# On $p$-adic Measures for Quaternionic Modular Forms

Yubo Jin  Department of Mathematical Sciences
Durham University
South. Rd.
Durham, DH1 3LE, U.K<EMAIL_ADDRESS>
###### Abstract.

The purpose of this paper is to study the special values of the standard
$L$-functions for quaternionic modular forms using the doubling method. We
obtain an integral representation for the $L$-function twisted by a character
following the work of Böcherer-Schmidt ([4]) on Siegel modular forms. As an
application, we construct the $p$-adic measure interpolating certain special
$L$-values.

###### Key words and phrases:

Quaternionic modular forms, Standard $L$-functions, Eisenstein series,
$p$-adic measures

###### 1991 Mathematics Subject Classification:

11F33, 11F55, 11F67

###### Contents

  1. 1 Introduction
  2. 2 Quaternionic Modular Forms and $L$-functions
  3. 3 Eisenstein Series and the Doubling Method
  4. 4 Differential Operators and Classical Reformulations
  5. 5 The Main Analytic Result
  6. 6 Special $L$-values and the $p$-adic Measure

##  1\. Introduction

In this paper, we study the special $L$-values of quaternionic modular forms
and construct the $p$-adic measure. The main strategy is to study the
$L$-functions utilizing an integral representation involving the Siegel
Eisenstein series and the properties of special $L$-values then follow from
the explicit Fourier expansion of Siegel Eisenstein series. For Siegel modular
forms there are two main approaches to obtain the integral representation and
constructing $p$-adic measures. One is the Rankin-Selberg method (e.g. [8]),
the other is the doubling method (e.g. [4]). We will mainly follow the
doubling method in [4] to get the integral representation for twisted
$L$-functions.

Let
$\mathbb{B}=\mathbb{Q}\oplus\mathbb{Q}\zeta\oplus\mathbb{Q}\xi\oplus\mathbb{Q}\zeta\xi$
be a quaternion algebra with
$\zeta^{2}=\alpha,\xi^{2}=\beta,\zeta\xi=-\xi\zeta$ and
$\alpha,\beta\in\mathbb{Z}$. The modular forms studied in this paper are
defined on the algebraic group

$G=\\{g\in\mathrm{SL}_{2n}(\mathbb{B}):g\phi
g^{\ast}=\phi\\},\phi=\left[\begin{array}[]{cc}0&\epsilon\\\
1_{n}&0\end{array}\right],$

with $\epsilon=\pm 1$. Let $K$ be a maximal compact subgroup of
$G(\mathbb{R})$. It is well known that $G(\mathbb{R})/K$ is a Rimannian
manifold and is actually Hermitian if
$\epsilon=-1,\mathbb{B}\otimes_{\mathbb{Q}}\mathbb{R}\cong\mathbb{H}$ or
$\epsilon=1,\mathbb{B}\otimes_{\mathbb{Q}}\mathbb{R}\cong M_{2}(\mathbb{R})$.
Here $\mathbb{H}$ is the Hamilton quaternion. We will restrict ourselves to
above two cases in this paper which we refer as Case I and Case II. More
precisely the symmetric space defined in Case I is the Type D domain in [14]
and the Type C domain in Case II. When $G(\mathbb{R})/K$ is a Hermitian
symmetric space, the algebraicity of modular forms makes sense (see [16]).
Indeed, the algebraicity for quaternionic modular forms of Case I is already
studied in our previous paper [7]. But in this paper we only consider the case
that the associated symmetric space has even degree, i.e. is a ‘tube’ domain.
Instead of defining algebraic modular forms using CM points as in [7, Section
5.2], we have an easier description via Fourier expansions. In particular, we
have a well-defined action of $\mathrm{Aut}(\mathbb{C})$ on modular forms in
terms of the action on the Fourier coefficients. This will allow us to obtain
a more precise algebraic result in Proposition 6.2. That is we can not only
show that the special $L$-values are algebraic up to some constants but also
describe the action of $\mathrm{Aut}(\mathbb{C})$.

After reviewing the preliminary of quaternionic modular forms and their
standard $L$-functions in Section 2, the main computations are carried out in
Section 3, 4, 5. Let $f:\mathcal{H}_{n}\to\mathbb{C}$ be a cusp form on degree
$n$ upper half plane. We will define a function
$\mathfrak{g}:\mathcal{H}_{n}\times\mathcal{H}_{n}\to\mathbb{C}$ deriving form
the pullbacks of certain Siegel Eisenstein series on $\mathcal{H}_{2n}$ along
$\mathcal{H}_{n}\times\mathcal{H}_{n}\to\mathcal{H}_{2n}$. In order to
construct the $p$-adic measure we need to define the function $\mathfrak{g}$
such that its Fourier coefficients have natrual $p$-adic interpolations, i.e.
satisfying the Kummer congruence (Proposition 6.3). The idea of the
construction of $\mathfrak{g}$ is same as the one in [4]. Indeed, it is same
as the twisted Eisenstein series there except for the slight difference for
local sections at bad places away from $p$. However, the global computations
of the inner product $\langle\mathfrak{g}(z,-\overline{w}),f(z)\rangle$ in [4]
do not trivially generalize to our case since the maximal order $\mathcal{O}$
of $\mathbb{B}$ is not a principal ideal domain in general. For example, the
coset decomposition in [4, Proposition 2.1] does not work in our case.
Therefore we proceed our computations adelically which is also inspired by
[15, 23]. The adelic formulas (3.16),(3.21) are reformulated in classcial
language and the differential operators are applied in Section 4. The
differential operator constructed in this paper is similar to the one used in
[4] for Siegel modular forms which is different to the one we used in [7] so
the holomorphic projection is avoided. After a final slight modification of
$\mathfrak{g}$, we summarize all the analytic results in Section 5. More
precisely, with $\mathfrak{g}(z,w)$ defined by (5.2),(5.3) whose Fourier
expansion is computed explicitly in Proposition 5.1, we obtain an inner
product formula of the form

$\displaystyle\left\langle\left\langle\mathfrak{g}(z,-\overline{w}),f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right]\right.\right\rangle^{z}_{\Gamma^{0}(N^{2}p)},f\left|{}_{\kappa}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right.\right\rangle^{w}_{\Gamma^{0}(N^{2}p)}$
$\displaystyle=$ $\displaystyle(\text{certain constants})\cdot\langle
f,f\rangle_{\Gamma^{0}(Np)}L_{Np}(t,f,\chi)E_{p}(t,\chi)$

for $t=2n+1,...,\kappa$ where $L_{Np}(s,f,\chi)$ is the partial $L$-function
with Euler factors at places dividing $Np$ removed and $E_{p}(s,\chi)$ the
modified $p$-Euler factor defined by (5.4).

As an application, we discuss the properties of special values in the final
section. The analytic property (Proposition 6.1) and algebraic property
(Proposition 6.2) can be obtained immediately from the inner product formula
(5.5) and the Fourier expansion in proposition 5.1. For the $p$-adic
interpolation, we follow the strategy in [8] by showing that the Kummer
congruences hold for those special $L$-values. We state our main result for
the $p$-adic interpolation below.

######  Theorem 1.1.

(Theorem 6.4,6.5) Let $f\in S_{\kappa}(\Gamma_{0}(N))$ be an eigenform and
$0\neq f_{0}\in S_{\kappa}(\Gamma_{0}(Np))$ its $p$-stabilisation for a fixed
odd prime $p$ with $p\nmid N$. We assume all primes $l\nmid N$ split in
$\mathbb{B}$ and $f$ is $p$-ordinary. Denote the $p$-Satake parameter of
$f_{0}$ as $\beta_{1,p},...,\beta_{2n,p}$ and for any integer $L|p^{\infty}$
write $\alpha(L)$ for the eigenvalue of $f_{0}$ under the action of $U(L)$:
$f_{0}|U(L)=\alpha(L)f_{0}$. Then there is a unique $p$-adic measure $\mu$ on
$\mathbb{Z}_{p}^{\times}$ such that:
(1) (Case I) For $t=2n+1,...,\kappa$ and primitive Dirichlet character $\chi$
of $p$-power conductor $c_{\chi}$ we have

$\int_{\mathbb{Z}_{p}^{\times}}\chi(x)x^{-t+2n}d\mu=(\text{certain
constants})\cdot\frac{E_{p}(t,\chi)L_{Np}(f,t,\chi)}{G(\chi)^{n}\langle
f_{0},f_{0}\rangle}.$

(2) (Case II) For $t=2n+1,...,\kappa$ and primitive Dirichlet character $\chi$
of $p$-power conductor $c_{\chi}$ with $\chi(-1)=(-1)^{t}$ we have

$\displaystyle\int_{\mathbb{Z}_{p}^{\times}}\chi(x)x^{-t+2n}d\mu=(\text{certain
constants})\cdot
g(\chi)\frac{1-\overline{\chi(p)}p^{t-2n-1}}{1-\chi(p)p^{2n-t}}\frac{E_{p}(t,\chi)L_{Np}(f,t,\chi)}{G(\chi)^{n}\langle
f_{0},f_{0}\rangle}.$

Here $g(\chi)$ is the usual Gauss sum for Dirichlet character $\chi$ and
$G(\chi)$ is the ‘quaternionic Gauss sum’ of $\chi$ defined by (5.8).

We remark that the $p$-adic aspects of the quaternionic modular forms are not
as well understood as those of Siegel modular forms and Hermitian modular
forms. For example, one can expect the result in [15] can be generalized to
modular forms studied here. That is one may hope to construct a $p$-adic
$L$-function for ‘ordinary families of quaternionic modular forms’. Though the
analytic computations in [15] can be generalized to quaternionic case, the
Hida theory as in [11] has not established in our case since the Shimura
varieties involved in quaternionic case are not of PEL-type. The $p$-adic
modular forms as well as the Hida theory for quaternionic Shimura varieties
will be explored in author’s subsequent research.

##  2\. Quaternionic Modular Forms and $L$-functions

In this section we review basics of quaternionic modular forms and their
(standard) $L$-functions. The references for this kind of modular forms
includes [6, 13] for Case I and [10, 21] for Case II. For preliminaries on
quaternion algebras the reader can refer to [24].

###  2.1. Quaternion algebra and algebraic groups

We start by fixing some notations for quaternion algebras and our algebraic
groups.

A quaternion algebra is a central simple algebra of dimension four over
$\mathbb{Q}$. Picking up a basis, we can write it in the form

$\mathbb{B}=\mathbb{Q}\oplus\mathbb{Q}\zeta\oplus\mathbb{Q}\xi\oplus\mathbb{Q}\zeta\xi,$

where

$\zeta^{2}=\alpha,\xi^{2}=\beta,\zeta\xi=-\xi\zeta,$

with $\alpha,\beta$ nonzero squarefree integers. The main involution of
$\mathbb{B}$ is given by

$\overline{\cdot}:\mathbb{B}\to\mathbb{B}:a+b\zeta+c\xi+d\zeta\xi\mapsto\overline{a+b\zeta+c\xi+d\zeta\xi}=a-b\zeta-c\xi-d\zeta\xi.$

We remind the reader that we will also use the same notation
$\overline{\cdot}$ for usual complex involution if its meaning is clear from
the context. The trace and norm are defined by
$\mathrm{tr}(x)=x+\overline{x},N(x)=x\overline{x}$ for $x\in\mathbb{B}$. As
usual we denote $M_{n}(\mathbb{B})$ as a set of $n\times n$ matrices with
entries in $\mathbb{B}$. For $X\in M_{n}(\mathbb{B})$, we write
$X^{\ast}=\text{${}^{t}\\!\overline{X}$},\hat{X}=(X^{\ast})^{-1}$ for the
conjugate transpose and its inverse (if makes sense). The determinant $\det$
and trace $\mathrm{tr}$ for matrices in $M_{n}(\mathbb{B})$ will mean the
reduced norm and reduced trace. We use the same notations $\det,\mathrm{tr}$
for usual determinant and trace of matrices whose entries are complex numbers.
Denote

$\mathrm{GL}_{n}(\mathbb{B})=\\{g\in M_{n}(\mathbb{B}):\det(g)\neq
0\\},\quad\mathrm{SL}_{n}(\mathbb{B})=\\{g\in M_{n}(\mathbb{B}):\det(g)=1\\}.$

Let $\mathbb{A}$ be the adele ring of $\mathbb{Q}$. For a place $v$ we mean
either a finite place corresponding to a prime or the archimedean place
$\infty$. The set of finite places is denoted as $\mathbf{h}$. We write
$\mathbb{A}=\mathbb{A}_{\mathbf{h}}\mathbb{R}$ with finite adeles
$\mathbb{A}_{\mathbf{h}}$ and $x=x_{\mathbf{h}}x_{\infty}$ with
$x\in\mathbb{A},x_{\mathbf{h}}\in\mathbb{A}_{\mathbf{h}},x_{\infty}\in\mathbb{R}$.
Fix embeddings $\overline{\mathbb{Q}}\to\overline{\mathbb{Q}}_{v}$ and set
$\mathbb{B}_{v}=\mathbb{B}\otimes_{\mathbb{Q}}\mathbb{Q}_{v}$. The previous
definition of trace, norm, determinant naturally extends locally and
adelically. Fix a maximal order $\mathcal{O}$ of $\mathbb{B}$ and set
$\mathcal{O}_{v}=\mathcal{O}\otimes_{\mathbb{Z}}\mathbb{Z}_{v}$. For a place
$v$ we call $v$ splits if $\mathbb{B}_{v}\cong M_{2}(\mathbb{Q}_{v})$. If this
is the case we fix an identification
$\mathfrak{i}_{v}:\mathbb{B}_{v}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M_{2}(\mathbb{Q}_{v})$
and assume $\mathfrak{i}_{v}(\mathcal{O}_{v})=M_{2}(\mathbb{Z}_{v})$ for
finite places. Write
$\mathbb{B}_{\infty}=\mathbb{B}\otimes_{\mathbb{Q}}\mathbb{R}$ for $v=\infty$.
We call $\mathbb{B}$ is definite if $\mathbb{B}_{\infty}=\mathbb{H}$ is non-
split and indefinite if otherwise. Here
$\mathbb{H}=\mathbb{R}\oplus\mathbb{R}\mathbf{i}\oplus\mathbb{R}\mathbf{j}\oplus\mathbb{R}\mathbf{ij}$
with $\mathbf{i}^{2}=\mathbf{j}^{2}=-1,\mathbf{ij=-ji}$ is the Hamilton
quaternion algebra.

We consider following two algebraic groups.
(Case I) $\mathbb{B}$ is definite.

$G:=G(\mathbb{Q}):=G_{n}(\mathbb{Q}):=\\{g\in\mathrm{SL}_{2n}(\mathbb{B}):gJ_{n}g^{\ast}=J_{n}\\},\quad
J_{n}=\left[\begin{array}[]{cc}0&-1_{n}\\\ 1_{n}&0\end{array}\right].$

(Case II) $\mathbb{B}$ is indefinite.

$G:=G(\mathbb{Q}):=G_{n}(\mathbb{Q}):=\\{g\in\mathrm{SL}_{2n}(\mathbb{B}):gI_{n}g^{\ast}=I_{n}\\},\quad
I_{n}=\left[\begin{array}[]{cc}0&1_{n}\\\ 1_{n}&0\end{array}\right].$

We also define (connected) similitude groups
(Case I)

$\tilde{G}:=\\{g\in\mathrm{GL}_{2n}(\mathbb{B}):gJ_{n}g^{\ast}=\nu(g)J_{n},\det(g)=\nu(g)^{n},\nu(g)\in\mathbb{Q}^{+}\\}.$

(Case II)

$\tilde{G}:=\\{g\in\mathrm{GL}_{2n}(\mathbb{B}):gI_{n}g^{\ast}=\nu(g)I_{n},\det(g)=\nu(g)^{n},\nu(g)\in\mathbb{Q}^{+}\\}.$

Fix an integral ideal $\mathfrak{n}=(N)$ of $\mathcal{O}$ generated by
$N=\prod_{v}p_{v}^{n_{v}}\in\mathbb{Z}$, we define open compact subgroups
$K_{0}(\mathfrak{n})$ and $K^{0}(\mathfrak{n})$ of
$G(\mathbb{A}_{\mathbf{h}})$ by

$K_{0}(\mathfrak{n}):=\prod_{v}K_{v}=\prod_{v}\left\\{\gamma:=\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\in
G(\mathcal{O}_{v}):\gamma\equiv\left[\begin{array}[]{cc}\ast&\ast\\\
0_{n}&\ast\end{array}\right]\text{ mod }p_{v}^{n_{v}}\right\\},$
$K^{0}(\mathfrak{n}):=\prod_{v}K_{v}:=\prod_{v}\left\\{\gamma=\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\in
G(\mathcal{O}_{v}):\gamma\equiv\left[\begin{array}[]{cc}\ast&0_{n}\\\
\ast&\ast\end{array}\right]\text{ mod }p_{v}^{n_{v}}\right\\}.$

Denote $\Gamma_{0}(N):=G(\mathbb{Q})\cap K_{0}(\mathfrak{n})$,
$\Gamma^{0}(N):=G(\mathbb{Q})\cap K^{0}(\mathfrak{n})$. We have

(2.1)
$G(\mathbb{A})=G(\mathbb{Q})G(\mathbb{R})K_{0}(\mathfrak{n})=G(\mathbb{Q})G(\mathbb{R})K^{0}(\mathfrak{n}).$

In Case II, this simply follows from the strong approximation. In Case I, the
strong approximation is not necessarily hold but for subgroups
$K_{0}(\mathfrak{n}),K^{0}(\mathfrak{n})$ above relation holds by weak
approximation and [6, Lemma 3.2].

###  2.2. Symmetric spaces

Define $S:=S(\mathbb{Q}):=\\{X\in M_{n}(\mathbb{B}):X^{\ast}=-\epsilon X\\}$
for the (additive) algebraic group of hermitian matrices (resp. skew-hermitian
matrices) with $\epsilon=-1$ (resp. $\epsilon=1$) in Case I (resp. Case II).
We use $S^{+}$ (resp. $S_{+}$) denote the subgroup of $S$ consisting of
positive definite (resp. positive) elements. The symmetric space is defined as

$\mathcal{H}:=\mathcal{H}_{n}:=\\{z=x+iy\in
S(\mathbb{R})\otimes_{\mathbb{R}}\mathbb{C}:x\in S(\mathbb{R}),y\in
S^{+}(\mathbb{R})\\},$

with the action of $\tilde{G}(\mathbb{R})$ given by

$g=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\times
z\mapsto(az+b)(cz+d)^{-1}.$

Set $\mu(g,z):=cz+d$ and $j(g,z):=\det[\mu(g,z)]$ for the automorphy factor.
Put $\delta(z):=\det(y)^{1/2}$ so $\delta(gz)=|j(g,z)|^{-1}\delta(z)$.

For Case I, the symmetric space is same as the quaternionic upper half plane
in [6]. Denote $z_{0}:=i\cdot 1_{n}$ for the origin of $\mathcal{H}$.

For Case II, since $\mathbb{B}$ is indefinite we have an isomorphism
$\mathbb{B}_{\infty}\cong M_{2}(\mathbb{R})$. For example, without lossing
generality, write
$\mathbb{B}_{\infty}=\mathbb{R}\oplus\mathbb{R}\zeta\oplus\mathbb{R}\xi\oplus\mathbb{R}\zeta\xi$
with $\zeta^{2}=1,\xi^{2}=-1,\zeta\xi=-\xi\zeta$. The isomorphism can be given
by

$\mathfrak{i}:\mathbb{B}_{\infty}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M_{2}(\mathbb{R}),\quad
x=a+b\zeta+c\xi+d\zeta\xi\mapsto\left[\begin{array}[]{cc}a+b&c+d\\\
-c+d&a-b\end{array}\right].$

Clearly $\mathfrak{i}(x^{\ast})=-J_{1}\text{${}^{t}\\!\mathfrak{i}(x)$}J_{1}$.
Extending this isomorphism entries by entries we obtain

$\mathfrak{i}:M_{n}(\mathbb{B})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M_{2n}(\mathbb{R}),\quad
X=(x_{ij})\mapsto(\mathfrak{i}(x_{ij})).$

We have
$\mathfrak{i}(X^{\ast})=-J_{n}^{\prime}\text{${}^{t}\\!\mathfrak{i}(X)$}J_{n}^{\prime}$
where $J_{n}^{\prime}=\mathrm{diag}[J_{1},...,J_{1}]$ with $n$ copies. This
induces the identification

$\iota:G(\mathbb{R})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{Sp}(2n)(\mathbb{R})=\\{g\in\mathrm{GL}_{4n}(\mathbb{R}):gJ_{2n}\text{${}^{t}\\!g$}=J_{2n}\\},$

given by
$g\mapsto\mathrm{diag}[1,-J_{n}^{\prime}]\mathfrak{i}(g)\mathrm{diag}[1,J_{n}^{\prime}]$.
It is well known that $\mathrm{Sp}(2n)(\mathbb{R})$ acts on Siegel upper half
plane

$\mathfrak{H}_{2n}=\\{z=x+iy\in
M_{2n}(\mathbb{R})\otimes_{\mathbb{R}}\mathbb{C}:\text{${}^{t}\\!z$}=z,y>0\\},$

by

$g=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]\times
z\mapsto(az+b)(cz+d)^{-1}.$

We fix an identification

$\iota:\mathcal{H}_{n}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathfrak{H}_{2n},z=x+iy\mapsto\mathfrak{i}(x)J_{n}^{\prime}+i\mathfrak{i}(y)J_{n}^{\prime},$

satisfying $\iota(gz)=\iota(g)\iota(z)$ for $g\in
G(\mathbb{R}),z\in\mathcal{H}_{n}$. The origin $z_{0}$ of $\mathcal{H}_{n}$ is
chosen as $z_{0}:=i\cdot\xi$ so that $\iota(z_{0})=i\cdot 1_{2n}$.

###  2.3. Definition of modular forms

Fix an integer $N$ and $\kappa\in\mathbb{Z}$. The group $\tilde{G}$ acts on
function $f:\mathcal{H}\to\mathbb{C}$ by

$(f|_{\kappa}\gamma)(z)=\det(\gamma)^{\kappa/2}j(\gamma,z)^{-\kappa}f(\gamma
z),\quad\gamma\in\tilde{G},z\in\mathcal{H}.$

######  Definition 2.1.

A holomorphic function $f:\mathcal{H}\to\mathbb{C}$ is called a modular form
for the congruence subgroup $\Gamma$ of weight $\kappa$ if

$f|_{\kappa}\gamma=f\text{ for all
}\gamma=\left[\begin{array}[]{cc}a_{\gamma}&b_{\gamma}\\\
c_{\gamma}&d_{\gamma}\end{array}\right]\subset\Gamma.$

Space of such functions is denoted as $M_{\kappa}(\Gamma)$.

Denote $e(z):=\mathrm{exp}(2\pi iz)$ for $z\in\mathbb{C}$. Set
$\lambda:=\frac{1}{2}\mathrm{tr}$ for Case I and $\lambda:=\mathrm{tr}$ for
Case II and $\lambda(x+iy):=\lambda(x)+i\lambda(y)$. For $f\in
M_{\kappa}(\Gamma_{0}(N))$ and $\gamma\in G$, there is a Fourier expansion of
the form

$(f|_{\kappa}\gamma)(z)=\sum_{\tau\in S_{+}}c(\tau,\gamma,f)e(\lambda(\tau
z)).$

We say that $f$ is a cusp form if $c(\tau,\gamma,f)=0$ for any $\gamma$ unless
$\tau\in S^{+}$. The subspace of cusp forms is denoted as
$S_{\kappa}(\Gamma_{0}(N))$. When $\gamma=1$ we omit $\gamma$ in the fomula
and write

$f(z)=\sum_{\tau\in\Lambda}c(\tau)e(\lambda(\tau z)),$

where more precisely, $\Lambda=\\{\tau\in S:\lambda(\tau s)\in\mathbb{Z}\text{
for any }s\in S(\mathcal{O})\\}$ is a lattice in $S_{+}$. This condition is
equivalent to saying that $\tau_{ii}\in\mathbb{Z}$ and
$2\tau_{ij}\in\mathcal{O}^{\\#}$. Here $\mathcal{O}^{\\#}$ is the dual lattice
of $\mathcal{O}$ respect to $\lambda$.

Let

$\mathbf{d}z=\left\\{\begin{array}[]{cc}\delta(z)^{-2n+1}dz&\text{ Case I,
}\\\ \delta(z)^{-2n-1}dz&\text{ Case II.}\end{array}\right.$

be the invariant differential of $\mathcal{H}$. Here $dz=dxdy$ is the usual
Euclidean measure. For $f,h\in M_{\kappa}(\Gamma_{0}(N))$ the Petersson inner
product is defined as

(2.2) $\langle f,g\rangle:=\langle
f,h\rangle_{\Gamma_{0}(N)}:=\int_{\Gamma_{0}(N)\backslash\mathcal{H}}f(z)\overline{h(z)}\delta(z)^{2k}\mathbf{d}z,$

whenever the integral converges. For example, the integral converges when one
of $f,h$ is a cusp form.

We also need the adelic definition of modular forms. Let $\mathfrak{n}=(N)$ be
the integral ideal in $\mathcal{O}$ generated by $N$. An (adelic) modular form
is defined as a holomorphic function $\mathbf{f}:G(\mathbb{A})\to\mathbb{C}$
satisfying

$\mathbf{f}(\alpha gkk_{\infty})=j(k_{\infty},z_{0})\mathbf{f}(g),$

for $\alpha\in G,k\in K,k_{\infty}\in K_{\infty}$. Here $K$ is an open compact
subgroup of $G(\mathbb{A}_{\mathbf{h}})$ and $K_{\infty}$ is the maximal
compact subgroup of $G(\mathbb{R})$ fixing the origin $z_{0}$. Denote
$\mathcal{M}_{\kappa}(K)$ for the space of such functions and
$\mathcal{S}_{\kappa}(K)$ be the subspace of cusp forms. We mainly consider
the group
$\Gamma_{0}(N),\Gamma^{0}(N),K_{0}(\mathfrak{n}),K^{0}(\mathfrak{n})$ in this
paper. By (2.1), two definitions of modular forms are related by

$\mathcal{M}_{\kappa}(K_{0}(\mathfrak{n}))\cong
M_{\kappa}(\Gamma_{0}(N)),\quad\mathcal{S}_{\kappa}(K_{0}(\mathfrak{n}))\cong
S_{\kappa}(\Gamma_{0}(N)),\quad\mathbf{f}\leftrightarrow f$

with $\mathbf{f}(g)=j(g_{\infty},z_{0})^{-k}f(g_{\infty}z_{0})$. Same
relations hold for changing $\Gamma_{0}(N)$ to $\Gamma^{0}(N)$ and
$K_{0}(\mathfrak{n})$ to $K^{0}(\mathfrak{n})$. For
$\mathbf{f,h}\in\mathcal{M}_{\kappa}(K_{0}(\mathfrak{n}))$ we can also define
the Petersson inner product as an integral (if converges)

(2.3)
$\langle\mathbf{f},\mathbf{h}\rangle:=\langle\mathbf{f},\mathbf{h}\rangle:=\int_{G(\mathbb{A})}\mathbf{f}(g)\overline{\mathbf{h}(g)}\mathbf{d}g.$

Here $\mathbf{d}g$ is the Haar measure of $G(\mathbb{A})$ such that the volume
of $K_{0}(\mathfrak{n})$ is $1$ and $dg_{\infty}=\mathbf{d}(g_{\infty}z_{0})$.
If $f,h$ are the classical modular form correspond to $\mathbf{f,h}$ then
(2.2),(2.3) are related by

(2.4) $\langle\mathbf{f,h}\rangle=\langle f,h\rangle.$

The similar relation also holds for $K^{0}(\mathfrak{n})$.

###  2.4. Hecke operators and $L$-functions

Fix $N\in\mathbb{Z}$ as above and assume that all primes $l\nmid N$ splits in
$\mathbb{B}$. Let

$\mathfrak{M}=\mathrm{GL}_{n}(\mathbb{B}_{\mathbf{h}})\cap
M_{n}(\mathcal{O}_{\mathbf{h}}),\quad\mathfrak{Q}=\\{\mathrm{diag}[\hat{w},w],w\in
M\\},\quad\mathfrak{X}=K_{0}(\mathfrak{n})\mathfrak{Q}K_{0}(\mathfrak{n}).$

The Hecke algebra $\mathbb{T}=\mathbb{T}(K_{0}(\mathfrak{n}),\mathfrak{X})$ is
defined as the $\mathbb{Q}$-algebra generated by double cosets
$[K_{0}(\mathfrak{n})\xi K_{0}(\mathfrak{n})]$ with $\xi\in\mathfrak{X}$.
Given $\mathbf{f}\in\mathcal{M}_{\kappa}(K_{0}(\mathfrak{n}))$ the Hecke
operator $[K_{0}(\mathfrak{n})\xi K_{0}(\mathfrak{n})]$ acts on $\mathbf{f}$
by

(2.5) $(\mathbf{f}|[K_{0}(\mathfrak{n})\xi K_{0}(\mathfrak{n})])(g)=\sum_{y\in
Y}\mathbf{f}(gy^{-1}),$

where $Y$ is a finite subset of $G_{\mathbf{h}}$ such that

$K_{0}(\mathfrak{n})\xi K_{0}(\mathfrak{n})=\bigcup_{y\in
Y}K_{0}(\mathfrak{n})y,\quad y=\left[\begin{array}[]{cc}a_{y}&b_{y}\\\
c_{y}&d_{y}\end{array}\right].$

This coset decomposition can be given explicitly in the following lemma whose
proof is same as [22, Lemma 19.2, 19.4] and [6, Lemma 6.1].

######  Lemma 2.2.

Let $\xi=\mathrm{diag}[\hat{w},w]\in\mathfrak{Q}_{v}$.
(1) If $v|\mathfrak{n}$, then

$K_{v}\xi K_{v}=\bigcup_{d,b}K_{v}\left[\begin{array}[]{cc}\hat{d}&\hat{d}b\\\
0&d\end{array}\right],$

with
$d\in\mathrm{GL}_{n}(\mathcal{O}_{v})\backslash\mathrm{GL}_{n}(\mathcal{O}_{v})w\mathrm{GL}_{n}(\mathcal{O}_{v})$,
$b\in S(\mathcal{O}_{v})/d^{\ast}S(\mathcal{O}_{v})d$.
(2) If $v\nmid\mathfrak{n}$, then

$K_{v}\xi K_{v}=\bigcup_{a,s}K_{v}\left[\begin{array}[]{cc}a&s\hat{a}\\\
0&\hat{a}\end{array}\right],$

with
$a\in\mathrm{GL}_{n}(\mathcal{O}_{v})\backslash\mathrm{GL}_{n}(\mathbb{Q}_{v}),s\in
S(\mathbb{Q}_{v})/S(\mathcal{O}_{v})$.

Let $\mathbf{f}\in\mathcal{S}_{\kappa}(K_{0}(\mathfrak{n}))$ be an eigenform
such that $\mathbf{f}|[K_{0}(\mathfrak{n})\xi
K_{0}(\mathfrak{n})]=\lambda_{\mathbf{f}}(\xi)\mathbf{f}$ for all
$\xi\in\mathfrak{X}$. Let
$\chi:\mathbb{Q}^{\times}\backslash\mathbb{A}^{\times}\to\mathbb{C}^{\times}$
be a Hecke character with $\chi^{\ast}:\mathbb{Z}\to\mathbb{C}$ its associate
Dirichlet character. The (standard) $L$-function is defined as

(2.6)
$L(s,\mathbf{f},\chi)=\Lambda^{2n}_{N}(s,\chi)\sum_{\begin{subarray}{c}\xi\in
K_{0}(\mathfrak{n})\backslash\mathfrak{X}/K_{0}(\mathfrak{n})\\\
\xi=\mathrm{diag}[\hat{w},w]\end{subarray}}\lambda_{\mathbf{f}}(\xi)\chi^{\ast}(\det(w))\det(w)^{-s}$

with
(Case I)

(2.7) $\Lambda^{2n}_{N}(s,\chi)=\prod_{i=0}^{2n-1}L_{N}(2s-2i,\chi^{2}),$

(Case II)

(2.8)
$\Lambda^{2n}_{N}(s,\chi)=L_{N}(s,\chi)\prod_{i=1}^{2n}L_{N}(2s-2i,\chi^{2}).$

Here the subscript $N$ in $L$-function means the Euler factors at $l|N$ are
removed. For $\chi=1$, we simply denote
$L(s,\mathbf{f}):=L(s,\mathbf{f},1),\Lambda_{N}^{2n}(s):=\Lambda_{N}^{2n}(s,1)$.

######  Proposition 2.3.

The $L$-function has an Euler product expression

$L(s,\mathbf{f},\chi)=\prod_{l}L^{(l)}(s,\mathbf{f},\chi),$

with $L^{(l)}(s,\mathbf{f},\chi)$ given by
(Case I)

$\displaystyle\prod_{i=1}^{2n}\left((1-\alpha_{i,l}\chi^{\ast}(l)l^{2n-1-s})(1-\alpha_{i,l}^{-1}\chi^{\ast}(l)l^{2n-1-s})\right)^{-1}$
$\displaystyle l\nmid N,$
$\displaystyle\prod_{i=1}^{2n}(1-\beta_{i,l}\chi^{\ast}(l)l^{2n-2-s})^{-1}$
$\displaystyle l|N,l\text{ split },$
$\displaystyle\prod_{i=1}^{n}(1-\delta_{i,l}\chi^{\ast}(l)l^{2n-1-s})^{-1}$
$\displaystyle l|N,l\text{ nonsplit },$

(Case II)

$\displaystyle(1-l^{2n-s}\chi^{\ast}(l))^{-1}$
$\displaystyle\prod_{i=1}^{2n}\left((1-\alpha_{i,l}\chi^{\ast}(l)l^{2n-s})(1-\alpha_{i,l}^{-1}\chi^{\ast}(l)l^{2n-s})\right)^{-1}$
$\displaystyle l\nmid N,$
$\displaystyle\prod_{i=1}^{2n}(1-\beta_{i,l}\chi^{\ast}(l)l^{2n-s})^{-1}$
$\displaystyle l|N,l\text{ split },$
$\displaystyle\prod_{i=1}^{n}(1-\delta_{i,l}\chi^{\ast}(l)l^{2n-3-s})^{-1}$
$\displaystyle l|N,l\text{ nonsplit },$

for those Satake parameters
$\alpha^{\pm}_{1,l},...,\alpha^{\pm}_{2n,l};\beta_{1,l},...,\beta_{2n,l};\delta_{1,l},...,\delta_{n,l}$
of $\mathbf{f}$.

###### Proof.

If $p$ splits in $\mathbb{B}$ we identify the group $G_{n}(\mathbb{Q}_{v})$
with orthogonal group $\mathrm{SO}(n,n)(\mathbb{Q}_{v})$ in Case I and
symplectic group $\mathrm{Sp}(2n)(\mathbb{Q}_{v})$ in Case II. We also
identify the local Hecke algebra $\mathbb{T}_{v}$ with the local Hecke algebra
of orthogonal and symplectic group which can be defined in a same way as
above. The Satake map is chosen as in [20, Section 16] for $l\nmid N$, [6,
Proposition 6.3] for $l$ nonsplit and $l$ split in Case I, [18, Theorem 2.9]
for $l$ in Case II. Then the computation is same as [22, Theorem 19.8]. For
Case I, this is also calculated in [6, Proposition 6.3]. However, there is a
typo in that computation. Let $l$ split and keep the notation and $\omega_{0}$
there, we need to calculate

$\sum_{d\in\mathrm{GL}_{2n}(\mathbb{Z}_{l})\backslash\mathrm{GL}_{2n}(\mathbb{Q}_{l})\cap
M_{2n}(\mathbb{Z}_{l})}\omega_{0}(\mathrm{GL}_{2n}(\mathbb{Z}_{l})d)[S^{\prime}(\mathbb{Z}_{l}):d^{\ast}S^{\prime}(\mathbb{Z}_{l})d]|\det(d)|_{l}^{s}.$

Here

$S^{\prime}=\\{h\in M_{2n}(\mathbb{Q}_{p}):h^{\ast}=-h\\},$

so the index
$[S^{\prime}(\mathbb{Z}_{l}):d^{\ast}S^{\prime}(\mathbb{Z}_{l})d]=|\det(d)|^{-2n+1}_{l}$
as calculated in [20, lemma 13.2]. The sum then equals
$\prod_{i=1}^{2n}(1-t_{i}l^{2n-2-s})^{-1}$ by of [20, lemma 16.3]. Similarly,
for $p$ nonsplit the term $|\det(d)|_{v}^{-(2n-1)}$ in that proof should be
$|\det(d)|_{v}^{-(2n+1)}$. ∎

By strong approximation of $\mathrm{GL}_{n}(\mathbb{B})$, the representatives
of $K_{0}(\mathfrak{n})\backslash\mathfrak{X}/K_{0}(\mathfrak{n})$ can be
taken in $G(\mathbb{Q})\cap\mathfrak{X}$. For such $\xi\in
G(\mathbb{Q})\cap\mathfrak{X}$ we have $K_{0}(\mathfrak{n})\xi
K_{0}(\mathfrak{n})\cap G(\mathbb{Q})=\Gamma_{0}(N)\xi\Gamma_{0}(N)$. Its
action on $f\in M_{\kappa}(\Gamma_{0}(N))$ is given by

(2.9) $f|[\Gamma_{0}(N)\xi\Gamma_{0}(N)]=\sum_{y\in Y}f|_{\kappa}y,$

if

$\Gamma_{0}(N)\xi\Gamma_{0}(N)=\bigcup_{y\in Y}\Gamma_{0}(N)y,\quad
y=\left[\begin{array}[]{cc}a_{y}&b_{y}\\\ c_{y}&d_{y}\end{array}\right].$

Let $f\in S_{\kappa}(\Gamma_{0}(N))$ be an eigenform and $\chi^{\ast}$ a
Dirichlet character corresponds to the Hecke character $\chi$ one similarly
define the $L$-function $L(s,f,\chi^{\ast})$ such that
$L(s,f,\chi^{\ast})=L(s,\mathbf{f},\chi)$.

We can also extend the definition of Hecke operators to similitude groups. For
example, for an integer $M$, we consider another Hecke operator $U(M)$ given
by double coset

$\Gamma_{0}(N)\left[\begin{array}[]{cc}1&0\\\
0&M\end{array}\right]\Gamma_{0}(N).$

It can be decomposed as

$\Gamma_{0}(N)\left[\begin{array}[]{cc}1&0\\\
0&M\end{array}\right]\Gamma_{0}(N)=\bigcup_{\mathfrak{T}\in
S(\mathcal{O})/MS(\mathcal{O})}\Gamma_{0}(N)\left[\begin{array}[]{cc}1&\mathfrak{T}\\\
0&M\end{array}\right].$

The action of $U(M)$ on $M_{\kappa}(\Gamma_{0}(N))$ is defined as

(2.10) $(f|U(M))(z)=M^{-\frac{n(n+1)}{2}}\sum_{\mathfrak{T}\in
S(\mathcal{O})/MS(\mathcal{O})}f(M^{-1}(z+\mathfrak{T})).$

Write the Fourier expansion of $f$ as

$f(z)=\sum_{\tau}c(\tau,y)e(\lambda(\tau x)),$

with $z=x+iy$. Then

(2.11) $(f|U(M))(z)=\sum_{\tau}c(M\tau,M^{-1}y)e(\lambda(\tau x)).$

###  2.5. $p$-stabilisation

Let $f\in S_{\kappa}(\Gamma_{0}(N))$ be an eigenform. Fix a prime $p\nmid N$,
we are going to construct an eigenform $f_{0}\in S_{\kappa}(\Gamma_{0}(Np))$
such that $f_{0}|U(p)=\alpha_{0}(p)f$ for some $\alpha_{0}$ and $f_{0}$ has
same eigenvalues with $f$ for Hecke operators away from $p$. The following
construction is taken from [8] for Siegel modular forms. Let $v$ be the place
corresponding to $p$. We can extend the definition of Hecke algebra to
similitude groups. For simplicity we write $\tilde{G}_{v}$ for such local
group after identified with orthogonal or symplectic group and write
$K_{v}=\tilde{G}_{v}\cap\mathrm{GL}_{2n}(\mathbb{Z}_{v})$. Let
$\mathbb{H}_{p}$ be the local Hecke algebra generated by double coset
$[K_{v}\xi K_{v}]$ for $\xi\in\tilde{G}_{v}$. The action of $\mathbb{H}_{p}$
on $\mathcal{M}_{\kappa}(K_{0}(\mathfrak{n}))$ is defined similarly as before.
There is a well known Satake isomorphism

$\omega:\mathbb{H}_{p}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{Q}[t^{\pm}_{0},...,t_{2n}^{\pm}]^{W_{2n}}.$

Here $W_{2n}$ is the Weyl group generated by permutation group $S_{2n}$ and
mappings:

$t_{0}\mapsto t_{0}t_{j},t_{j}\mapsto t_{j}^{-1},t_{i}\mapsto t_{i},(1\leq
i\leq 2n,i\neq j).$

For example, this can be constructed as in [8, 2.1.7] so that the restriction
to $\mathbb{T}_{p}$ is same as the one in the proof of Proposition 2.3.
Consider the polynomial $\tilde{Q}(X)\in\mathbb{Q}[t_{0},...,t_{2n}][X]$:

$\tilde{Q}(X)=(1-t_{0}z)\prod_{r=1}^{2n}\prod_{1\leq i_{1}\leq...\leq
i_{r}\leq 2n}(1-t_{0}t_{i_{1}}...t_{i_{r}}X).$

Then there exists a polynomial
$Q(X)=\sum_{i=0}^{2^{2n}}(-1)^{i}T_{i}X^{i}\text{ with
}T_{i}\in\mathbb{H}_{p}$ such that $\omega(Q(X))=\tilde{Q}(X)$. This can be
decomposed as

$Q(X)=\left(\sum_{i=1}^{2^{2n}-1}V_{i}(p)X^{i}\right)(1-U(p)X),V_{i}(p)=\sum_{j=0}^{i}(-1)^{j}T_{j}U(p)^{i-j}.$

Take $\mathbf{f}\in\mathcal{S}_{\kappa}(K_{0}(\mathfrak{n}))$ be an eigenform
corresponding to $f\in S_{\kappa}(\Gamma_{0}(N))$ with Satake parameters
$\alpha_{0,p},...,\alpha_{2n,p}$. Put

$\mathbf{f}_{0}^{\prime}=\sum_{i=1}^{2^{2n}-1}\alpha_{0,p}^{-i}\mathbf{f}|V_{i}(p),$

then
$\mathbf{f}_{0}^{\prime}\in\mathcal{S}_{\kappa}(K_{0}(\mathfrak{n}p^{2^{2n}-1}))$
is an eigenform satisfying
$\mathbf{f}_{0}^{\prime}|U(p)=\alpha_{0,p}\mathbf{f}_{0}^{\prime}$ and has
same eigenvalues as $\mathbf{f}$ for Hecke operators away from $p$. This
determines an eigenform $f_{0}^{\prime}\in S(\Gamma_{0}(Np^{2^{2n}-1}))$ with
same properties. Define

$f_{0}=\sum_{\gamma\in\Gamma_{0}(Np^{2^{2n}-1})\backslash\Gamma_{0}(Np)}f_{0}^{\prime}\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
Np^{2^{2n}-1}&0\end{array}\right]\gamma\left[\begin{array}[]{cc}0&\epsilon\\\
Np&0\end{array}\right]\right..$

Then $f_{0}\in S_{\kappa}(\Gamma_{0}(Np))$ and
$f_{0}|U(p)=\alpha_{0,p}f_{0}$.We call such $f_{0}$ the $p$-stabilisation of
$f$. The eigenform $f$ is called $p$-ordinary if $|\alpha_{0,p}|_{p}=1$.

######  Remark 2.4.

The construction of $f_{0}$ above is not necessarily nonzero as pointed out in
[8]. In the sequel we shall always apply the assumption that $f_{0}\neq 0$.
There is another construction of $p$-stabilisation in [4]. We do not use the
method there because we do not have the ‘Andrianov-identity’ ([4, (9.6)]) in
our setting.

##  3\. Eisenstein Series and the Doubling Method

In this section, we use the doubling method to study $L$-functions. We define
an Siegel Eisenstein series on $G_{2n}$ and consider its pull-back along
$G_{n}\times G_{n}\to G_{2n}$. Its Petersson inner product against our
eigenform gives an integral representation for the $L$-functions twisted by a
Hecke character of $p$-power conductor. The Eisenstein series is defined such
that its Fourier expansion has a natural $p$-adic interpolation. In
particular, we use a different construction when we twist the $L$-function by
a trivial character. This will result in the $p$-modification when
constructing $p$-adic measures.

###  3.1. Siegel Eisenstein series and its Fourier expansion

We start by reviewing the Eisenstein series defined in [6, 21]. Denote
$P:=P_{n}\subset G_{n}$ be the Siegel parabolic subgroup consisting elements
of the form $\left[\begin{array}[]{cc}\ast&\ast\\\
0_{n}&\ast\end{array}\right]$. Fix $\kappa\in\mathbb{Z}$ and an odd prime $p$
splits in $\mathbb{B}$. Let $\chi$ be a Hecke character of conductor
$p^{\mathfrak{c}}$ and assume $\chi_{\infty}(-1)=(-1)^{\kappa}$ in Case II. In
Case I we do not need this assumption since $\det(g)\in\mathbb{R}_{+}$ for
$g\in\mathrm{GL}_{n}(\mathbb{H})$. Let $\mathfrak{m}=(M)$ be an integral ideal
of $\mathcal{O}$ such that $v_{p}(M)=2\mathfrak{c}$. Let
$\mu(g,s):=\prod_{v}\mu_{v}(g,s):G(\mathbb{A})\times\mathbb{C}\to\mathbb{C}$
be a function such that:
(1)
$\mu_{\infty}(g,s)=j(g_{\infty},z_{0})^{-\kappa}|j(g_{\infty},z_{0})|_{\infty}^{\kappa-s}$,
(2) $\mu_{v}(g,s)=\chi_{v}(\det d_{p})^{-1}|\det d_{p}|_{v}^{-s}$ if
$v\nmid\mathfrak{m}$,
(3) For $v|\mathfrak{m}$, set $\mu_{v}(g,s)=\chi_{v}(\det
d_{p})^{-1}\chi_{v}(\det d_{k})|\det d_{p}|_{v}^{-s}$ if $g=qJk$ with $q\in
P(\mathbb{Q}_{v}),k\in K_{0}(\mathfrak{m})$ and $\mu_{v}(g,s)=0$ if $g$ is not
of this form.
We define a Siegel Eisenstein series by

(3.1)
$\mathbf{E}(g,s):=\mathbf{E}_{\kappa}(g,s;\chi,\mathfrak{m})=\sum_{\gamma\in
P\backslash G}\mu(\gamma g,s).$

Clearly if we set $E(z,s)=j(g_{z},z_{0})^{\kappa}\mathbf{E}(g_{z},s)$ for
$z=g_{z}\cdot z_{0}$ then

(3.2) $E(z,s)=E_{\kappa}(z,s;\chi,\mathfrak{m})=\sum_{\gamma\in P\cap
J\Gamma_{0}(M)J^{-1}\backslash J\Gamma_{0}(M)}\chi_{\mathfrak{m}}(\det
c_{\gamma})^{-1}\delta(z)^{s-\kappa}|_{\kappa}\gamma.$

It converges absolutely when $\kappa>2n-1$ in Case I and $\kappa>2n+1$ in Case
II. The Fourier expansions of $E(z,s)$ are discussed in [6, Section 3],[21,
Section 5] and the special values of those confluent hypergeometric functions
are calculated in [17]. We need to assume that all $l\nmid M$ splits in
$\mathbb{B}$ so that the Eisenstein series has a nice Fourier expansion. We
are interested in the special values at $s=\kappa$, write

(3.3)
$\Lambda^{n}_{M}(\kappa,\chi)E_{\kappa}(z,\kappa,\chi)=\sum_{h\in\Lambda_{n}}a(h,y,\kappa,\chi)e(\lambda(hz)),$

with $\Lambda_{n}=\\{\tau\in S_{n}:\lambda(\tau s)\in\mathbb{Z}\text{ for any
}s\in S(\mathcal{O})\\}$. When $h>0$ we have
(Case I)

(3.4) $\displaystyle a(h,y,\kappa,\chi)$
$\displaystyle=A_{\kappa}(n)\det(h)^{\kappa-\frac{2n-1}{2}}\prod_{l\nmid
M}P_{l}(\chi^{\ast}(l)l^{-\kappa}),$ $\displaystyle A_{\kappa}(n)$
$\displaystyle=2^{2-2n}(2\pi
i)^{2n\kappa}(4D_{\mathbb{B}})^{-\frac{n(n-1)}{2}}\Gamma_{n}(2\kappa)^{-1},$
$\displaystyle\Lambda_{M}^{n}(\kappa,\chi)$
$\displaystyle=\prod_{i=0}^{n-1}L_{M}(2\kappa-2i,\chi^{2}),\quad\Gamma_{n}(s)=\pi^{n(n-1)}\prod_{i=0}^{n-1}\Gamma(s-2i).$

(Case II)

(3.5) $\displaystyle a(h,y,\kappa,\chi)$
$\displaystyle=A_{\kappa}(n)\det(h)^{\kappa-\frac{2n+1}{2}}L_{M}(\kappa-n,\chi\rho_{h})\prod_{l\nmid
M}P_{l}(\chi^{\ast}(l)l^{-\kappa}),$ $\displaystyle A_{\kappa}(n)$
$\displaystyle=2^{1-\frac{2n+1}{2}}(2\pi
i)^{2n\kappa}(D_{\mathbb{B}})^{-\frac{n(n+1)}{2}}\Gamma_{2n}(\kappa)^{-1},$
$\displaystyle\Lambda_{M}^{n}(\kappa,\chi)$
$\displaystyle=L_{M}(\kappa,\chi)\prod_{i=1}^{n}L_{M}(2\kappa-2i,\chi^{2}),\quad\Gamma_{n}(s)=\pi^{\frac{n(n-1)}{4}}\prod_{i=0}^{n-1}\Gamma(s-i/2).$

Here $D_{\mathbb{B}}$ is the discriminant of $\mathbb{B}$, i.e. the product of
non-split primes; $P_{l}\in\mathcal{O}[X]$ are polynomials such that for all
primes $l$, $P_{l}=1$ if $\det(2h)\in\mathbb{Z}_{l}^{\times}$; $\rho_{h}$ is
the character corresponding to $\mathbb{Q}(c^{1/2})/\mathbb{Q}$ with
$c=(-1)^{n}\det(h)$.

######  Proposition 3.1.

Keep the notation as above and $\sigma\in\mathrm{Aut}(\mathbb{C})$. Assume
$h>0$ then
(1) (Case I)

(3.6)
$\left(\frac{a(h,y,\kappa,\chi)}{A_{\kappa}(n)}\right)^{\sigma}=\frac{a(h,y,\kappa,\chi^{\sigma})}{A_{\kappa}(n)}.$

(2) (Case II)

(3.7) $\left(\frac{a(h,y,\kappa,\chi)}{A_{\kappa}(n)(\pi
i)^{\kappa-n}g(\chi\rho_{h})}\right)^{\sigma}=\frac{a(h,y,\kappa,\chi^{\sigma})}{A_{\kappa}(n)(\pi
i)^{\kappa-n}g(\chi^{\sigma}\rho^{\sigma}_{h})}.$

Here $g(\chi)$ is the Gauss sum for $\chi$.

###### Proof.

The proposition follows from the explicit formulas for $a(h,y,\kappa,\chi)$.
For Case II we are using the fact that

$\left(\frac{L(k,\chi)}{(\pi
i)^{k}g(\chi)}\right)^{\sigma}=\frac{L(k,\chi^{\sigma})}{(\pi
i)^{k}g(\chi^{\sigma})}.$

∎

###  3.2. The integral representation: nontrivial twist

We consider the following embedding

$\displaystyle G_{n}\times G_{n}$ $\displaystyle\to G_{2n},$
$\displaystyle\left[\begin{array}[]{cc}a_{1}&b_{1}\\\
c_{1}&d_{1}\end{array}\right]\times\left[\begin{array}[]{cc}a_{2}&b_{2}\\\
c_{2}&d_{2}\end{array}\right]$
$\displaystyle\mapsto\left[\begin{array}[]{cccc}a_{1}&0&b_{1}&0\\\
0&a_{2}&0&b_{2}\\\ c_{1}&0&d_{1}&0\\\ 0&c_{2}&0&d_{2}\end{array}\right].$

We then view $G_{n}\times G_{n}$ as a subgroup of $G_{2n}$ and denote
$\gamma_{1}\times\gamma_{2}$ as the image of $(\gamma_{1},\gamma_{2})$ in
$G_{2n}$ under the embedding. We can also define an embedding for symmetric
spaces

$\mathcal{H}_{n}\times\mathcal{H}_{n}\to\mathcal{H}_{2n},\quad z\times
w\mapsto\left[\begin{array}[]{cc}z&0\\\ 0&w\end{array}\right].$

This embedding is compatible with the embedding of groups in the sense that
$(\gamma_{1}\times\gamma_{2})(z\times w)=\gamma_{1}z\times\gamma_{2}w$. To
ease the notation, we keep the subscript $2n$ and omit the subscript $n$ if it
is clear from the context.

Let $\mathbf{f}\in S_{\kappa}(K_{0}(\mathfrak{n}p))$ be an eigenform with
$(\mathfrak{n},p)=1$ and $\chi$ is a nontrivial Hecke character with conductor
$p^{\mathfrak{c}}>1$. Take $\mathfrak{m}=\mathfrak{n}^{2}p^{2\mathfrak{c}}$ in
last subsection and define a function on $G(\mathbb{A})\times G(\mathbb{A})$
(for $s\in\mathbb{C}$) by

(3.8)
$\mathcal{E}(g,h,s):=\mathcal{E}_{\kappa}(g,h,s;\chi,\mathfrak{n}^{2}p^{2\mathfrak{c}})=\sum_{\gamma\in
P_{2n}\backslash G_{2n}}\tilde{\mu}(\gamma(g\times h),s),$

where:
(1) $\tilde{\mu}_{v}(g,s)=\mu_{v}(g,s)$ for $v\nmid\mathfrak{n}p$,
(2) For $v|\mathfrak{n}$, set

(3.9)
$\tilde{\mu}_{v}(g,s)=\mu_{v}\left(g\left[\begin{array}[]{cccc}1_{n}&0&0&|\mathfrak{n}|_{v}\\\
0&1_{n}&-\epsilon|\mathfrak{n}|_{v}&0\\\ 0&0&1_{n}&0\\\
0&0&0&1_{n}\end{array}\right],s\right),$

(3) For $v=p$, set

(3.10)
$\tilde{\mu}_{p}(g,s)=\sum_{x\in\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p})}\chi_{p}(\det
x)^{-1}\mu_{p}\left(g\left[\begin{array}[]{cccc}1_{n}&0&0&\frac{x}{p^{\mathfrak{c}}}\\\
0&1_{n}&\frac{-\epsilon x^{\ast}}{p^{\mathfrak{c}}}&0\\\ 0&0&1_{n}&0\\\
0&0&0&1_{n}\end{array}\right],s\right).$

Then
$\mathcal{E}(g,h,s)\in\mathcal{M}_{\kappa}(K_{0}(\mathfrak{n}^{2}p^{2\mathfrak{c}}))\otimes\mathcal{M}_{\kappa}(K_{0}(\mathfrak{n}^{2}p^{2\mathfrak{c}}))$
(holomorphic at $s=\kappa$). That is

$\mathcal{E}(gk_{1},hk_{2},s)=\mathcal{E}(g,h,s)\text{ for }k_{1},k_{2}\in
K_{0}(\mathfrak{n}^{2}p^{2\mathfrak{c}}).$

######  Remark 3.2.

In next section, we will define the classical counterpart
$\mathfrak{E}(z,w,s)$ of $\mathcal{E}_{\kappa}(g,h,s)$. One see that the
function $\mathcal{E}_{\kappa}$ constructed here is same as the twisted
Eisenstein series constructed in [4, Section 2] except the sections at
$v|\mathfrak{n}$. At such places, both in [4] and here the local sections
$\tilde{\mu}_{v}$ are chosen such that it only contribute constants in inner
product formula, i.e. all Euler factors at $v|\mathfrak{n}$ are removed (see
lemma 3.5). Hence the inner product fomular obtained here in Proposition 3.16,
3.21 is same as the one in [4] up to constants.

Denote $h^{c}=\left[\begin{array}[]{cc}-1_{n}&0\\\
0&1_{n}\end{array}\right]h\left[\begin{array}[]{cc}-1_{n}&0\\\
0&1_{n}\end{array}\right]$ and consider the integral

(3.11) $\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})}\mathcal{E}(g,h^{c},s)\overline{\mathbf{f}(g)}\mathbf{d}g.$

We compute this integral using the well known ‘doubling method’ strategy (see
for example [19, 22]). That is the function $\mathcal{E}$ can be decomposed as
a sum according to the coset decomposition of $P_{2n}(\mathbb{Q})\backslash
G_{2n}(\mathbb{Q})/G(\mathbb{Q})\times G(\mathbb{Q})$. By the cuspidality of
$\mathbf{f}$, only the coset given by

$\tau=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&1&1&0\\\
-\epsilon&0&0&1\end{array}\right],$

with $\epsilon=-1$ in Case I and $\epsilon=1$ in Case II can contribute to
above integral (3.11). In this case we obtain

$\eqref{22}=\int_{G(\mathbb{A})}\tilde{\mu}\left(\tau\left(g\times\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right]\right),s\right)\overline{\mathbf{f}(hg)}\mathbf{d}g.$

We discuss the above integral case by case in following lemmas.

######  Lemma 3.3.

Assume $\mathrm{Re}(s)>4n+1-\kappa$ then

(3.12)
$\int_{G(\mathbb{R})}\tilde{\mu}_{\infty}\left(\tau\left(g\times\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right]\right),s\right)\overline{\mathbf{f}(hg)}\mathbf{d}g=c_{\kappa}(s)\overline{\mathbf{f}(h)}$

with
(Case I)

$c_{\kappa}(s)=\pi^{n(2n-1)}c_{\kappa}^{\prime}(s),$

(Case II)

$c_{\kappa}(s)=\frac{2^{2n}\pi^{n(2n+1)}}{({\kappa}+s-2n+1)({\kappa}+s-2n+3)...({\kappa}+s+2n-1)}c_{\kappa}^{\prime}(s),$
$c^{\prime}_{\kappa}(s)=\frac{\Gamma({\kappa}+s-2n+2)\Gamma({\kappa}+s-2n+4)...\Gamma({\kappa}+s-2)}{\Gamma({\kappa}+s+1)\Gamma({\kappa}+s+2)...\Gamma({\kappa}+s+2n-1)}.$

###### Proof.

Denote $z=g\cdot z_{0}$ and $f(z)=j(g,z_{0})^{\kappa}\mathbf{f}(g)$. The
integral in (3.12) can be rewritten as

$\int_{\mathcal{H}}\det(z+z_{0})^{-\kappa}|\det(z+z_{0})|_{\infty}^{\kappa-s}\delta(z)^{\kappa+s}\overline{j(h,z)^{-\kappa}f(hz)}\mathbf{d}z.$

This kind of integral is calculated by the same method as [20, Propostion
A.2.9]. We remind the reader that our function $\delta$ here is actually
$\delta^{1/2}$ there. There exists a constant $c_{\kappa}(s)$ independent of
$f$ such that above integral equals

$c_{\kappa}(s)\overline{j(h,z_{0})^{-\kappa}f(hz_{0})}=c_{\kappa}(s)\overline{\mathbf{f}(h)}.$

The constant $c_{\kappa}(s)$ is simply determined by taking $z=0,f=1$. That is

$c_{\kappa}(s)=\int_{\mathcal{H}}\delta(z)^{{\kappa}+s}\mathbf{d}z=\int_{\mathcal{H}}\delta(z)^{\kappa+s-2n-\epsilon}dz.$

By changing our symmetric space to bounded symmetric space by Cayley
transform, this constant $c_{\kappa}(s)$ is calculated in [12, Theorem 2.4.1]
for Case I and [12, Theorem 2.3.1] for Case II. ∎

To ease the notation, we only discuss Case I in following two lemmas and omit
the same proof for Case II.

######  Lemma 3.4.

Let $v\nmid\mathfrak{m}$ corresponds to a prime $l$, then

(3.13) $\int_{G(\mathbb{Q}_{l})}\mu_{l}(\tau(g\times
J),s)\overline{\mathbf{f}(hg)}\mathbf{d}g=\Lambda^{2n}_{l}(s,\chi)^{-1}L^{(l)}(s,\mathbf{f},\chi)\overline{\mathbf{f}(h)}.$

###### Proof.

Recall the Cartan decomposition

$G(\mathbb{Q}_{l})=\coprod_{\begin{subarray}{c}e_{1},...,e_{n}\in\mathbb{Z}\\\
0\leq e_{1}\leq...\leq
e_{n}\end{subarray}}G(\mathcal{O}_{l})\mathrm{diag}[k^{-1}_{e_{1},...,e_{n}},k_{e_{1},...,e_{n}}]G(\mathcal{O}_{l}),k_{e_{1},...,e_{n}}=\mathrm{diag}[l^{e_{1}},...,l^{e}_{n}].$

Denote the Hecke operator corresponds to
$\mathrm{diag}[k^{-1}_{e_{1},...,e_{n}},k_{e_{1},...,e_{n}}]$ by
$T_{e_{1},...,e_{n}}$. The integral on the left hand side of (3.13) can be
written as

$\displaystyle\sum_{\begin{subarray}{c}e_{1},...,e_{n}\in\mathbb{Z}\\\ 0\leq
e_{1}\leq...\leq
e_{n}\end{subarray}}\mu_{l}(\tau(\mathrm{diag}[k_{e_{1},...,e_{n}}^{-1},k_{e_{1},...,e_{n}}]\times
J))\sum_{k\in
G(\mathcal{O}_{v})\mathrm{diag}[k_{e_{1},...,e_{n}}^{-1},k_{e_{1},...,e_{n}}]G(\mathcal{O}_{v})/G(\mathcal{O}_{v})}\overline{\mathbf{f}(hk)}$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}e_{1},...,e_{n}\in\mathbb{Z}\\\ 0\leq
e_{1}\leq...\leq e_{n}\end{subarray}}\chi_{l}(\det
k_{e_{1},...,e_{n}})\det(k_{e_{1},...,e_{n}})^{-s}\overline{T_{e_{1},...,e_{n}}\mathbf{f}(h)}$
$\displaystyle=$
$\displaystyle\Lambda^{2n}_{v}(s,\chi)^{-1}L^{(l)}(s,\mathbf{f},\chi)\overline{\mathbf{f}(h)}.$

∎

######  Lemma 3.5.

(1) Denote $\mathbb{Q}_{\mathfrak{n}}=\prod_{v|\mathfrak{n}}\mathbb{Q}_{v}$
and $\mathcal{O}_{\mathfrak{n}}=\prod_{v|\mathfrak{n}}\mathcal{O}_{v}$ then

(3.14)
$\int_{G(\mathbb{Q}_{\mathfrak{n}})}\tilde{\mu}_{\mathfrak{n}}(\tau(g\times
J),s)\overline{\mathbf{f}(hg)}\mathbf{d}g=\chi_{\mathfrak{n}}(\mathfrak{n})^{-2}\mathfrak{n}^{2ns}\mathrm{vol}(K^{1}_{\mathfrak{n}})\overline{\mathbf{f}(h\mathrm{diag}[\mathfrak{n}^{-1},\mathfrak{n}]J)}.$

Here, for any integer $\mathfrak{n}$ we denote

$K^{1}_{\mathfrak{n}}=\left\\{\gamma=\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]\in G(\mathcal{O}_{\mathfrak{n}}):a,d\equiv 1\text{ mod
}\mathfrak{n},c\equiv 0\text{ mod }\mathfrak{n}^{2}\right\\}.$

(2)

(3.15) $\displaystyle\int_{G(\mathbb{Q}_{p})}\tilde{\mu}_{p}(\tau(g\times
J),s)\overline{\mathbf{f}(hg)}\mathbf{d}g$ $\displaystyle=$ $\displaystyle
p^{2\mathfrak{c}ns}\mathrm{vol}(\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p}))\mathrm{vol}(K_{p^{\mathfrak{c}}}^{1})\overline{\mathbf{f}(h\mathrm{diag}[p^{-\mathfrak{c}},p^{\mathfrak{c}}]J)}.$

###### Proof.

Note that

$\tilde{\mu}_{\mathfrak{n}}(\tau(g\times
J),s)=\mu_{\mathfrak{n}}(\tau(gJ^{-1}\mathrm{diag}[\mathfrak{n},\mathfrak{n}^{-1}]\times
1)\tau^{-1}d_{\mathfrak{n}}^{-1}J,s)$

with $d_{\mathfrak{n}}=\mathrm{diag}[\mathfrak{n}\cdot
1_{n},1_{n},\mathfrak{n}^{-1}\cdot 1_{n},1_{n}]$. Changing variables of the
integral in (3.14) and consider

$\int_{G(\mathbb{Q}_{\mathfrak{n}})}\mu(\tau(g\times
1)\tau^{-1}d_{\mathfrak{n}}^{-1}J,s)\overline{\mathbf{f}(hg\mathrm{diag}[\mathfrak{n}^{-1},\mathfrak{n}]J)}\mathbf{d}g.$

Let $g=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]$, then

$\tau(g\times
1)\tau^{-1}d_{\mathfrak{n}}^{-1}J=\left[\begin{array}[]{cccc}b\mathfrak{n}&0&-a\mathfrak{n}^{-1}&b\\\
0&0&0&-1\\\ d\mathfrak{n}&0&-c\mathfrak{n}^{-1}&d-1\\\
b\mathfrak{n}&1&-(a-1)\mathfrak{n}^{-1}&b\end{array}\right].$

By the definition of $\mu$, the integrand is nonzero unless $d$ is invertible
and

$\left[\begin{array}[]{cc}d\mathfrak{n}&0\\\
b\mathfrak{n}&1\end{array}\right]^{-1}\left[\begin{array}[]{cc}-c\mathfrak{n}^{-1}&d-1\\\
(-a+1)\mathfrak{n}^{-1}&b\end{array}\right]\in
M_{2n}(\mathcal{O}_{\mathfrak{n}}),$

which implies $g\in K^{1}_{\mathfrak{n}}$. Hence the integral equals

$\int_{K^{1}_{\mathfrak{n}}}\chi_{\mathfrak{n}}(\mathfrak{n})^{-2}\mathfrak{n}^{2ns}\overline{\mathbf{f}(hg\mathrm{diag}[\mathfrak{n}^{-1},\mathfrak{n}]J)}\mathbf{d}g=\chi_{\mathfrak{n}}(\mathfrak{n})^{-2}\mathfrak{n}^{2ns}\mathrm{vol}(K^{1}_{\mathfrak{n}})\overline{\mathbf{f}(h\mathrm{diag}[\mathfrak{n}^{-1},\mathfrak{n}]J)}.$

For (2) note that

$\tilde{\mu}_{p}(\tau(g\times
1),s)=\sum_{x\in\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p})}\chi(\det
x)^{-1}\mu_{p}(\tau(gJ^{-1}\mathrm{diag}[p^{\mathfrak{c}},p^{-\mathfrak{c}}]\times
1)\tau^{-1}d_{p}^{-1}J,s),$

with
$d_{p}=\mathrm{diag}[p^{\mathfrak{c}}x^{-1},1_{n},p^{-\mathfrak{c}}x,1_{n}]$.
Changing variables of the integral in (3.15) we obtain

$\displaystyle\sum_{x\in\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p})}\chi(\det
x)^{-1}\int_{G(\mathbb{Q}_{p})}\mu_{p}(\tau(g\times
1)\tau^{-1}d_{\mathfrak{n}}^{-1}J,s)\overline{\mathbf{f}(hg\mathrm{diag}[p^{-\mathfrak{c}},p^{\mathfrak{c}}]J)}\mathbf{d}g$
$\displaystyle=$
$\displaystyle\sum_{x\in\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p})}\chi(\det
x)^{-1}\cdot p^{\mathfrak{c}ns}\mathrm{vol}(K_{p^{\mathfrak{c}}}^{1})\chi(\det
x)\overline{\mathbf{f}(h\mathrm{diag}[p^{-\mathfrak{c}},p^{\mathfrak{c}}]J)}$
$\displaystyle=$ $\displaystyle
p^{2\mathfrak{c}ns}\mathrm{vol}(\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p}))\mathrm{vol}(K_{p^{\mathfrak{c}}}^{1})\overline{\mathbf{f}(h\mathrm{diag}[p^{-\mathfrak{c}},p^{\mathfrak{c}}]J)}.$

∎

Denote

$\Omega^{\mathfrak{c}}_{\kappa}(s)=\mathfrak{n}^{2ns}p^{2\mathfrak{c}ns}\chi_{\mathfrak{n}}(\mathfrak{n})^{-2}\mathrm{vol}(K_{\mathfrak{np^{\mathfrak{c}}}}^{1})\mathrm{vol}(\mathrm{GL}_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}\mathrm{GL}_{n}(\mathcal{O}_{p}))c_{\kappa}(s)$

for the product of all constants in above lemmas. We summarize above
discussions in the following proposition.

######  Proposition 3.6.

Let $\mathbf{f}\in S_{\kappa}(K_{0}(\mathfrak{n}p))$ be an eigenform with
$(\mathfrak{n},p)=1$ and $\chi$ is a nontrivial Hecke character of conductor
$p^{\mathfrak{c}}>1$. Then

(3.16) $\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})}\mathcal{E}(g,h^{c},s)\overline{\mathbf{f}\left(g\left[\begin{array}[]{cc}0&\epsilon\mathfrak{n}^{-1}p^{-\mathfrak{c}}\\\
\mathfrak{n}p^{\mathfrak{c}}&0\end{array}\right]\right)}\mathbf{d}g=\frac{\Omega_{\kappa}^{\mathfrak{c}}(s)}{\Lambda^{2n}_{\mathfrak{n}p}(s,\chi)}L_{\mathfrak{np}}(s,\mathbf{f},\chi)\overline{\mathbf{f}\left(h\right)}.$

###  3.3. The modification at $p$

Recall that $p$ is a fixed odd prime split in $\mathbb{B}$, say
$p\mathcal{O}=\mathfrak{pp}^{\prime}$. Let $\mathbf{f}\in
S_{\kappa}(K_{0}(\mathfrak{n}p))$ be an eigenform with $(\mathfrak{n},p)=1$
and $\chi=1$ is the trivial character.

For $0\leq i\leq n$ and $r=0$ or $1$ denote $T(p_{i,r})$ for the Hecke
operator given by double coset

$[K_{0}(p)\xi K_{0}(p)],\quad\xi=\mathrm{diag}[\hat{w},w],\quad
w=\left[\begin{array}[]{ccc}1_{n-i-1}&0&0\\\ 0&\mathfrak{p}^{r}&0\\\
0&0&p\cdot 1_{i}\end{array}\right].$

Suppose there is a double coset decomposition

$\mathrm{GL}_{n}(\mathcal{O}_{p})w\mathrm{GL}_{n}(\mathcal{O}_{p})=\coprod_{j}\mathrm{GL}_{n}(\mathcal{O}_{p})\delta_{rij}.$

Define

(3.17) $\mathcal{E}(g,h,s,i,r)=\sum_{\gamma\in P_{2n}\backslash
G_{2n}}\mu^{i,r}(\gamma(g\times h),s)$

with $\mu_{v}^{i,r}=\tilde{\mu}_{v}$ as in last subsection for $v\neq p$ and

(3.18) $\mu_{p}^{i,r}(g,s)=\sum_{j}\sum_{x\in
pM_{n}(\mathcal{O}_{p})\hat{\delta}_{rij}/pM_{n}(\mathcal{O}_{p})}\mu_{p}\left(g\left[\begin{array}[]{cccc}1_{n}&0&0&\frac{x}{p}\\\
0&1_{n}&-\frac{\epsilon x^{\ast}}{p}&0\\\ 0&0&1_{n}&0\\\
0&0&0&1_{n}\end{array}\right],s\right).$

######  Lemma 3.7.

Let $\lambda_{2i+r}$ be the Hecke eigenvalues of $\mathbf{f}$ under
$T(p_{i,r})$, i.e. $T(p_{i,r})\mathbf{f}=\lambda_{2i+r}\mathbf{f}$. Then

(3.19)
$\displaystyle\int_{G(\mathbb{Q}_{p})}\mu_{p}^{i,r}\left(\tau\left(g\times\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right]\right),s\right)\overline{\mathbf{f}\left(hg\left[\begin{array}[]{cc}0&\epsilon
p^{-1}\\\ p&0\end{array}\right]\right)}\mathbf{d}g$ $\displaystyle=$
$\displaystyle
p^{s(2i+r)}L^{(p)}(s,\mathbf{f})\lambda_{2n-2i-r}\overline{\mathbf{f}\left(h\right)}.$

###### Proof.

Again we only prove for Case I and omit the same proof for Case II. Note that

$\left[\begin{array}[]{cccc}1&0&0&\frac{x}{p}\\\ 0&1&\frac{x^{\ast}}{p}&0\\\
0&0&1&0\\\ 0&0&0&1\end{array}\right]=J^{-1}d_{p}\tau_{x}^{-1}d_{p}^{-1}J,$

with

$\tau_{x}=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&1&0&0\\\ 0&x&1&0\\\
x^{\ast}&0&0&1\end{array}\right],d_{p}=\left[\begin{array}[]{cccc}p&0&0&0\\\
0&1&0&0\\\ 0&0&p^{-1}&0\\\ 0&0&0&1\end{array}\right].$

We need to compute

(3.20) $\int_{G(\mathbb{Q}_{p})}\sum_{j,x}\mu_{p}(\tau(g\times
1)\tau_{x}^{-1}d_{x}^{-1}J)\overline{\mathbf{f}(hg)}\mathbf{d}g.$

Let $g=\left[\begin{array}[]{cc}a&b\\\ c&d\end{array}\right]$, then

$\tau(g\times
1)\tau_{x}^{-1}d_{x}^{-1}J=\left[\begin{array}[]{cccc}pb&0&-p^{-1}a&bx\\\
0&0&0&-1\\\ pd&0&-p^{-1}c&dx-1\\\
pb&1&-p^{-1}(a-x^{\ast})&bx\end{array}\right].$

We need $d$ to be invertible and

$\left[\begin{array}[]{cc}pd&0\\\
pb&1\end{array}\right]^{-1}\left[\begin{array}[]{cc}-p^{-1}c&dx^{\ast}-1\\\
-p^{-1}(a-x)&bx^{\ast}\end{array}\right]=\left[\begin{array}[]{cc}-p^{-2}d^{-1}c&p^{-1}x-p^{-1}d^{-1}\\\
p^{-1}x^{\ast}-p^{-1}\hat{d}&bd^{-1}\end{array}\right]\in
M_{2n}(\mathcal{O}_{p}).$

That is

$bd^{-1}\in M_{n}(\mathcal{O}_{p}),d^{-1}c\in
p^{2}M_{n}(\mathcal{O}_{p}),d^{-1}\in x+pM_{n}(\mathcal{O}_{p}).$

There is a permutation $j\mapsto j^{\prime}$ such that $\hat{d}$ runs through
$\mathrm{GL}_{n}(\mathbb{B}_{p})\cap
pM_{n}(\mathcal{O}_{p})\delta_{rij^{\prime}}^{-1}$ for fixed $j$. Hence,
taking the sum over $j,x$, (3.20) can be written as

$\sum_{g}|\det(pd)|_{p}^{-s}\overline{\mathbf{f}(hg)}\mathbf{d}g.$

with the sum taking over

$g\in K_{0}(p^{2})\mathrm{diag}[\hat{d},d]K_{0}(p^{2})\text{ with
}\hat{d}\in\mathrm{GL}_{n}(\mathbb{B}_{p})\cap pM_{n}(\mathcal{O}_{p})w^{-1}.$

Equivalently, we take the sum of $g$ over

$K_{0}(p^{2})\mathrm{diag}[\hat{d},d]K_{0}(p^{2})\cdot
K_{0}(p^{2})\mathrm{diag}[p\hat{w},p^{-1}w]K_{0}(p^{2})\text{ with
}\hat{d}\in\mathrm{GL}_{n}(\mathbb{B}_{p})\cap M_{n}(\mathcal{O}_{p}).$

Let $T_{d}$ be the Hecke operator given by the double coset
$K_{0}(p^{2})\mathrm{diag}[d^{\ast},d^{-1}]K_{0}(p^{2})$ and $T_{w}$ given by
the double coset
$K_{0}(p^{2})\mathrm{diag}[p^{-1}w^{\ast},pw^{-1}]K_{0}(p^{2})$. Then (3.20)
equals

$\displaystyle\sum_{\begin{subarray}{c}\xi\in
K_{0}(p^{2})\backslash\mathfrak{X}/K_{0}(p^{2})\\\
\xi=\mathrm{diag}[d^{\ast},d^{-1}]\end{subarray}}|\det(d)|_{p}^{-s}|\det(w)|_{p}^{-s}\overline{\mathbf{f}|T_{d}|T_{w}(h)}$
$\displaystyle=$ $\displaystyle
L^{(p)}(s,\mathbf{f})|\det(w)|_{p}^{-s}\lambda_{2n-2i-r}\overline{\mathbf{f}(h)}$

as desired. ∎

Combining the computations for integrals outside $p$ in last subsection we
obtain the following proposition.

######  Proposition 3.8.

Let $\mathbf{f}\in S_{\kappa}(K_{0}(\mathfrak{n}p))$ be an eigenform with
$(\mathfrak{n},p)=1$. Write $\beta_{1,p},...,\beta_{2n,p}$ be its Satake
parameters at $p$. Then

(3.21) $\displaystyle\int_{G(\mathbb{Q})\backslash
G(\mathbb{A})}\sum_{\begin{subarray}{c}0\leq i\leq n\\\
r=0,1\end{subarray}}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\mathcal{E}(g,h^{c},s,i,r)\overline{\mathbf{f}\left(g\left[\begin{array}[]{cc}0&\epsilon\mathfrak{n}^{-1}p^{-1}\\\
\mathfrak{n}p&0\end{array}\right]\right)}\mathbf{d}g$ $\displaystyle=$
$\displaystyle\frac{p^{-2n^{2}-2n+2ns}\Omega^{0}_{\kappa}(s)}{\Lambda_{\mathfrak{n}p}^{2n}(s)}\prod_{i=1}^{2n}(1-\beta_{i,p}p^{-s+2n+\epsilon})L_{\mathfrak{n}}(s,\mathbf{f})\overline{\mathbf{f}(h)}.$

###### Proof.

The integral can be written as

$\displaystyle\frac{\Omega_{\kappa}(s)}{\Lambda_{\mathfrak{n}p}^{2n}(s)}\left(\sum_{i,r}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\lambda_{2n-2i-r}(p^{-s})^{2n-2i-r}\right)L_{\mathfrak{n}}(s,\mathbf{f})\overline{\mathbf{f}(h)}$
$\displaystyle=$
$\displaystyle\frac{p^{-2n^{2}-2n+2ns}\Omega^{0}_{\kappa}(s)}{\Lambda_{\mathfrak{n}p}^{2n}(s)}\left(\sum_{i=0}^{2n}(-1)^{i}p^{\frac{i(i-1)}{2}}\lambda_{i}(p^{-s+1})^{i}\right)L_{\mathfrak{n}}(s,\mathbf{f})\overline{\mathbf{f}(h)}.$

The proposition then follows by following two ‘Tamagawa-identities’ relating
the eigenvalues and Satake parameters.
(Case I)

$\sum_{i=0}^{2n}(-1)^{i}p^{\frac{i(i-1)}{2}}\lambda_{i}X^{i}=\prod_{i=1}^{2n}(1-\beta_{i,p}p^{2n-2}X),$

(Case II)

$\sum_{i=0}^{2n}(-1)^{i}p^{\frac{i(i-1)}{2}}\lambda_{i}X^{i}=\prod_{i=1}^{2n}(1-\beta_{i,p}p^{2n}X).$

Note that if we identify $G_{n}(\mathbb{Q}_{v})$ with orthogonal group
$\mathrm{SO}(2n,2n)(\mathbb{Q}_{v})$ in Case I and symplectic group
$\mathrm{Sp}(2n,\mathbb{Q}_{v})$ in Case II then the Hecke operator
$T(p_{i,r})$ can be identified with the one for
$\mathrm{SO}(2n,2n)(\mathbb{Q}_{v})$ or $\mathrm{Sp}(2n,\mathbb{Q}_{v})$ given
by double coset

$[K_{0}(p)\xi
K_{0}(p)],\xi=\mathrm{diag}[\hat{w},w],w=\left[\begin{array}[]{cc}1_{2n-2i-r}&0\\\
0&p\cdot 1_{2i+r}\end{array}\right].$

Here we are abusing the notation $K_{0}(p)$ for subgroup of
$\mathrm{SO}(2n,2n)(\mathbb{Z}_{v})$ or $\mathrm{Sp}(2n,\mathbb{Z}_{v})$
consisting elements of the form $\left[\begin{array}[]{cc}a&b\\\
c&d\end{array}\right]$ with $c\equiv 0\text{ mod }p$. Then above two identites
can be proved by using [22, Lemma 19.13] and explicit description of Satake
map as chosen in [6, Proposition 6.3] for Case I and [21, Theorem 2.9] for
Case II. ∎

##  4\. Differential Operators and Classical Reformulations

###  4.1. Differential Operators

We first recall some results in [2, 4]. Let $\mathfrak{H}_{n}$ be the Siegel
upper half plane with the action of $\mathrm{Sp}(n)(\mathbb{R})$. There are
compatible doubling maps

$\mathrm{Sp}(n)(\mathbb{R})\times\mathrm{Sp}(n)(\mathbb{R})\to\mathrm{Sp}(2n)(\mathbb{R}),\quad\mathfrak{H}_{n}\times\mathfrak{H}_{n}\to\mathfrak{H}_{2n}.$

For any $\alpha\in\mathbb{C}$ there is a differential operator
$\mathcal{D}_{n,\alpha}$ such that for any integer $\kappa>n$ and any
holomorphic function $f$ and $\gamma\in\mathrm{Sp}(n)(\mathbb{R})$ we have

(4.1) $\displaystyle\mathcal{D}_{n,\kappa}(f|V)$
$\displaystyle=(\mathcal{D}_{n,\kappa}f)|V,$
$\displaystyle\mathcal{D}_{n,\kappa}(f|_{\kappa}1\times\gamma)$
$\displaystyle=(\mathcal{D}_{\kappa}f)|_{\kappa+1}1\times\gamma,$
$\displaystyle\mathcal{D}_{n,\kappa}(f|_{\kappa}\gamma\times 1)$
$\displaystyle=(\mathcal{D}_{\kappa}f)|_{\kappa+1}\gamma\times 1.$

Here $V$ is the operator such that

$(f|V)\left(\left[\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{array}\right]\right)=f\left(\left[\begin{array}[]{cc}z_{4}&z_{3}\\\
z_{2}&z_{1}\end{array}\right]\right).$

Now let

$G(\mathbb{R})\times G(\mathbb{R})\to
G_{2n}(\mathbb{R}),\quad\mathcal{H}\times\mathcal{H}\to\mathcal{H}_{2n},$

be compatible doubling maps as in Section 3. We are going to construct
differential operators $\mathfrak{D}_{\alpha}:=\mathfrak{D}_{n,\alpha}$ on
$\mathcal{H}_{2n}$ having similar transformation properties as (4.1). In Case
II we have $\mathcal{H}_{2n}\cong\mathfrak{H}_{4n}$ so the pullback of
$\mathcal{D}_{2n,\alpha}$ along
$\iota:\mathcal{H}_{2n}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathfrak{H}_{4n}$
has desired properties. We then focused on the Case I. We keep the notations
in [2]. For more details on matrix calculus the reader can refer to [9,
III.6]. Let $A$ be a $n\times n$ matrix, for $0\leq h\leq n$ and two subsets
$a=\\{a_{1}<a_{2}<...<a_{h}\\}$ and $b=\\{b_{1}<b_{2}<...<b_{h}\\}$ of
$N=\\{1,...,n\\}$, denote $|A|_{b}^{a}$ be the determinant of the $h\times h$
matrix whose rows and columns are given by $a$ and $b$ in their natural order.
Let $A^{[h]}$ be a $\left(\begin{array}[]{c}n\\\
h\end{array}\right)\times\left(\begin{array}[]{c}n\\\ h\end{array}\right)$
matrix whose entries are $h$-minors of $A$ and $\mathrm{Ad}^{[p]}A$ its
adjoint. We also use the notation $\sqcap$ defined in [2, Section 2.1] whose
properties are recorded as follow.
(1) $A^{[h]}=A\sqcap...\sqcap A$ ($h$-copies),
(2) $(A+B)^{[h]}=\sum_{p+q=h}\left(\begin{array}[]{c}h\\\
q\end{array}\right)A^{[p]}\sqcap B^{[q]}$,
(3) $A^{[p+q]}(C\sqcap D)=(A^{[p]}C)\sqcap(A^{[p]}D)$,
(4) $(C\sqcap D)A^{[p+q]}=(CA^{[p]})\sqcap(DA^{[q]})$,
(5) $(\mathrm{Ad}^{[p]}A)A^{[p]}=\det(A)1$.

We view $M_{n}(\mathbb{H})$ as subspace of $\mathbb{R}^{4n^{2}}$ with usual
Euclidean measure. Over $\mathcal{H}_{n}$ we have basic differential operators
$\partial_{ij}=\frac{\partial}{\partial z_{ij}}$ where
$\frac{\partial}{\partial z_{ij}}=\frac{\partial}{\partial
x_{ij}}+i\frac{\partial}{\partial y_{ij}}$ with $z_{ij}=x_{ij}+iy_{ij}$,
$1\leq i,j\leq n$. Put these differential operators into a matrix
$\partial=(\partial_{ij})$. For $z=x+iy\in
M_{n}(\mathbb{H})\otimes_{\mathbb{R}}\mathbb{C}$ we formally set
$z^{\ast}=x^{\ast}+iy^{\ast}$ so $z^{\ast}=z$ if $z\in\mathcal{H}_{n}$. For
$\mathfrak{Z}\in\mathcal{H}_{2n}$ and $\partial$ on $\mathcal{H}_{2n}$ we
write them as

$\mathfrak{Z}=\left[\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{array}\right],\,\partial=\left[\begin{array}[]{cc}\partial_{1}&\partial_{2}\\\
\partial_{3}&\partial_{4}\end{array}\right],\,z_{3}=z_{2}^{\ast},\,\partial_{3}=\partial_{2}^{\ast}.$

Let

$f_{\mathfrak{T}}(z)=e^{\lambda(\mathfrak{T}\mathfrak{Z})}=e^{\lambda(\tau_{1}z_{1}+\tau_{3}z_{2}+\tau_{2}z_{3}+\tau_{4}z_{4})},\quad\mathfrak{T}=\left[\begin{array}[]{cc}\tau_{1}&\tau_{2}\\\
\tau_{3}&\tau_{4}\end{array}\right]=\mathfrak{T}^{\ast}.$

Our computation will based on the following basic identities

$\partial_{1}^{[a]}f_{\mathfrak{T}}(z)=\text{${}^{t}\\!\tau$}_{1}^{[a]},\,\partial_{4}^{[a]}f_{\mathfrak{T}}(z)=\text{${}^{t}\\!\tau$}_{4}^{[a]},\,\partial_{2}^{[a]}f_{\mathfrak{T}}(z)=\text{${}^{t}\\!\tau$}_{3}^{[a]},\,\partial_{3}^{[a]}f_{\mathfrak{T}}(z)=\text{${}^{t}\\!\tau$}_{2}^{[a]}.$

We now start the construction. The construction is not completely same to the
one for symplectic case in [2] so we repeat all the details here. The main
difference comes from the involution of quaternion algebras. Consider $f$ of
the form

$f(\mathfrak{Z})=g(z_{2},z_{3},z_{4})e^{\lambda(\tau
z_{1})},\,\tau=\tau^{\ast},\,\det(\tau)\neq 0.$

For nonnegative integers $a,b,p$ with $a+b+p=n$, put

(4.2) $\displaystyle\delta(p,a,b)$
$\displaystyle=\text{${}^{t}\\!z$}_{3}^{[a]}\partial_{4}^{[a]}\sqcap((1_{n}^{[p]}\sqcap\text{${}^{t}\\!z$}_{3}^{[b]}\partial_{3}^{[b]})(\mathrm{Ad}^{[b+p]}\partial_{1})\partial_{2}^{[b+p]})$
$\displaystyle+z_{2}^{[a]}\text{${}^{t}\\!\partial_{4}$}^{[a]}\sqcap((1_{n}^{[p]}\sqcap
z_{2}^{[b]}\text{${}^{t}\\!\partial$}_{2}^{[b]})(\mathrm{Ad}^{[b+p]}\text{${}^{t}\\!\partial$}_{1})\text{${}^{t}\\!\partial_{3}$}^{[b+p]}).$

Then

$\displaystyle\delta(p,a,b)f$
$\displaystyle=\left(\text{${}^{t}\\!\tau$}^{[a+b]}\text{${}^{t}\\!z_{3}$}^{[a+b]}(\partial_{4}^{[a]}\sqcap\partial_{3}^{[b]}\text{${}^{t}\\!\tau$}^{-[b]}\partial_{2}^{[b]})\sqcap\partial_{2}^{[p]}\right.$
$\displaystyle\left.+\tau^{[a+b]}z_{2}^{[a+b]}\text{${}^{t}\\!(\partial_{4}^{[a]}\sqcap\partial_{3}^{[b]}\text{${}^{t}\\!\tau$}^{-[b]}\partial_{2}^{[b]})\sqcap\text{${}^{t}\\!\partial_{3}$}^{[p]}$}\right)f.$

For nonnegative integer $p,q$ such that $p+q=n$ put

$\Delta(p,q)=\sum_{a+b=q}(-1)^{b}\left(\begin{array}[]{c}q\\\
b\end{array}\right)\delta(p,a,b).$

By straightforward computation

$\Delta(p,q)f=\left(\partial_{2}^{[p]}\sqcap\text{${}^{t}\\!\tau$}^{[q]}\text{${}^{t}\\!z$}_{3}^{[q]}(\partial_{4}-\partial_{3}\text{${}^{t}\\!\tau$}^{-1}\partial_{2})^{[q]}+\text{${}^{t}\\!\partial_{3}$}^{[p]}\sqcap\tau^{[q]}z_{2}^{[q]}\text{${}^{t}\\!(\partial_{4}-\partial_{3}\text{${}^{t}\\!\tau$}^{-1}\partial_{2})^{[q]}$}\right)f.$

######  Lemma 4.1.

Let $V$ be the operator given by

$(f|V)\left(\left[\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{array}\right]\right)=f\left(\left[\begin{array}[]{cc}z_{4}&z_{3}\\\
z_{2}&z_{1}\end{array}\right]\right),$

then

(4.3) $\Delta(p,q)(f|V)=(\Delta(p,q)f)|V.$

###### Proof.

It suffices to prove for certain ‘test functions’

$f_{\mathfrak{T}}(\mathfrak{Z})=e^{\lambda(\mathfrak{T}\mathfrak{Z})}=e^{\lambda(\tau_{1}z_{1}+\tau_{2}z_{3}+\tau_{3}z_{2}+\tau_{4}z_{4})},\,\mathfrak{T}=\left[\begin{array}[]{cc}\tau_{1}&\tau_{2}\\\
\tau_{3}&\tau_{4}\end{array}\right].$

By straightforward computation

$(\Delta(p,q)f)|V=\left(\text{${}^{t}\\!\tau$}_{3}^{[p]}\sqcap\text{${}^{t}\\!\tau$}_{1}^{[q]}\text{${}^{t}\\!z$}_{2}^{[q]}(\text{${}^{t}\\!\tau$}_{4}-\text{${}^{t}\\!\tau$}_{2}\text{${}^{t}\\!\tau$}_{1}^{-1}\text{${}^{t}\\!\tau$}_{3})^{[q]}+\tau_{2}^{[p]}\sqcap\tau_{1}^{[q]}z_{3}^{[q]}(\tau_{4}-\tau_{3}\tau_{1}^{-1}\tau_{2})^{[q]}\right)f.$

Obviously,
$(f_{\mathfrak{T}}|V)(\mathfrak{Z})=e^{\lambda(\tau_{1}z_{4}+\tau_{2}z_{2}+\tau_{3}z_{3}+\tau_{4}z_{4})}$,
so

$\Delta(p,q)(f|V)=\left(\text{${}^{t}\\!\tau$}_{2}^{[p]}\sqcap\text{${}^{t}\\!\tau$}_{4}^{[q]}\text{${}^{t}\\!z$}_{3}^{[q]}(\text{${}^{t}\\!\tau$}_{1}-\text{${}^{t}\\!\tau$}_{3}\text{${}^{t}\\!\tau$}_{4}^{-1}\text{${}^{t}\\!\tau$}_{2})^{[q]}+\tau_{3}^{[p]}\sqcap\tau_{4}^{[q]}z_{2}^{[q]}(\tau_{1}-\tau_{2}\tau_{4}^{-1}\tau_{3})^{[q]}\right)f.$

The lemma then follows from properties of $\sqcap$ and the fact $A^{[p]}\sqcap
B^{[q]}=\text{${}^{t}\\!A$}^{[p]}\sqcap\text{${}^{t}\\!B$}^{[q]}$ if $p+q=n$.
∎

The following lemma can be proved exactly the same as [3, Section 2] for
unitary case.

######  Lemma 4.2.

(1) Let $\tau=\tau^{\ast}$ then

(4.4)
$\partial_{i}^{[q]}\det(\tau)^{s}=C_{q}(s)\det(\tau)^{s}\cdot\text{${}^{t}\\!\tau$}^{[-q]}.$

with
$C_{q}(s)=s(s+2)...(s+2q-2)=\frac{\Gamma_{q}(s+2q-2)}{\Gamma_{q}(s+2q-3)}$.
(2) For a holomorphic function $h$ on $\mathcal{H}_{2n}$, we have

(4.5) $\displaystyle\partial_{4}^{[q]}(h(-z_{4}^{-1})\det(z_{4})^{s})$
$\displaystyle=\det(z_{4})^{s}\sum_{a+b=q}(-1)^{a}\left(\begin{array}[]{c}q\\\
a\end{array}\right)C_{a}(s-2q+2)\text{${}^{t}\\!z$}_{4}^{[-q]}$
$\displaystyle\times(\text{${}^{t}\\!z$}_{4}^{[-a]}\sqcap(\partial_{4}^{[b]}h)(-z_{4}^{-1}))\text{${}^{t}\\!z$}_{4}^{-[q]}.$

######  Proposition 4.3.

Let $f=f_{\mathfrak{T}}$ be the test function with $\det(\tau_{1})\neq 0$ then

(4.6) $\displaystyle\Delta(p,q)(f|_{k}1\times
J_{n})=(-1)^{n}\sum_{a+b=q}\left(\begin{array}[]{c}q\\\
a\end{array}\right)C_{a}(k-2n-2q+2)$ $\displaystyle\times$
$\displaystyle\left(\overline{\tau_{1}z_{2}-\tau_{2}}^{[p]}\sqcap\text{${}^{t}\\!\tau$}_{1}^{[q]}\text{${}^{t}\\!z$}_{3}^{[q]}(1^{[a]}\sqcap\text{${}^{t}\\!z$}_{4}^{-[b]}\text{${}^{t}\\!($}\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2}-\tau_{4})^{[b]})\right.$
$\displaystyle+$
$\displaystyle\left.(\tau_{1}z_{2}-\tau_{2})^{[p]}\sqcap\tau_{1}^{[q]}z_{2}^{[q]}(1^{[a]}\sqcap
z_{4}^{-[b]}(\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2}-\tau_{4})^{[b]})\right)f|_{k+1}1\times
J_{n}.$

###### Proof.

Obviously

$(1\times
J_{n})\mathfrak{Z}=\left[\begin{array}[]{cc}z_{1}-z_{2}z_{4}^{-1}z_{3}&z_{2}z_{4}^{-1}\\\
z_{4}^{-1}z_{3}&-z_{4}^{-1}\end{array}\right],$ $(f|_{k}1\times
J_{n})(\mathfrak{Z})=\det(z_{4})^{-k}e^{\lambda(\tau_{1}(z_{1}-z_{2}z_{4}^{-1}z_{3})+\tau_{2}z_{4}^{-1}z_{3}+\tau_{3}z_{2}z_{4}^{-1}-\tau_{4}z_{4}^{-1})}.$

First note that for $\tau=\tau^{\ast}$ with $\det(\tau)\neq 0$ we have

$(\partial_{4}+\partial_{3}\text{${}^{t}\\!\tau$}^{-1}\partial_{2})^{[q]}\det(z_{4})^{-n}e^{\lambda(\tau
z_{2}z_{4}^{-1}z_{2}^{\ast})}=0.$

Indeed, write $\mathfrak{Z}=\mathfrak{X}+i\mathfrak{Y}$ with
$\mathfrak{X,Y}\in S(\mathbb{R})$. Without lose of generality, we replace
$\mathfrak{Z}$ by

$\mathfrak{Y}=\mathfrak{Y}^{\ast}=\left[\begin{array}[]{cc}y_{1}&y_{2}\\\
y_{3}&y_{4}\end{array}\right]>0,y_{3}^{\ast}=y_{2}.$

Note that

$\pi^{-2n^{2}}\int_{M_{n}(\mathbb{H})}e^{\lambda(x^{\ast}y_{4}x)}dx=\det(y_{4})^{-n}.$

Changing variables $x\mapsto x+y_{4}^{-1}y_{2}\tau^{1/2}$ we have

$\det(y)^{-n}e^{\lambda(\tau(y_{2}y_{4}^{-1}y_{2}^{\ast}))}=\pi^{-2n^{2}}\int_{M_{n}(\mathbb{H})}e^{-\lambda(x^{\ast}y_{4}x+\tau^{1/2}y_{2}x+x^{\ast}y_{2}^{\ast}\tau^{1/2})}dx.$

Applying
$(\partial_{4}+\partial_{3}\text{${}^{t}\\!\tau$}^{-1}\partial_{2})^{[q]}$ we
obtain

$\pi^{-2n^{2}}\int_{M_{n}(\mathbb{H})}\sum_{a+b=q}\left(\begin{array}[]{c}q\\\
a\end{array}\right)(-\overline{x}\text{${}^{t}\\!x$})^{[a]}\sqcap(\overline{x}\text{${}^{t}\\!x$})^{[b]}e^{-\lambda(x^{\ast}y_{4}x+\tau^{1/2}y_{2}x+x^{\ast}y_{2}^{\ast}\tau^{1/2})}dx=0.$

Decompose $f|_{k}1\times
J_{n}=g(z_{2},z_{3},z_{4})h(z_{4})e^{\lambda(t_{1}z_{1})}$ with

$\displaystyle g(z_{2},z_{3},z_{4})$
$\displaystyle=\det(z_{4})^{-n}e^{\lambda(-\tau_{1}(z_{2}-\tau_{1}^{-1}\tau_{2})z_{4}^{-1}(z_{2}-\tau_{1}^{-1}\tau_{2})^{\ast}}),$
$\displaystyle h(z_{4})$
$\displaystyle=\det(z_{4})^{n-k}e^{\lambda((\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2}-\tau_{4})z_{4}^{-1})}.$

Note that

$(\partial_{4}-\partial_{3}\text{${}^{t}\\!\tau$}_{1}^{-1}\partial_{2})^{[q]}(gh)=\sum_{a+b=q}\left(\begin{array}[]{c}q\\\
a\end{array}\right)((\partial_{4}-\partial_{3}\text{${}^{t}\\!\tau$}_{1}^{-1}\partial_{2})^{[a]}g)\sqcap(\partial_{4}^{[b]}h).$

The sum vanishes unless $a=0,b=q$. We then have

$\displaystyle(\partial_{4}-\partial_{3}\text{${}^{t}\\!\tau$}_{1}^{-1}\partial_{2})^{[q]}f|_{k}1\times
J_{n}$
$\displaystyle=(-1)^{q}\det(z_{4})^{n-k}\sum_{a+b=q}\left(\begin{array}[]{c}q\\\
a\end{array}\right)C_{a}(k-2n-2q+2)\text{${}^{t}\\!z_{4}$}^{-[q]}$
$\displaystyle\times(\text{${}^{t}\\!z_{4}$}^{-[a]}\sqcap\text{${}^{t}\\!($}\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2}-\tau_{4})^{[b]})\text{${}^{t}\\!z_{4}$}^{-[q]}e^{\lambda(\tau(1\times
J_{n})\mathfrak{Z})}.$

Note that

$\displaystyle\partial_{2}^{[p]}g$
$\displaystyle=(-1)^{p}\cdot\text{${}^{t}\\!(\tau_{1}z_{2}-\tau_{2})^{\ast[p]}$}\text{${}^{t}\\!z_{4}$}^{-[p]}g,$
$\displaystyle\text{${}^{t}\\!\partial$}_{3}^{[p]}g$
$\displaystyle=(-1)^{p}(\tau_{1}z_{2}-\tau_{2})^{[p]}z_{4}^{-[p]}g.$

The proposition then follows from properties of $\sqcap$. ∎

Setting

$\mathfrak{D}_{\kappa}=\sum_{p+q=n}\lambda_{\kappa}(q)\Delta(p,q),$

we will determine the coefficient $\lambda_{\kappa}(q)$ such that

$\mathfrak{D}_{\kappa}(f|_{\kappa}1\times
J_{n})=(\mathfrak{D}_{\kappa}f)|_{\kappa+1}1\times J_{n}.$

Let $f=f_{\tau}$ be the test function with $\det(\tau_{1})\neq 0$ as above. By
above proposition, we can write

$\displaystyle\mathfrak{D}_{\kappa}(f|_{\kappa}1\times J_{n})$
$\displaystyle=\sum_{p+q=n}\sum_{c+d=p}\sum_{a+b=q}(-1)^{a+b+d+n}\lambda_{\kappa}(q)\left(\begin{array}[]{c}p\\\
c\end{array}\right)\left(\begin{array}[]{c}q\\\
a\end{array}\right)C_{a}(-k+2n+2b)$
$\displaystyle\times\left(\text{${}^{t}\\!($}\tau_{1}z_{3})^{[c+q]}(1^{[a+c]}\sqcap\text{${}^{t}\\!z$}_{4}^{-[b]}\text{${}^{t}\\!($}\tau_{4}-\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2})^{[b]})\sqcap\text{${}^{t}\\!\tau$}_{2}^{[d]}\right.$
$\displaystyle+\left.(\tau_{1}z_{2})^{[c+q]}(1^{[a+c]}\sqcap
z_{4}^{-[b]}(\tau_{4}-\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2})^{[b]})\sqcap\tau_{2}^{[d]}\right)f|_{\kappa+1}1\times
J_{n}.$

Denote $t=a+c$ and put

$a_{\kappa}(t,b,d)=\sum_{a+c=t}(-1)^{a+b+d+n}\lambda_{\kappa}(a+b)\left(\begin{array}[]{c}a+b\\\
a\end{array}\right)\left(\begin{array}[]{c}c+d\\\
c\end{array}\right)C_{a}(-\kappa+2n+b).$

Then we have

$\displaystyle\mathfrak{D}_{\kappa}(f|_{\kappa}1\times J_{n})$
$\displaystyle=\sum_{t+b+d=n}a_{\kappa}(t,b,d)\left((\text{${}^{t}\\!\tau$}_{1}\text{${}^{t}\\!z$}_{3})^{[t+b]}(1^{[t]}\sqcap\text{${}^{t}\\!z$}_{4}^{[-b]}\text{${}^{t}\\!($}\tau_{4}-\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2})^{[b]})\sqcap\text{${}^{t}\\!\tau$}_{3}^{[d]}\right.$
$\displaystyle+\left.(\tau_{1}z_{2})^{[t+b]}(1^{[t]}\sqcap
z_{4}^{-[b]}(\tau_{4}-\tau_{2}^{\ast}\tau_{1}^{-1}\tau_{2})^{[b]})\sqcap\tau_{2}^{[d]}\right)f|_{{\kappa}+1}1\times
J_{n}.$

One computes that

$\displaystyle(\mathfrak{D}_{\kappa}f)|_{{\kappa}+1}1\times J_{n}$
$\displaystyle=\sum_{p+q=n}\lambda_{\kappa}(q)\left((\text{${}^{t}\\!\tau$}_{1}\text{${}^{t}\\!z$}_{3}\text{${}^{t}\\!z$}_{4}^{-1})^{[q]}\text{${}^{t}\\!($}\tau_{4}-\tau_{3}\tau_{1}^{-1}\tau_{2})^{[q]}\sqcap\text{${}^{t}\\!\tau$}_{3}^{[p]}\right.$
$\displaystyle\left.+(\tau_{1}z_{2}z_{4}^{-1})^{[q]}(\tau_{4}-\tau_{3}\tau_{1}^{-1}\tau_{2})^{[q]}\sqcap\tau_{2}^{[p]}\right)f|_{{\kappa}+1}1\times
J_{n}.$

Therefore, in order to have the desired property, we must have
$a_{\kappa}(0,q,p)=\lambda_{\kappa}(q)$ and $a_{\kappa}(t,b,d)=0$ otherwise.
By straightforward computation,

$\lambda_{\kappa}(q)=\left(\begin{array}[]{c}n\\\
q\end{array}\right)\tilde{C}_{q}(-{\kappa}+2n),\tilde{C}_{q}(s)=\prod_{\begin{subarray}{c}0\leq
p\leq n\\\ p\neq q\end{subarray}}C_{p}(s),$

has the desired property. We then define

(4.7)
$\mathfrak{D}_{\kappa}=\frac{1}{2}\sum_{p+q=n}\left(\begin{array}[]{c}n\\\
q\end{array}\right)\tilde{C}_{q}(-\kappa+2n)\Delta(p,q),$

and summarize our result in the following proposition.

######  Proposition 4.4.

In both Case I and Case II, there is a differential operator
$\mathfrak{D}_{\alpha}$ on $\mathcal{H}_{2n}$ with $\alpha\in\mathbb{C}$ such
that for any integer $\kappa>2n$ and any holomorphic function $f$ and
$\gamma\in G(\mathbb{R})$ we have

(4.8) $\displaystyle\mathfrak{D}_{\kappa}(f|V)$
$\displaystyle=(\mathfrak{D}_{\kappa}f)|V,$
$\displaystyle\mathfrak{D}_{\kappa}(f|_{\kappa}1\times\gamma)$
$\displaystyle=(\mathfrak{D}_{\kappa}f)|_{\kappa+1}1\times\gamma,$
$\displaystyle\mathfrak{D}_{\kappa}(f|_{\kappa}\gamma\times 1)$
$\displaystyle=(\mathfrak{D}_{\kappa}f)|_{\kappa+1}\gamma\times 1.$

###### Proof.

It suffices to prove these formulas for test function as above and generators
of $G(\mathbb{R})$. The first and second formula follows easily from our
previous discussion and the third formula follows from the first and second. ∎

######  Remark 4.5.

The above construction also works for unitary case if we replace
$C_{q}(-\kappa+2n)$ by $C_{q}(-\kappa+n)$ and $C_{a}(s)=s(s+1)...(s+a-1)$.

For $s\in\mathbb{C}$ and an integer $v\geq 1$ we define

$\mathfrak{D}_{s}^{v}=\mathfrak{D}_{s+v-1}\circ...\circ\mathfrak{D}_{s},\mathring{\mathfrak{D}}_{s}^{v}=(\mathfrak{D}_{s}^{v})|_{z_{2}=0}.$

For $\mathfrak{T}=-\epsilon\mathfrak{T}^{\ast}$ with $\epsilon=-1$ in Case I
and $\epsilon=1$ in Case II, we define a polynomial $\mathfrak{P}_{s}^{v}$ in
the entries $\mathfrak{T}_{ij}$ of $\mathfrak{T}$ by

$\mathring{\mathfrak{D}_{s}^{v}}(e^{\lambda(\mathfrak{T}\mathfrak{Z})})=\mathfrak{P}_{s}^{v}(\mathfrak{T})e^{\lambda(\tau_{1}z_{1}+\tau_{4}z_{4})},\mathfrak{T}=\left[\begin{array}[]{cc}\tau_{1}&\tau_{2}\\\
\tau_{3}&\tau_{4}\end{array}\right].$

$\mathring{\mathfrak{D}}_{s}^{v}$ is a homogeneous polynomial in
$\partial_{ij}$ with at most one term free of $\partial_{1}$ and
$\partial_{4}$, namely
$c_{s}^{v}(\partial_{2}^{[n]}+\text{${}^{t}\\!\partial$}_{3}^{[n]})^{v}$ with
certain constant $c_{s}^{v}$. To determine the constant, we observe that
(Case I)

$\mathfrak{D}_{s}(\det(z_{2}+z_{3})^{v})=C_{n}(v)\tilde{C}_{n}(-s+2n)\det(z_{2}+z_{3})^{v-1},$

(Case II)

$\mathfrak{D}_{s}(\det(z_{2}+z_{3})^{v})=C_{2n}(v)\tilde{C}_{2n}(-s+n)\det(z_{2}+z_{3})^{v-1}.$

Here $C_{a}(s)=s(s+\frac{1}{2})...(s+\frac{n-1}{2})$ in Case II. We remind the
reader that we are keep using the notation in [2] which is slightly different
to the one in [4]. We then have
(Case I)

$\mathring{\mathfrak{D}}_{s}^{v}(\det(z_{2}+z_{3})^{v})=\prod_{i=0}^{v-1}C_{n}(v-i)\tilde{C}_{n}(-s+i+2n).$

(Case II)

$\mathring{\mathfrak{D}}_{s}^{v}(\det(z_{2}+z_{3})^{v})=\prod_{i=0}^{v-1}C_{2n}(v-i)\tilde{C}_{2n}(-s+i+n).$

Therefore, $c_{s}^{v}=\prod_{i=0}^{v-1}\tilde{C}_{n}(-s+i+2n)$ in Case I and
$c_{s}^{v}=\prod_{i=0}^{v-1}\tilde{C}_{2n}(-s+i+n)$ in Case II. Let $p$ be an
odd prime. Then for
$\mathfrak{T}=\left[\begin{array}[]{cc}\tau_{1}&\tau_{2}\\\
\tau_{3}&\tau_{4}\end{array}\right]\in\Lambda$ such that
$2L^{-1}\tau_{1},2L^{-1}\tau_{4}\in M_{n}(\mathcal{O}^{\\#})$ with $L=p^{n}$ a
power of $p$, there is a congruence

(4.9) $\mathfrak{P}_{s}^{v}(\mathfrak{T})\equiv
c_{s}^{v}\det(\tau_{2})^{v}\text{ mod }L.$

###  4.2. Classical reformulations

Let $\mathbf{f}\in S_{\kappa}(K_{0}(\mathfrak{n}p))$ be an eigenform as in
Section 3. Denote $f\in S_{\kappa}(\Gamma_{0}(Np))$ be its corresponding
classical modular forms. Let $\chi\neq 1$ be a primitive Dirichlet character
of conductor $p^{\mathfrak{c}}$. Define

(4.10)
$\mathfrak{E}_{\kappa}(z,w,s):=j(g_{z},z_{0})^{\kappa}j(h_{z},z_{0})^{\kappa}\mathcal{E}_{\kappa}(g_{z},h_{z},s),$

for $z=g_{z}z_{0},w=h_{z}z_{0}$ be a function on
$\mathcal{H}\times\mathcal{H}$. Here $\mathcal{E}_{\kappa}$ is defined using
the Hecke character corresponds to $\chi$. Clearly
$\mathfrak{E}_{\kappa}(z,w,\kappa)\in
M_{\kappa}(\Gamma_{0}(N^{2}p^{2\mathfrak{c}}))\otimes
M_{\kappa}(\Gamma_{0}(N^{2}p^{2\mathfrak{c}}))$. Similarly, for $\chi=1$
define

(4.11)
$\mathfrak{E}_{\kappa}(z,w,s,i,r):=j(g_{z},z_{0})^{\kappa}j(h_{z},z_{0})^{\kappa}\mathcal{E}_{\kappa}(g_{z},h_{z},s,i,r).$

Then $\mathfrak{E}_{\kappa}(z,w,\kappa,i,r)\in
M_{\kappa}(\Gamma_{0}(N^{2}p^{2}))\otimes M_{\kappa}(\Gamma_{0}(N^{2}p^{2}))$.
The inner product formula in Proposition 3.16 and Proposition 3.21 can be
reformulated as

(4.12)
$\left\langle\mathfrak{E}_{\kappa}(z,-\overline{w},s),f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
N^{2}p^{2\mathfrak{c}}&0\end{array}\right](z)\right.\right\rangle_{\Gamma_{0}(N^{2}p^{2\mathfrak{c}})}^{z}=\frac{\Omega^{\mathfrak{c}}_{\kappa}(s)}{\Lambda^{2n}_{Np}(s,\chi)}L_{Np}(s,f,\chi)\overline{f(w)}$

for $\chi\neq 1$ and

(4.13) $\displaystyle\left\langle\sum_{\begin{subarray}{c}0\leq i\leq n\\\
r=0,1\end{subarray}}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\mathfrak{E}_{\kappa}(z,-\overline{w},s,i,r),f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
N^{2}p^{2}&0\end{array}\right](z)\right.\right\rangle^{z}_{\Gamma_{0}(Np)}$
$\displaystyle=$
$\displaystyle\frac{p^{-2n^{2}-2n+2ns}\Omega^{0}_{\kappa}(s)}{\Lambda^{2n}_{Np}(s)}\prod_{i=1}^{2n}(1-\beta_{i,p}p^{-s+2n+\epsilon})L_{N}(s,f)\overline{f(w)}.$

Let $v\geq 0$. For $\chi\neq 1$, define

(4.14)
$\mathfrak{E}_{\kappa}^{v}(z,w,s):=\delta(z)^{s-\kappa+v}\delta(w)^{s-\kappa+v}\mathfrak{D}_{s}^{v}\left(\delta(z)^{-s+\kappa-v}\delta(w)^{-s+\kappa-v}\mathfrak{E}_{\kappa-v}(z,w,s)\right).$

For $\chi=1$ define

(4.15)
$\mathfrak{E}_{\kappa}^{v}(z,w,s,i,r):=\delta(z)^{s-\kappa+v}\delta(w)^{s-\kappa+v}\mathfrak{D}_{s}^{v}\left(\delta(z)^{-s+\kappa-v}\delta(w)^{-s+\kappa-v}\mathfrak{E}_{\kappa-v}(z,w,s,i,r)\right).$

######  Proposition 4.6.

Assume $\kappa>2n$ and $f\in S_{\kappa}(\Gamma_{0}(Np))$ is an eigenform.
Denote
(Case I)

$\Omega^{\mathfrak{c}}_{\kappa,v}(s)=\prod_{i=0}^{v-1}\tilde{C}_{n}(-s+i+2n)C_{n}(-s-i)\Omega^{\mathfrak{c}}_{\kappa}(s),$

(Case II)

$\Omega^{\mathfrak{c}}_{\kappa,v}(s)=\prod_{i=0}^{v-1}\tilde{C}_{2n}(-s+i+n)C_{2n}(-s-i)\Omega^{\mathfrak{c}}_{\kappa}(s).$

(1) Let $\chi$ be a nontrivial Hecke character of conductor
$p^{\mathfrak{c}}$, then

(4.16)
$\left\langle\mathfrak{E}^{v}_{\kappa}(z,-\overline{w},s),f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
N^{2}p^{2\mathfrak{c}}&0\end{array}\right](z)\right.\right\rangle_{\Gamma_{0}(N^{2}p^{2\mathfrak{c}})}^{z}=\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(s)}{\Lambda^{2n}_{Np}(s,\chi)}L_{Np}(s,f,\chi)\overline{f(w)}.$

(2) Let $\chi=1$, then

(4.17) $\displaystyle\left\langle\sum_{\begin{subarray}{c}0\leq i\leq n\\\
r=0,1\end{subarray}}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\mathfrak{E}^{v}_{\kappa}(z,-\overline{w},s,i,r),f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
N^{2}p^{2}&0\end{array}\right](z)\right.\right\rangle^{z}_{\Gamma_{0}(Np)}$
$\displaystyle=$
$\displaystyle\frac{p^{-2n^{2}-2n+2ns}\Omega^{0}_{\kappa,v}(s)}{\Lambda^{2n}_{Np}(s)}\prod_{i=1}^{2n}(1-\beta_{i,p}p^{-s+2n+\epsilon})L_{N}(s,f)\overline{f(w)}.$

###### Proof.

Similar to the computations in [2, Section 4] we need to apply the
differential operators to

$f_{s}(\mathfrak{Z})=\det(z_{1}+z_{2}+z_{3}+z_{4})^{-s},\quad\mathfrak{Z}=\left[\begin{array}[]{cc}z_{1}&z_{2}\\\
z_{3}&z_{4}\end{array}\right]$

which is
(Case I)

$\mathfrak{D}_{s}^{v}f_{s}=\prod_{i=0}^{v-1}\tilde{C}_{n}(-s+i+2n)C_{n}(-s-i)f_{s+v},$

(Case II)

$\mathfrak{D}_{s}^{v}f_{s}=\prod_{i=0}^{v-1}\tilde{C}_{2n}(-s+i+n)C_{2n}(-s-i)f_{s+v}.$

Hence, up to those constants

$\mathfrak{D}_{s}^{v}(j(g,z)^{-\kappa+v}|j(g,z)|^{\kappa-
v-s})=j(g,z)^{-\kappa}|j(g,z)|^{\kappa-s}$

and the proposition can be easily obtained from (4.12),(4.13). ∎

##  5\. The Main Analytic Result

Let $p\nmid N$ be an odd prime split in $\mathbb{B}$ as in previous sections.
Let $f\in S_{\kappa}(\Gamma_{0}(Np))$ be an eigenform for all Hecke operators
and also an eigenform of $U(L)$ for any $L|p^{\infty}$, i.e
$f|U(L)=\alpha(L)f$. Suppose $\chi\neq 1$ is a nontrivial primitive Dirichlet
character of conductor $p^{\mathfrak{c}}$. We rewrite the inner product
formula in (4.16) as

(5.1)
$\displaystyle\left\langle\mathfrak{E}^{v}_{\kappa}(z,-\overline{w},s)\left|{}^{z}_{\kappa}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p^{2\mathfrak{c}}\end{array}\right]\right.\left|{}^{w}_{\kappa}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p^{2\mathfrak{c}}\end{array}\right]\right.,f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right](z)\right.\right\rangle_{\Gamma^{0}(N^{2}p^{2\mathfrak{c}})}^{z}$
$\displaystyle=$
$\displaystyle\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(s)}{\Lambda^{2n}_{Np}(s,\chi)}L_{Np}(s,f,\chi)\overline{f(w)}.$

This is a Petersson inner product of level $N^{2}p^{2\mathfrak{c}}$. We are
going to express it as a Petersson inner product of level $N^{2}p$. For
$f,g\in M_{\kappa}(\Gamma^{0}(N^{2}p^{2\mathfrak{c}}))$, there is a trace
operator

$\mathrm{Tr}f:=\sum_{\gamma}f|_{\kappa}\gamma,\mathrm{Tr}g:=\sum_{\gamma}g|_{\kappa}\gamma,$

where $\gamma$ runs over
$\Gamma^{0}(N^{2}p^{2\mathfrak{c}})\backslash\Gamma^{0}(N^{2}p)$ such that
$\mathrm{Tr}f,\mathrm{Tr}g\in M_{\kappa}(\Gamma^{0}(N^{2}p))$. Moreover, we
have

$\langle
f,g\rangle_{\Gamma^{0}(N^{2}p^{2\mathfrak{c}})}=\left\langle\mathrm{Tr}f,\mathrm{Tr}g\right\rangle_{\Gamma^{0}(N^{2}p)}.$

In particular, the representatives can be chosen as

$\left\\{\left[\begin{array}[]{cc}1&N^{2}pT\\\
0&1\end{array}\right]:T=\epsilon T^{\ast}\in M_{n}(\mathcal{O})\text{ mod
}p^{2\mathfrak{c}-1}\right\\}.$

By the fact

$\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p^{2\mathfrak{c}}\end{array}\right]\left[\begin{array}[]{cc}1&N^{2}pT\\\
0&1\end{array}\right]=\left[\begin{array}[]{cc}1&T\\\
0&p^{2\mathfrak{c}-1}\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
0&Np\end{array}\right],$

we have

$\mathrm{Tr}\left(f\left|{}_{\kappa}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p^{2\mathfrak{c}}\end{array}\right]\right.\right)=f|U(p^{2\mathfrak{c}-1})\left|{}_{\kappa}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right..$

For any $p$-power integer $L$ with $p^{\mathfrak{c}}|L$, define

(5.2) $\displaystyle\mathfrak{g}(z,w):=\mathfrak{g}(z,w,\kappa,v,\chi,L)$
$\displaystyle=$
$\displaystyle\Lambda_{Np}^{2n}(\kappa-v,\chi)\mathfrak{E}_{\kappa}^{v}(z,w,\kappa-v)|^{z}U(L^{2})|^{w}U(L^{2})\left|{}_{\kappa}^{z}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right.\left|{}_{\kappa}^{w}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right.,$

and similarly for $\chi=1$ define

(5.3) $\displaystyle\mathfrak{g}(z,w):=\mathfrak{g}(z,w,\kappa,v,1,L)$
$\displaystyle=$
$\displaystyle\sum_{i,r}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\Lambda_{Np}^{2n}(\kappa-v)$
$\displaystyle\times$
$\displaystyle\mathfrak{E}^{v}_{\kappa}(z,w,\kappa-v,i,r)|^{z}U(L^{2})|^{w}U(L^{2})\left|{}_{\kappa}^{z}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right.\left|{}_{\kappa}^{w}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right..$

Denote

(5.4)
$E_{p}(s,\chi)=\prod_{i=1}^{2n}\frac{1-\beta_{i,p}^{-1}\chi(p)p^{s-2n-\epsilon}}{1-\beta_{i,p}\chi(p)p^{2n-s+\epsilon-1}},$

so $E_{p}(s,\chi)=1$ unless $\chi=1$. Then by above discussions and
straightforward computations we have

(5.5)
$\displaystyle\left\langle\left\langle\mathfrak{g}(z,-\overline{w}),f\left|{}_{\kappa}\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right]\right.\right\rangle^{z}_{\Gamma^{0}(N^{2}p)},f\left|{}_{\kappa}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right.\right\rangle^{w}_{\Gamma^{0}(N^{2}p)}$
$\displaystyle=$
$\displaystyle\Omega^{\mathfrak{c}}_{\kappa,v}(\kappa-v)\frac{\alpha(pL^{4})}{\alpha(p^{2\mathfrak{c}})}\langle
f,f\rangle_{\Gamma^{0}(Np)}L_{Np}(\kappa-v,f,\chi)E_{p}(\kappa-v,\chi).$

We end up this section by recording the Fourier expansion of
$\mathfrak{g}(z,w)$ in the following proposition.

######  Proposition 5.1.

Let $\kappa,v\in\mathbb{Z}$ with $\kappa-v>2n$ and $\chi$ a primitive
Dirichlet character of conductor $p^{\mathfrak{c}}$. Denote
$\mathfrak{g}(z,w)$ for the function defined by (5.2) or (5.3). Then
$\mathfrak{g}(z,w)$ has Fourier expansion
(Case I)

(5.6) $\displaystyle\mathfrak{g}(z,w)$
$\displaystyle=p^{\mathfrak{c}n(n-1)}G(\chi)^{n}(N^{2}p)^{-2\kappa
n}\sum_{\mathfrak{T}}\mathfrak{P}_{\kappa-v}^{v}(\mathfrak{T})\chi^{-1}(\det(2\tau_{2}))e(2N^{-1}\lambda(\tau_{2}))$
$\displaystyle\times
A_{\kappa-v}(2n)\det(\mathfrak{T})^{\kappa-v-\frac{4n-1}{2}}\prod_{l\nmid
Np}P(\chi(l)l^{-\kappa+v})e(\lambda((N^{2}p)^{-1}(\tau_{1}z+\tau_{4}w))),$

(Case II)

(5.7) $\displaystyle\mathfrak{g}(z,w)$
$\displaystyle=p^{\mathfrak{c}n(n-1)}G(\chi)^{n}(N^{2}p)^{-2\kappa
n}\sum_{\mathfrak{T}}\mathfrak{P}_{\kappa-v}^{v}(\mathfrak{T})\chi^{-1}(\det(2\tau_{2}))e(2N^{-1}\lambda(\tau_{2}))$
$\displaystyle\times
A_{\kappa-v}(2n)\det(\mathfrak{T})^{\kappa-v-\frac{4n+1}{2}}\prod_{l\nmid
Np}P(\chi(l)l^{-\kappa+v})$ $\displaystyle\times
L_{Np}(\kappa-v-2n,\chi\rho_{\mathfrak{T}})e(\lambda((N^{2}p)^{-1}(\tau_{1}z+\tau_{4}w))).$

Here $\mathfrak{T}=\left[\begin{array}[]{cc}L^{2}\tau_{1}&\tau_{2}\\\
\tau_{3}&L^{2}\tau_{4}\end{array}\right]\in\Lambda_{2n}^{+}$ with
$\tau_{1},\tau_{4}\in\Lambda_{n},2\tau_{2}\in M_{n}(\mathcal{O}^{\\#})$,
$p\nmid\det(\mathfrak{T})$ and

(5.8)
$G(\chi)=\sum_{x\in\mathcal{O}_{p}/p^{\mathfrak{c}}\mathcal{O}_{p}}\chi(N(x))e(p^{-\mathfrak{c}}\lambda(x)).$

###### Proof.

Assume $\chi\neq 1$. For $\tau\in\Lambda_{2n}$, we write it as
$\tau=\left[\begin{array}[]{cc}\tau_{1}&\tau_{2}\\\
-\epsilon\tau_{2}^{\ast}&\tau_{4}\end{array}\right]$ with
$\tau_{1},\tau_{4}\in\Lambda_{n},\tau_{2}\in M_{n}(\mathcal{O})$. By our
definition of $\mathfrak{E}_{\kappa}(z,w,\kappa)$, it has a Fourier expansion
of the form

$\displaystyle\Lambda_{Np}^{2n}(\kappa,\chi)\mathfrak{E}_{\kappa}(z,w,\kappa)$
$\displaystyle=\sum_{\tau\in\Lambda_{2n}}a(\tau,\mathrm{diag}[y,v],\kappa,\chi)e(\lambda(hz))$
$\displaystyle\times\sum_{x\in
M_{n}(\mathcal{O}_{p})/p^{\mathfrak{c}}M_{n}(\mathcal{O}_{p})}\chi(\det
x)^{-1}e\left(\frac{2\lambda(\tau_{2}x)}{p^{\mathfrak{c}}}\right)e\left(\frac{2\lambda(\tau_{2})}{N}\right),$

where $z=x+iy,w=u+iv$ and $a(\tau,\mathrm{diag}[y,v],\kappa,\chi)$ are the
Fourier coefficients of $E_{\kappa}(z\times w,\kappa,\chi)$ as in (3.3). The
second line comes from the contributions of $\tilde{\mu}_{v}$ with $v|Np$. It
is nonzero unless $\det\tau_{2}$ is coprime to $p$. Hence, after applying the
operator $U(L^{2})$ we see that only those
$\mathfrak{T}=\left[\begin{array}[]{cc}L^{2}\tau_{1}&\tau_{2}\\\
-\epsilon\tau_{2}^{\ast}&\tau_{4}\end{array}\right]$ with $\det\tau_{2}$
coprime to $p$ can contribute to the Fourier expansions. In particular
$\mathfrak{T}>0$ with $\det\mathfrak{T}$ coprime to $p$. Therefore, by the
explicit formulas for $a(\tau,\mathrm{diag}[y,v],\kappa,\chi)$ in (3.4),(3.5)
we obtain the first part of the proposition.

For $\chi=1$, note that

$\displaystyle\Lambda_{Np}^{2n}(\kappa,\chi)\sum_{\begin{subarray}{c}0\leq
i\leq n\\\
r=0,1\end{subarray}}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\mathfrak{E}_{\kappa}(z,w,\kappa,i,r)$
$\displaystyle=$
$\displaystyle\sum_{\tau\in\Lambda_{2n}}a(\tau,\mathrm{diag}[y,v],\kappa,\chi)e(\lambda(hz))e(2N^{-1}\lambda(\tau_{2}))$
$\displaystyle\times$ $\displaystyle\sum_{\begin{subarray}{c}0\leq i\leq n\\\
r=0,1\end{subarray}}(-1)^{r}p^{\frac{(2i+r)(2i+r-1)}{2}-2n(2i+r)}\sum_{j}\sum_{x\in
M_{n}(\mathcal{O})\hat{\delta}_{rij}/M_{n}(\mathcal{O}_{p})}e(2\lambda(\tau_{2}x)).$

The second line comes from the contributions of $\mu^{i,r}_{p}$ and equals $1$
if $\det\tau_{2}\in\mathbb{Z}_{p}^{\times}$ and $0$ if otherwise. Hence, only
those $\mathfrak{T}\in\Lambda_{2n}^{+}$ with $\det\mathfrak{T}$ coprime to $p$
contributes to the Fourier expansion of $\mathfrak{g}(z,w)$ and the
proposition follows from the explicit formulas (3.4),(3.5). ∎

##  6\. Special $L$-values and the $p$-adic Measure

Let $f\in S_{\kappa}(\Gamma_{0}(N))$ be an eigenform with $\kappa>2n$. We make
following two assumptions:
(a) all primes $l\nmid N$ splits in $\mathbb{B}$ and $p$ is a fixed odd prime
with $p\nmid N$ (thus also splits in $\mathbb{B}$),
(b) $f$ is $p$-ordinary with nonzero $p$-stabilisation $f_{0}\in
S_{\kappa}(\Gamma_{0}(Np))$ and $f_{0}|U(L)=\alpha(L)f_{0}$ for any integer
$L$ with $L|p^{\infty}$.

In previous sections, we interpolate the special $L$-values of $f$ by inner
product formula (5.5) relate to Eisenstein series. The properties of these
special values then follows from the explicit formulas of the Fourier
expansion obtained in Proposition 5.1. In this section, we summarize these
properties and construct the $p$-adic measure interpolating these special
values.

Denote the $p$-Satake parameters of $f_{0}$ as $\beta_{1,p},...,\beta_{2n,p}$.
Let $2n+1\leq t\leq\kappa$ be an integer and $v=\kappa-t$. Since the Fourier
coefficients of $\mathfrak{g}(z,w)$ is holomorphic we have the following
analytic property.

######  Proposition 6.1.

Let $\chi$ be a primitive Dirichlet character of conductor $p^{\mathfrak{c}}$.
Then

$E_{p}(t,\chi)L_{Np}(t,f,\chi)$

has meromorphic continuation to whole complex plane and is holomorphic at
$t=2n+1,...,\kappa$.

The algebraicity of special $L$-values for Case I is already studied in our
previous paper [7]. In this paper we are restrict ourselves to the case that
our algebraic group is given by a totally isotropic hermitian form so the
associated symmetric space will be a ‘tube’ domain. Under this restriction the
algebraicity of modular forms can be described in terms of Fourier expansion
while for the general case discussed in [7, Section 5.2] we are using the
description of CM points or Fourier-Jacobi expansions. The advantage of using
the description via Fourier expansion is that we can now describe the explicit
action of $\sigma\in\mathrm{Aut}(\mathbb{C})$ as in the following proposition.

######  Proposition 6.2.

For any $\sigma\in\mathrm{Aut}(\mathbb{C})$ we have
(Case I)

(6.1)
$\displaystyle\left(\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(t)}{G(\chi)^{n}A_{t}(2m)}\frac{E_{p}(t,\chi)L_{Np}(f,t,\chi)}{\langle
f_{0},f_{0}\rangle}\right)^{\sigma}=\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(t)}{G(\chi^{\sigma})^{n}A_{t}(2n)}\frac{E_{p}(t,\chi)L_{Np}(f^{\sigma},t,\chi^{\sigma})}{\langle
f_{0}^{\sigma},f_{0}^{c\sigma c}\rangle},$

(Case II)

(6.2)
$\displaystyle\left(\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(t)}{G(\chi)^{n}A_{t}(2m)}\frac{E_{p}(t,\chi)L_{Np}(f,t,\chi)}{g(\chi)\langle
f_{0},f_{0}\rangle}\right)^{\sigma}=\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(t)}{G(\chi^{\sigma})^{n}A_{t}(2n)}\frac{E_{p}(t,\chi)L_{Np}(f^{\sigma},t,\chi^{\sigma})}{g(\chi^{\sigma})\langle
f_{0}^{\sigma},f_{0}^{c\sigma c}\rangle}.$

for $t=2n+1,...,\kappa$. Here $f^{c}$ is the complex conjugation of $f$.

This can be proved by the same method as in the appendix of [4] using the
Fourier expansion of $\mathfrak{g}(z,w)$ in Proposition 5.1 and algebraic
properties of Fourier coefficients in Proposition 3.1. As remarked in [4],
since only the Fourier coefficients of maximal rank involved in the Fourier
expansion of $\mathfrak{g}(z,w)$ we actually obtain a better result that
allowing us to discuss the special values below the absolute convergence bound
of Eisenstein series. More precisely, here we are assuming $\kappa>2n$ while
in [7, Theorem 6.3] we need $\kappa>4n-1$ (note that $n$ there is $2n$ here).

We now start the construction of the $p$-adic measure. Let
$\mathbb{C}_{p}=\widehat{\overline{\mathbb{Q}}}_{p}$ be the completion of
$\overline{\mathbb{Q}}_{p}$, and fix an embedding
$\overline{\mathbb{Q}}\to\mathbb{C}_{p}$. The $p$-adic absolute value
$|\cdot|_{p}$ naturally extends to $\mathbb{C}_{p}$, denote

$\mathcal{O}_{\mathbb{C}_{p}}=\\{x\in\mathbb{C}_{p}:|x|_{p}\leq 1\\}.$

A $p$-adic distribution on $\mathbb{Z}_{p}^{\times}$ is a continuous linear
functional
$\mu:C^{\infty}(\mathbb{Z}_{p}^{\times},\mathbb{C}_{p})\to\mathbb{C}_{p}$. To
construct a $p$-adic distribution it is enough to define $\mu$ on all sets of
form $a+L\mathbb{Z}_{p}$ satisfying

$\mu(a+L\mathbb{Z}_{p})=\sum_{b=0}^{p-1}\mu(a+bL+pL\mathbb{Z}_{p}).$

For such $\mu$ and a locally constant function $f$, we denote
$\mu(f)=\int_{\mathbb{Z}_{p}^{\times}}fd\mu$. A $p$-adic distribution is
called a $p$-adic measure if it is bounded.

We will follow the strategy of constructing $p$-adic measures in [8]. That is
we first define a $p$-adic distribution interpolate the special $L$-values.
Then check the distribution satisfying the Kummer congruences so the $p$-adic
measure exist. Finally, since the $p$-adic measure is uniquely determined by
its value at all primitive Dirichlet character of $p$-power conductor, we
conclude that the $p$-adic distribution we constructed is the desired $p$-adic
measure. We recall the criterion of Kummer congruences as follow.

######  Proposition 6.3.

Let $\\{f_{i}\\}\subset
C(\mathbb{Z}_{p}^{\times},\mathcal{O}_{\mathbb{C}_{p}})$ be a dense system of
continuous functions, $\\{a_{i}\\}\subset\mathcal{O}_{\mathbb{C}_{p}}$ a
system of elements. Then the existence of p-adic measure
$\mu:C(\mathbb{Z}_{p}^{\times},\mathcal{O}_{\mathbb{C}_{p}})\to\mathcal{O}_{\mathbb{C}_{p}}$
such that

$\int_{\mathbb{Z}_{p}^{\times}}f_{i}d\mu=a_{i}$

is equivalent to the following Kummer congruences: for an arbitrary choice of
elements $b_{i}\in\mathbb{C}_{p}$ almost all of which vanish

$\sum_{i}b_{i}f_{i}(y)\in p^{m}\mathcal{O}_{\mathbb{C}_{p}}\text{ for all
}y\in\mathbb{Z}_{p}^{\times}\Longrightarrow\sum_{i}b_{i}a_{i}\in
p^{m}\mathcal{O}_{\mathbb{C}_{p}}.$

Recall that $\epsilon=-1$ in Case I and $\epsilon=1$ in Case II. For a
function $\mathfrak{g}(z,w)$ on $M_{k}(\Gamma^{0}(N^{2}p),\psi)\otimes
M_{k}(\Gamma^{0}(N^{2}p))$ let

$\mathcal{F}(\mathfrak{g})=\frac{\left\langle\left\langle\mathfrak{g}(z,-\overline{w}),f_{0}\left|{}_{k}\left[\begin{array}[]{cc}0&\epsilon\\\
1&0\end{array}\right]\right.\right\rangle^{w}_{\Gamma^{0}(N^{2}p)},f_{0}\left|{}_{k}\left[\begin{array}[]{cc}1&0\\\
0&N^{2}p\end{array}\right]\right.\right\rangle^{z}_{\Gamma^{0}(N^{2}p)}}{\langle
f_{0},f_{0}\rangle^{2}_{\Gamma_{0}(N^{2}p)}}.$

In particular, taking $\mathfrak{g}$ as in (5.2),(5.3) we obtain the desired
special $L$-values by (5.5).

We first discuss Case I. Define

$\mathcal{H}^{t}_{a,L}(z,w)=p^{2\kappa
n}A_{t}(2n)^{-1}\sum_{\chi}\chi(a)c_{\chi}^{-n(n-1)}G(\chi)^{-n}\cdot\mathfrak{g}(z,w,\kappa,v,\chi,L),$

where $\chi$ runs over all characters of conductor $c_{\chi}|L$. It is clear
that $\alpha(L)^{-4}\mathcal{F}(\mathcal{H}^{t}_{a,L})$ is independent of $L$.
Thus if we set

$\mu_{t}(a+L\mathbb{Z}_{p}):=\frac{p}{(p-1)L}\alpha(L)^{-4}\mathcal{F}(\mathcal{H}^{t}_{a,L}),$

then

$\sum_{b=0}^{p-1}\mu_{t}(a+bp^{n}+pL\mathbb{Z}_{p})=p\cdot\frac{p}{(p-1)pL}\alpha(pL)^{-4}\mathcal{F}(\mathcal{H}_{a+bp^{n},pL}^{t})=\mu_{t}(a+L\mathbb{Z}_{p}).$

Therefore, $\mu_{t}$ is a $p$-adic distribution on $\mathbb{Z}_{p}^{\times}$
such that

$\displaystyle\int_{\mathbb{Z}_{p}^{\times}}\chi d\mu_{t}$
$\displaystyle=\sum_{a\in(\mathbb{Z}/L\mathbb{Z})^{\times}}\chi(a)\mu_{t}(a+L\mathbb{Z}_{p})$
$\displaystyle=p^{2\kappa
n}A_{t}(2n)^{-1}c_{\chi}^{-n(n-1)}G(\chi)^{-n}\alpha(L)^{-4}\cdot\mathcal{F}(\mathfrak{g}(z,w,\kappa,v,\chi))$
$\displaystyle=p^{2kn}c_{\chi}^{-n(n-1)}G(\chi)^{-n}\alpha(p)\alpha(c_{\chi})^{-2}\frac{\Omega^{\mathfrak{c}}_{k,v}(t)}{A_{t}(2n)}\frac{E_{p}(t,\chi)L_{Np}(f_{0},t,\chi)}{\langle
f_{0},f_{0}\rangle}.$

These distribution indeed take algebraic values and we view them as elements
in $\mathbb{C}_{p}$. By the Fourier expansion of $\mathfrak{g}(z,w)$ in
Proposition 5.1, we have

(6.3)
$\mathcal{H}_{a,L}^{t}(z,w)=\sum_{\tau_{1},\tau_{4}\in\Lambda_{n}^{+}}\alpha_{a,L}(\tau_{1},\tau_{4})e(2N^{-1}\lambda(\tau_{2}))e(\lambda((N^{2}p)^{-1}(\tau_{1}z+\tau_{4}w))),$

with

(6.4) $\alpha_{a,L}(\tau_{1},\tau_{4})=N^{-4\kappa
n}\sum_{\tau_{2}}\mathfrak{P}_{t}^{v}(\mathfrak{T})\sum_{\chi}\chi(a\det(2\tau_{2})^{-1})\prod_{l\nmid
Np}P(\chi(l)l^{-t}).$

Here $\mathfrak{T}=\left[\begin{array}[]{cc}L^{2}\tau_{1}&\tau_{2}\\\
\tau_{2}^{\ast}&L^{2}\tau_{4}\end{array}\right]$ and
$p\nmid\det(\mathfrak{T})$ so these Fourier coefficients are actually
$p$-integral. Put
$\tilde{\mathcal{H}}_{a,L}^{t}(z,w)=a^{t-2n}\kappa_{t}\mathcal{H}_{a,L}^{t}(z,w)$
with $\kappa_{t}=\frac{c_{2n+1}^{\kappa-2n-1}}{c_{t}^{v}}$ and
$\tilde{\mu}_{t}(a+L\mathbb{Z}_{p})=\frac{p}{(p-1)L}\alpha(L)^{-4}\mathcal{F}(\tilde{H}_{a,L}^{t})$,
then $\tilde{\mathcal{H}}_{a,L}^{t}(z,w)\equiv
a^{t-2n-1}\cdot\tilde{\mathcal{H}}_{a,L}^{2n+1}(z,w)\text{ mod }L$ and
$d\tilde{\mu}_{t}(x)=x^{2n+1-t}d\tilde{\mu}_{2n+1}(x)$ by congruence (4.9).
Set

$\mu(a+L\mathbb{Z}_{p})=\frac{p}{(p-1)L}\alpha(L)^{-4}\mathcal{F}(\tilde{\mathcal{H}}_{a,L}^{2n+1}).$

######  Theorem 6.4.

There is a unique $p$-adic measure $\mu$ on $\mathbb{Z}_{p}^{\times}$ such
that for primitive Dirichlet character $\chi$ of $p$-power conductor
$c_{\chi}$ we have

(6.5)
$\int_{\mathbb{Z}_{p}^{\times}}\chi(x)x^{-t+2n}d\mu=\frac{\alpha(p)}{\alpha(c_{\chi}^{2})}\frac{p^{2\kappa
n}\kappa_{t}}{c_{\chi}^{n(n-1)}G(\chi)^{n}}\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(t)}{A_{t}(2n)}\frac{E_{p}(t,\chi)L_{Np}(f,t,\chi)}{\langle
f_{0},f_{0}\rangle}.$

for $t=2n+1,...,\kappa$.

###### Proof.

The interpolation properties are satisfied by our construction. Since the
Fourier coefficients of $\tilde{H}_{a,L}^{2n+1}$ is $p$-integral, by the same
argument as [4, Lemma 9.7], $\mathcal{F}(\tilde{H}_{a,L}^{2n+1})$ takes values
in $\mathcal{O}_{\mathbb{C}_{p}}$ up to a constant $C$. That is
$C^{-1}\cdot\mu$ defines a measure
$C(\mathbb{Z}_{p}^{\times},\mathcal{O}_{\mathbb{C}_{p}})\to\mathcal{O}_{\mathbb{C}_{p}}$.
It suffices to check the Kummer congruences for $C^{-1}\cdot\mu$ or
equivalently for $\mu_{t}$. Let $\\{b_{i}\\}\subset\mathbb{C}_{p}$ be a system
of elements with almost of all vanish and assume

$\sum_{i}b_{i}\chi_{i}(x)\in p^{m}\mathcal{O}_{\mathbb{C}_{p}}\text{ for all
}x\in\mathbb{Z}_{p}^{\times}.$

We need to show that

$\sum_{i}b_{i}\int_{\mathbb{Z}_{p}^{\times}}\chi_{i}(x)d\mu_{t}\in
p^{m}\mathcal{O}_{\mathbb{C}_{p}}.$

This equals

$\mathcal{F}\left(p^{2\kappa
n}A_{t}(2n)^{-1}\alpha(L)^{-4}\sum_{i}b_{i}c_{\chi_{i}}^{-n(n-1)}G(\chi_{i})^{-n}\cdot\mathfrak{g}(z,w)\right).$

Taking the Fourier expansion of function under the bracket as (6.3), its
Fourier coefficients equals

$\displaystyle\alpha_{a,L}(\tau_{1},\tau_{4})$ $\displaystyle=N^{-4\kappa
n}\alpha(L)^{-4}\sum_{\tau_{2}}\mathfrak{P}_{t}^{v}(\mathfrak{T})\det(\mathfrak{T})^{t-\frac{4n-1}{2}}$
$\displaystyle\times\sum_{i}b_{i}\chi_{i}(a\det(2\tau_{2})^{-1})\cdot\prod_{l\nmid
Np}P(\chi_{i}(l)l^{-t}).$

The theorem follows since the first line in above formula is $p$-integral as
$\alpha(L)$ is a $p$-adic unit by the ordinarity assumption. ∎

We now extend above discussion to Case II. As in symplectic case we need
assumption on the parity of characters. In Case I the parity assumption is not
necessary due to the fact that $\mathrm{det}(g)\in\mathbb{R}_{+}$ for
$g\in\mathrm{GL}_{n}(\mathbb{H})$ (see [6, Remark 3.1]).

######  Theorem 6.5.

There is a unique $p$-adic measure $\mu$ on $\mathbb{Z}_{p}^{\times}$ such
that for primitive Dirichlet character $\chi$ of $p$-power conductor
$c_{\chi}$ and $\chi(-1)=(-1)^{t}$ we have

(6.6) $\displaystyle\int_{\mathbb{Z}_{p}^{\times}}\chi(x)x^{-t+2n}d\mu$
$\displaystyle=2(2\pi
i)^{t-2n}(t-2n-1)!\frac{\alpha(p)}{\alpha(c_{\chi}^{2})}\frac{p^{2\kappa
n}\kappa_{t}}{c_{\chi}^{n(n+1)-t}G(\chi)^{n}}$ $\displaystyle\times
g(\chi)\frac{1-\overline{\chi(p)}p^{t-2n-1}}{1-\chi(p)p^{2n-t}}\frac{\Omega^{\mathfrak{c}}_{\kappa,v}(t)}{A_{t}(2n)}\frac{E_{p}(t,\chi)L_{Np}(f_{0},t,\chi)}{\langle
f_{0},f_{0}\rangle}.$

Here $g(\chi)$ is the Gauss sum for Dirichlet character $\chi$.

###### Proof.

The proof is similar to Case I but note that the Fourier coefficient of
$\mathfrak{g}(z,w)$ has a term of Dirichlet $L$-functions. For $2n+1\leq
t\leq\kappa$, define

$\displaystyle\mathcal{H}_{a,L}^{t}(z,w)$ $\displaystyle=p^{2\kappa
n}A_{t}(2n)^{-1}\sum_{\chi}\chi(a)c_{\chi}^{-n(n-1)}G(\chi)^{-n}$
$\displaystyle\times 2(2\pi
i)^{t-2n}(t-2n-1)!c_{\chi}^{t-2n}g(\chi)\frac{1-\overline{\chi(p)}p^{t-2n-1}}{1-\chi(p)p^{2n-t}}\cdot\mathfrak{g}(z,w),$

and

$\mu_{t}(a+L\mathbb{Z}_{p})=\frac{p}{(p-1)L}\alpha(L)^{-4}\mathcal{F}(\mathcal{H}_{a,L}^{t}).$

Then $\mu_{t}$ is a $p$-adic distribution on $\mathbb{Z}_{p}^{\times}$ such
that

$\displaystyle\int_{\mathbb{Z}_{p}^{\times}}\chi d\mu_{t}$
$\displaystyle=2(2\pi i)^{n+t}(n+t-1)!\alpha(p)\alpha(c_{\chi})^{-2}p^{2\kappa
n}c_{\chi}^{-n(n-1)}G(\chi)^{-n}$ $\displaystyle\times
c_{\psi\chi^{\circ}\chi}^{t-2n}g(\psi\chi^{\circ}\chi)\frac{1-\overline{\chi(p)}p^{t-2n-1}}{1-\chi(p)p^{2n-t}}\frac{\Omega_{\kappa,v}(t)}{A_{t}(2n)}\frac{E_{p}(t,\chi)L_{Np}(f_{0},t,\chi)}{\langle
f_{0},f_{0}\rangle}.$

Take the Fourier expansion of $\mathcal{H}_{a,L}^{t}(z,w)$ as in Case I. Its
Fourier coefficent can be written as

$\displaystyle\alpha_{a,L}(\tau_{1},\tau_{4})$ $\displaystyle=N^{-4\kappa
n}\sum_{\tau_{2}}\det(\mathfrak{T})^{t-\frac{4n+1}{2}}\mathfrak{P}_{t}^{v}(\mathfrak{T})\sum_{\chi}\chi(a\det(2\tau_{2}))^{-1}\prod_{l\nmid
Np}P(\chi(l)l^{-t})$ $\displaystyle\times 2(2\pi
i)^{t-2n}(t-2n-1)!c_{\chi}^{t-2n}g(\chi)\left(1-\overline{\chi(p)}p^{t-2n-1}\right)L_{N}(t-2n,\chi\rho_{\mathfrak{T}}).$

We now check that the distribution $\mu_{l}$ satisfies Kummer congruences. Let
$\\{b_{i}\\}\subset\mathbb{C}_{p}$ be a system of elements with almost of all
vanish and assume

$\sum_{i}b_{i}\chi_{i}(x)\in p^{m}\mathcal{O}_{\mathbb{C}_{p}}\text{ for all
}x\in\mathbb{Z}_{p}^{\times}.$

We need to show that

$\sum_{i}b_{i}\int_{\mathbb{Z}_{p}^{\times}}\chi_{i}(x)d\mu_{l}\in
p^{m}\mathcal{O}_{\mathbb{C}_{p}}.$

This equals

$\mathcal{F}\left(2(2\pi i)^{t-2n}(t-2n-1)!p^{2\kappa
n}A_{t}(2n)^{-l}\alpha(L)^{-4}\sum_{i}b_{i}\chi_{i}(a)c_{\chi_{i}}^{-n}G(\chi_{i})^{-n}\right.$
$\left.c_{\chi_{i}}^{t-2n}g(\chi_{i})\frac{1-\overline{\chi_{i}(p)}p^{t-2n-1}}{1-\chi_{i}(p)p^{2n-t}}\cdot\mathfrak{g}(z,w)\right).$

Taking the Fourier expansion of function under the bracket, its Fourier
coefficients equals

$\displaystyle\alpha_{a,L}(\tau_{1},\tau_{4})$ $\displaystyle=N^{-4\kappa
n}\alpha(L)^{-4}\sum_{\tau_{2}}\mathfrak{P}_{t}^{v}(\mathfrak{T})\det(\mathfrak{T})^{t-\frac{4n+1}{2}}$
$\displaystyle\times\sum_{i}b_{i}(1-\overline{\chi_{i}(p)}p^{t-2n-1})L(1-t+2n,\overline{\chi_{i}\rho_{\mathfrak{T}}})\chi_{i}(a\det(4\tau_{2}))^{-1}$
$\displaystyle\times\prod_{l\nmid
Np}P(\chi_{i}(l)l^{-t})\cdot\prod_{l|N}\left(1-\chi_{i}\rho_{\mathfrak{T}}(l)l^{2n-t}\right).$

Here we are using the functional equation of Dirichlet $L$-functions. The sum
over $i$ is in $p^{m}\mathcal{O}_{\mathbb{C}_{p}}$ by the Kummer congruences
for Dirichlet $L$-functions (see [4, Section 8] for $p$-adic measure attached
to Dirichlet $L$-function). Gluing all these $\mu_{t}$ together as in Case I
by using the congruence properties of $\mathfrak{P}_{t}^{v}(\mathfrak{T})$, we
then obtain the desired $p$-adic measure $\mu$. ∎

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