# Derivations of Siegel modular forms from connections

Enlin Yang Department of Mathematical Science, Tsinghua University, Beijing,
P. R. China 100084<EMAIL_ADDRESS>and  Linsheng Yin Department of
Mathematical Science, Tsinghua University, Beijing, P. R. China 100084
<EMAIL_ADDRESS>
###### Abstract.

We introduce a method in differential geometry to study the derivative
operators of Siegel modular forms. By determining the coefficients of the
invariant Levi-Civita connection on a Siegel upper half plane, and further by
calculating the expressions of the differential forms under this connection,
we get a non-holomorphic derivative operator of the Siegel modular forms. In
order to get a holomorphic derivative operator, we introduce a weaker notion,
called modular connection, on the Siegel upper half plane than a connection in
differential geometry. Then we show that on a Siegel upper half plane there
exists at most one holomorphic modular connection in some sense, and get a
possible holomorphic derivative operator of Siegel modular forms.

###### Key words and phrases:

Levi-Civita connection, Siegel modular form, differential operator

## Introduction

In this paper, we introduce a differential geometric method to study the
derivative operators of Siegel modular forms, which, theoretically, may be
applied to the study of the derivative operators of any automorphic form. Our
idea comes from the observation on the two derivative operators of the
classical modular forms constructed by combinations. It is well-known [13]
that if $f$ is a modular forms of weight $2k$, then
$D_{k}f:=\frac{df}{dz}-\frac{\sqrt{-1}k}{y}f$ is a non-holomorphic modular
forms of weight $2k+2$, and $D_{k}f:=\frac{df}{dz}-\sqrt{-1}kG_{2}(z)f$, due
to J.-P. Serre [9], is a holomorphic modular forms of weight $2k+2$, where
$G_{2}(z)$ is the Eisenstein series of weight 2. We notice that the first
operator can be constructed by the Levi-Civita connection corresponding to the
invariant metric in the classical upper half plane, but the second can not be
constructed from any connection. However, if we loosen some condition in the
definition of the connection and define a concept called modular connection,
we can get Serre’s holomorphic derivative from the unique holomorphic modular
connection on the upper half plane. In this paper we extend these results to
Siegel upper half planes and Siegel modular forms. We determine the
coefficients of the Levi-Civita connection corresponding to the invariant
metric in a Siegel upper half plane, and compute the expressions of the
differential forms under the connection, which give us a non-holomorphic
derivative operator of Siegel modular forms. Our main results are as follows.

Let $\mathbb{H}_{g}$ be the Siegel upper plane of degree $g$,
$\\{dZ_{ij}:1\leq i,j\leq g\\}$ a series of coordinates on $\mathbb{H}_{g}$,
$\Gamma_{g}=\mbox{\rm Sp}(2g,\mathbb{Z})$ the full Siegel modular group which
acts on $\mathbb{H}_{g}$ naturally, $M_{k}=M_{k}(\Gamma_{g})$ the vector space
of the classical (or scalar-valued) Siegel modular forms of weight $k$,
$\widetilde{M}_{k}=\widetilde{M}_{k}(\Gamma_{g})$ the
$\mathbb{C}^{\infty}$-Siegel modular forms of weight $k$. Put

$\frac{\partial}{\partial Z}=(\partial_{ij})_{g\times
g}\qquad\text{and}\qquad\partial_{ij}=\frac{1}{2^{1-\delta(i,j)}}\cdot\frac{\partial}{\partial
Z_{ij}},$

where $\delta(i,j)=1$ if $i=j$ and $\delta(i,j)=0$ if $i\neq j$.

######  Theorem 0.1 (See theorem 2.7).

Let $f\in M_{2k}(\Gamma_{g})$. Then

$\det\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]f\right)\in\widetilde{M}_{2gk+2}(\Gamma_{g}).$

Here $\Gamma_{g}$ can be replaced by any congruence subgroup and the weight
$2k$ can also be replaced by any positive integer, with a little modification
of our proofs.

To get a holomorphic derivative operator, we introduce the notion of modular
connections on the Siegel upper half plane, whose condition is weaker than the
classical definition of the connections. Then we show the following result.

######  Theorem 0.2 (See theorem 2.9).

Any symmetric $g\times g$ matrix $G(Z)=(G_{ij}(Z))$ consisting of
$\mathbb{C}^{\infty}$ functions on $\mathbb{H}_{g}$, which satisfies the
transformation formula

$(CZ+D)^{-1}G(\gamma(Z))=G(Z)\cdot(CZ+D)^{t}+2C^{t}$

for any $\gamma=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\mbox{\rm Sp}(2g,\mathbb{Z})$, gives a unique modular
connection $\mathbb{D}$ such that for any $\mathbb{C}^{\infty}$-function $f$
on $\mathbb{H}_{g}$

$\mathbb{D}(dZ_{rs})=-\sum_{i,j=1}^{g}G_{ij}dZ_{si}dZ_{rj}\quad\text{and}\quad\mathbb{D}(f(\det(dZ)^{k})=\mbox{\rm
Tr}\left(\left[\frac{\partial}{\partial Z}-kG\right]fdZ\right)(\det(dZ))^{k},$

and thus gives a derivative operator $M_{2k}\rightarrow\widetilde{M}_{2kg+2}$
by $f\mapsto\det\left(\left[\frac{\partial}{\partial Z}-kG\right]f\right)$.
Furthermore, there exists at most one holomorphic symmetric matrix $G$ to
satisfy the transformation formula. If such a $G$ exists, the operator
corresponding to $G$ is holomorphic.

In the classical case of $g=1$, the function $\sqrt{-1}G_{2}(z)$ is the unique
holomorphic function on the upper half plane satisfying the condition, which
gives Serre’s derivative. But when $g\geq 2$ we are not able to construct such
a matrix function $G$.

H. Maass has constructed a non-holomorphic derivative operator of Siegel
modular forms by invariant differential operators. For Siegel modular forms
$f$ of weight $k$, Maass ([7], P317) defines the operator

$D_{k}f(Z)=\det(Y)^{\kappa-k-1}\det\left(\frac{\partial}{\partial
Z}\right)[\det(Y)^{k+1-\kappa}f(Z)],$

where $\kappa=(g+1)/2$ and the determinant of $\frac{\partial}{\partial Z}$ is
taken first, and shows that the differential operator $D_{k}$ acts on the
$\mathbb{C}^{\infty}$-Siegel modular forms and maps $\widetilde{M}_{k}$ to
$\widetilde{M}_{k+2}$. We do not know the relation between our operator in
Theorem 0.1 and Maass’. Compared to our operator, $D_{k}$ is linear with
respect to $f$. Moreover, our operator is a combination of degree 1 partial
derivatives of $f$, but $D_{k}$ is a combination of degree $g$ partial
derivatives. G. Shimura [8] considers the compositions
$D_{r}^{k}=D_{r+2k-2}\cdots D_{r+2}D_{r}$ of Maass’ operator, which maps
$\widetilde{M}_{r}$ to $\widetilde{M}_{r+2k}$. For our operator one can also
consider the compositions and then construct the Rankin-Cohen brackets. We
wish that Maass’ operator could be got in this way.

The paper is organized as follows. In section one, we introduce the concept of
modular connection on a Siegel upper plane, and show several lemmas on it. In
section two, we compute the expressions of the differential forms under the
modular connection, and prove the two theorems above. Finally in section
three, we show Lemma 2.1 which explicitly gives the connection coefficients of
the Levi-Civita connection on a Siegel upper half plane.

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 we introduce the notion of modular connection on a Siegel upper
half plane, and show several lemmas about it.

###  1.1. Connections in differential geometry

For the backgrounds and notations on differential geometry, especially on
connections, we refer to the books [2] and [4]. Here we just recall some basic
definitions and results on connections. Suppose $E$ is a $q$-dimensional real
vector bundle on a smooth manifold $M$, and $\Gamma(E)$ is the set of smooth
sections of $E$ on $M$. Let $T^{*}(M)$ be the cotangent space of $M$. A
connection on the vector bundle $E$ is a map

$D:\quad\Gamma(E)\longrightarrow\Gamma(T^{*}(M)\otimes E),$

which satisfies the following conditions

  1. (1)

For any $s_{1},s_{2}\in\Gamma(E)$,

$D(s_{1}+s_{2})=D(s_{1})+D(s_{2}).$

  2. (2)

For any $s\in\Gamma(E)$ and any $\alpha\in\mathbb{C}^{\infty}(M)$,

$D(\alpha s)=d\alpha\otimes\alpha D(s).$

If $M$ has a generalized Riemannian metric $G=\sum_{i,j}g_{ij}du^{i}du^{j}$,
by the fundamental theorem of Riemannian geometry, $M$ has a unique torsion-
free and metric-compatible connection, called Levi-Civita connection of $M$.
The coefficients $\Gamma_{ij}^{k}$ of the Levi-Civita connection are given by

$\Gamma_{ij}^{k}=\frac{1}{2}\sum_{l}g^{kl}\left(\frac{\partial
g_{il}}{\partial u^{j}}+\frac{\partial g_{jl}}{\partial u^{i}}-\frac{\partial
g_{ij}}{\partial u^{l}}\right),$ (1.1)

where $g^{ij}$ are elements of the matrix $(g^{ij}):=(g_{ij})^{-1}$.

The following lemma is useful in the application of connections to automorphic
forms.

######  Lemma 1.1.

Let $\Gamma$ be a group, $(M,G)$ a Riemannian manifold and $D$ the Levi-Civita
connection on $M$. If $\Gamma$ has a smooth left action on $M$ such that
$G(\sigma_{\star}X,\sigma_{\star}Y)=G(X,Y)$ for all $\sigma\in\Gamma,X,Y\in
T(M)$, then

$\sigma D=D\sigma\qquad(\sigma\in\Gamma).$

Moreover, if $M$ is a complex manifold such that $\Gamma$ maps $(r,s)$ forms
to $(r,s)$ forms, and put $D=D^{1,0}+D^{0,1}$, where $D^{1,0}$ is the
holomorphic part, then for $\sigma\in\Gamma$

$\sigma D^{1,0}=D^{1,0}\sigma\quad\text{ and }\quad\sigma
D^{0,1}=D^{0,1}\sigma.$

###### Proof.

$D$ is the unique torsion free connection which preserves the Riemannian
metric $G$. Since $G$ is $\Gamma$-invariant, the connection
$\sigma^{-1}D\sigma$ also preserves the Riemannian metric and is torsion free
for any $\sigma\in\Gamma$, hence $\sigma D=D\sigma$. For more detail, see
([10], P35). ∎

###  1.2. Seigel upper half plane

We first fix some notations. The Siegel upper half plane of degree $g\geq 1$
is defined to be the $g(g+1)/2$ dimensional open complex variety

$\mathbb{H}_{g}:=\\{Z=X+\sqrt{-1}Y\in M(g,\mathbb{C})\mid Z^{t}=Z,Y>0\\}.$

Write $Z=(Z_{ij})$. Set $\Omega=\\{(i,j)\mid 1\leq i\leq j\leq g\\}$ with the
dictionary order. If $I=(i,j)\in\Omega$, we define $Z_{I}:=Z_{ij}$. Fix a
series of coordinates $\\{dZ_{I},d\bar{Z}_{I}\mid I\in\Omega\\}$ on
$\mathbb{H}_{g}$. The symplectic group of degree $g>0$ over $\mathbb{R}$ is
the group

$\mbox{\rm Sp}(2g,\mathbb{R})=\left\\{M\in
GL(2g,\mathbb{R})\,\middle|\,MJM^{t}=J\right\\},$

where $J=\left(\begin{array}[]{cc}0&I_{g}\\\ -I_{g}&0\end{array}\right)$. We
usually write an element of $\mbox{\rm Sp}(2g,\mathbb{R})$ in the form
$\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)$, where $A,B,C$ and $D$
are $g\times g$ blocks. The symplectic group $\mbox{\rm Sp}(2g,\mathbb{R})$
acts on $\mathbb{H}_{g}$ by the rule:

$\gamma(Z):=(AZ+B)(CZ+D)^{-1},\quad
Z\in\mathbb{H}_{g},\quad\gamma=\left(\begin{array}[]{cc}A&B\\\
C&D\end{array}\right)\in\mbox{\rm Sp}(2g,\mathbb{R}).$

By Maass ([6], P98), $d(\gamma
Z)=(ZC^{t}+D^{t})^{-1}dZ(CZ+D)^{-1}:=(d\tilde{Z}_{ij})$. Let

$(d\tilde{Z}_{11},d\tilde{Z}_{12},\cdots,d\tilde{Z}_{1g},\cdots,\cdots,d\tilde{Z}_{gg})=(dZ_{11},dZ_{12},\cdots,dZ_{1g},\cdots,\cdots,dZ_{gg})\cdot
S(\gamma,Z)$

where $S:=S(\gamma,Z)$ is a $\frac{g(g+1)}{2}\times\frac{g(g+1)}{2}$ matrix of
holomorphic functions on $\mbox{\rm Sp}(2g,\mathbb{Z})\times\mathbb{H}_{g}$.

From Lemma 1.1, one can see that the connection matrix $\omega$ consisting of
the connection coefficients of the Levi-Civita connection associated to the
invariant metric $ds^{2}=\mbox{\rm Tr}(Y^{-1}dZ\cdot Y^{-1}d\bar{Z})$ given by
Siegel ([6], P8) on the Siegel upper plane $\mathbb{H}_{g}$ satisfies

$\gamma(\omega)=-S^{-1}\cdot dS+S^{-1}\cdot\omega\cdot S$

for all $\gamma\in\mbox{\rm Sp}(2g,\mathbb{R})$. Refer also to the proof of
Lemma 1.5 below. But in the studying of modular forms, we only need that the
equality holds for all $\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$, the Siegel
modular group. So we need to introduce a weaker notion to study modular forms.

Now we recall the definition of Siegel modular forms, for more details, see
[1] and [3].

######  Definition 1.2.

A (classical) Siegel modular form of weight $k$ (and degree $g$) is a
holomorphic function $f:\mathbb{H}_{g}\rightarrow\mathbb{C}$ such that

$f(\gamma(Z))=\det(CZ+D)^{k}f(Z)$

for all $\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\\\
\end{array}\right)\in\mbox{\rm Sp}(2g,\mathbb{Z})$ (with the usual
holomorphicity requirement at $\infty$ when $g=1$).

###  1.3. Modular connections

The notations are the same as those above.

######  Definition 1.3 (Modular Connection Coefficients (MCC)).

The modular connection coefficient on $\mathbb{H}_{g}$ is a series of
$\mathbb{C}^{\infty}$-functions $\\{\Gamma_{IJ}^{K}\mid I,J,K\in\Omega\\}$
such that for all $\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$,

$\gamma(\omega)=-S^{-1}\cdot dS+S^{-1}\cdot\omega\cdot S,\quad\text{or}\quad
S\cdot\gamma(\omega)=\omega\cdot S-dS,$

where $\omega=(\omega_{I}^{J})$ and
$\omega_{I}^{J}=\sum_{K\in\Omega}\Gamma_{IK}^{J}dZ_{K}$. Here $I$ and $J$ are
the row and column indices respectively. When $\\{\Gamma_{IJ}^{K}\\}$ are
holomorphic, we call it holomorphic MCC (HMCC). The matrix $\omega$ is called
the modular connection matrix.

In the following, $\mathbb{C}^{\infty}(\mathbb{H}_{g})$ is the set of
$\mathbb{C}^{\infty}$ functions on $\mathbb{H}_{g}$, and
$\text{Hol}(\mathbb{H}_{g})$ is the set of holomorphic functions on
$\mathbb{H}_{g}$.

######  Definition 1.4 (Modular Connection).

Let $\\{\Gamma_{IJ}^{K}\\}$ be a MCC (resp. HMCC) on $\mathbb{H}_{g}$ and
$\Omega^{\infty}$ be the commutative
$\mathbb{C}^{\infty}(\mathbb{H}_{g})$-algebra (resp.
$\text{Hol}(\mathbb{H}_{g})$-algebra) generated by $\\{dZ_{I}\\}_{I\in\Omega}$
with the relations $dZ_{I}dZ_{J}=dZ_{J}dZ_{I}$ for any $I,J\in\Omega$. The
linear operator

$D:\Omega^{\infty}\longrightarrow\Omega^{\infty}$

is uniquely defined by the following two relations

$D(dZ_{K})=-\sum_{I,J\in\Omega}\Gamma_{IJ}^{K}dZ_{I}dZ_{J}$

and

$D(fdZ_{K_{1}}dZ_{K_{2}}\cdots dZ_{K_{r}})=df\cdot dZ_{K_{1}}\cdots
dZ_{K_{r}}+\sum_{i=1}^{r}fdZ_{K_{1}}dZ_{K_{2}}\cdots D(dZ_{K_{i}})\cdots
dZ_{K_{r}},$

and we call it the modular connection associated to $\\{\Gamma_{IJ}^{K}\\}$.

One can easily show that $D(dZ_{K})=-\sum_{I\in\Omega}\omega_{I}^{K}\cdot
dZ_{I}$ and

$(D(dZ_{11}),D(dZ_{12}),\cdots,\cdots,\cdots,D(dZ_{gg}))=-(dZ_{11},dZ_{12},\cdots,\cdots,\cdots,dZ_{gg})\cdot\omega$

When $\\{\Gamma_{IJ}^{K}\\}$ is holomorphic, we also call $D$ a holomorphic
modular connection. Compared with the definition of connections in
differential geometry, except for the weaker conditions, the modular
connection also ignore the part on $\\{d\bar{Z}_{I}\\}_{I\in\Omega}$.

###  1.4. Basic lemmas on modular connections

The following two lemmas are basic to our application of modular connections
to the Siegel modular forms. For the modular connections, we have the similar
result to Lemma 1.1.

######  Lemma 1.5.

Let $D$ be a modular connection on $\mathbb{H}_{g}$. Then $\gamma D=D\gamma$
for any $\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$. Moreover, if $f$ is a Siegel
modular forms of weight $2k$, then $D\left(f(\det(dZ))^{k}\right)$ is
invariant under the action of $\Gamma_{g}=\mbox{\rm Sp}(2g,\mathbb{Z})$.

###### Proof.

Let $\alpha=(dZ_{11},\cdots,dZ_{1g},dZ_{22},\cdots,dZ_{2g},\cdots,dZ_{gg})$.
Then $D\alpha=-\alpha\omega$. On one side,

$\gamma(D\alpha)=-\gamma(\alpha)\gamma(\omega)=-\alpha
S(-S^{-1}dS\cdot+S^{-1}\cdot\omega\cdot S)=\alpha(dS-\omega\cdot S).$

On the other side,

$D(\gamma\alpha)=D(\alpha\cdot S)=D(\alpha)\cdot S+\alpha\cdot
dS=-\alpha\omega\cdot S+\alpha dS=\alpha(-\omega\cdot S+dS).$

For $f\in M_{2k}(\Gamma_{g})$, we see $f(\det(dZ))^{k}$ is invariant under the
action of $\Gamma_{g}$, and so is $D(f(\det(dZ))^{k})$. ∎

The following lemma directly from Lemma 1.1 gives a modular connection.

######  Lemma 1.6.

The holomorphic part $D^{1,0}$ of the Levi-Civita connection associated to the
invariant metric $ds^{2}=\mbox{\rm Tr}(Y^{-1}dZ\cdot Y^{-1}d\bar{Z})$ on the
Siegel upper plane $\mathbb{H}_{g}$ is a modular connection.

We will give the explicit expression of $D^{1,0}(f\det(dZ)^{k})$ in
Proposition 2.4.

###  1.5. Modular connections in the classical case

Let us consider the classical upper half plane $\mathbb{H}=\mathbb{H}_{1}$ to
look for what condition of the coefficient
$\Gamma:=\Gamma_{(1,1),(1,1)}^{(1,1)}$ of a modular connection should satisfy.
Let $\omega=\Gamma dz$. One can easily check that for $\gamma\in\mbox{\rm
SL}(2,\mathbb{Z})$

$\gamma(\omega)=-S^{-1}dS+S^{-1}\omega
S\Longleftrightarrow\frac{\gamma(\Gamma)}{(cz+d)^{2}}=\Gamma+\frac{2c}{cz+d}.$

Recall that ([5], P113)

$\frac{1}{(cz+d)^{2}}\cdot\frac{\sqrt{-1}}{\text{Im}(\gamma
z)}=\frac{\sqrt{-1}}{\text{Im}(z)}+\frac{2c}{cz+d}\quad\text{and}\quad\frac{\sqrt{-1}G_{2}(\gamma
z)}{(cz+d)^{2}}=\sqrt{-1}G_{2}(z)+\frac{2c}{cz+d},$

where

$G_{2}(z)=\frac{1}{2\pi}\left(\sum_{n\not=0}\frac{1}{n^{2}}+\sum_{m\not=0}\sum_{n\in\mathbb{Z}}\frac{1}{(mz+n)^{2}}\right).$

So $\frac{\sqrt{-1}}{y}$ and $\sqrt{-1}G_{2}(z)$ give us two modular
connections on $\mathbb{H}$, which we denote by $D_{1}$ and $D_{2}$
respectively. The later is holomorphic. We have

$D_{1}(fdz)=\left(\frac{df}{dz}-\frac{\sqrt{-1}}{y}f\right)dzdz\quad\text{and}\quad
D_{2}(fdz)=\left(\frac{df}{dz}-\sqrt{-1}G_{2}(z)f\right)dzdz.$

By the expressions of $D_{1}(f(dz)^{k})$ and $D_{2}(f(dz)^{k})$ and by Lemma
1.5, we have

######  Corollary 1.7.

Let $f$ be a modular form of weight $2k$. Then
$\frac{df}{dz}-\frac{\sqrt{-1}k}{y}f$ and $\frac{df}{dz}-\sqrt{-1}kG_{2}(z)f$
are modular forms of weight $2k+2$. They are non-holomorphic and holomorphic,
respectively.

In fact, the modular connection $D_{1}$ comes from the Levi-Civita connection
$D$ associated to the invariant metric $ds^{2}=\frac{dz\,d\bar{z}}{y^{2}}$.

######  Lemma 1.8.

$D(dz)=-\frac{\sqrt{-1}}{y}dz\,dz\quad\text{and}\quad
D(d\bar{z})=\frac{\sqrt{-1}}{y}d\bar{z}\,d\bar{z}$. So the coefficients of $D$
give the modular connection $D_{1}$.

###### Proof.

Since $ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}=\frac{dz\,d\bar{z}}{y^{2}}$, using
the coordinates ${dz,d\overline{z}}$ and the equality (1.1), we have
$\Gamma_{1,1}^{1}=\frac{\sqrt{-1}}{y},\ \Gamma_{2,1}^{1}=\Gamma_{1,2}^{1}=0,\
\Gamma_{2,1}^{2}=\Gamma_{1,2}^{2}=0$ and
$\Gamma_{2,2}^{2}=-\frac{\sqrt{-1}}{y}$. ∎

The connection $D_{2}$ is not from differential geometry. It is unique.

######  Lemma 1.9 (Uniqueness Lemma).

$\sqrt{-1}G_{2}(z)$ is the unique holomorphic function $\Gamma$ satisfying

$\frac{\gamma(\Gamma)}{(cz+d)^{2}}=\Gamma+\frac{2c}{cz+d}~{}~{}\text{for
all}~{}~{}\gamma=\left(\begin{array}[]{cc}a&b\\\ c&d\\\
\end{array}\right)\in\mbox{\rm SL}(2,\mathbb{Z}),$

and so $D_{2}$ is the unique holomorphic modular connection on $\mathbb{H}$.

###### Proof.

We have
$\gamma(\sqrt{-1}G_{2}(z)-\Gamma)=(cz+d)^{2}(\sqrt{-1}G_{2}(z)-\Gamma)$ for
all $\gamma\in\mbox{\rm SL}(2,\mathbb{Z})$. So $\sqrt{-1}G_{2}(z)-\Gamma$ is a
modular form of weight 2 and hence must be zero ([5], P117). ∎

We will generalize these results to $\mathbb{H}_{g}$.

##  2\. Derivative Operators of Siegel Modular Forms

In this section, we first state the result determining the coefficients of the
invariant Levi-Civita connection on a Siegel upper half plane, whose proof we
put in the last section, then we compute the expressions of the differential
forms under this connection. Finally we get a non-holomorphic derivative
operator and a possible holomorphic derivative operator.

###  2.1. Coefficients of Levi-Civita connection

The notations are the same as those in section 1. For $I=(i,j)\in\Omega$, put

$N(I)=\frac{(i-1)(2g-i)}{2}+j,$

which gives a one to one and order keeping correspondence between $\Omega$ and
$\\{1,2,\cdots,\frac{g(g+1)}{2}\\}$. Write $u^{N(I)}=Z_{ij}$ and
$u^{N(g,g)+N(i,j)}=\overline{Z}_{ij}$. For $Z=X+\sqrt{-1}Y\in\mathbb{H}_{g}$,
let $R=(R_{ij})=Y^{-1}$. For $1\leq s\leq g$, we set

$\Omega_{s}=\\{(1,s),(2,s),\cdots,(s,s),(s,s+1),\cdots,(s,g)\\}\subset\Omega.$

Let $K=(r,s)\in\Omega$. Assume that the elements of $Z$ in the column
including $Z_{K}=Z_{rs}$ are

$u^{a_{1}}=Z_{1s},u^{a_{2}}=Z_{2s},\cdots,u^{a_{s}}=Z_{ss},u^{a_{s+1}}=Z_{s+1,s}=Z_{s,s+1},\cdots,u^{a_{g}}=Z_{gs}=Z_{sg},$

and the elements in the row including $Z_{rs}$ are

$u^{b_{1}}=Z_{r1}=Z_{1r},u^{b_{2}}=Z_{r2}=Z_{2r},\cdots,u^{b_{r}}=Z_{rr},u^{b_{r+1}}=Z_{r,r+1},\cdots,u^{b_{g}}=Z_{rg}.$

For $I\times J\in\Omega_{s}\times\Omega_{r}$, assume $Z_{I}=u^{a_{i}}$ and
$Z_{J}=u^{b_{j}}$. Similarly do it for $J\times
I\in\Omega_{s}\times\Omega_{r}$. We define

$\Gamma_{IJ}^{K}\,(=\Gamma_{JI}^{K}):=\left\\{\begin{array}[]{ll}\frac{\sqrt{-1}R_{ij}}{2^{(1-\delta(r,s))(1-\delta(a_{i},b_{j}))}}&\mbox{
if }I\times J\text{ or }J\times I\in\Omega_{s}\times\Omega_{r}\\\ 0&\text{ if
}I\times J\text{ and }J\times
I\not\in\Omega_{s}\times\Omega_{r},\end{array}\right.$ (2.1)

where $\delta(r,s)$ denotes the Kronecker delta symbol. For example,
$\Gamma_{(1,i)(1,j)}^{(1,1)}=\sqrt{-1}R_{ij}$ and $\Gamma_{IJ}^{(1,1)}=0$ if
$I$ or $J\not\in\Omega_{1}$.

Notice that if we use the coordinates $\\{u^{a_{i}},u^{b_{j}}\\}$, then the
coefficients of the Levi-Civita connection satisfy

${}_{Z}\\!\Gamma_{IJ}^{K}={}_{u}\\!\Gamma_{N(I)N(J)}^{N(K)}.$

In this paper we will use these two kinds of coordinates alternately. The
proof of the following lemma is long and complicated. For the convenience of
the reader, we put it in the last section.

######  Lemma 2.1.

The coefficients $\\{\Gamma_{IJ}^{K}\\}$ defined in the equality (2.1) give
the Levi-Civita connection on $\mathbb{H}_{g}$ associated to the invariant
metric $ds^{2}=\mbox{\rm Tr}(Y^{-1}dZ\,Y^{-1}d\bar{Z})$, and hence give a
modular connection, which we denote by $D$.

###  2.2. Expression of differential forms under $D$

We first compute $D(dZ_{K})$.

######  Lemma 2.2.

Let $K=(r,s)\in\Omega$. We have

$D(dZ_{K})=-\sqrt{-1}(dZ_{s1},dZ_{s2},\cdots,dZ_{sg})Y^{-1}\cdot(dZ_{r1},dZ_{r2},\cdots,dZ_{rg})^{t}.$

###### Proof.

The notations are as above. If $r=s$, then $\Omega_{r}=\Omega_{s}$. By Lemma
2.1, we have

$\Gamma_{IJ}^{K}=\Gamma_{JI}^{K}=\left\\{\begin{array}[]{ll}\sqrt{-1}R_{ij}&\mbox{
if }I\text{ and }J\in\Omega_{r}\\\ 0&\text{ if }I\text{ or
}J\not\in\Omega_{r},\end{array}\right.$

Hence,

$\displaystyle D(dZ_{K})$ $\displaystyle=$
$\displaystyle-\sum_{I,J\in\Omega}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-\sum_{I,J\in\Omega_{r}}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-\sum_{i,j=1}^{g}\sqrt{-1}R_{ij}dZ_{si}dZ_{rj}$
$\displaystyle=$
$\displaystyle-\sqrt{-1}(dZ_{s1},dZ_{s2},\cdots,dZ_{sg})Y^{-1}\cdot(dZ_{r1},dZ_{r2},\cdots,dZ_{rg})^{t}.$

If $r\not=s$, we assume $r<s$. Then $\Omega_{r}\bigcap\Omega_{s}=\\{(r,s)\\}$
and
$(\Omega_{r}\times\Omega_{s})\bigcap(\Omega_{s}\times\Omega_{r})=\\{(r,s)\times(r,s)\\}$.
Put $A=(\Omega_{r}\times\Omega_{s})\bigcup(\Omega_{s}\times\Omega_{r})$ and
$B=(\Omega_{r}\times\Omega_{s})\bigcap(\Omega_{s}\times\Omega_{r})$. We have
again by Lemma 2.1,

$\Gamma_{IJ}^{K}=\Gamma_{JI}^{K}=\left\\{\begin{array}[]{lll}\frac{\sqrt{-1}R_{ij}}{2^{1-\delta(a_{i},b_{j})}}=\frac{\sqrt{-1}R_{ij}}{2}&\mbox{
if }I\times J\in A\setminus B\\\ \
\frac{\sqrt{-1}R_{ij}}{2^{1-\delta(a_{i},b_{j})}}=\sqrt{-1}R_{ij}&\mbox{ if
}I\times J\in B.\\\ \ 0&\text{ if }I\times J\not\in A\end{array}\right.$

Hence,

$\displaystyle
D(dZ_{K})=-\sum_{I,J\in\Omega}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-\sum_{I\times J\in
A}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}$ $\displaystyle=$
$\displaystyle-2\sum_{I\in\Omega_{r}\atop
J\in\Omega_{s}}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}+\sum_{I\in\Omega_{r}\bigcap\Omega_{s}\atop
J\in\Omega_{r}\bigcap\Omega_{s}}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}=-2\sum_{I\times
J\in\Omega_{r}\times\Omega_{s}-B}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}-\sum_{I\times
J\in B}\Gamma_{I,J}^{K}dZ_{I}dZ_{J}$ $\displaystyle=$
$\displaystyle-\sum_{i,j=1}^{g}\sqrt{-1}R_{ij}dZ_{si}dZ_{rj}=-\sqrt{-1}(dZ_{s1},dZ_{s2},\cdots,dZ_{sg})Y^{-1}\cdot(dZ_{r1},dZ_{r2},\cdots,dZ_{rg})^{t}.$

The case $s<r$ is similar. ∎

######  Proposition 2.3.

$D(\det(dZ))=-\sqrt{-1}\mbox{\rm Tr}(Y^{-1}dZ)\det(dZ)$.

###### Proof.

Put $\alpha_{i}=(dZ_{i1},dZ_{i2},\cdots,dZ_{ig})$ and
$\beta_{j}=(dZ_{1j},dZ_{2j},\cdots,dZ_{gj})^{t}$. By Lemma 2.2,
$D(d(Z_{ij}))=-\sqrt{-1}\alpha_{i}Y^{-1}\beta_{j}$ and
$D(\alpha_{i})=-\sqrt{-1}\alpha_{i}Y^{-1}dZ$. Thus

$D(\det(dZ))=-\sqrt{-1}\left\\{\det\left(\begin{array}[]{c}\alpha_{1}Y^{-1}dZ\\\
\alpha_{2}\\\ \vdots\\\
\alpha_{g}\end{array}\right)+\det\left(\begin{array}[]{c}\alpha_{1}\\\
\alpha_{2}Y^{-1}dZ\\\ \vdots\\\
\alpha_{g}\end{array}\right)+\cdots+\det\left(\begin{array}[]{c}\alpha_{1}\\\
\vdots\\\ \alpha_{g-1}\\\ \alpha_{g}Y^{-1}dZ\end{array}\right)\right\\}.$

Let $A=Y^{-1}dZ=(A_{ij})$ and $dZ[i,j]$ be the algebraic cofactor of $dZ$ at
the position $(i,j)$. By the formula above, we have

$\displaystyle\sqrt{-1}D(\det(dZ))=\sum_{k,j,i=1}^{n}dZ_{ki}\cdot A_{ij}\cdot
dZ[k,j]=\sum_{j,i=1}^{n}A_{ij}\sum_{k=1}^{n}dZ_{ik}\cdot dZ[k,j]$
$\displaystyle=$
$\displaystyle\sum_{j,i=1}^{n}A_{ij}\delta(i,j)\det(dZ)=\mbox{\rm
Tr}(A)\det(dZ)=\mbox{\rm Tr}(Y^{-1}dZ)\det(dZ).$

∎

Put

$\frac{\partial}{\partial Z}=(\partial_{ij})_{g\times
g},\qquad\partial_{ij}=\frac{1+\delta(i,j)}{2}\cdot\frac{\partial}{\partial
Z_{ij}}$

as in the introduction. Then for any $\mathbb{C}^{\infty}$-function $f$ on
$\mathbb{H}_{g}$,

$df=\sum_{1\leq i\leq j\leq g}\frac{\partial f}{\partial
Z_{ij}}dZ_{ij}=\sum_{i=1}^{g}\sum_{j=1}^{g}\partial_{ij}fdZ_{ij}=\mbox{\rm
Tr}\left(\frac{\partial}{\partial Z}f\cdot dZ\right)$

######  Proposition 2.4.

For any $\mathbb{C}^{\infty}$-function $f$ on $\mathbb{H}_{g}$, we have

$D\left(f\det(dZ)^{k}\right)=\mbox{\rm Tr}\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)\det(dZ)^{k}.$

###### Proof.

Since $df=\mbox{\rm Tr}(\frac{\partial}{\partial Z}f\cdot dZ)$, we have, by
Proposition 2.3,

$\displaystyle D\left(f\det(dZ)^{k}\right)$ $\displaystyle=$ $\displaystyle
df\cdot\det(dZ)^{k}+f\cdot D((\det(dZ)^{k})=(df-\sqrt{-1}kf\mbox{\rm
Tr}(Y^{-1}dZ))\det(dZ)^{k}$ $\displaystyle=$ $\displaystyle\mbox{\rm
Tr}\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)\det(dZ)^{k}.$

∎

In the following, $\Omega^{i}_{\mathbb{H}_{g}}$ is the sheaf of holomorphic
$i$-forms on $\mathbb{H}_{g}$. Recall that a section of
$\Omega^{1}_{\mathbb{H}_{g}}$ can be written as $\mbox{\rm Tr}(GdZ)$, where
$G$ is a symmetric matrix of holomorphic functions on $\mathbb{H}_{g}$.

######  Proposition 2.5.

For any section $\mbox{\rm Tr}(GdZ)\in\Omega_{\mathbb{H}_{g}}^{1}$, where $G$
is a symmetric matrix of holomorphic functions on $\mathbb{H}_{g}$, we have

$D(\mbox{\rm Tr}(GdZ))=\mbox{\rm
Tr}\left\\{\left[\left(\frac{\partial}{\partial Z}\right)^{t}\otimes
G\right]\cdot[dZ\otimes dZ]\right\\}-\sqrt{-1}\mbox{\rm Tr}\left(GdZ\cdot
Y^{-1}dZ\right),$

where $\otimes$ is the Kronecker product of matrices.

To show this proposition, we need the following lemma.

######  Lemma 2.6.

We have

  1. (1)

Let $A=(a_{ij})_{n\times n},B=(b_{ij})_{n\times n},C=(c_{ij})_{n\times
n},D=(d_{ij})_{n\times n}$. Then

$\mbox{\rm Tr}((A\otimes B)(C\otimes
D))=\sum_{i,j,k,l=1}^{n}a_{ij}b_{kl}c_{lk}d_{ji}=\mbox{\rm Tr}((A\otimes
C)(B\otimes D)),$

  2. (2)

$d(\mbox{\rm Tr}(GdZ))=\mbox{\rm Tr}\left(\left(\left(\frac{\partial}{\partial
Z}\right)^{t}\otimes G\right)\cdot(dZ\otimes dZ)\right)$ for a symmetric
matrix $G=(G_{ij})$ of functions.

###### Proof.

The proof of (1) is easy. We only show (2). By (1),

$\displaystyle d(\mbox{\rm Tr}(GdZ))$ $\displaystyle=$ $\displaystyle
d\left(\sum_{i,j=1}^{g}G_{ij}dZ_{ij}\right)=\sum_{i,j=1}^{g}\sum_{k,l=1}^{g}\partial_{kl}G_{ij}\cdot
dZ_{kl}dZ_{ji}$ $\displaystyle=$ $\displaystyle\mbox{\rm
Tr}\bigg{(}\big{(}(\partial_{kl})^{t}\otimes(G_{ij})\big{)}\cdot(dZ\otimes
dZ)\bigg{)}.$

∎

###### Proof of Proposition 2.5.

As
$dZ=(\alpha_{1}^{t},\alpha_{2}^{t},\cdots,\alpha_{g}^{t})^{t}=(\beta_{1},\beta_{2},\cdots,\beta_{g})$,
we have

$\displaystyle D(\mbox{\rm Tr}(GdZ))-\mbox{\rm
Tr}\left\\{\left[\left(\frac{\partial}{\partial Z}\right)^{t}\otimes
G\right]\cdot[dZ\otimes dZ]\right\\}=\mbox{\rm Tr}\left(GD(dZ)\right)$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{g}G_{ij}D(dZ_{ij})=-\sqrt{-1}\sum_{i,j=1}^{g}G_{ij}d(\alpha_{i})Y^{-1}d(\beta_{j})=-\sqrt{-1}\mbox{\rm
Tr}\left(GdZY^{-1}dZ\right).$

∎

###  2.3. Derivative operators

If $0\neq f\in M_{2k}(\Gamma_{g})$, we have, by Proposition 2.4 and Lemma 1.5,

$\mbox{\rm Tr}\left(\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)\det(dZ)^{k}$

is invariant under the action of $\Gamma_{g}$. Let
$\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\in\Gamma_{g}$.
Since $\gamma(\det(dZ))^{k}=\frac{\det(dZ)^{k}}{\det(CZ+D)^{2k}}$ and
$\gamma(f)=\det(CZ+D)^{2k}f$, we have

$\mbox{\rm Tr}\left(\frac{1}{f}\left[\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right]fdZ\right)$

is invariant under $\Gamma_{g}$. Put
$h:=\frac{1}{f}\left(\frac{\partial}{\partial Z}-\sqrt{-1}kY^{-1}\right)f$.
Then $h$ is a symmetric matrix of functions on $\mathbb{H}_{g}$, and
$\mbox{\rm Tr}(hdZ)$ is invariant under the action of $\Gamma_{g}$. Since for
any $\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\\\
\end{array}\right)\in\Gamma_{g}$, $d(\gamma Z)=(ZC^{t}+D^{t})^{-1}\cdot
dZ\cdot(CZ+D)^{-1}$, we have (see ([3], P210))

$h(\gamma Z)=(CZ+D)h(Z)(ZC^{t}+D^{t}).$

Thus $\det(h)=\det\left(\frac{1}{f}\left(\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right)f\right)$ is a non-holomorphic Siegel modular form
of weight 2. Let $\widetilde{M}_{k}(\Gamma_{g})$ be the
$\mathbb{C}^{\infty}$-Siegel modular forms of weight $k$ as in the
introduction. Finally we get

######  Theorem 2.7.

If $f\in M_{2k}(\Gamma_{g})$, then $\det\left(\left(\frac{\partial}{\partial
Z}-\sqrt{-1}kY^{-1}\right)f\right)\in\widetilde{M}_{2kg+2}(\Gamma_{g})$.

Let $f\in M_{2r}(\Gamma_{g})$ and $h\in M_{2s}(\Gamma_{g})$. We have

$\displaystyle D(f\det(dZ)^{r})\cdot h\det(dZ)^{s}-f\det(dZ)^{r}\cdot
D(h\det(dZ)^{s})$ $\displaystyle=$ $\displaystyle\mbox{\rm
Tr}\left(\left[h\frac{\partial}{\partial Z}f-f\frac{\partial}{\partial
Z}h\right]dZ\right)\det(dZ)^{r+s}$

is invariant under $\Gamma_{g}$. The same consideration as above gives us

$\det\left(h\frac{\partial}{\partial Z}f-f\frac{\partial}{\partial
Z}h\right)\in M_{2(r+s)g+2}.$

We can continue this construction to find combinations of higher derivatives
of $f$ and $h$ which are modular. By setting $[f,h]_{0}:=fh$,
$[f,h]_{1}:=\det(h\frac{\partial}{\partial Z}f-f\frac{\partial}{\partial
Z}h)$, and so on, one would get the Rankin-Cohen brackets.

###  2.4. The unique theorem

We first show a lemma.

######  Lemma 2.8.

Let $\gamma=\left(\begin{array}[]{cc}A&B\\\ C&D\end{array}\right)\in\mbox{\rm
Sp}(2g,\mathbb{Z})$. Then

$(CZ+D)^{-1}\sqrt{-1}(\text{Im}\gamma(Z))^{-1}=\sqrt{-1}(\text{Im}(Z))^{-1}\cdot(CZ+D)^{t}+2C^{t}.$

###### Proof.

Since Im$(\gamma(Z))=((C\bar{Z}+D)^{t})^{-1}Y(CZ+D)^{-1}$ (see [1]) and
$Y^{t}=Y$, we have

$\displaystyle(\text{Im}\gamma(Z))^{-1}$ $\displaystyle=$
$\displaystyle(CZ+D)Y^{-1}\cdot(C\bar{Z}+D)^{t}=(CZ+D)Y^{-1}\cdot(CZ+D-2\sqrt{-1}CY)^{t}$
$\displaystyle=$
$\displaystyle(CZ+D)Y^{-1}\cdot(CZ+D)^{t}-(CZ+D)2\sqrt{-1}C^{t},$

and thus the result. ∎

######  Theorem 2.9.

For any symmetric $g\times g$ matrix $G=(G_{ij})$ consisting of
$\mathbb{C}^{\infty}$ (or holomorphic) functions on $\mathbb{H}_{g}$ which
satisfies the transformation formula

$(CZ+D)^{-1}\gamma(G)=G\cdot(CZ+D)^{t}+2C^{t},$

there exists a unique modular connection $\mathbb{D}$ such that

$\mathbb{D}(dZ_{rs})=-\sum_{i,j=1}^{g}G_{ij}dZ_{si}dZ_{rj}\quad{and}\quad\mathbb{D}(f(\det(dZ)^{k})=\mbox{\rm
Tr}\left(\left[\frac{\partial}{\partial Z}-kG\right]fdZ\right)(\det(dZ))^{k},$

and thus $G$ gives a derivative operator
$M_{2k}\rightarrow\widetilde{M}_{2kg+2}$ by
$f\mapsto\det\left(\left[\frac{\partial}{\partial Z}-kG\right]f\right)$.
Furthermore, there exists at most one holomorphic symmetric matrix $G$ to
satisfy the transformation formula. If such a $G$ exists, the operator
corresponding to $G$ is holomorphic.

###### Proof.

One notes that in the definition of modular connection coefficients
$\Gamma_{IJ}^{K}$, we only need the transformation law
$\gamma(\omega)=-S^{-1}\cdot dS+S^{-1}\cdot\omega\cdot S$ for
$\gamma\in\mbox{\rm Sp}(2g,\mathbb{Z})$. If $G$ has the same transformation
law as $\sqrt{-1}(\text{Im}(Z))^{-1}$, then we can use the same method in
Lemma 2.1 to construct $\\{\Gamma_{IJ}^{K}\\}$ and to calculate the
expressions of the differential forms under $\mathbb{D}$. The same discussion
as in subsection 2.3 tells us $\det\left(\left[\frac{\partial}{\partial
Z}-kG\right]f\right)\in\widetilde{M}_{2kg+2}$.

On the uniqueness, let $G$ and $\tilde{G}$ be two holomorphic matrices to
satisfy the transformation formula. Then

$(CZ+D)^{-1}(G(\gamma Z)-\tilde{G}(\gamma Z))=(G(Z)-\tilde{G}(Z))(CZ+D)^{t}.$

So $\mbox{\rm
Tr}\\{(G-\tilde{G})dZ\\}\in(\Omega_{\mathbb{H}_{g}}^{1})^{\Gamma_{g}}$. In
[11] and [12] R. Weissauer proved that if $v$ is not of the form
$[u]:=ug-\frac{1}{2}u(u-1)$, then
$(\Omega_{\mathbb{H}_{g}}^{v})^{\Gamma_{g}}=0$. If $g\geq 2$, one gets
$(\Omega_{\mathbb{H}_{g}}^{1})^{\Gamma_{g}}=0$, and thus $0=\mbox{\rm
Tr}\\{(G-\tilde{G})dZ\\}=2\sum\limits_{i<j}(G_{ij}-\tilde{G}_{ij})dZ_{ji}+\sum\limits_{i=1}^{g}(G_{ii}-\tilde{G}_{ii})dZ_{ii}$.
Hence $G_{ij}=\tilde{G}_{ij}$ for all $1\leq i,j\leq g$. The case $g=1$ has
been proved in Lemma 1.9. ∎

######  Question 2.10.

Does there exist such a $G$? If so, how to construct it?

##  3\. Explicit Construction of the Levi-Civita Connection

In this section we give the proof of Lemma 2.1. We denote $I,J,K,L,\cdots$ the
elements in $\Omega$, $\ i,j,k,l,r,s,\cdots$ the elements in
$\\{1,2,\cdots,g\\}$ and $\alpha,\beta,\gamma,\delta,\epsilon$ the elements in
$\\{1,2,\cdots,g(g+1)/2\\}$.

###  3.1. Riemannian metric

Let $R:=(R_{ij})_{g\times g}=Y^{-1}$. Then

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\mbox{\rm Tr}(R\cdot
dZ\cdot R\cdot d\overline{Z})=\sum_{i\leq j}\sum_{r\leq
s}2^{2-\delta(i,j)-\delta(r,s)}\times\frac{R_{ir}R_{js}+R_{jr}R_{is}}{2}dZ_{ij}d\overline{Z}_{rs},$

and thus the Riemannian metric matrix associated to $ds^{2}=\mbox{\rm
Tr}(Y^{-1}dZ\cdot Y^{-1}d\overline{Z})$ is given by

$G=\left(\begin{array}[]{cc}0&W\\\ W&0\\\ \end{array}\right)$

where

$W=(W_{IJ})_{I,J\in\Omega},\quad
W_{IJ}=\frac{R_{ir}R_{js}+R_{jr}R_{is}}{2^{\delta(i,j)+\delta(r,s)}},~{}~{}\text{
if }I=(i,j)\text{ and }J=(r,s).$

######  Lemma 3.1.

The inverse $W^{-1}$ of $W$ is given by

$M:=(M_{IJ})_{I,J\in\Omega},\quad
M_{IJ}=Y_{ir}Y_{js}+Y_{jr}Y_{is},~{}~{}\text{ if }I=(i,j)\text{ and }J=(r,s).$

###### Proof.

We need to show that for any $I=(i,j)\in\Omega$ and $K=(p,q)\in\Omega$,

$\sum_{J\in\Omega}M_{IJ}W_{JK}=\delta(I,K).$

By direct computations, we have

$\displaystyle\sum_{1\leq r\leq s\leq
g}M_{(i,j),(r,s)}W_{(r,s),(p,q)}=\sum_{1\leq r\leq s\leq
g}(Y_{ir}Y_{js}+Y_{jr}Y_{is})\frac{R_{rp}R_{sq}+R_{sp}R_{rq}}{2^{\delta(r,s)+\delta(p,q)}}$
$\displaystyle=$ $\displaystyle\sum_{1\leq r\leq
g}2^{1-\delta(p,q)}Y_{ir}Y_{jr}R_{rp}R_{rq}+\sum_{1\leq r<s\leq
g}(Y_{ir}Y_{js}+Y_{jr}Y_{is})\frac{R_{rp}R_{sq}+R_{sp}R_{rq}}{2^{\delta(p,q)}}$
$\displaystyle=$ $\displaystyle 2^{-\delta(p,q)}\sum_{1\leq r\leq
g}\sum_{1\leq s\leq g}(Y_{ir}Y_{js}R_{rp}R_{sq}+Y_{jr}Y_{is}R_{rp}R_{sq})$
$\displaystyle=$ $\displaystyle 2^{-\delta(p,q)}\sum_{1\leq r\leq
g}(Y_{ir}R_{rp}\delta(j,q)+Y_{jr}R_{rp}\delta(i,q))$ $\displaystyle=$
$\displaystyle
2^{-\delta(p,q)}\\{\delta(i,p)\delta(j,q)+\delta(j,p)\delta(i,q)\\}=\delta(I,K).$

Notice that, in the last three steps, we have used the equality $\sum_{1\leq
r\leq g}Y_{ir}R_{rp}=\delta(r,p)$, which comes from $R=Y^{-1}$. ∎

###  3.2. Connection Coefficients

As before, we put $u^{N(i,j)}=Z_{ij}$, $u^{N(g,g)+N(i,j)}=\overline{Z}_{ij}$,

$G=\left(\begin{array}[]{cc}0&W\\\ W&0\\\
\end{array}\right)\quad\text{and}\quad\widehat{G}:=G^{-1}=\left(\begin{array}[]{cc}0&M\\\
M&0\\\ \end{array}\right).$

By the Equality (1.1), we have

$\Gamma_{\alpha,\beta}^{\gamma}=\sum_{1\leq\rho\leq
g(g+1)}\frac{1}{2}\widehat{G}_{\gamma,\rho}\left(\frac{\partial
G_{\alpha,\rho}}{\partial u^{\beta}}+\frac{\partial G_{\beta,\rho}}{\partial
u^{\alpha}}-\frac{\partial G_{\alpha,\beta}}{\partial u^{\rho}}\right).$

Assume $0\leq\alpha,\beta,\gamma\leq\frac{g(g+1)}{2}$, then
$G_{\alpha,\beta}=0$. We have:

$\Gamma_{\alpha,\beta}^{\gamma}=\sum_{1\leq\rho\leq
g(g+1)}\frac{1}{2}\widehat{G}_{\gamma,\rho}\left(\frac{\partial
G_{\alpha,\rho}}{\partial u^{\beta}}+\frac{\partial G_{\beta,\rho}}{\partial
u^{\alpha}}\right).$

Hence for $I,J,K\in\Omega$

$\Gamma_{I,J}^{K}=\sum_{L\in\Omega}\frac{1}{2}M_{K,L}\left(\frac{\partial
W_{I,L}}{\partial Z_{J}}+\frac{\partial W_{J,L}}{\partial Z_{I}}\right).$

Again, notice that

$\sum_{L\in\Omega}M_{K,L}W_{I,L}=\delta(K,I)\ \text{ and }\
\sum_{L\in\Omega}M_{K,L}W_{J,L}=\delta(K,J).$

Do partial derivatives on both sides with respect to $Z_{J}$ and $Z_{I}$
respectively, we have

$\sum_{L\in\Omega}M_{K,L}\frac{\partial W_{I,L}}{\partial
Z_{J}}+\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{J}}W_{I,L}=0,$

and

$\sum_{L\in\Omega}M_{K,L}\frac{\partial W_{J,L}}{\partial
Z_{I}}+\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{I}}W_{J,L}=0.$

Finally we get

$\Gamma_{I,J}^{K}=-\frac{1}{2}\left(\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{J}}W_{I,L}+\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{I}}W_{J,L}\right).$

If $I=(i,j),J=(r,s),K=(p,q)$ and $L=(a,b)\in\Omega$, then
$M_{I,J}=Y_{ir}Y_{js}+Y_{jr}Y_{is}$ and

$\displaystyle\frac{\partial M_{K,L}}{\partial
Z_{J}}=\frac{\partial(Y_{pa}Y_{qb}+Y_{qa}Y_{pb})}{\partial Z_{J}}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2}\left\\{\sigma_{(p,a),(r,s)}Y_{qb}+\sigma_{(q,b),(r,s)}Y_{pa}+\sigma_{(q,a),(r,s)}Y_{pb}+\sigma_{(p,b),(r,s)}Y_{qa}\right\\}$

Here we define:

$\sigma_{(p,a),(r,s)}=\left\\{\begin{array}[]{ll}1,&\mbox{ if
}Z_{pa}=Z_{rs},\\\ 0,&\mbox{ if }Z_{pa}\neq Z_{rs}.\end{array}\right.$

One should notice the difference of the notation above with the notation
$\delta_{(p,a),(r,s)}:=\delta((p,a),(r,s))=\delta(p,r)\delta(a,s)$. These two
notations have the following relations:

$\sigma_{(p,a),(r,s)}=\delta_{(p,a),(r,s)}+\delta_{(p,a),(s,r)}-\delta_{(p,a),(r,s)}\cdot\delta_{(p,a),(s,r)}.$

Using the equality
$W_{IJ}=\frac{R_{ir}R_{js}+R_{jr}R_{is}}{2^{\delta(i,j)+\delta(r,s)}}$ and
others above, we have

$\displaystyle\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial
Z_{J}}W_{I,L}=-\frac{\sqrt{-1}}{2}\sum_{L=(a,b)\in\Omega}\\{\sigma_{(p,a),(r,s)}Y_{qb}+\sigma_{(q,b),(r,s)}Y_{pa}$
$\displaystyle+\sigma_{(q,a),(r,s)}Y_{pb}+\sigma_{(p,b),(r,s)}Y_{qa}\\}W_{I,L}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2^{1+\delta(i,j)}}\Bigg{\\{}\sum_{1\leq a\leq
g}\sigma_{(p,a),(r,s)}\delta(q,j)R_{ia}+\sum_{1\leq b\leq
g}\sigma_{(p,b),(r,s)}\delta(q,i)R_{jb}$ $\displaystyle+\sum_{1\leq a\leq
g}\sigma_{(q,a),(r,s)}\delta(p,i)R_{ja}+\sum_{1\leq b\leq
g}\sigma_{(q,b),(r,s)}\delta(p,j)R_{ib}\Bigg{\\}}.$

While

$\displaystyle\sum_{1\leq a\leq g}\sigma_{(p,a),(r,s)}\delta(q,j)R_{ia}$
$\displaystyle=$ $\displaystyle\sum_{1\leq a\leq
g}\\{\delta(p,r)\delta(a,s)+\delta(p,s)\delta(a,r)$
$\displaystyle-\delta(p,r)\delta(a,s)\delta(p,s)\delta(a,r)\\}\delta(q,j)R_{ia}$
$\displaystyle=$
$\displaystyle\delta(q,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{is}\\},$
$\displaystyle\sum_{1\leq b\leq g}\sigma_{(p,b),(r,s)}\delta(q,i)R_{jb}$
$\displaystyle=$
$\displaystyle\delta(q,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{js}\\},$
$\displaystyle\sum_{1\leq a\leq g}\sigma_{(q,a),(r,s)}\delta(p,i)R_{ja}$
$\displaystyle=$
$\displaystyle\delta(p,i)\\{\delta(q,r)R_{js}+\delta(q,s)R_{jr}-\delta(q,r)\delta(q,s)R_{js}\\},$
$\displaystyle\sum_{1\leq b\leq g}\sigma_{(q,b),(r,s)}\delta(p,j)R_{ib}$
$\displaystyle=$
$\displaystyle\delta(p,j)\\{\delta(q,r)R_{is}+\delta(q,s)R_{ir}-\delta(q,r)\delta(q,s)R_{is}\\}.$

Combining these equalities together, we have

$\displaystyle\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{J}}W_{I,L}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2^{1+\delta(i,j)}}\\{\delta(q,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{is}\\}$
$\displaystyle+\delta(q,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{js}\\}$
$\displaystyle+\delta(p,i)\\{\delta(q,r)R_{js}+\delta(q,s)R_{jr}-\delta(q,r)\delta(q,s)R_{js}\\}$
$\displaystyle+\delta(p,j)\\{\delta(q,r)R_{is}+\delta(q,s)R_{ir}-\delta(q,r)\delta(q,s)R_{is}\\}\\}$

Similarly, we have

$\displaystyle\sum_{L\in\Omega}\frac{\partial M_{K,L}}{\partial Z_{I}}W_{J,L}$
$\displaystyle=$
$\displaystyle-\frac{\sqrt{-1}}{2^{1+\delta(r,s)}}\\{\delta(q,s)\\{\delta(p,i)R_{rj}+\delta(p,j)R_{ri}-\delta(p,i)\delta(p,j)R_{rj}\\}$
$\displaystyle+\delta(q,r)\\{\delta(p,i)R_{sj}+\delta(p,j)R_{si}-\delta(p,i)\delta(p,j)R_{sj}\\}$
$\displaystyle+\delta(p,r)\\{\delta(q,i)R_{sj}+\delta(q,j)R_{si}-\delta(q,i)\delta(q,j)R_{sj}\\}$
$\displaystyle+\delta(p,s)\\{\delta(q,i)R_{rj}+\delta(q,j)R_{ir}-\delta(q,i)\delta(q,j)R_{rj}\\}\\}$

Finally we get

######  Lemma 3.2.

$\displaystyle\Gamma_{I,J}^{K}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{J}}W_{I,L}+\sum_{L\in\Omega}\frac{\partial
M_{K,L}}{\partial Z_{I}}W_{J,L}\right)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2^{2+\delta(i,j)}}\\{\delta(q,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{is}\\}$
$\displaystyle+\delta(q,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{js}\\}$
$\displaystyle+\delta(p,i)\\{\delta(q,r)R_{js}+\delta(q,s)R_{jr}-\delta(q,r)\delta(q,s)R_{js}\\}$
$\displaystyle+\delta(p,j)\\{\delta(q,r)R_{is}+\delta(q,s)R_{ir}-\delta(q,r)\delta(q,s)R_{is}\\}\\}$
$\displaystyle+\frac{\sqrt{-1}}{2^{2+\delta(r,s)}}\\{\delta(q,s)\\{\delta(p,i)R_{rj}+\delta(p,j)R_{ri}-\delta(p,i)\delta(p,j)R_{rj}\\}$
$\displaystyle+\delta(q,r)\\{\delta(p,i)R_{sj}+\delta(p,j)R_{si}-\delta(p,i)\delta(p,j)R_{sj}\\}$
$\displaystyle+\delta(p,r)\\{\delta(q,i)R_{sj}+\delta(q,j)R_{si}-\delta(q,i)\delta(q,j)R_{sj}\\}$
$\displaystyle+\delta(p,s)\\{\delta(q,i)R_{rj}+\delta(q,j)R_{ir}-\delta(q,i)\delta(q,j)R_{rj}\\}\\}$

Using the lemma 3.2 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 $\Gamma_{I,J}^{K}=\sqrt{-1}R_{1,1}$;

  * •

if $j=1,s\neq 1$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{1,s}$;

  * •

if $j\neq 1,s\neq 1$, then
$\Gamma_{I,J}^{K}=\sqrt{-1}R_{j,s}=\sqrt{-1}R_{s,j}$.

In general case, if $K=(p,q),I=(i,j),J=(r,s)$, and if both $(Z_{ij},Z_{rs})$
and $(Z_{rs},Z_{ij})$ do not belong to
$\\{Z_{1p},Z_{2p},\cdots,Z_{gp}\\}\times\\{Z_{1q},Z_{2q},\cdots,Z_{gq}\\}$,
then all terms in the last equality of lemma 3.2 are zero, hence
$\Gamma_{I,J}^{K}=\Gamma_{J,I}^{K}=0$.

If $p=q$, then

$\displaystyle\Gamma_{I,J}^{(p,p)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2^{1+\delta(i,j)}}\\{\delta(p,i)\\{\delta(p,r)R_{js}+\delta(p,s)R_{jr}-\delta(p,r)\delta(p,s)R_{jr}\\}$
$\displaystyle+\delta(p,j)\\{\delta(p,r)R_{is}+\delta(p,s)R_{ir}-\delta(p,r)\delta(p,s)R_{ir}\\}\\}$
$\displaystyle+\frac{\sqrt{-1}}{2^{1+\delta(r,s)}}\\{\delta(p,r)\\{\delta(p,i)R_{sj}+\delta(p,j)R_{si}-\delta(p,i)\delta(p,j)R_{sj}\\}$
$\displaystyle+\delta(p,s)\\{\delta(p,i)R_{rj}+\delta(p,j)R_{ri}-\delta(p,i)\delta(p,j)R_{rj}\\}\\}.$

If $Z_{ij}=Z_{ji}$ and $Z_{rs}=Z_{sr}$ 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
$\Gamma_{I,J}^{K}=\sqrt{-1}R_{js}$

  * •

If $i=s=p$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{jr}$.

  * •

If $j=r=p$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{is}$.

  * •

If $j=s=p$, then $\Gamma_{I,J}^{K}=\sqrt{-1}R_{ir}$.

If $p<q$, we may assume that $Z_{ij}(i\leq j)$ belong to the same row with
$Z_{pq}$ and $Z_{rs}(r\leq s)$ belongs to the same column with $Z_{pq}$ (Other
cases can be proved in the same way). Then $i=p\leq j$, $r\leq s=q$ and

$\displaystyle\Gamma_{I,J}^{K}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2^{2+\delta(i,j)}}(\delta(q,j)\delta(p,r)R_{is}+R_{jr}+\delta(p,j)R_{ir})$
$\displaystyle+\frac{\sqrt{-1}}{2^{2+\delta(r,s)}}(\delta(p,r)\delta(q,j)R_{si}+R_{rj}+\delta(q,r)R_{sj}).$

  * •

If $i=p=j\leq q$ and $r<s=q$, then

$\Gamma_{I,J}^{K}=\frac{\sqrt{-1}}{8}(R_{jr}+R_{ir})+\frac{\sqrt{-1}}{4}R_{rj}=\frac{\sqrt{-1}}{2}R_{jr}.$

  * •

If $i=p=j$ and $r=s=q$, then $\Gamma_{I,J}^{K}=\frac{\sqrt{-1}}{2}R_{jr}$.

  * •

If $i=p<j$ and $r<s=q$, then

$\displaystyle\Gamma_{I,J}^{K}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{4}(\delta(q,j)\delta(p,r)R_{is}+R_{jr})+\frac{\sqrt{-1}}{4}(\delta(p,r)\delta(q,j)R_{is}+R_{rj})$
$\displaystyle=$
$\displaystyle\frac{\sqrt{-1}}{2}(\delta(q,j)\delta(p,r)R_{pq}+R_{jr})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\sqrt{-1}R_{pq}&\mbox{ if }j=q\text{
and }r=p,\\\ \frac{\sqrt{-1}}{2}R_{jr}&\text{ otherwise}.\end{array}\right.$

  * •

If $i=p<j$ and $r=s=q$, then $\Gamma_{I,J}^{K}=\frac{\sqrt{-1}}{2}R_{jr}$.

At last, we complete the proof of Lemma 2.1.

## References

  * [1] A. Andrianov, Introduction to Siegel Modular Forms and Dirichlet Series, Springer 2008.
  * [2] S. S. Chern, W. H. Chen, K. S. Lam, Lectures on Differential Geometry, World Scientific, 2000.
  * [3] G. van der Geer, Siegel Modular Forms and Their Applications, In ”The 1-2-3 of Modular Forms : Lectures at a Summer School in Nordfjordeid, Norway” (K. Ranestad, ed.), Universitext, Springer-Verlag, Berlin, 2008, 181-245.
  * [4] J. Jost, Riemannian Geometry and Geometric Analysis, Springer(Universitext), Berlin, 1988.
  * [5] N. Koblitz, Introduction to Elliptic curves and Modular Forms, Springer-Verlag, New York, 1984.
  * [6] H. Maass, Lectures on Siegel’s Modular Functions, Tate Institute of Fundamental Research, Bombay, 1955.
  * [7] H. Maass, Siegel’s Modular Forms and Dirichlet Series, LNM 216, Berlin-Heidelberg New York, 1971.
  * [8] G. Shimura, Arithmetic of Differential Operators on Symmetric Domains, Duke Math. J. 48, 1981, 813-843.
  * [9] J.-P.Serre,Congruences Et Formes Modulaires, Seminaire Bourbaki, 24e annee,1971/72,n416.
  * [10] Paul-Emile Paradan, Symmetric Spaces of the Non-compact Type: Lie Groups, http://math.univ-lyon1.fr/~ remy/smf_ sec_ 18_ 02.pdf
  * [11] R. Weissauer, Vektorwertige Siegelsche Modulformen Kleinen Gewichtes, J. Reine Angew. Math. 343, 1983 , 184-202.
  * [12] R. Weissauer, Divisors of the Siegel Modular Variety, LNM 1240, Berlin-Heidelberg New York, 1987, 304-324.
  * [13] D. Zagier, Elliptic Modular Forms and Their Applications, In ”The 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway” (K. Ranestad, ed.), Universitext, Springer-Verlag, Berlin, 2008, 1-103.