆ ,
0 0 0 0 0 0 0 0 − 0 2
∈
there does exist M Ωˆ such that M < Z >= 0 as desired by the first assertion of
0 ∈ Eˆ2 0 0
our Theorem. To prove the second assertion, we first take an arbitrary element Z Hˆ
2
∈
and put Z = L 1 < Z >. Then by the first assertion we can take an M Ωˆ such
0 − 0 ∈ Eˆ2
that M < Z >= 0. Now by taking M = LM L 1, which is clearly in Ωˆ , we can obtain
0 0 0 − 2
M < Z >= iI by using the action of L to both sides of M < Z >= 0 . This proves that
0 0
Z is in the orbit of iI Hˆ as desired. And thus the proof of Theorem 2 is complete.
2
∈
We now take a step furthertoconsider the stability groupof a pointZ in Eˆ underthe
0 2
action of Ωˆ , and then that of a point Z in Hˆ under the action of Ωˆ . For any Z Eˆ ,
Eˆ2 2 2 0 ∈ 2
we use S(1) to denote its stability group in Ωˆ , that is, we define
Z0 Eˆ2
S(1) := M Ωˆ M < Z >= Z . (17)
Z0 0 ∈ Eˆ2 0 0 0
n (cid:12) o
In particular, we have (cid:12)
(cid:12)
S(1) := M Ωˆ M < 0>= 0 . (18)
0 0 ∈ Eˆ2 0
n (cid:12) o
By Theorem 2 we know that there exists an M(cid:12)(cid:12) 1 ∈ ΩˆEˆ2 such that M1 < 0 >= Z0, so the
latter set in (18) does has general meaning as the former set in (17). More precisely, we
have the following proposition.
Proposition 1. Let M be an element in Ωˆ such that M < 0>= Z . Then we have
1 Eˆ2 1 0
S(1) = M S(1)M 1 = M M M 1 M S(1) . (19)
Z0 1 0 1− 1 0 1− 0 ∈ 0
n (cid:12) o
In particular, the right hand side of (19) is irrelevant to(cid:12)the choice of M with M < 0 >=
1 1
Z .
0
(1)
Proof. ThisisjustadirectconsequenceofthedefinitionofthestabilitygroupS together
Z0
with the definition of group action.
(1)
Based on Proposition 1, to understand more about the stability group S , one only
Z0
(1)
needs to know more about S . As for this, we have the following
0
Proposition 2. Let S(1) be defined as in (18). Then we have
0
A 0
S(1) = M = 0 A C(2,2), A tA¯ = 1, qA = εA q
0 0 0 A¯0 0 ∈ 0 0 0 0
(cid:26) (cid:18) (cid:19)(cid:12) (cid:27)
(cid:12)
= M0 = A00 A¯00 (cid:12)(cid:12) A0 = ε((ξξ11+ξξ22))//22 ε((ξξ11−+ξξ22))//22 ,ξ1, ξ2 ∈ C, |ξ1| = |ξ2|= 1 .
(cid:26) (cid:18) (cid:19)(cid:12) (cid:18) − (cid:19) (cid:27)
(cid:12) 10
(cid:12)
(cid:12)