derivatives of siegel modular forms and modular connections
TRANSCRIPT
manuscripta math. © Springer-Verlag Berlin Heidelberg 2014
Enlin Yang · Linsheng Yin
Derivatives of Siegel modular forms and modularconnections
Received: 15 February 2014 / Accepted: 5 June 2014
Abstract. We introduce a method in differential geometry to study the derivative opera-tors of Siegel modular forms. By determining the coefficients of the invariant Levi–Civitaconnection on a Siegel upper half plane, and further by calculating the expressions of thedifferential forms under this connection, we get a non-holomorphic derivative operator ofthe Siegel modular forms. In order to get a holomorphic derivative operator, we introduce aweaker notion, called modular connection, on the Siegel upper half plane. Then we show thaton a Siegel upper half plane there exists at most one holomorphic Sp(2g, Z)-modular con-nection in some sense, and get a possible holomorphic derivative operator of Siegel modularforms.
Introduction
In this paper, we introduce a differential geometric method to study the derivative operatorsof Siegel modular forms. Recently, our method is applied to study the derivations of Siegel–Jacobi forms [19]. Our idea comes from the observation on two derivative operators ofclassical modular forms constructed by combinations. It is well-known in [20] that if f is
a modular forms of weight 2k, then D1 f := d fdz −
√−1ky f is a non-holomorphic modular
forms of weight 2k+2, and D2 f := d fdz −√−1kG2(z) f , due to J.-P. Serre, is a holomorphic
modular forms of weight 2k +2, where G2(z) is the Eisenstein Series of weight 2. We noticethat the first operator can be constructed by the Levi–Civita connection corresponding tothe invariant metric of the classical upper half plane. In this paper, we define a conceptcalled modular connection. Then we can get Serre’s holomorphic derivative from the uniqueholomorphic SL(2, Z)-modular connection. These results can be generalized to Siegel upperhalf plane and Siegel modular forms. First we determine the coefficients of the Levi–Civitaconnection corresponding to the invariant metric of the Siegel upper half plane, then wecompute the expressions of the differential forms under the connection. Finally we geta non-holomorphic derivative operator of Siegel modular forms. Our main results are asfollows.
E. Yang (B) · L. Yin: Department of Mathematical Science, Tsinghua University, Beijing,100084, People’s Republic of China. e-mail: [email protected]
L. Yine-mail: [email protected]
present addressE. Yang: Department of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan
Mathematics Subject Classification (2000):11F46, 70G45
This paper was partially supported by NSFC No. 11271212.
DOI: 10.1007/s00229-014-0687-5
E. Yang, L. Yin
Let Hg be the Siegel upper half plane of degree g, {d Zi j : 1 ≤ i, j ≤ g} a series ofcoordinates on Hg , and Γg = Sp(2g, Z) the full Siegel modular group of degree g whichacts on Hg naturally. For any congruence subgroup Γ of Γg , let Mk = Mk(Γ ) be the vectorspace of the classical (or scalar-valued) Siegel modular forms of weight k with respect to Γ ,˜Mk = ˜Mk(Γ ) the C
∞-Siegel modular forms of weight k with respect to Γ . Put
∂
∂ Z= (∂i j )g×g and ∂i j = 1
21−δ(i, j)· ∂
∂ Zi j,
where δ(i, j) = 1 if i = j and δ(i, j) = 0 if i �= j .
Theorem 1 (See theorem 3 and corollary 2). Let f ∈ Mk(Γ ). Then
det
([
∂
∂ Z−
√−1k
2Y −1
]
f
)
∈ ˜Mgk+2(Γ ).
To get the holomorphic derivative operators, we introduce the notion of modular connec-tions on the Siegel upper half plane, whose condition is weaker than the classical definitions.Then we show the following result.
Theorem 2 (See theorem 4 and corollary 2). Let Γ be a subgroup of Sp(2g, R). Anysymmetric g×g matrix G = (Gi j )of C∞-functions on Hg, which satisfies the transformationformula
(C Z + D)−1γ (G) = G · (C Z + D)t + 2Ct ,
for any γ =(
A BC D
)
∈ Γ , gives a Γ -modular connection D such that for any C∞-function
f on Hg
D(d Zrs) = −g∑
i, j=1
Gi j d Zsi d Zr j
and
D( f (det(d Z)k) = {d f − kTr(Gd Z) f } (det(d Z))k .
If Γ is a congruence subgroup of Γg and if f ∈ Mk(Γ ), then for such a G, we have
det
([
∂
∂ Z− kG
2
]
f
)
∈ ˜Mgk+2(Γ ).
Furthermore, if Γ = Γg is the full Siegel modular group, then there exists at most oneholomorphic matrix G satisfying the transformation formula.
In the classical case of g = 1, the function√−1G2(z) is the unique holomorphic
function on the upper half plane satisfying the condition, which gives Serre’s derivative. Butwhen g ≥ 2 we don’t know how to construct such a holomorphic matrix function G.
H. Maass has constructed a non-holomorphic derivative operator of Siegel modularforms by invariant differential operators. For a Siegel modular form f of weight k, he ([13],P317) defines the operator
Dk f (Z) = det(Y )κ−k−1 det
(
∂
∂ Z
)
[det(Y )k+1−κ f (Z)],
Derivatives of Siegel modular forms and modular connections
where κ = (g + 1)/2 and the determination of ∂∂ Z is taken first, and shows that the differ-
ential operator Dk acts on the C∞-Siegel modular forms and maps ˜Mk to ˜Mk+2. We do not
know the relation between our operator and Maass’s. Compared to our operator in Theorem1, Dk is linear with respect to f . Moreover, our operator is a combination of degree 1 partialderivatives of f , but Dk is a combination of degree g partial derivatives. G. Shimura [15]considers the compositions Dk
r = Dr+2k−2 · · · Dr+2 Dr of Maass’s operator, which maps˜Mr to ˜Mr+2k . For our operator one can also consider the compositions and then construct theRankin-Cohen brackets. However in this paper we only construct the first bracket. For Siegelmodular forms and Jacobi forms, the Rankin-Cohen type differential operators have beenstudied by many authors(cf. Böcherer [2,3], Ibukiyama [6,9]). Ibukiyama [9] uses invari-ant pluri-harmonic polynomials to characterize Rankin–Cohen type differential operatorsdefined on n-tuples of Siegel modular forms with values in spaces of vector-valued Siegelmodular forms.
The paper is organized as follows. In section one, we introduce the concept of modularconnection on Siegel upper half plane, and show several lemmas on it which are basic ondetermining the coefficients of the Levi–Civita connection on the Siegel upper half plane andthe derivative operator of Siegel modular forms. In section two, we compute the expressionsof the differential forms under the modular connections, and get the derivative operatorof Siegel modular forms. We also show the uniqueness theorem on holomorphic modularconnections in this section. Finally in section three, we show Lemma 6 which explicitlydetermine the connection coefficients of the Levi–Civita connection on Siegel upper halfplane.
Our calculations in sections 2 and 3 are tested by matlab in the cases g = 2 and g = 3.
1. Modular connections
In this section we first recall the definition of connections in differential geometry. Then weintroduce the notion of modular connection on Siegel upper half plane, and show severallemmas about it.
1.1. Connections in differential geometry
For the backgrounds and notations in differential geometry, especially on connections, werefer to the books [5] and [10]. Here we just recall some basic definitions and results onconnections. Suppose E is a q-dimensional real vector bundle on a smooth manifold M , andΓ (E) is the set of smooth sections of E on M . Let T ∗(M) be the cotangent space of M . Aconnection on the vector bundle E is a map
D : Γ (E) −→ Γ (T ∗(M) ⊗ E),
which satisfies the following conditions
1. For any s1, s2 ∈ Γ (E),
D(s1 + s2) = D(s1) + D(s2).
2. For any s ∈ Γ (E) and any α ∈ C∞(M),
D(αs) = dα ⊗ s + αD(s).
E. Yang, L. Yin
If M has a generalized Riemannian metric G =∑i, j gi j dui du j , by the fundamental theo-rem of Riemannian geometry, M has a unique torsion-free and metric-compatible connection,called Levi–Civita connection of M . The coefficients Γ k
i j of the Levi-Civita connection aregiven by
Γ ki j =
q∑
�=1
1
2gk�
(
∂gi�
∂u j+ ∂g j�
∂ui− ∂gi j
∂u�
)
, (1)
where gi j are elements of the matrix (gi j ) := (gi j )−1.
The following lemma is useful in its application to automorphic forms although theproof is easy.
Lemma 1. Let Γ be a group, (M, G) a Riemannian manifold and D the Levi-Civita con-nection on M. If Γ has a smooth left action on M such that G(σX, σY ) = G(X, Y ) forall σ ∈ Γ, X, Y ∈ T (M), then
σ D = Dσ (σ ∈ Γ ).
Moreover, if M is a complex manifold such that Γ maps (r, s) forms to (r, s) forms, and putD = D1,0 + D0,1, where D1,0 is the holomorphic part, then for σ ∈ Γ
σ D1,0 = D1,0σ and σ D0,1 = D0,1σ.
Proof. D is the unique torsion free connection which preserves the Riemannian metric G.Since G is Γ -invariant, the connection σ−1 Dσ also preserves the Riemannian metric andis torsion free for any σ ∈ Γ , hence σ D = Dσ . For more detail, see ([14], P35).
1.2. Siegel upper half plane
We first fix some notations. The Siegel upper half plane of degree g ≥ 1 is defined to be theg(g + 1)/2 dimensional open complex variety
Hg :={
Z = X + √−1Y ∈ M(g, C) | Zt = Z , Y > 0}
.
Write Z = (Zi j ) , Y = 12√−1
(Z − Z̄) and d Z = (d Zi j ). Set Ω = {(i, j) | 1 ≤ i ≤ j ≤ g}with the dictionary order. If I = (i, j) ∈ Ω , we define Z I := Zi j . Fix a series of coordinates{d Z I , d Z̄ I | I ∈ Ω} on Hg . The symplectic group of degree g > 0 over R is the group
Sp(2g, R) = {M ∈ GL(2g, R)∣
∣M J Mt = J}
,
where J =(
0 Ig−Ig 0
)
. We usually write an element of Sp(2g, R) in the form
(
A BC D
)
,
where A, B, C and D are g × g blocks. The Siegel modular group Γg := Sp(2g, Z) ⊂Sp(2g, R) acts on Hg by the rule:
γ (Z) := (AZ + B)(C Z + D)−1, Z ∈ Hg, γ =(
A BC D
)
∈ Sp(2g, Z).
By Maass ([12], P98), d(γ Z) = (ZCt + Dt )−1d Z(C Z + D)−1 := (d Z̃i j ). Let
(d Z̃11, d Z̃12, · · · , d Z̃1g, · · · , · · · , d Z̃gg)
= (d Z11, d Z12, · · · , d Z1g, · · · , · · · , d Zgg) · S(γ, Z)
Derivatives of Siegel modular forms and modular connections
where S := S(γ, Z) is a g(g+1)2 × g(g+1)
2 matrix of holomorphic functions on Sp(2g, Z)×Hg .
From Lemma 1, one can see that the connection matrix ω consisting of the connec-tion coefficients of the Levi-Civita connection associated to the invariant metric ds2 =Tr(Y −1d Z · Y −1d Z̄) given by Siegel ([12], P8) on the Siegel upper plane Hg satisfies
γ (ω) = −S−1 · d S + S−1 · ω · S
for all γ ∈ Sp(2g, R), and see also the proof of Lemma 2 below. But in the studying ofmodular forms, we only need that the equality holds for all γ ∈ Sp(2g, Z). So we need tointroduce a weaker notion to study modular forms.
Now we recall the definition of Siegel modular forms, for more details, see [1] and [16].
Definition 1. Let Γ be a congruence subgroup of Γg . A (classical) Siegel modular form ofweight k (and degree g) with respect to Γ is a holomorphic function f : Hg → C such that
f (γ (Z)) = det(C Z + D)k f (Z)
for all γ =(
A BC D
)
∈ Γ (with the usual holomorphicity requirement at ∞ when g = 1).
We denote by Mk = Mk(Γ ) the vector space of the classical Siegel modular forms of weightk with respect to Γ .
1.3. Modular connections
The notations are all the same as above.
Definition 2 (Modular Connection Coefficients (MCC)). LetΓ be a subgroup of Sp(2g, R).The Modular Connection Coefficient on Hg with respect to Γ is a series of C
∞-functions{Γ K
I J | I, J, K ∈ Ω} such that for all γ ∈ Γ ,
γ (ω) = −S−1 · d S + S−1 · ω · S, or S · γ (ω) = ω · S − d S,
where ω = (ωJI ) and ωJ
I = ∑
K∈Ω
Γ JI K d ZK . Here I and J are the row and column indices
respectively. When {Γ KI J } are holomorphic, we call it holomorphic MCC (HMCC). The
matrix ω is called the modular connection matrix.
In the following, C∞(Hg) is the set of C
∞-functions on Hg , and Hol(Hg) is the set ofholomorphic functions on Hg .
Definition 3 (Modular Connection). Let Γ be a subgroup of Sp(2g, R). Let {Γ KI J } be a
MCC (or HMCC) on Hg with respect to Γ and Ω∞ be the commutative C∞(Hg)-algebra
(resp. Hol(Hg)-algebra) generated by {d Z I }I∈Ω with the relations d Z I d Z J = d Z J d Z Ifor any I, J ∈ Ω . The linear operator
D : Ω∞ −→ Ω∞
is uniquely defined by the following two relations
D( f d ZK ) =∑
I∈Ω
∂ f
∂ Z Id Z I d ZK − f
∑
I,J∈Ω
Γ KI J d Z I d Z J
E. Yang, L. Yin
and
D( f d ZK1 d ZK2 · · · d ZKr ) = d f · d ZK1 · · · d ZKr
+r∑
i=1
f d ZK1 d ZK2 · · · D(d ZKi ) · · · d ZKr ,
and we call it the Γ -modular connection associated to {Γ KI J }.
One can easily show that D(d ZK ) = −∑I∈Ω ωKI · d Z I and
(D(d Z11), D(d Z12), · · · , · · · , · · · , D(d Zgg))
= −(d Z11, d Z12, · · · , · · · , · · · , d Zgg) · ω
When {Γ KI J } is holomorphic, we also call D a holomorphic Γ -modular connection. Com-
pared with the definition of connections in differential geometry, besides the weaker condi-tions, the modular connection also ignores the part on {d Z̄ I }.
1.4. Basic lemmas on modular connections
The following two lemmas are basic to our application of modular connections to the Siegelmodular forms. For modular connections, we have similar result to Lemma 1.
Lemma 2. Let Γ be a subgroup of Sp(2g, R). Let D be a Γ -modular connection on Hg.Then γ D = Dγ for any γ ∈ Γ . Moreover, if Γ is a congruence subgroup of Sp(2g, Z)
and f is a Siegel modular form of weight 2k with respect to Γ , then D( f · (det(d Z))k) isinvariant under the action of Γ .
Proof. Let α = (d Z11, · · · , d Z1g, d Z22, · · · , d Z2g, · · · , d Zgg). Then Dα = −αω. Onone side,
γ (Dα) = −γ (α)γ (ω) = −αS(−S−1d S · +S−1 · ω · S)
= α(d S − ω · S).
On the other side,
D(γ α) = D(α · S) = D(α) · S + α · d S = −αω · S + αd S = α(−ω · S + d S).
For f ∈ M2k(Γ ), we see f · (det(d Z)k) is invariant under Γ , and so is D( f · (det(d Z))k).
The following lemma directly from Lemma 1 gives a modular connection.
Lemma 3. The holomorphic part D1,0 of the Levi-Civita connection associated to the in-variant metric ds2 = Tr(Y −1d Z ·Y −1d Z̄) on the Siegel upper half space Hg is a Γ -modularconnection D for any subgroup Γ of Sp(2g, R).
We will explicitly compute the expression of D( f · det(d Z)k) in Proposition 2.
Derivatives of Siegel modular forms and modular connections
1.5. Modular connections on the upper half plane
Let’s consider the classical upper half plane H = H1 to look for which conditions the
coefficient Γ(1,1)(1,1),(1,1)
of a SL(2, Z)-modular connection should satisfy. Let ω = T dz. Onecan easily check that for γ ∈ SL(2, Z)
γ (ω) = −S−1d S + S−1ωS ⇐⇒ γ (T )
(cz + d)2 = T + 2c
cz + d.
Recall that ([11], P113)
1
(cz + d)2 ·√−1
Im(γ z)=
√−1
Im(z)+ 2c
cz + dand
1
(cz + d)2 · √−1G2(γ z)
= √−1G2(z) + 2c
cz + d,
where
G2(z) =∑
n �=0
1
n2 +∑
m �=0
∑
n∈Z
1
(mz + n)2 .
So√−1
y and√−1G2(z) give us two SL(2, Z)-modular connections on the upper half plane
H, denoted by D1 and D2 respectively. The later modular connection D2 is holomorphic.We have
D1( f dz) =(
d f
dz−
√−1
yf
)
dzdz and D2( f dz) =(
d f
dz− √−1G2(z) f
)
dzdz.
By the Lemma 2, we have
Corollary 1. Let f be a modular form of weight 2k. Then d fdz −
√−1ky f is a non-holomorphic
modular form of weight 2k + 2, and d fdz − √−1kG2(z) f is a holomorphic modular form of
weight 2k + 2.
In fact, the modular connection D1 is a SL(2, R)-modular connection and comes from theLevi–Civita connection D associated to the invariant metric
ds2 = dx2 + dy2
y2 = dz dz̄
y2 .
Lemma 4. D(dz) = −√−1
y dz dz, and D(dz̄) =√−1
y dz̄ d z̄. So the coefficients of Dgive the modular connection D1.
Proof. Since ds2 = dx2+dy2
y2 = dz dz̄y2 , using the coordinates dz, dz and the formula (1),
we have Γ 11,1 =
√−1y , Γ 1
2,1 = Γ 11,2 = 0, Γ 2
2,1 = Γ 21,2 = 0 and Γ 2
2,2 = −√−1
y .
The SL(2, Z)-modular connection D2 is unique.
Lemma 5 (Uniqueness Lemma).√−1G2(z) is the unique holomorphic function T satis-
fying
γ (T )
(cz + d)2 = T + 2c
cz + d, for all γ =
(
a bc d
)
∈ SL(2, Z),
and so there exists a unique holomorphic SL(2, Z)-modular connection on H.
E. Yang, L. Yin
Proof. We have γ (√−1G2(z) − T ) = (cz + d)2(
√−1G2(z) − T ) for all γ ∈ SL(2, Z).So
√−1G2(z) − T is a modular form of weight 2 and hence must be zero ([11], P117).
We will generalize these results to Hg .
2. The derivative operator of Siegel modular forms
In this section, we first state a result about the coefficients of the invariant Levi–Civitaconnection on a Siegel upper half plane, whose proof is given in the last section, then wecompute the expression of the differential forms under this connection. Finally we get anon-holomorphic derivative operator.
2.1. Coefficients of Levi–Civita connection
The notations are same as in section 1. For I = (i, j) ∈ Ω , put |I | = i + j and
N (I ) = (i − 1)(2g − i)
2+ j,
which gives a one to one and order keeping correspondence between Ω and
{1, 2, · · · ,g(g+1)
2 }. Write uN (I ) = Zi j and uN (g,g)+N (i, j) = Zi j . For Z = X +√−1Y ∈Hg , let R := (Ri j ) := Y −1. For 1 ≤ s ≤ g, we put
Ωs = {(1, s), (2, s), · · · , (s, s), (s, s + 1), · · · , (s, g)} ⊂ Ω.
Let K = (r, s) ∈ Ω . Assume that the elements of Z in the column including ZK = Zrsare
ua1 = Z1s , ua2 = Z2s , · · · , uas = Zss , uas+1 = Zs+1,s = Zs,s+1, · · · , uag = Zgs = Zsg,
and the elements in the row including Zrs are
ub1 = Zr1 = Z1r , ub2 = Zr2 = Z2r , · · · , ubr = Zrr , ubr+1 = Zr,r+1, · · · , ubg = Zrg .
For I × J ∈ Ωs ×Ωr , assume Z I = uai and Z J = ub j . Similarly do it for J × I ∈ Ωs ×Ωr .We define
Γ KI J (= Γ K
J I ) :={
√−1Ri j
2(1−δ(r,s))(1−δ(ai ,b j ))if I × J or J × I ∈ Ωs × Ωr
0 if I × J and J × I �∈ Ωs × Ωr ,(2)
where δ(r, s) denotes the Kronecker delta symbol. For example, Γ (1,1)(1,i)(1, j) = √−1Ri j and
Γ(1,1)I J = 0 if I or J �∈ Ω1.
Notice that if we use the coordinates {uai , ub j }, then the coefficients of the Levi–Civitaconnection satisfy
ZΓKI J = uΓ
N (K )N (I )N (J )
.
In this paper we will use these two kinds of notations alternately. The proof of the followinglemma is long and complicated. For the convenience of the reader, we put it in the lastsection.
Lemma 6. The coefficients {Γ KI J } defined above give the Levi-Civita connection on Hg
associated to the invariant metric ds2 = Tr(Y −1d Z Y −1d Z̄), and hence give a Γ -modularconnection for any subgroup Γ of Sp(2g, R), which we denote by D.
Derivatives of Siegel modular forms and modular connections
2.2. Expression of differential forms under D
We first compute D(d ZK ).
Lemma 7. Let K = (r, s) ∈ Ω . We have
D(d ZK ) = −√−1(d Zs1, d Zs2, · · · , d Zsg)Y −1 · (d Zr1, d Zr2, · · · , d Zrg)t .
Proof. The notations are as above. If r = s, then Ωr = Ωs . By Lemma 6 and formula (2),we have
Γ KI J = Γ K
J I ={√−1Ri j if I and J ∈ Ωr
0 if I or J �∈ Ωr .
By definition, we have uai = Z I and ub j = Z J . Hence,
D(d ZK ) =−∑
I,J∈Ω
Γ KI,J d Z I d Z J =−
∑
I,J∈Ωr
Γ KI,J d Z I d Z J =−
g∑
i, j=1
√−1Ri j d Zsi d Zr j
= −√−1(d Zs1, d Zs2, · · · , d Zsg)Y −1 · (d Zr1, d Zr2, · · · , d Zrg)t .
If r �= s, we assume r < s. Then Ωr⋂
Ωs = {(r, s)} and (Ωr × Ωs)⋂
(Ωs × Ωr ) ={(r, s) × (r, s)}. Put A = (Ωr × Ωs)
⋃
(Ωs × Ωr ) and B = (Ωr × Ωs)⋂
(Ωs × Ωr ). Wehave again by Lemma 6 and formula (2),
Γ KI J = Γ K
J I =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
√−1Ri j
21−δ(ai ,b j )=
√−1Ri j2 if I × J ∈ A \ B
√−1Ri j
21−δ(ai ,b j )= √−1Ri j if I × J ∈ B.
0 if I × J �∈ A
Hence,
D(d ZK ) = −∑
I,J∈Ω
Γ KI,J d Z I d Z J = −
∑
I×J∈A
Γ KI,J d Z I d Z J
= −2∑
I∈ΩrJ∈Ωs
Γ KI,J d Z I d Z J +
∑
I∈Ωr⋂
ΩsJ∈Ωr
⋂
Ωs
Γ KI,J d Z I d Z J
= −2∑
I×J∈Ωr ×Ωs−B
Γ KI,J d Z I d Z J −
∑
I×J∈B
Γ KI,J d Z I d Z J
= −g∑
i, j=1
√−1Ri j d Zsi d Zr j
= −√−1(d Zs1, d Zs2, · · · , d Zsg)Y −1 · (d Zr1, d Zr2, · · · , d Zrg)t .
Proposition 1. D(det(d Z)) = −√−1Tr(Y −1d Z) det(d Z).
E. Yang, L. Yin
Proof. Put αi = (d Zi1, d Zi2, · · · , d Zig) and β j = (d Z1 j , d Z2 j , · · · , d Zgj )t . By
Lemma 7, D(d(Zi j )) = −√−1αi Y −1β j and D(αi ) = −√−1αi Y −1d Z . Thus
D(det(d Z)) = −√−1
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
det
⎛
⎜
⎜
⎜
⎝
α1Y −1d Zα2...
αg
⎞
⎟
⎟
⎟
⎠
+ det
⎛
⎜
⎜
⎜
⎝
α1α2Y −1d Z
...
αg
⎞
⎟
⎟
⎟
⎠
+ · · · + det
⎛
⎜
⎜
⎜
⎝
α1...
αg−1αgY −1d Z
⎞
⎟
⎟
⎟
⎠
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
.
Let A = Y −1d Z = (Ai j ) and d Z [i, j] be the algebraic cofactor of d Z at the position (i, j).By the above formula, we have
√−1D(det(d Z)) =n∑
k, j,i=1
d Zki · Ai j · d Z [k, j] =n∑
j,i=1
Ai j
n∑
k=1
d Zik · d Z [k, j]
=n∑
j,i=1
Ai j δ(i, j) det(d Z) = Tr(A) det(d Z) = Tr(Y −1d Z) det(d Z).
Put
∂
∂ Z= (∂i j )g×g, ∂i j = 1 + δ(i, j)
2· ∂
∂ Zi j
same as in the introduction. Then for any smooth function f on Hg ,
d f =∑
1≤i≤ j≤g
∂ f
∂ Zi jd Zi j =
g∑
i=1
g∑
j=1
∂i j f d Zi j = Tr
(
∂
∂ Zf · d Z
)
Proposition 2. For any smooth function f on Hg, we have
D(
f det(d Z)k)
= Tr
([
∂
∂ Z− √−1kY −1
]
f d Z
)
det(d Z)k .
Proof. We have d f = Tr( ∂∂ Z f · d Z). By Proposition 1,
D(
f det(d Z)k)
= d f · det(d Z)k + f(
D(det(d Z)k)
= d f · det(d Z)k + k f det(d Z)k−1 D(det(d Z))
= (d f − √−1k f Tr(Y −1d Z)) det(d Z)k
= Tr
(
∂
∂ Zf · d Z − √−1k f Y −1d Z
)
det(d Z)k
= Tr
([
∂
∂ Z− √−1kY −1
]
f d Z
)
det(d Z)k .
In the following, Ω iHg
is the sheaf of holomorphic i-forms on Hg . Recall that a section
of Ω1Hg
can be written as Tr( f d Z), where f is a symmetric matrix of holomorphic functions
on Hg .
Derivatives of Siegel modular forms and modular connections
Proposition 3. For any section Tr( f d Z) ∈ Ω1Hg
, where f is a symmetric matrix of holo-
morphic functions on Hg, we have
D(Tr( f d Z)) = Tr
{[
(
∂
∂ Z
)t⊗ f
]
· [d Z ⊗ d Z ]}
− √−1Tr(
f d Z · Y −1d Z)
,
where ⊗ is the Kronecker product of matrices.
To show this proposition, we need the following lemma.
Lemma 8. We have
1. Let A = (ai j )n×n, B = (bi j )n×n, C = (ci j )n×n, D = (di j )n×n. Then
Tr((A ⊗ B)(C ⊗ D)) =n∑
i, j,k,l=1
ai j bkl clkd ji = Tr((A ⊗ C)(B ⊗ D)),
2. d(Tr( f d Z)) = Tr
((
(
∂∂ Z
)t ⊗ f
)
· (d Z ⊗ d Z)
)
for a symmetric matrix f = ( fi j ) of
functions.
Proof. The proof of (1) is easy. We only show (2). By (1),
d(Tr( f d Z)) = d
⎛
⎝
g∑
i, j=1
fi j d Zi j
⎞
⎠ =g∑
i, j=1
g∑
k,l=1
∂kl fi j · d Zkl d Z ji
= Tr
(
(
(∂kl )t ⊗ ( fi j )
) · (d Z ⊗ d Z)
)
.
Proof (Proof of Proposition 3). As d Z = (αt1, αt
2, · · · , αtg)t = (β1, β2, · · · , βg), we have
D(Tr( f d Z)) − Tr
{[
(
∂
∂ Z
)t⊗ f
]
· [d Z ⊗ d Z ]}
= Tr ( f D(d Z))
=g∑
i, j=1
fi j D(d Zi j ) = −√−1g∑
i, j=1
fi j d(αi )Y−1d(β j ) = −√−1Tr
(
f d ZY −1d Z)
.
2.3. Derivative operators
Let Γ be a congruence subgroup of Γg = Sp(2g, Z). If 0 �= f ∈ M2k(Γ ), then byProposition 2 and Lemma 2
Tr
([
∂
∂ Z− √−1kY −1
]
f d Z
)
det(d Z)k
is invariant under the action ofΓ . Letγ =(
A BC D
)
∈ Γ . Sinceγ (det(d Z))k = det(d Z)k
det(C Z+D)2k
and γ ( f ) = det(C Z + D)2k f , we have
Tr
(
1
f
[
∂
∂ Z− √−1kY −1
]
f d Z
)
E. Yang, L. Yin
is invariant under Γ . Put h := 1f
(
∂∂ Z − √−1kY −1
)
f . Then h is a symmetric matrix of
functions on Hg , and Tr(hd Z) is invariant under the action of Γ . But for any γ = ( A BC D
) ∈Γ , d(γ Z) = (ZCt + Dt )d Z · (C Z + D)−1, we have (see ([16], P210))
h(γ Z) = (C Z + D)h(Z)(ZCt + Dt ).
Therefore, det(h) = det(
1f
(
∂∂ Z − √−1kY −1
)
f)
transforms like a Siegel modular form
of weight 2. Hence det(
1f
(
∂∂ Z − √−1kY −1
)
f)
is a non-holomorphic Siegel modular
form of weight 2 with respect to Γ . let ˜Mk(Γ ) be the C∞-Siegel modular forms of weight
k as in the introduction. Finally we get
Theorem 3. Let Γ be a congruence subgroup of Γg = Sp(2g, Z). If f ∈ M2k(Γ ), then
det
((
∂
∂ Z− √−1kY −1
)
f
)
∈ ˜M2kg+2(Γ ).
Theorem 3 is also true for odd weights.
Corollary 2. Let Γ be a congruence subgroup of Γg = Sp(2g, Z). If f ∈ Mk(Γ ), then
det
((
∂
∂ Z−
√−1k
2Y −1
)
f
)
∈ ˜Mkg+2(Γ ).
Proof. For f ∈ Mk(Γ ), we have f 2 ∈ M2k(Γ ). Similar to the proof of Proposition 2, wehave
D( f 2 · det(d Z)k) = 2 f · d f · det(d Z)k + f 2 · D(det(d Z)k)
= 2 f · Tr
([
∂
∂ Z−
√−1k
2Y −1
]
f d Z
)
det(d Z)k .
Now by the same reason as Theorem 3, we get that det((
∂∂ Z −
√−1k2 Y −1
)
f)
∈ ˜Mkg+2(Γ ).
If f ∈ M2r (Γ ) and h ∈ M2s(Γ ), then
D( f det d Zr ) · h det d Zs − f det d Zr · D(h det d Zs)
= Tr
([
h∂
∂ Zf − f
∂
∂ Zh
]
d Z
)
det d Zr+s
is invariant under Γ , hence
det
(
h∂
∂ Zf − f
∂
∂ Zh
)
∈ M2(r+s)g+2(Γ ).
We can continue this construction to find combinations of higher derivatives of f and hwhich are modular. By setting [ f, h]0 := f h, [ f, h]1 := det(h ∂
∂ Z f − f ∂∂ Z h), and so on,
one would get the Rankin-Cohen brackets.
Derivatives of Siegel modular forms and modular connections
2.4. The uniqueness theorem
We first show a lemma.
Lemma 9. Let γ =(
A BC D
)
∈ Sp(2g, Z). Then
(C Z + D)−1√−1(Imγ (Z))−1 = √−1Y −1 · (C Z + D)t + 2Ct .
Proof. Since Im(γ (Z)) = ((C Z̄ + D)t )−1Y (C Z + D)−1 (see [1]) and Y t = Y , we have
Imγ (Z)−1 = (C Z + D)Y −1 · (C Z̄ + D)t = (C Z + D)Y −1 · (C Z + D − 2√−1CY )t
= (C Z + D)Y −1 · (C Z + D)t − (C Z + D)2√−1Ct ,
and thus we get the result.
Theorem 4. Let Γ be a subgroup of Sp(2g, R). Any symmetric g × g matrix G = (Gi j ) ofC
∞-functions on Hg, which satisfies the transformation formula
(C Z + D)−1γ (G) = G · (C Z + D)t + 2Ct ,
for any γ =(
A BC D
)
∈ Γ , gives a Γ -modular connection D such that for any C∞-function
f on Hg
D(d Zrs) = −g∑
i, j=1
Gi j d Zsi d Zr j
and
D( f (det(d Z)k) = {d f − kTr(Gd Z) f } (det(d Z))k .
If Γ is a congruence subgroup of Γg and if f ∈ M2k(Γ ), then for such a G, we have
det
([
∂
∂ Z− kG
]
f
)
∈ ˜M2gk+2(Γ ).
Furthermore, if Γ = Γg is the full Siegel modular group, then there exists at most oneholomorphic matrix G satisfying the transformation formula.
Proof. One notes that in the definition of modular connection coefficients Γ KI J , we only
need the transformation law γ (ω) = −S−1 · d S + S−1 · ω · S for γ ∈ Γ . If G has thesame transformation law as
√−1(Im(Z)−1), then we can use the same method in Lemma6 to construct {Γ K
I J } and to calculate the expressions of the differential forms under D. If fbelongs to M2k(Γ ), then
Tr
([
∂
∂ Z− kG
]
f d Z
)
det(d Z)k
is invariant under the action of Γ by Lemma 2. Hence by a similar reason as Theorem 3, wehave det(( ∂
∂ Z − kG) f ) ∈ ˜M2kg+2(Γ ).
If Γ = Γg , let G and G̃ be two holomorphic matrices satisfying the transformationformula, then
(C Z + D)−1(G(γ Z) − G̃(γ Z)) = (G(Z) − G̃(Z))(C Z + D)t
E. Yang, L. Yin
for all γ ∈ Γ . So Tr{(G − G̃)d Z} ∈ (Ω1Hg
)Γg . In [17] and [18] R. Weissauer proved that
if v is not of the form [u] := ug − 12 u(u − 1), then (Ωv
Hg)Γg = 0. If g ≥ 2, one gets
(Ω1Hg
)Γg = 0, and thus G = G̃. The case g = 1 has been proved in Lemma 5.
Problem 1. Does there exist such G? If so, how to construct it?
Due to the referees comments, we make the following two remarks.
Remark 1. The uniqueness theorem for holomorphic modular connection may not be truefor congruence subgroups. For example, consider the space S2(Γ0(11)) of cusp forms ofweight 2 for the congruence subgroup
Γ0(11) ={(
a bc d
)
∈ SL(2, Z) | c ≡ 0 mod 11
}
.
Then dimS2(Γ0(11)) = 1 (cf. [11], P130). Let G2(z) be the Eisenstein Series of weight 2given in Sect. 1.5. Choose any nonzero element Φ(z) ∈ S2(Γ0(11)), then
√−1(G2(z) +Φ(z)) determines a holomorphic Γ0(11)-modular connection on the upper half plane. Hencethe holomorphic Γ0(11)-modular connection is not unique in this case.
One can also consider similar settings in the cases of Hilbert–Siegel modular forms andSiegel–Jacobi forms. But one needs much more complicated calculations for Siegel–Jacobiforms(cf. [19]).
Remark 2. In the case g = 1. Let f be a modular form of weight 2k. Then d fdz −
√−1ky f
can be constructed from Lie-algebra in the following way: Let R = 12
( 1√−1√−1 −1
)
be an
element in the complexification of the Lie algebra gl(2, R). The element R acts on the spaceMask(H1) of Maass forms of weight 2k on the upper half plane H1 (See [4], P137). By ([4],Proposition 2.2.5), the operator defined by R can be identified with the so-called Maassraising operator:
Rk = 2√−1y
∂
∂z+ k.
Moreover, Rk maps Mask(H1) to Mask+1(H1). For the modular form f , Φ f := yk f (z) is
a Maass form of weight 2k. Then Rk(Φ f ) = 2√−1yk+1(
d fdz −
√−1ky f ) is a Maass form
of weight 2k + 2. Hence the operator d fdz −
√−1ky f can be constructed from the element R.
However, the Serre’s derivation d fdz −√−1kG2(z) f (cf. Corollary 1) can not be constructed
in this way.In the case g > 1, since our operators given in Theorem 3 and Theorem 4 are not linear,
they cannot be constructed form Lie algebra point of view either. Finally, we remark that thecenter of the universal enveloping algebra of Sp(2g, R) is generated by g Casimir operatorsΔ2,Δ4, · · · , Δ2g (cf. [7], Proposition 2.3.3). These operators don’t change the weight ofSiegel Maass forms.
3. Explicit construction of the Levi–Civita connection
In this section we give the proof of Lemma 6. We use I, J, K , L , · · · to describe the elementsof Ω, i, j, k, l, r, s, · · · to denote the elements of {1, 2, · · · , g} and α, β, γ, δ, ε to denotethe elements of {1, 2, · · · , g(g + 1)/2}.
Derivatives of Siegel modular forms and modular connections
3.1. Riemann-metric
Let R := (Ri j )g×g = Y −1. Then
ds2 = Tr(d Z · R · d Z · R) =g∑
i=1
g∑
j=1
g∑
r=1
g∑
s=1
Rir R jsd Zi j d Zrs
=g∑
i=1
g∑
r=1
R2ir d Zii d Zrr +
g∑
i=1
∑
r<s2Rir Risd Zii d Zrs
+g∑
r=1
g∑
i< j
2Rir R jr d Zi j d Zrr +∑
i< j
∑
r<s2(Rir R js + R jr Ris)d Zi j d Zrs
=∑
i≤ j
∑
r≤s
22−δ(i, j)−δ(r,s) × Rir R js + R jr Ris
2d Zi j d Zrs
Thus the Riemann-metric matrix associated to ds2 = Tr(d Z · Y −1 · d Z · Y −1) is given by
G =(
0 WW 0
)
where
W = (WI J )I,J∈Ω, WI J = Rir R js + R jr Ris
2δ(i, j)+δ(r,s), if I = (i, j) and J = (r, s).
Lemma 10. The inverse W−1 of W is given by
M := (MI J )I,J∈Ω, MI J = Yir Y js + Y jr Yis , if I = (i, j) and J = (r, s).
Proof. We need to show that for any I = (i, j) ∈ Ω and K = (p, q) ∈ Ω ,∑
J∈Ω
MI J WJ K = δ(I, K ).
By direct computations, we have
∑
1≤r≤s≤g
M(i, j),(r,s)W(r,s),(p,q) =∑
1≤r≤s≤g
(Yir Y js + Y jr Yis)Rr p Rsq + Rsp Rrq
2δ(r,s)+δ(p,q)
=∑
1≤r≤g
21−δ(p,q)Yir Y jr Rr p Rrq +∑
1≤r<s≤g
(Yir Y js + Y jr Yis)Rr p Rsq + Rsp Rrq
2δ(p,q)
= 2−δ(p,q)[
2∑
1≤r≤g
Yir Y jr Rr p Rrq +∑
1≤r<s≤g
(Yir Y js Rr p Rsq + Y jr Yis Rr p Rsq )
+∑
1≤r<s≤g
(Yir Y js Rsp Rrq + Y jr Yis Rsp Rrq )]
= 2−δ(p,q)∑
1≤r≤g
∑
1≤s≤g
(Yir Y js Rr p Rsq + Y jr Yis Rr p Rsq )
= 2−δ(p,q)∑
1≤r≤g
(Yir Rr pδ( j, q) + Y jr Rr pδ(i, q))
= 2−δ(p,q){δ(i, p)δ( j, q) + δ( j, p)δ(i, q)}.
E. Yang, L. Yin
Notice that, in the last three steps, we have used the equality
∑
1≤r≤g
Yir Rr p = δ(r, p),
which comes from R = Y −1. Since i ≤ j and p ≤ q , by an easy calculation, we get:
2−δ(p,q){δ(i, p)δ( j, q) + δ( j, p)δ(i, q)} = δ(I, K ).
3.2. Connection Coefficients
As before, we put uN (i, j) = Zi j and uN (g,g)+N (i, j) = Zi j ,
G =(
0 WW 0
)
, ̂G := G−1 =(
0 MM 0
)
.
By the formula (1), we have
Γγα,β =
∑
1≤ρ≤g(g+1)
1
2̂Gγ,ρ
(
∂Gα,ρ
∂uβ+ ∂Gβ,ρ
∂uα− ∂Gα,β
∂uρ
)
.
If 0 ≤ α, β, γ ≤ g(g+1)2 , then Gα,β = 0. We have:
Γγα,β =
∑
1≤ρ≤g(g+1)
1
2̂Gγ,ρ
(
∂Gα,ρ
∂uβ+ ∂Gβ,ρ
∂uα
)
.
Hence
Γ KI,J =
∑
L∈Ω
1
2MK ,L
(
∂WI,L
∂ Z J+ ∂WJ,L
∂ Z I
)
.
Again, notice that
∑
L∈Ω
MK ,L WI,L = δ(K , I ) and∑
L∈Ω
MK ,L WJ,L = δ(K , J ).
Do partial derivatives on both sides with respect to Z J and Z I respectively, we have
∑
L∈Ω
MK ,L∂WI,L
∂ Z J+∑
L∈Ω
∂ MK ,L
∂ Z JWI,L = 0,
and
∑
L∈Ω
MK ,L∂WJ,L
∂ Z I+∑
L∈Ω
∂ MK ,L
∂ Z IWJ,L = 0.
Finally we get
Γ KI,J = −1
2
⎛
⎝
∑
L∈Ω
∂ MK ,L
∂ Z JWI,L +
∑
L∈Ω
∂ MK ,L
∂ Z IWJ,L
⎞
⎠ .
Derivatives of Siegel modular forms and modular connections
If I = (i, j), J = (r, s) and L = (a, b) ∈ Ω , then MI,J = Yir Y js + Y jr Yis and
∂ MK ,L
∂ Z J= ∂(YpaYqb + YqaYpb)
∂ Z J
= −√−1
2
{
σ(p,a),(r,s)Yqb + σ(q,b),(r,s)Ypa + σ(q,a),(r,s)Ypb + σ(p,b),(r,s)Yqa}
Here we define:
σ(p,a),(r,s) ={
1, if Z pa = Zrs ,
0, if Z pa �= Zrs .
One should notice the difference of the notation above with the notation δ(p,a),(r,s) :=δ((p, a), (r, s)) = δ(p, r)δ(a, s). These two notations have the following relations:
σ(p,a),(r,s) = δ(p,a),(r,s) + δ(p,a),(s,r) − δ(p,a),(r,s) · δ(p,a),(s,r).
Using the equality WI J = Rir R js+R jr Ris
2δ(i, j)+δ(r,s) and others above, we have
∑
L∈Ω
∂ MK ,L
∂ Z JWI,L = −
√−1
2
∑
L=(a,b)∈Ω
{σ(p,a),(r,s)Yqb + σ(q,b),(r,s)Ypa
+ σ(q,a),(r,s)Ypb + σ(p,b),(r,s)Yqa}WI,L
= −√−1
2
∑
L=(a,b)∈Ω
{σ(p,a),(r,s)Yqb + σ(q,b),(r,s)Ypa + σ(q,a),(r,s)Ypb +
+σ(p,b),(r,s)Yqa} × Ria R jb + R ja Rib
2δ(i, j)+δ(a,b)
= −√−1
2
∑
1≤a≤g
21−δ(i, j){σ(p,a),(r,s)Yqa Ria R ja + σ(q,a),(r,s)Ypa Ria R ja}
+∑
1≤a<b≤g
2−δ(i, j){σ(p,a),(r,s)Yqb Ria R jb + σ(p,b),(r,s)Yqa Ria R jb
+ σ(p,b),(r,s)Yqb R ja Rib + σ(p,b),(r,s)Yqa R ja Rib + σ(q,a),(r,s)Ypb Ria R jb
+ σ(q,b),(r,s)Ypa Ria R jb + σ(q,a),(r,s)Ypb R ja Rib + σ(q,b),(r,s)Ypa R ja Rib}
= −√−1
21+δ(i, j)
∑
1≤a≤g
∑
1≤b≤g
{σ(p,a),(r,s)Yqb Ria R jb + σ(p,b),(r,s)Yqa Ria R jb
+σ(q,a),(r,s)Ypb R ja Rib + σ(q,b),(r,s)Ypa R ja Rib}
= −√−1
21+δ(i, j)
⎧
⎨
⎩
∑
1≤a≤g
σ(p,a),(r,s)δ(q, j)Ria +∑
1≤b≤g
σ(p,b),(r,s)δ(q, i)R jb
+∑
1≤a≤g
σ(q,a),(r,s)δ(p, i)R ja +∑
1≤b≤g
σ(q,b),(r,s)δ(p, j)Rib
⎫
⎬
⎭
.
E. Yang, L. Yin
While∑
1≤a≤g
σ(p,a),(r,s)δ(q, j)Ria =∑
1≤a≤g
{δ(p, r)δ(a, s) + δ(p, s)δ(a, r)
−δ(p, r)δ(a, s)δ(p, s)δ(a, r)}δ(q, j)Ria
= δ(q, j){δ(p, r)Ris + δ(p, s)Rir − δ(p, r)δ(p, s)Ris},∑
1≤b≤g
σ(p,b),(r,s)δ(q, i)R jb = δ(q, i){δ(p, r)R js + δ(p, s)R jr − δ(p, r)δ(p, s)R js},∑
1≤a≤g
σ(q,a),(r,s)δ(p, i)R ja = δ(p, i){δ(q, r)R js + δ(q, s)R jr − δ(q, r)δ(q, s)R js},∑
1≤b≤g
σ(q,b),(r,s)δ(p, j)Rib = δ(p, j){δ(q, r)Ris + δ(q, s)Rir − δ(q, r)δ(q, s)Ris}.
Combining these equalities together, we have
∑
L∈Ω
∂ MK ,L
∂ Z JWI,L = −
√−1
21+δ(i, j){δ(q, j){δ(p, r)Ris + δ(p, s)Rir − δ(p, r)δ(p, s)Ris}
+ δ(q, i){δ(p, r)R js + δ(p, s)R jr − δ(p, r)δ(p, s)R js}+ δ(p, i){δ(q, r)R js + δ(q, s)R jr − δ(q, r)δ(q, s)R js}+ δ(p, j){δ(q, r)Ris + δ(q, s)Rir − δ(q, r)δ(q, s)Ris}}
Similarly, we have
∑
L∈Ω
∂ MK ,L
∂ Z IWJ,L = −
√−1
21+δ(r,s){δ(q, s){δ(p, i)Rr j + δ(p, j)Rri − δ(p, i)δ(p, j)Rr j }
+ δ(q, r){δ(p, i)Rsj + δ(p, j)Rsi − δ(p, i)δ(p, j)Rsj }+ δ(p, r){δ(q, i)Rsj + δ(q, j)Rsi − δ(q, i)δ(q, j)Rsj }+ δ(p, s){δ(q, i)Rr j + δ(q, j)Rir − δ(q, i)δ(q, j)Rr j }}
Finally we get
Lemma 11.
Γ KI,J = −1
2
⎛
⎝
∑
L∈Ω
∂ MK ,L
∂ Z JWI,L +
∑
L∈Ω
∂ MK ,L
∂ Z IWJ,L
⎞
⎠
=√−1
22+δ(i, j){δ(q, j){δ(p, r)Ris + δ(p, s)Rir − δ(p, r)δ(p, s)Ris}
+ δ(q, i){δ(p, r)R js + δ(p, s)R jr − δ(p, r)δ(p, s)R js}+ δ(p, i){δ(q, r)R js + δ(q, s)R jr − δ(q, r)δ(q, s)R js}+ δ(p, j){δ(q, r)Ris + δ(q, s)Rir − δ(q, r)δ(q, s)Ris}}+
√−1
22+δ(r,s){δ(q, s){δ(p, i)Rr j + δ(p, j)Rri − δ(p, i)δ(p, j)Rr j }
+δ(q, r){δ(p, i)Rsj + δ(p, j)Rsi − δ(p, i)δ(p, j)Rsj }+ δ(p, r){δ(q, i)Rsj + δ(q, j)Rsi − δ(q, i)δ(q, j)Rsj }+ δ(p, s){δ(q, i)Rr j + δ(q, j)Rir − δ(q, i)δ(q, j)Rr j }}
Derivatives of Siegel modular forms and modular connections
Using the lemma 11 above, one can easily show that in the case K = (p, q) = (1, 1), I =(i, j) = (1, j), J = (r, s) = (1, s), we have
– if j = s = 1,then Γ KI,J = √−1R1,1;
– if j = 1, s �= 1,then Γ KI,J = √−1R1,s ;
– if j �= 1, s �= 1,then Γ KI,J = √−1R j,s = √−1Rs, j .
In general case, if K = (p, q), I = (i, j), J = (r, s), and if both (Zi j , Zrs) and (Zrs , Zi j )
do not belong to {Z1p, Z2p, · · · , Zgp} × {Z1q , Z2q , · · · , Zgq }, then all terms in the last
equality of lemma 11 are zero, hence Γ KI,J = Γ K
J,I = 0.If p = q , then
Γ(p,p)I,J =
√−1
21+δ(i, j){δ(p, i){δ(p, r)R js + δ(p, s)R jr − δ(p, r)δ(p, s)R jr }
+δ(p, j){δ(p, r)Ris + δ(p, s)Rir − δ(p, r)δ(p, s)Rir }}+
√−1
21+δ(r,s){δ(p, r){δ(p, i)Rsj + δ(p, j)Rsi − δ(p, i)δ(p, j)Rsj }
+δ(p, s){δ(p, i)Rr j + δ(p, j)Rri − δ(p, i)δ(p, j)Rr j }}If Zi j = Z ji and Zrs = Zsr belong to the same row or column with Z pp , then i = r = p,or i = s = p, or j = r = p, or j = s = p.
– If i = r = p, one can use the formula above to show that Γ KI,J = √−1R js
– If i = s = p, then Γ KI,J = √−1R jr .
– If j = r = p, then Γ KI,J = √−1Ris .
– If j = s = p, then Γ KI,J = √−1Rir .
If p < q , we may assume that Zi j (i ≤ j) belong to the same row with Z pq andZrs(r ≤ s) belongs to the same column with Z pq (Other cases can be proved in the sameway). Then i = p ≤ j , r ≤ s = q and
Γ KI,J =
√−1
22+δ(i, j)(δ(q, j)δ(p, r)Ris + R jr + δ(p, j)Rir )
+√−1
22+δ(r,s)(δ(p, r)δ(q, j)Rsi + Rr j + δ(q, r)Rsj ).
– If i = p = j ≤ q and r < s = q , then
Γ KI,J =
√−1
8(R jr + Rir ) +
√−1
4Rr j =
√−1
2R jr .
– If i = p = j and r = s = q , then Γ KI,J =
√−12 R jr .
– If i = p < j and r < s = q , then
Γ KI,J =
√−1
4(δ(q, j)δ(p, r)Ris + R jr ) +
√−1
4(δ(p, r)δ(q, j)Ris + Rr j )
=√−1
2(δ(q, j)δ(p, r)Rpq + R jr )
={
Rpq if j = q and r = p,√−12 R jr otherwise.
– If i = p < j and r = s = q , then Γ KI,J =
√−12 R jr .
E. Yang, L. Yin
At last, we complete the proof of Lemma 6.
Acknowledgements. The authors are grateful to the referees for helpful remarks and sug-gestions.
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