Heisenberg Groups, Theta Functions and the Weil Representation

0.9 MB·English

by  Jae-Hyun Yang

HEISENBERG GROUPS, THETA FUNCTIONS AND
THE WEIL REPRESENTATION
9
0
0 JAE-HYUNYANG
2

y
a
M Table of Contents

2
1
1. Introduction

]
T 2. The Heisenberg Group H(n,m)
R
N
3. Theta Functions
.
h
t 4. Induced Representations
a
m
5. Schr¨odinger Representations
[
6. Fock Representations

1
v 7. Lattice Representations
5
6 8. The Coadjoint Orbits of H(n,m)
8 R
1 9. Hermite Operators
.
5
10. Harmonic Analysis on H(n,m) H(n,m)
0 Z \ R
9
11. The Symplectic Group
0
:
v 12. Some Geometry on Siegel Space
i
X 13. The Weil Representation
r
a 14. Covariant Maps for the Weil Representation
15. Theta Series with Quadratic Forms
16. Theta Series in Spherical Harmonics
17. Relation between Theta Series and the Weil Representation
18. Spectral Theory on the Abelian Variety
References
This work was partially supported by the Max-Planck-Institute fu¨r Mathematik and
InhaUniversity.
1
2 JAE-HYUNYANG
1. Introduction
A certain nilpotent Lie group plays an important role in the study of the
foundations of quantum mechanics (cf.[30]and[41]) and the study of theta
functions (see[4],[5],[14],[27],[28],[31],[39],[42]and[43]).
For any positive integers m and n, we consider the Heisenberg group
H(n,m) := (λ,µ,κ) λ,µ R(m,n), κ R(m,m), κ+µtλ symmetric
R | ∈ ∈
n o
endowed with the following multiplication law
(λ,µ,κ) (λ′,µ′,κ′)= (λ+λ′,µ+µ′,κ+κ′+λtµ′ µtλ′).
◦ −
(n,m)
The Heisenberg group H is embedded in the symplectic group Sp(m+
R
n,R) via the mapping
I 0 0 tµ
n
λ I µ κ
H(n,m) (λ,µ,κ)  m  Sp(m+n,R).
R ∋ 7−→ 0 0 In tλ ∈
−
0 0 0 Im 
 
 
ThisHeisenberggroupisa2-stepnilpotentLiegroupandisimportantinthe
(n,m)
study of smooth compactification of Siegel moduli spaces. In fact, H
R
is obtained as the unipotent radical of the parabolic subgroup of the ra-
tional boundary component F (cf.[6]pp.122-123,[29]p.21or[52]p.36). In
n
the case m = 1, the study on this Heisenberg group was done by many
mathematicians, e.g., P. Cartier[4], J. Igusa[14], D. Mumford[27], [28] and
many analysts(cf.[2]) explicitly. For the case m > 1, the multiplication law
is alittle different from that of theHeisenberg group which is usually known
and needs much more complicated computation than the case m = 1.
(n,m)
The aim of this paper is to investigate the Heisenberg group H in
R
more detail. In the previous papers [42] and [43], the author decomposed
theL2-spaceL2 H(n,m) H(n,m) with respectto therightregular represen-
Z \ R
tation of H(n,m)(cid:16)explicitly and r(cid:17)elated the study of H(n,m) to that of theta
R R
(n,m) (n,m)
functions, where H denotes the discrete subgroup of H consisting
Z R
(n,m)
of integral elements. We need to investigate H for the study of Jacobi
R
forms(cf.[52],[57]), degeneration of abelian varieties(cf.[6]) and so on.
This paper is organized as follows. In Section 2, we introduce the new
(n,m)
multiplication on H which will be useful in the subsequent sections.
R
(n,m)
And we find the Lie algebra of H and obtain the commutation relation
R
(n,m)
for H . In Section 3, we give an explicit description of theta functions
R
due to J. Igusa(cf. [14] or [27]) and identify the theta functions with the
(n,m)
smooth functions on H satisfying some conditions. The results of this
R
section will be used later. In Section 4, using the Mackey decomposition of
a locally compact group(cf.[24]), we introduce the induced representations
HEISENBERG GROUPS, THETA FUNCTIONS AND THE WEIL REPRESENTATION 3
(n,m) (n,m)
of H and compute the unitary dual of H . In Section 5, we realize
R R
(n,m) (n,m)
theSchro¨dinger representation of H as thethe representation of H
R R
induced by the one-dimensional unitary character of a certain subgroup of
H(n,m). In Section 6, we consider the Fock representation UF,M, of
R HF,M
(n,m)
H . We prove that for a positive definite symmetric half-integral matrix
R (cid:0) (cid:1)
of degree m, UF,M is unitarily equivalent to the Schro¨dinger representa-
M
tion US,M. We also find an orthonormal basis for the representation space
. This section is mainly based on the papers [31, 32, 45]. In Section
F,M
H
7, we prove that for any positive definite symmetric, half-integral matrix of
(n,m)
degree m, the lattice representation π of H is unitarily equivalent to
M R
the (det 2 )n-multiples of the Schro¨dinger representation US,M. We give
M
a relation between the lattice representation π and theta functions. This
M
section is based on the paper [46]. In Section 8, we find the coadjoint orbits
(n,m)
of H . And we describe explicitly the connection between the coad-
R
(n,m)
joints orbits and the irreducible unitary representations of H following
R
the work of A. Kirillov(cf. [16],[17]and[18]). In Section 9, considering the
Schro¨dinger representation US,Im, L2 R(m,n),dξ , we study the Hermite
operators and the Hermite functions. We prove that Hermite functions
(cid:0) (cid:0) (cid:1)(cid:1)
defined in this section form an orthonormal basis for L2 R(m,n),dξ and
eigenfunctions for Hermite operators, the Fourier transform and the Fourier
(cid:0) (cid:1)
cotransform. We mention that Hermitian functions are used to construct
non-holomorphic modular forms of half-integral weight(cf.[43]). Implic-
(n,m)
itly the study of the Heisenberg group H implies that the confluent
R
hypergeometric equations (in this case, the Hermite equation) are related
to the study of automorphic forms. In Section 10, we investigate the ir-
reducible components of L2 H(n,m) H(n,m) . We describe the connection
Z \ R
among these irreducible com(cid:16)ponents, the Sc(cid:17)hro¨dinger representations, the
Fock representations and the lattice representations explicitly. We also pro-
vide the orthonormal bases for the representation spaces respectively. A
decomposition of L2(Γ G) for a general nilpotent Lie group G and a dis-
\
crete subgroupΓ of G was dealt by C. C. Moore(cf.[26]). In Section 11, we
briefly review the symplectic group and its action on the Siegel upper half
plane to be needed in the subsequent sections. We construct the universal
covering group of the symplectic group. In Section 12, we present some
properties of the geometry on the Siegel upper half plane which are used
in the subsequent sections. In Section 13, we study the Weil representation
associated to a positive definite symmetric real matrix of degree m. We
describe the explicit actions for the Weil representation. We describe the
results on the Weil representation which were obtained by Kashiwara and
Vergne[15]. In Section 14, we construct the covariant maps for the Weil
representation. In Section 15, we review various type of theta series asso-
ciated to quadratic forms. In Section 16, we discuss the theta series with
4 JAE-HYUNYANG
harmonic coefficients. Pluriharmonic polynomials play an important role in
the study of the Weil representation. We prove that the theta series with
pluriharmonic polynomials as coefficients are a modular form for a suitable
congruence subgroup of the Siegel modular group. This section is mainly
based on the book [28]. In Section 17, we investigate the relation between
the Weil representation and the theta series. We construct modular forms
using the covariant maps for the Weil representation. In Section 18, we
discuss the spectral theory on the principally polarized abelian variety A
Ω
attached to an element of the Siegel upper half plane. We decompose the
L2-space of A into irreducibles explicitly. We refer to [47] for more detail.
Ω
Finally I would like to mention that a Heisenberg group was paid to an
attention by some differential geometers, e.g., M. L. Gromov, in the sense
of a parabolic geometry. A Heisenberg group is regarded as a principal fibre
bundle over an Euclidean space with a vector space or a circle as fibres and
may be also regarded as the boundary of a complex ball. The geometry of
this group is quite different from that of an Euclidean space.
Notations: We denote by Z, R and C the ring of integers, the field of real
numbers, and the field of complex numbers respectively. C∗ denotes the
multiplicative group consisting of all nonzero complex numbers. C∗ denotes
1
the multiplicative group consisting of all complex numbers z with z = 1.
| |
Sp(n,R) denotes the symplectic group of degree n. H denotes the Siegel
n
upper half plane of degree n. The symbol “:=” means that the expression
on the right is the definition of that on the left. We denote by Z+ the set of
all positive integers. F(k,l) denotes the set of all k l matrices with entries
×
in a commutative ring F. For any M F(k,l), tM denotes the transposed
∈
matrix of M. For a complex matrix A, A denotes the complex conjugate of
A. The diagonal matrix with entries a , ,a on the diagonal position is
1 n
···
denoted by diag(a , ,a ). For A F(k,k), σ(A) denotes the trace of A.
1 n
··· ∈
For A F(k,l) and B F(k,k), we set B[A]= tABA. I denotes the identity
k
∈ ∈
matrix of degreek. For apositive integer m, Sym(m,K) denotes the vector
space consisting of all symmetric m m matrices with entries in a field K.
×
If H is a complex matrix or a complex bilinear form on a complex vector
space,ReH andImH denotetherealpartofH andtheimaginarypartofH
respectively. If X is a space, (X), C(X) and C∞(X) denotes the Schwarz
S c
space of infinitely differentiable functions on X that are rapidly decreasing
at infinity, the space of all continuous functions on X and the vector space
consisting of all compactly supported and infinitely differentiable functions
on X respectively.
HEISENBERG GROUPS, THETA FUNCTIONS AND THE WEIL REPRESENTATION 5
Z(m,n) = J = (J ) Z(m,n) J 0 for all k,a ,
≥0 ka ∈ | ka ≥
n o
J = J ,
k,a
| |
k,a
X
J ǫ =(J , ,J 1, ,J ),
ka 11 ka mn
± ··· ± ···
J! =J ! J ! J !.
11 ka mn
··· ···
For ξ = (ξ ) R(m,n) or C(m,n) and J = (J ) Z(m,n), we denote
ka ∈ ka ∈ ≥0
ξJ = ξJ11ξJ12 ξJka ξJmn.
11 12 ··· ka ··· mn
6 JAE-HYUNYANG
Table of Symbols
Section 2: , , , α , α∗, , , S , X0, X , X , D0 , D ,
A A S λ λ Oκˆ Oyˆ κˆ kl ka lb kl ka
D , Z0, Y+, Y−, E•, R , P , Q
lb b kl ka lb b kl b kl ka lb b
A
Section 3: Ωb, RMΩ , ϑ(S) B (Ω,W), Qξ(W), J(ξ,W), l(ξ), HerQ,
(cid:20) (cid:21)
ψ(ξ), L(Q,l,ψ), SymQ, Th(H,ψ,L), χ , q , H ,
S,Ω,A,B S,Ω S,Ω
ψ , L , A , RΩ, Θ, Th(H , ψ ,L ), RΩ , Θ ,
S,Ω Ω S,Ω S S,Ω S,Ω Ω S,A,B A,B
J , J˜ , , , ϕ
S,Ω,A,B S,Ω,A,B S,Ω X f
A L
Section 4: U , ( , ) , , k , s , T , T ,χ
σ H σ g g κˆ xˆ,yˆ xˆ
H
Section 5: G,K, k , s , U , U , σc, c, , , Φ , dU (X), f
g g σc c H H Hσc Hc c c c,J
Section 6: P , Q , A, J, J , V+, V−, T, H, V , G , z0, z1,
ka lb C ∗ C
R+, R−, z+, z−, UF,c, F,c, δ , , ( , ) , Λ, Λ ,
c F,c F,c f
H H
∆, ∆ , dµ(W), , Φ (W),κ(W,W′), f ,
ψ m,n J M
H k k
( , ) , dµ (W), Φ (W), k(U,W), (W,W′), k (U,W),
M M M,J M
I
I (W,W′), US,M, I , h , A (U,W), dUF,M(X)
M M J M
Section 7: L∗B, ΓL, ΓL∗B, Z0, φk,l, φM,q, πM,q, HM, T, φM,α,
, ϑ ,q , ϕ , , π , π , H ,
HM,α M,α M M,qM HM,qM M,qM M,qM M,qM
E , F , F , ϑ
φ φ Ω,φ Ω,φ
Section 8: g ,g∗, F(a,b,c), Ad∗, Ω , Ω , (G), G, B , ad∗, G ,
G a,b c O F g F
g , , radB , Ω , X, B , π , k, χ , π , π , χ , π1,
F F F ΩF a,b c,kb c,k c c c
, C∞(G), C∞(g), C(g∗), (G/ ), π1, L2(G/Z,χ ),
CFg c c e S Z c c
TC L2 R(m,n),dξ , HS L2 R(m,n),dξ , Ad∗ , ω , χ ,
K b,c b,c
p(Ω )
(cid:0)c (cid:0) (cid:1)(cid:1) (cid:0) (cid:0) (cid:1)(cid:1)
Section 9: dU (X), A+ , A−, C , f , f , h , H , P , ∂ , U(X),
Im ka lb kl 0 J J ka J ka
A+, A−, c , d , b
k,p k,p k,p
A A
Section 10: dξ , , , Φ(M) α (Ω ), Γ , H(M) α , ρ, R(c),
Ω,M T L J 0 |· G Ω 0
(cid:20) (cid:21) (cid:20) (cid:21)
A
f(M), Φ(M), ∆ , H (ξ), ϑ , H(M) α (Ω )
Ω,J Ω,α Ω,M J M,α,J J 0 |·
(cid:20) (cid:21)
HEISENBERG GROUPS, THETA FUNCTIONS AND THE WEIL REPRESENTATION 7
Section 11: Sp(n,R), J , Γ , (Γ ) , Γ (q), Γ , Γ (q),Ω∗, X∗, Y∗,
n n n Ω n ϑ,n n,0
dM , T (H ), (V,B), L⊥, Sp(B), τ(L ,L ,L ), Λ, Λ, U,
Ω Ω n 1 2 3
τ(L ,L , ,L ), U(L ,u ;U,L ), π : Λ Λ, L , S^p(B) ,
1 2 ··· k 1 1 2 −→ ∗ e ∗
E, W (E,L ), (V,ε), δ(A), g , ξ (L ,ε ),(L ,ε ) , L+, s (g),
2 M,L 1 1 2 2 L
e
s (L1,ε1),(L2,ε2) , s∗, Sp(B)∗, c∗(cid:0)(g1,g2), ϕ(g,n),(cid:1)Mp(B)∗
Section 12: d(cid:0)s2, dvn, R(Ω0,Ω(cid:1)1), ρ(Ω0,Ω1), Dn, Ψ, T, G∗, SU(n,n),
e
P+, K , ds2, ∆ , K, sp(n,R), k, p, ψ, δ, T , Pol(T ), Φ,
∗ ∗ ∗ n n
f , Φ, , , ∂ . dµ ,
Z n n n n n
P R R F
Section 13: U , GJ, UM, R , α (M ,M ), J(M,Ω), J∗(M,Ω), Sp(n,R) ,
c c c c 1 2 ∗
R , s (M), Mp(n,R), ω , t , d , σ , T (t ), A (d ), B (σ ),
c c c b a n c b c a c n
O(m), (σ,V ), L2 R(m,n);σ , ω (σ), O\(m), Σ , K := U(n),
e σ c m
K, (Hn,Vτ), Tτ,(cid:0)Φτ, H, τ(cid:1)(σ), H(σ), σ∗, Fσ
O
Section 14: F(c), J (M,Ω)
m
b
Section 15: A(S,T), S, ϑ (Ω), h(Ω), ϑ (Ω), ϑ(Ω;a,b), a,b ,
S S;A,B
{ }
a
γ , Ce, ν(γ), t , k , ∆(n)(Ω), ν (γ)
⋄ b S n S
(cid:18) (cid:19)
α α α
Section 16: ϑ (Ω,Z), χ (N), P , ϑ (Ω,Z),
S β β m,n S,P β
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
α
ϑ (Ω,Z), ϑ (Ω), P(∂), P , P,Q , H(S),
S,P S,P β N h i
(cid:20) (cid:21)
I, h , T = (t ), P , H(S) , I , f , P , H(S)⊥,
ij kl m,n R R A,B A,B
α ^ 1 k
O(S), P(Z), ϑW β , GL(n,C), L2, M, L2, Hm,n(ρ)
(cid:20) (cid:21)
Section 17: (π,Vπ),eF, θ, Θ(Ω), ΘM(Ω), ϑ, FM, ωfM, f
Section 18: H C(m,n), H , Γ , E , F (Ω), L , Ω , A , ∆ ,
n n,m n,m kj kj Ω ♭ Ω n,m
× b
, ∆ , L2(A ), ImΩ, E (Z), ds2, ΓJ, H(n,m), f ,
Fn,m Ω Ω Ω;A,B Ω Z || ||Ω
dv , (f,g) , L2(T), E (W), ∆ , Φ
Ω Ω A,B T Ω
8 JAE-HYUNYANG
2. The Heisenberg Group
For any two positive integer m and n, we let
H(n,m) = (λ,µ,κ) λ,µ R(m,n), κ R(m,m), κ+µtλ symmetric
R ∈ ∈
n (cid:12) o
the Heisenberg group endowed with the following multiplication law
(cid:12)
(cid:12)
(2.1) (λ,µ,κ) (λ ,µ ,κ ) := (λ+λ ,µ+µ ,κ+κ +λtµ µtλ ).
0 0 0 0 0 0 0 0
◦ −
(n,m)
We observe that H is a 2-step nilpotent Lie group. It is easy to see that
R
(n,m)
the inverse of an element (λ,µ,κ) H is given by
∈ R
(λ,µ,κ)−1 = ( λ, µ, κ+λtµ µtλ).
− − − −
Now we put
(2.2) [λ,µ,κ] = (0,µ,κ) (λ,0,0) = (λ,µ,κ µtλ).
◦ −
(n,m)
Then H may be regarded as a group equipped with the following mul-
R
tiplication
(2.3) [λ,µ,κ] [λ ,µ ,κ ]= [λ+λ ,µ+µ ,κ+κ +λtµ +µ tλ].
0 0 0 0 0 0 0 0
⋄
(n,m)
The inverse of [λ,µ,κ] H is given by
∈ R
[λ,µ,κ]−1 = [ λ, µ, κ+λtµ+µtλ].
− − −
We set
(2.4) = [0,µ,κ] H(n,m) µ R(m,n), κ= tκ R(m,m) .
A ∈ R ∈ ∈
n (cid:12) o
Then is a commutative normal su(cid:12)bgroup of H(n,m). Let be the Pontra-
A (cid:12) R A
jagin dualof , i.e., the commutative group consisting of all unitary charac-
A
ters of . Then is isomorphic to the additive group R(m,bn) Sym(m,R)
A A ×
via
(2.5) a,aˆ :b= e2πiσ(µˆtµ+κˆκ), a= [0,µ,κ] , aˆ = (µˆ,κˆ) .
h i ∈ A ∈ A
We put
b
(2.6) = [λ,0,0] H(n,m) λ R(m,n) = R(m,n).
S ∈ R ∈ ∼
n (cid:12) o
Then acts on as follows:
(cid:12)
S A (cid:12)
(2.7) α ([0,µ,κ]) := [0,µ,κ+λtµ+µtλ], α = [λ,0,0] .
λ λ
∈S
(n,m)
It is easy to see that the Heisenberg group H , is isomorphic to the
R ⋄
semidirect product GH := ⋉ of and(cid:16) whose m(cid:17)ultiplication is given
S A A S
by
(λ,a) (λ ,a ) = (λ+λ ,a+α (a )), λ,λ , a,a .
0 0 0 λ 0 0 0
· ∈ S ∈ A
On the other hand, acts on by
S A
(2.8) α∗(aˆ) := (µˆ+2κˆλ,κˆ), [λ,0,0] , a = (µˆ,κˆ) .
λ b ∈S ∈ A
Then we have the relation α (a),aˆ = a,α∗(aˆ) for all a and aˆ .
h λ i h λ i ∈Ab ∈ A
b
HEISENBERG GROUPS, THETA FUNCTIONS AND THE WEIL REPRESENTATION 9
We have two types of -orbits in .
S A
Type I. Let κˆ Sym(m,R) with κˆ = 0. The -orbit of aˆ(κˆ) := (0,κˆ)
∈ b6 S ∈ A
is given by
b
(2.9) := (2κˆλ,κˆ) λ R(m,n) = R(m,n).
κˆ ∼
O ∈ A ∈
n (cid:12) o
(cid:12)
b b
(cid:12)
Type II. Let yˆ R(m,n). The -orbit of aˆ(yˆ) := (yˆ,0) is given by
yˆ
∈ S O
(2.10) := (yˆ,0)b= aˆ(yˆ).
yˆ
O { }
We have b
=
κˆ yˆ
A  O   O 
κˆ∈Sy[m(m,R) [ yˆ∈R[(m,n)
b  b   b 
as a set. The stabilizer of at aˆ(κˆ)= (0,κˆ) is given by
κˆ
S S
(2.11) = 0 .
κˆ
S { }
And the stabilizer of at aˆ(yˆ) = (yˆ,0) is given by
yˆ
S S
(2.12) = [λ,0,0] λ R(m,n) = = R(m,n).
Syˆ ∈ S ∼
n (cid:12) o
(cid:12)
The following matrices
(cid:12)
0 0 0 0
0 0 0 1(E +E )
X0 := 2 kl lk , 1 k l m,
kl 0 0 0 0 ≤ ≤ ≤
0 0 0 0 
 
 
0 0 0 0
E 0 0 0
Xka := 0ka 0 0 tEka, 1 ≤ k ≤ m, 1≤ a≤ n,
−
 0 0 0 0 
 
0 0 0 tE 
lb
0 0 E 0
Xlb :=0 0 0lb 0 , 1 ≤ l ≤ m, 1≤ b ≤ n
0 0 0 0 
b  
 
(n,m) (n,m)
form a basis of the Lie algebra of the real Heisenberg group H .
HR R
Here E denotes the m m matrix with entry 1 wherethe k-th row and the
kl
×
l-th column meet, all other entries 0 and E (resp. E ) denotes the m n
ka lb
×
matrix with entry 1 where the k-th(resp. the l-th) row and the a-th(resp.
the b-th) column meet, all other entires 0. By an easy calculation, we see
10 JAE-HYUNYANG
that the following vector fields
∂
D0 := , 1 k m,
kl ∂κ ≤ ≤
kl
k m
∂ ∂ ∂
D := µ + µ , 1 k m, 1 a n,
ka pa pa
∂λ − ∂κ ∂κ  ≤ ≤ ≤ ≤
ka pk kp
p=1 p=k+1
X X
 
l m
∂ ∂ ∂
D := + λ + λ , 1 k m, 1 a n
lb pb pb
∂µ  ∂κ ∂κ  ≤ ≤ ≤ ≤
lb pl lp
p=1 p=l+1
X X
fobrmabasisfortheLiealgebraofleft-invariantvectorfieldsontheLiegroup
(n,m)
H .
R
Lemma 2.1. We have the following Heisenberg commutation relations
[D0 ,D0 ] =[D0 ,D ]= [D0 ,D ] = 0,
kl st kl sa kl sa
[D ,D ] =[D ,D ] = 0,
ka lb ka lb
b
[D ,D ] =2δ D0 ,
ka lb bab bkl
where 1 k,l,s,t m, 1 a,b n and δ denotes the Kronecker delta
ab
≤ ≤ b ≤ ≤
symbol.
Proof. The proof follows from a straightforward calculation. (cid:3)
We put
Z0 := √ 1D0 , 1 k l m,
kl − − kl ≤ ≤ ≤
1
Y+ := (D +√ 1D ), 1 k m, 1 a n,
ka 2 ka − ka ≤ ≤ ≤ ≤
1
Y− := (D √ 1Dˆb ), 1 l m, 1 b n.
lb 2 lb− − lb ≤ ≤ ≤ ≤
Then it is easy to see that the vector fields Z0, Y+, Y− form a basis of the
kl ka lb
(n,m)
complexification of the real Lie algebra .
HR
Lemma 2.2. We have the following commutation relations
[Z0,Z0] = [Z0,Y+] = [Z0,Y−]= 0,
kl st kl sa kl sa
[Y+,Y+] = [Y−,Y−] = 0,
ka lb ka lb
[Y+,Y−] = δ Z0,
ka lb ab kl
where 1 k,l,s,t m and 1 a,b n.
≤ ≤ ≤ ≤
Proof. It follows immediately from Lemma 2.1. (cid:3)
We let E• := E +E for 1 k l m. We put
kl kl lk ≤ ≤ ≤
R (r) := exp 2rX0 = (0,0,rE• ), r R,
kl kl kl ∈
Psa(x) := exp(cid:0)xXsa (cid:1)= (xEsa,0,0), x R,
∈
Qtb(y) := exp(cid:0)yXtb(cid:1)= (0,yEtb,0), y R,
∈
(cid:0) (cid:1)
b