# On vector valued automorphic forms and the standard $L$-function attached to
them

Oliver Stein  Fakultät für Informatik und Mathematik
Ostbayerische Technische Hochschule Regensburg
Galgenbergstrasse 32
93053 Regensburg
Germany<EMAIL_ADDRESS>
###### Abstract.

We develop a theory of vector valued automorphic forms associated to the Weil
representation $\omega_{f}$ and corresponding to vector valued modular forms
transforming with the “finite” Weil representation $\rho_{L}$. For each prime
$p$ we determine the structure of a vector valued spherical Hecke algebra
depending on $\omega_{f}$, which acts on the space of automorphic forms. We
finally define a standard $L$-function of such an automorphic form an prove
that it can be continued meromorphically to the whole complex $s$-plane.

###### 2020 Mathematics Subject Classification:

11F27 11F25 11F41 11F66

##  1\. Introduction

An extremely important aspect of modular forms is their connection to
$L$-functions, which in turn play a key role in number theory and beyond.
There are several ways to associate an $L$-function to a modular form. Two
examples of $L$-functions are relevant for the present paper. Both are
associated to a common Hecke eigenform $f$. The first one is defined in terms
of the eigenvalues $\lambda_{D}(f)$ of a family of Hecke operators $T(D)$,
that is, $f\mid T(D)=\lambda_{D}(f)f$. It is given by

(1.1) $D(s,f)=\sum_{D}\lambda_{D}(f)\det(D)^{-s}.$

The second one is constructed as an Euler product in terms of the Satake
parameters $\alpha_{i,p}$ of $f$, where $p$ is a prime. It can be written in
the form $L(s,f)=\prod_{p<\infty}L_{p}(s,f)$ with

$L_{p}(s,f)=R(\alpha_{i,p},p^{s}),$

where $R(\alpha_{i,p},p^{s})$ is a rational expression depending on the
parameters $\alpha_{i,p}$ and the prime power $p^{s}$. One way to obtain
$R(\alpha_{i,p},p^{s})$, is to express the formal local power series
$\sum_{D}T(D)X^{r(D)}$ as a rational function in $X$ as was done in [Bo2],
[BM], [BoSP] or [Sh1].

Vector valued modular forms transforming with the Weil representation play an
important role any many recent papers. Most of the classical theory of modular
forms has been established for this type of modular forms over the past years.
However, still not much is known about associated Dirichlet series. In [BS], a
zeta function of the form (1.1) was introduced. In [St1] the analytic
properties of a slightly more general zeta function were investigated. So far,
to the best of my knowledge, there has been no standard $L$-function defined
for this type of modular forms. The main objective of the present paper is to
define such a standard $L$-function assigned to a vector valued Hecke
eigenform $f$ and to prove that it is meromorphic on the complex $s$-plane. To
achieve this, we mainly follow Murase ([Mu1], $\S$ 4), Bouganis and Marzec
([BM], Chapter 7) and Böcherer and Schulze-Pillot ([BoSP], I.$\S$2). As in
[Mu1], we first develop a theory of vector valued automorphic forms for the
Weil representation and prove that the space of these forms is isomorphic to
the space of vector valued cusp forms for the Weil representation. This may be
of interest in its own right. We then define for each prime $p$ a spherical
Hecke algebra depending on the Weil representation and an action of this
algebra on the space automorphic forms, which is compatible with the action of
Hecke operators (as defined in [BS]) on the space of cusp forms. For a common
automorphic eigenform $F$ of all elements $T_{k,l}$ of the before mentioned
spherical Hecke algebra we are then able to produce a result of the form

$\sum_{(k,l)}\lambda_{F,p}(T_{k,l})p^{-(k+l)}=R(\alpha_{1,p},\alpha_{2,p},p^{s}),$

where the Satake parameters are given by two unramified characters of the
$p$-adic numbers $\mathbb{Q}_{p}$. As was suggested in [BM] and [BoSP], we use
this identity to define a standard $L$-function of $F$. The last part of this
paper is concerned with the proof of the meromorphic continuation of this
$L$-function to the whole $s$-plane.

Let us describe the content of the paper in more detail. To this end, let
$(L,(\cdot,\cdot))$ be an even lattice of even rank $m$ and type
$(b^{+},b^{-})$ with (even) signature $\operatorname{sig}(L)=b^{+}-b^{-}$ and
level $N$. Associated to the bilinear form $(\cdot,\cdot)$ there is a
quadratic form $q$. The modulo 1 reduction of $(\cdot,\cdot)$ and $q$ defines
a bilinear form and quadratic form, respectively, on the discriminant form
$D=L^{\prime}/L$. Here $L^{\prime}$ is the dual lattice of $L$. The Weil
representation $\rho_{L}$ is a representation of
$\Gamma=\operatorname{SL}_{2}(\mathbb{Z})$ on the group ring $\mathbb{C}[D]$.
As usual, denote with $\mathbb{Z}_{p}$ the ring of $p$-adic integers. As will
be explained later in the paper, $\rho_{L}$ is isomorphic to a finite
dimensional subrepresentation of the Weil representation
$\omega_{f}=\bigotimes_{p<\infty}\omega_{p}$ (originally defined by Weil
[Wei]) of $\operatorname{SL}_{2}(\widehat{\mathbb{Z}})$ on a space $S_{L}$
(isomorphic to $\mathbb{C}[D]$). Here $\widehat{\mathbb{Z}}$ is defined by
$\prod_{p<\infty}\mathbb{Z}_{p}$. We have the relation

$\rho_{L}(\gamma)=\omega_{f}(\gamma_{f})$

for all $\gamma\in\Gamma$, where $\gamma_{f}$ is the projection of $\gamma$
into $\operatorname{SL}_{2}(\widehat{\mathbb{Z}})$. For the details see
Chapter 4.

For $\kappa\in\mathbb{Z}$, a vector valued modular form of weight $\kappa$ and
type $\rho_{L}$ is a holomorphic function $f:\mathbb{H}\rightarrow\mathbb{C}$,
which satisfies

$f(\gamma\tau)=(c\tau+d)^{\kappa}\rho_{L}(\gamma)f(\tau)$

for all $\gamma=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\Gamma$ and all $\tau$ in the complex upper half
plane $\mathbb{H}$, and is holomorphic at the cusp $\infty$. We denote the
space of all such functions with $M_{\kappa}(\rho_{L})$ and write
$S_{\kappa}(\rho_{L})$ for the subspace of cuspforms. Now let $\mathbb{A}$ be
the ring of adeles, $\mathcal{G}(\mathbb{Q})$ a subgroup of
$\operatorname{GL}_{2}(\mathbb{Q})^{+}$ and

$\mathcal{G}(\mathbb{A})=\sideset{}{{}^{\prime}}{\prod}_{p\leq\infty}\mathcal{Q}_{p}=\left\\{(g_{p})\in\prod_{p\leq\infty}\mathcal{Q}_{p}\;|\;g_{p}\in\mathcal{K}_{p}\text{
for almost all primes }p\right\\},$

where $\mathcal{Q}_{p}$ and $\mathcal{K}_{p}$ is a subgroup of
$GL_{2}(\mathbb{Q}_{p})$ and $\operatorname{GL}_{2}(\mathbb{Z}_{p})$,
respectively. We assign to $f$ a function
$F_{f}:\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A})\rightarrow\mathbb{C}$
by means of strong approximation for the group $\mathcal{G}(\mathbb{A})$. For
$g=g_{\mathbb{Q}}(g_{\infty}\times k)$ we put

$F_{f}(g)=\omega_{f}^{-1}(k)j(g_{\infty},i)^{-\kappa}f(g_{\infty}i).$

Here $g_{\mathbb{Q}}\in\mathcal{G}(\mathbb{Q})$,
$g_{\infty}\in\mathcal{Q}_{\infty}<\operatorname{GL}_{2}^{+}(\mathbb{R})$ and
$k\in\mathcal{K}=\prod_{p<\infty}\mathcal{K}_{p}$. In Proposition 6.3 and
Lemma 6.4 we will show that $F$ is a cuspidal vector valued automorphic form
of type $\omega_{f}$, which can be seen as a vector valued analogue of a
scalar valued cuspidal automorphic form. Moreover, Theorem 6.5 states that
space of these functions, satisfying further properties, is isomorphic to
$S_{\kappa}(\rho_{L})$. We denote this space with $A_{\kappa}(\omega_{f})$.
The inverse map can also be explicitly given: For $F\in
A_{\kappa}(\omega_{f})$ it can be proven that $f_{F}$, specified by

$\tau\mapsto f_{F}(\tau)=j(g_{\tau},i)^{\kappa}F(g_{\tau}\times 1_{f})$

with $g_{\tau}=\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right)\left(\begin{smallmatrix}y^{1/2}&0\\\
0&y^{-1/2}\end{smallmatrix}\right)$ and $\tau=g_{\tau}i=x+iy\in\mathbb{H}$, is
indeed an element of $S_{\kappa}(\rho_{L})$. Note that the definition of
$F_{f}$ has already occurred in the work of Werner ([We]). The function
$f_{F}$ can be found in Kudla’s paper [Ku]. However, as far as I know, a
theory of vector valued automorphic forms has not yet appeared in the
literature.

It is well known that for each prime $p$ the spherical Hecke algebra
$\mathcal{H}(\operatorname{GL}_{2}(\mathbb{Q}_{p})//\operatorname{GL}_{2}(\mathbb{Z}_{p}))$
of locally constant, compactly supported and complex-valued functions, which
additionally satisfy

$f(k_{1}gk_{2})=f(g)$

for all $k_{1},k_{2}\in\operatorname{GL}_{2}(\mathbb{Z}_{p})$ and all
$g\in\operatorname{GL}_{2}(\mathbb{Q}_{p})$, acts on the space of automorphic
forms. This action is compatible with action of Hecke operators on the space
of cusp forms (see for example [BP], [Ge] or [KL]). Werner introduced in [We]
an action on $A_{\kappa}(\omega_{f})$, which is compatible with the action of
the double coset $\Gamma\left(\begin{smallmatrix}p^{-1}&0\\\
0&1\end{smallmatrix}\right)\Gamma$ on $S_{\kappa}(\rho_{L})$. In this paper,
we extend Werner’s result. We define for each prime $p$ a spherical Hecke
algebra $\mathcal{H}(\mathcal{Q}_{p})//\mathcal{K}_{p},\omega_{p})$ of type
$\omega_{p}$ as follows: Let $L_{p}=L\otimes\mathbb{Z}_{p}$ and $S_{L_{p}}$ as
above, but associated to the $p$-adic lattice $L_{p}$. Note that
$S_{L}=\bigotimes_{p<\infty}S_{L_{p}}$ (see Chapter 4 for details). Then
$\mathcal{H}(\mathcal{Q}_{p})//\mathcal{K}_{p},\omega_{p})$ consists of all
locally constant and compactly supported functions
$f:\mathcal{Q}_{p}\rightarrow S_{L_{p}}$, which satisfy

$f(k_{1}gk_{2})=\omega_{p}(k_{1})\circ f(g)\circ\omega_{p}(k_{2})$

for all $k_{1},k_{2}\in\mathcal{K}_{p}$ and all $g\in\mathcal{Q}_{p}$. Hecke
algebras of this type are well known and studied in the literature ([BK],
[Ho], [He]). A “tool” to investigate the structure of Hecke algebras (for a
pair of groups $(G,K)$ with suitable properties), vector valued or not, is the
Satake map (see [Sa] or [Ca]), whose image is a Hecke algebra easier to
understand. Under the assumption that $L_{p}^{\prime}/L_{p}$ is anisotropic,
we determine a set of generators of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and connect it to a
scalar valued Hecke algebra by means of a variant of the classical Satake map,
which we adopt from [He] and which is suitable in our situation. Subsequently,
we define an action of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ on
$A_{\kappa}(\omega_{f})$, which can be interpreted as vector-valued analogue
of the versions in [BP] or [Mu1]. We show in Theorem 6.9 that this action is
compatible with the action of Hecke operator on $S_{\kappa}(\rho_{L})$.

Denote with $T_{k,l}$, $(k,l)\in\mathbb{Z}^{2}$ with $l\geq k\geq 0$ and
$k+l\in 2\mathbb{Z}$, a set of generators of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and $F\in
A_{\kappa}(\omega_{f})$ a common eigenform of all $T_{k,l}$. Following Arakawa
in [Ar], we transfer some results on spherical functions to the vector valued
setting and use them to prove

$\sum_{(k,l)\in\Lambda_{+}}\lambda_{F,p}(T_{k,l})p^{-s(k+l)}=\frac{1}{C(L_{p})}\frac{1+\chi_{F,p}^{(1)}(p)\chi_{F,p}^{(2)}(p)p^{-2s+1}}{(1-\chi_{F,p}^{(1)}(p^{2})p^{-2s+1})(1-\chi_{F,p}^{(2)}(p^{2})p^{-2s+1})},$

where $(\chi_{F,p}^{(1)},\chi_{F,p}^{(2)})$ is a pair of unramified characters
of $\mathbb{Q}_{p}$ and $C(L_{p})$ is some constant. As indicated above, this
identity gives rise to our definition of the local standard $L$-function
$L_{p}(s,F)$.

The last part of the paper deals with the analytic properties of $L(s,F)$.
Following [Ar] again, we find a direct relation between the standard
$L$-function and a zeta function $\mathcal{Z}(s,f)$ similar to the one in
(1.1). In [St1] a variant of the before mentioned zeta function was studied.
It turns out that it is a meromorphic function and satisfies a functional
equation. The property of being meromorphic carries over to $L(s,F)$, the
functional equation does unfortunately not. The local $L$-function
$L_{p}(s,F)$ possesses an integral representation of the form

(1.2) $\int_{\mathcal{Q}_{p}}\nu_{s+\frac{1}{2}}(g)\omega_{\chi}(g)dg,$

where $\nu_{s}$ is defined in (7.14) and $\omega_{\chi}$ is the zonal
spherical function associated to the pair of characters
$(\chi_{F,p}^{(1)},\chi_{F,p}^{(2)})$. The analytic properties of integrals of
this type (as a function of $s$) were subject of several papers, among them
[Ta] and [An]. The main difference between (1.2) and these integrals is that
the one considered in the cited papers are defined over a multiplicative group
of an algebra, which is not the case for (1.2). As consequence, the tools used
in [Ta] and [An] to prove a functional equation, are not at our disposal for
this task. It seems (to the best of my knowledge) unclear to date how they can
be replaced. It would be interesting to investigate this subject further and
to provide the means to prove a functional equation in the present situation.

##  2\. Notation

As usual, we let $e(z)$, $z\in\mathbb{C}$ be the abbreviation for $e^{2\pi
iz}$. For any prime $p\in\mathbb{Z}$ by $\mathbb{Q}_{p}$ we mean the field of
$p$-adic numbers and by $\mathbb{Z}_{p}$ its ring of $p$-adic integers;
$|\cdot|_{p}$ is the $p$-adic absolute value and
$\operatorname{ord}_{p}(\cdot)$ the $p$-adic valuation of $\mathbb{Q}_{p}$. We
write $\mathbb{A}$ for the adele ring and for $\mathbb{A}^{\times}$ for the
idele group. By $\mathbb{A}_{f}$ we mean the set of finite adeles. For any
ring $R$, as usual, $M_{2,2}(R)$ and $\operatorname{GL}_{2}(R)$ are the set of
$2\times 2$ matrices, the subgroup of invertible matrices in $M_{2,2}(R)$. The
following subgroups are important for this paper:

$\displaystyle\mathcal{G}(R)=\\{M\in
GL_{2}(R)\;|\;\det(M)\in(R^{\times})^{2}\\},$ $\displaystyle
D(R)=\\{\left(\begin{smallmatrix}r&0\\\
0&r^{-1}\end{smallmatrix}\right)\;|\;r\in R^{\times}\\},$ $\displaystyle
M(R)=\\{\left(\begin{smallmatrix}r_{1}&0\\\
0&r_{2}\end{smallmatrix}\right)\;|\;r_{1},r_{2}\in R^{\times}\\},$
$\displaystyle N(R)=\\{\left(\begin{smallmatrix}1&r\\\
0&1\end{smallmatrix}\right)\;|\;r\in R\\},$ $\displaystyle
U(R)=\\{\left(\begin{smallmatrix}1&0\\\ r&1\end{smallmatrix}\right)\,|\;r\in
R\\}\text{ and }$

$B(R)=M(R)\cdot N(R)=N(R)\cdot M(R)$. Throughout the paper we use the
following abbreviations for certain elements of these groups

$\displaystyle n\\_(c)=\left(\begin{smallmatrix}1&0\\\
c&1\end{smallmatrix}\right),\quad n(b)=\left(\begin{smallmatrix}1&b\\\
0&1\end{smallmatrix}\right),\quad m(s)=\left(\begin{smallmatrix}s&0\\\
0&s^{-1}\end{smallmatrix}\right),\quad
m(t_{1},t_{2})=\left(\begin{smallmatrix}t_{1}&0\\\
0&t_{2}\end{smallmatrix}\right)\text{ and }w=\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right).$

For $R=\mathbb{Q}_{p}$ or $R=\mathbb{Z}_{p}$ the groups listed below will be
frequently be utilized in this paper.

(2.1)
$\begin{split}&\mathcal{Q}_{p}=\\{M\in\operatorname{GL}_{2}(\mathbb{Q}_{p})\;|\;\det(M)\in(\mathbb{Q}_{p}^{\times})^{2}\\},\\\
&\mathcal{K}_{p}=\\{M\in\operatorname{GL}_{2}(\mathbb{Z}_{p})\;|\;\det(M)\in(\mathbb{Z}_{p}^{\times})^{2}),\\\
&\mathcal{M}_{p}=\\{M\in
M(\mathbb{Q}_{p})\;|\;\det(M)\in(\mathbb{Q}_{p}^{\times})^{2}\\},\\\
&\mathcal{D}_{p}=\mathcal{M}_{p}\cap\mathcal{K}_{p}\text{ and }\\\
&\mathcal{N}_{p}=\\{\left(\begin{smallmatrix}r_{1}&0\\\
0&r_{2}\end{smallmatrix}\right)\in
M(\mathbb{Z}_{p})\;|\;r_{1}\in(\mathbb{Z}_{p}^{\times})^{2},r_{2}=1\\}.\end{split}$

Moreover, the subgroup

(2.2) $\mathcal{K}_{0}(p)=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathcal{K}_{p}\;|\;c\equiv 0\bmod{p}\right\\}$

of $\mathcal{K}_{p}$ will be relevant. Since $\mathcal{Q}_{p}$ is locally
compact (see Lemma 5.3), we may fix a Haar measure on $\mathcal{Q}_{p}$ such
that $\int_{G\cap\mathcal{K}_{p}}dg=1$ for any of the groups
$G=\mathcal{Q}_{p},\mathcal{M}_{p}$ and $N(\mathbb{Q}_{p})$. We denote with
$\mu(K)$ the measure of any subgroup $K$ of $\mathcal{K}_{p}$.

For $x_{p}\in\mathcal{Q}_{p}$ let
$\iota_{p}(x_{p})=(\alpha_{q})_{q\leq\infty}\in\operatorname{GL}_{2}(\mathbb{A})$
with $\alpha_{q}=1_{q}$ for $q\not=p$ and $\alpha_{p}=x_{p}$.

For an arbitrary set $A$, by $\mathbbm{1}_{A}$ we mean the characteristic
function of $A$. Moreover, we will make frequently use of the following
subsets of $\mathbb{Z}^{2}$:

$\displaystyle\Lambda=\\{(k,l)\in\mathbb{Z}^{2}\;|\;k,l\geq 0\text{ and
}k+l\in 2\mathbb{Z}\\}\text{ and }$
$\displaystyle\Lambda_{+}=\\{(k,l)\in\Lambda\;|\;k\leq l\\}.$

Finally, as usual, we write
$\mathbb{H}=\\{\tau\in\mathbb{C}\;|\;\operatorname{Im}(\tau)>0\\}$ for the
complex upper half plane and $\left(\frac{\cdot}{d}\right)$ for the Legendre
symbol.

##  3\. Hecke operators $T(m(p^{k},p^{l}))$ and standard zeta-functions

In this section we briefly summarize some facts on lattices, discriminant
forms and the “finite” Weil representation. We also recall the definition of
vector valued modular forms for the Weil representation and some related
theory relevant for the present paper. Subsequently, we explain how to extend
the definition of the Hecke operator $T(p^{2l})^{*}$ in Definition 5.5 of [BS]
to the action of double cosets of the form $\Gamma m(p^{k},p^{l})\Gamma$. In
[BS] a standard zeta function associated to an eigenform of all Hecke
operators $T(m(n^{2},1)),\;n\in\mathbb{N}$ is introduced. Here, we consider a
variant of this zeta function and state some analytic properties based on the
investigations in [St1].

Let $L$ be a lattice of rank $m$ equipped with a symmetric $\mathbb{Z}$-valued
bilinear form $(\cdot,\cdot)$ such that the associated quadratic form

$q(x):=\frac{1}{2}(x,x),\quad x\in L,$

takes values in $\mathbb{Z}$. We assume that $m$ is even, $L$ is non-
degenerate and denote its type by $(b^{+},b^{-})$ and signature $b^{+}-b^{-}$
by $\operatorname{sig}(L)$. Note that $\operatorname{sig}(L)$ is also even. We
stick with these assumptions on $L$ for the rest of this paper unless we state
it otherwise. Further, let

$L^{\prime}:=\\{x\in V=L\otimes\mathbb{Q}\;:\;(x,y)\in\mathbb{Z}\quad\text{
for all }\;y\in L\\}$

be the dual lattice of $L$. Since $L\subset L^{\prime}$, the elementary
divisor theorem implies that $L^{\prime}/L$ is a finite group. We denote this
group by $D$. The modulo 1 reduction of both, the bilinear form
$(\cdot,\cdot)$ and the associated quadratic form, defines a
$\mathbb{Q}/\mathbb{Z}$-valued bilinear form $(\cdot,\cdot)$ with
corresponding $\mathbb{Q}/\mathbb{Z}$-valued quadratic form on $D$. We call
$D$ combined with $(\cdot,\cdot)$ a discriminant form or a quadratic module.
We call it anisotropic, if $q(\mu)=0$ holds only for $\mu=0$.

It is well known that any discriminant form can be decomposed into direct sum
of quadratic modules of the following form (cf. [BEF])

$\displaystyle\mathcal{A}_{p^{k}}^{t}=\left(\mathbb{Z}/p^{k}\mathbb{Z},\;\frac{tx^{2}}{p^{k}}\right),\;p>2,\quad\mathcal{A}_{2^{k}}^{t}=\left((\mathbb{Z}/2^{k}\mathbb{Z},\;\frac{tx^{2}}{2^{k+1}}\right),$
$\displaystyle\mathcal{B}_{2^{k}}=\left(\mathbb{Z}/2^{k}\mathbb{Z}\oplus\mathbb{Z}/2^{k}\mathbb{Z};\;\frac{x^{2}+2xy+y^{2}}{2^{k}}\right),\quad\mathcal{C}_{2^{k}}=\left(\mathbb{Z}/2^{k}\mathbb{Z}\oplus\mathbb{Z}/2^{k}\mathbb{Z};\;\frac{xy}{2^{k}}\right).$

The structure of anisotropic finite quadratic modules is well known: In
particular, for an odd prime $p$ each $p$-group $D_{p}$ of a discriminant form
$D$ can be either written as $\mathcal{A}_{p}^{t}$ or as a direct sum
$\mathcal{A}_{p}^{t}\oplus\mathcal{A}_{p}^{1}$. For further details we refer
to [BEF].

The Weil representation $\rho_{L}$ is a representation of
$\Gamma=\operatorname{SL}_{2}(\mathbb{Z})$ on the group ring $\mathbb{C}[D]$.
We denote the standard basis of $\mathbb{C}[D]$ by
$\\{\mathfrak{e}_{\lambda}\\}_{\lambda\in D}$. The standard scalar product on
$\mathbb{C}[D]$ is given by

(3.1) $\left\langle\sum_{\lambda\in
D}a_{\lambda}\mathfrak{e}_{\lambda},\sum_{\lambda\in
D}b_{\lambda}\mathfrak{e}_{\lambda}\right\rangle=\sum_{\lambda\in
D}a_{\lambda}\overline{b_{\lambda}}.$

As $\Gamma$ is generated by the matrices

(3.2) $S=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix},\quad T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},$

it is sufficient to define $\rho_{L}$ by the action on these generators.

######  Definition 3.1.

The representation $\rho_{L}$ of $\Gamma$ on $\mathbb{C}[D]$, defined by

(3.3)
$\begin{split}&\rho_{L}(T)\mathfrak{e}_{\lambda}:=e(bq(\lambda))\mathfrak{e}_{\lambda},\\\
&\rho_{L}(S)\mathfrak{e}_{\lambda}:=\frac{e(-\operatorname{sig}(L)/8)}{|D|^{1/2}}\sum_{\mu\in
D}e(-(\mu,\lambda))\mathfrak{e}_{\mu},\end{split}$

is called Weil representation.

Let $Z=\left(\begin{smallmatrix}-1&0\\\ 0&-1\end{smallmatrix}\right)$. The
action of $Z$ is given by

(3.4)
$\rho_{L}(Z)(\mathfrak{e}_{\lambda})=e(-\operatorname{sig}(L)/4)\mathfrak{e}_{-\lambda}.$

We denote by $N$ the level of the lattice $L$. It is the smallest positive
integer such that $Nq(\lambda)\in\mathbb{Z}$ for all $\lambda\in L^{\prime}$.
One can prove that the Weil representation $\rho_{L}$ is trivial on
$\Gamma(N)$, the principal congruence subgroup of level $N$. Therefore,
$\rho_{L}$ factors over the finite group

$\Gamma/\Gamma(N)\cong\operatorname{SL}_{2}(\mathbb{Z}/N\mathbb{Z}).$

For the rest of this paper we suppose that $N$ is odd.

The following Lemma describes the action of the Weil representation on
matrices of the form

$R_{d}=ST^{d}S^{-1}T^{a}ST^{d},$

where $ad\equiv 1\bmod{N}$. It es easily checked that
$R_{d}\equiv\left(\begin{smallmatrix}a&0\\\
0&d\end{smallmatrix}\right)\bmod{N}$.

######  Lemma 3.2.

For $a,d$ as above we have

(3.5)
$\rho_{L}(R_{d})\mathfrak{e}_{\lambda}=\frac{g_{d}(D)}{g(D)}\mathfrak{e}_{\lambda}.$

Here $g_{d}(D)$ denotes the Gauss sum

(3.6) $g_{d}(D)=\sum_{\lambda\in D}e(dq(\lambda))$

and $g(D)=g_{1}(D)$.

Notice that by Milgram’s formula we can evaluate $g(D)$ explicitly

$g(D)=\sqrt{|D|}e(\operatorname{sig}(L)/8).$

We now define vector-valued modular forms of type $\rho_{L}$. With respect to
the standard basis of $\mathbb{C}[D]$ a function
$f:\mathbb{H}\rightarrow\mathbb{C}[D]$ can be written in the form

$f(\tau)=\sum_{\lambda\in D}f_{\lambda}(\tau)\mathfrak{e}_{\lambda}.$

The following operator generalises the usual Petersson slash operator to the
space of all those functions. For $\kappa\in\mathbb{Z}$ define

(3.7)
$f\mid_{\kappa,L}\gamma=j(\gamma,\tau)^{-\kappa}\rho_{L}(\gamma)^{-1}f(\gamma\tau),$

where

$j(g,\tau)=\det(g)^{-1/2}(c\tau+d)$

is the usual automorphy factor if $g=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{GL}_{2}^{+}(\mathbb{R})$.

A holomorphic function $f:\mathbb{H}\rightarrow\mathbb{C}[D]$ is called a
modular form of weight $\kappa$ and type $\rho_{L}$ for $\Gamma$ if
$f\mid_{\kappa,L}\gamma=f$ for all $\gamma\in\Gamma$, and if $f$ is
holomorphic at the cusp $\infty$. Here the last condition means that all
Fourier coefficients $c(\lambda,n)$ with $n<0$ vanish. If in addition
$c(\lambda,n)=0$ for all $n=0$, we call the corresponding modular form a cusp
form. We denote by $M_{\kappa}(\rho_{L})$ the space of all such modular forms,
by $S_{\kappa}(\rho_{L})$ the subspace cusp forms. For more details see e.g.
[Br1] or [BS]. Note that formula (3.4) implies that
$M_{\kappa}(\rho_{L})=\\{0\\}$ unless

(3.8) $2\kappa\equiv\operatorname{sig}(L)\pmod{2}.$

Therefore, if the signature of $L$ is even, only non-trivial spaces of
integral weight can occur.

The Petersson scalar product on $S_{\kappa}(\rho_{L})$ is given by

(3.9) $(f,g)=\int_{\Gamma(1)\backslash\mathbb{H}}\langle
f(\tau),g(\tau)\rangle\operatorname{Im}\tau^{k}d\mu(\tau)$

where

$d\mu(\tau)=\frac{dx\;dy}{y^{2}}$

denotes the hyperbolic volume element.

In [BS] Hecke operators $T(d^{2})^{*}$ on $S_{\kappa}(\rho_{L})$ were
introduced. More generally, for

$\displaystyle\mathcal{G}(N)=$
$\displaystyle\left\\{M\in\operatorname{GL}_{2}^{+}(\mathbb{Q});\quad\exists
n\in\mathbb{Z}\text{ with }(n,N)=1\text{ such that }nM\in
M_{2}(\mathbb{Z})\right.$ $\displaystyle\left.\text{ and
}(\det(nM),N)=1\right\\},$
$\mathcal{Q}(N)=\left\\{(M,r)\in\mathcal{G}(N)\times(\mathbb{Z}/N\mathbb{Z})^{*};\quad\det(M)\equiv
r^{2}\pmod{N}\right\\}$

and $g\in\mathcal{Q}(N)$, the Hecke operator was defined in the usual way by
the action of double cosets $\Gamma g\Gamma$:

(3.10) $f\mid_{\kappa,L}T(g,r)=\det(g)^{l/2-1}\sum_{M\in\Gamma\backslash\Gamma
g\Gamma}f\mid_{\kappa,L}(M,r)$

In [BS] it is explained in detail how to extend $\rho_{L}$ to the group
$\mathcal{Q}(N)$. In this paper, we content ourselves with Hecke operators
defined by matrices $g\in\mathcal{G}(\mathbb{Q})$. Let

$\mathcal{G}^{N}(\mathbb{Q})=\\{g\in\mathcal{G}(\mathbb{Q})\;|\;(\det(g),N)=1\\}.$

Then $\mathcal{G}^{N}(\mathbb{Q})$ can be embedded homomorphically into
$\mathcal{Q}(N)$ by $g\mapsto(g,t\bmod{N})$, where $\det(g)=t^{2}$. Thus, we
identify any matrix $g\in\mathcal{G}^{N}(\mathbb{Q})$ with $(g,t\bmod{N})$ and
write throughout the whole paper only $\rho_{L}(g)$ or $T(g)$ instead of
$\rho_{L}(g,t)$ or $T(g,t)$, respectively.

Let $m(k,l)\in\mathcal{G}^{N}(\mathbb{Q})\cap M_{2}(\mathbb{Z})$ with $k|l$.
As already pointed out in [St1] in a similar situation, there is a close
relation between the Hecke operators $T(m(k,l))$ and $T(m(k^{-1},l^{-1}))$:
Clearly,

$\Gamma m(k,l)\Gamma=\Gamma m(l,k)\Gamma=m(kl,kl)\Gamma
m(k^{-1},l^{-1})\Gamma$

and thus

(3.11) $\begin{split}T(m(k^{-1},l^{-1}))&=\frac{g(D)}{g_{kl}(D)}T(m(k,l))\\\
&=T(m(k,l)).\end{split}$

For the last equation we have utilized that $kl$ is a square mod $N$ and
consequently $g_{kl}(D)=g(D)$. In the same vein, we obtain

(3.12)
$\begin{split}&T(m(k,l))=\frac{g_{k}(D)}{g(D)}T(m(\frac{l}{k},1)).\end{split}$

The factors $\frac{g(D)}{g_{kl}(D)}$ and $\frac{g_{k}(D)}{g(D)}$ result from
the action of $m(kl,kl)$ and $m(k,k)^{-1}$, respectively in the Weil
representation(cf. [BS], (3.5)). Observe that the very same definition of
$\rho_{L}(m(k,k)^{-1})$ and $\rho_{L}(m(kl,kl))$ is still meaningful even if
$(k,N)>1$.

As demonstrated in [BS], Chapter 5, (3.10) defines still a Hecke operator if
we assume $(k,N)>1$ and choose $\Gamma m(k^{2},1)\Gamma$ as double coset
acting on $S_{k,L}$. The goal of the subsequent remarks is to extend the Hecke
operator in this case to double cosets $\Gamma m(k,l)\Gamma$ and $\Gamma
m(k^{-1},l^{-1})\Gamma$ with $m(k,l),m(k^{-1},l^{-1})$ as above, but
$(kl,N)>1$. We make use of the same ideas as in [St1], Chap. 4. To this end,
we define

$\rho_{L}^{-1}(m(k,l))\mathfrak{e}_{\lambda}=\frac{g_{k}(D)}{g(D)}\rho_{L}^{-1}(m(1,\frac{l}{k}))\mathfrak{e}_{\lambda}$

and

(3.13)
$\rho_{L}^{-1}(m(k^{-1},l^{-1}))\mathfrak{e}_{\lambda}=\frac{g(D)}{g_{l}(D)}\rho_{L}^{-1}(m(\frac{l}{k},1))\mathfrak{e}_{\lambda}.$

In view of these identities, the extension to the corresponding double cosets
works the same way as in [BS], Chapter 5. Consequently, (3.12) is still valid
for $T(m(k^{-1},l^{-1}))$ and $T(m(k,l))$. Also, the arguments leading to
(3.11) are still applicable in the case $(kl,N)>1$. Thus,
$T(m(k^{-1},l^{-1}))$ and $T(m(k,l))$ are related by this identity. However,
the factor $\frac{g(D)}{g_{kl}(D)}$ is not equal to one in this case.

For the following remarks we assume that level $N$ of $L$ is additionally
square free. We keep this assumption whenever we are dealing with the zeta
function $\mathcal{Z}(s,f)$ (to be defined below). Let $f\in
S_{\kappa}(\rho_{L})$ be a simultaneous eigenform of all Hecke operators
$T(m(k^{2},1)),\;k\in\mathbb{N}$, with eigenvalues $\lambda_{f}(m(k^{2},1))$
(see Remark 6.1, ii) in [St1] for some details when such a cusp form exists).
Then the relations (3.11) and (3.12) immediately imply that $f$ is an
eigenform of all Hecke operators $T(m(k^{-1},l^{-1}))$ and $T(m(k,l))$ with
eigenvalues

(3.14)
$\lambda_{f}(m(k,l))=\frac{g_{k}(D)}{g(D)}\lambda_{f}(m(\frac{l}{k},1))$

and

(3.15)
$\lambda_{f}(m(k^{-1},l^{-1}))=\begin{cases}\lambda_{f}(m(k,l)),&(kl,N)=1,\\\
\frac{g(D)}{g_{kl}(D)}\lambda_{f}(m(k,l)),&(kl,N)>1.\end{cases}$

In [St1] the analytic properties of the standard zeta function

$Z(s,f)=\sum_{k\in\mathbb{N}}\lambda_{f}(m(k^{2},1))k^{-2s}$

were studied. By Theorem 5.6 in [BS] this series possesses an Euler product

(3.16) $Z(s,f)=\prod_{p}Z_{p}(s,f)$

with $Z_{p}(s,f)=\sum_{k\in\mathbb{N}}\lambda_{f}(m(p^{2k},1))p^{-2ks}.$ Let
us consider the more general local zeta function

$\mathcal{Z}_{p}(s,f)=\sum_{(k,l)\in\Lambda_{+}}\lambda_{f}(m(p^{-k},p^{-l}))p^{-s(k+l)}.$

By (3.15) we have

(3.17)
$\mathcal{Z}_{p}(s,f)=\begin{cases}\sum_{(k,l)\in\Lambda_{+}}\lambda_{f}(m(p^{k},p^{l}))p^{-s(k+l)},&(p,N)=1,\\\
\sum_{(k,l)\in\Lambda_{+}}\frac{g(D)}{g_{p^{k+l}}(D)}\lambda_{f}(m(p^{k},p^{l}))p^{-s(k+l)},&p\mid
N.\end{cases}$

Subsequently, we want to relate that the zeta functions introduced above,
which boils down to evaluate the expressions $\frac{g(D)}{g_{p^{k+l}}(D)}$ and
$\frac{g_{k}(D)}{g(D)}$ explicitly. By [We1], Lemma 5.8, we know that

(3.18) $n\mapsto\chi_{D}(n)=\frac{g_{n}(D)}{g(D)}$

is a quadratic character of $(\mathbb{Z}/N\mathbb{Z})^{\times}$. More
specifically,

(3.19)
$\frac{g(D)}{g_{n}(D)}=\frac{g_{n}(D)}{g(D)}=\left(\frac{n}{|D|}\right)e\left(\frac{(t-1)\text{oddity}(D)}{8}\right).$

A proof for the last equation can be found in [We1], Theorem 5.17. It is known
that if $|D|$ is odd, $\text{ oddity}(D)\equiv 0\bmod{8}$, see e. g. [We1],
Lemma 5.8 or [CS], Chap. 15, $\S$ 7\. Thus, $\frac{g(D)}{g_{n}(D)}$ simplifies
to

(3.20) $\chi_{D}(n)=\left(\frac{n}{|D|}\right)$

in this case. We now evaluate the quotient
$\frac{g(D)}{g_{p^{r}}(D)},\;r\in\mathbb{N}_{0},$ more explicitly for the case
$p\mid N$. Decomposing $D$ into $p$-groups $D_{p}$, we find

(3.21)
$\frac{g(D)}{g_{p^{r}}(D)}=\frac{g(D_{p})}{g_{p^{r}}(D_{p})}\prod_{q\nmid
N}\frac{g(D_{q})}{g_{p^{r}}(D_{q})}.$

By the assumption on $N$, the level of $D_{p}$ is $p$ and therefore

(3.22) $g_{p^{r}}(D_{p})=\begin{cases}|D_{p}|,&r>0,\\\
g(D_{p}),&r=0.\end{cases}$

Let $D_{p}^{\perp}$ be the orthogonal complement of $D_{p}$ in $D$ with
respect to $\langle\cdot,\cdot\rangle$. Then we clearly have $D=D_{p}\oplus
D_{p}^{\perp}$ and $(|D_{p}^{\perp}|,p)=1$. Since $(p,q)=1$, in view of (3.22)
the left-hand side of (3.21) can be written in the form

(3.23)
$\frac{e(\operatorname{sig}(D_{p})/8)}{|D_{p}|^{1/2}}\chi_{D_{p}^{\perp}}(p^{r}).$

Notice, if $p^{r}$ is a square, $g_{p^{r}}(D_{p}^{\perp})=g(D_{p}^{\perp})$
and $\chi_{D_{p}^{\perp}}(p^{r})=1$.

It follows for a “bad” prime $p$ dividing $N$ by means of (3.14) and (3.23)
that (3.17) becomes

$\displaystyle\mathcal{Z}_{p}(s,f)$
$\displaystyle=\frac{e(\operatorname{sig}(D_{p})/8)}{|D_{p}|^{1/2}}\sum_{l\in\mathbb{N}_{0}}\lambda_{f}(m(p^{2l},1))p^{-s2l}+\sum_{\begin{subarray}{c}(k,l)\in\Lambda_{+}\\\
k>0\end{subarray}}\chi_{D_{p}^{\perp}}(p^{k})\lambda_{f}(m(p^{l-k},1)p^{-s(k+l)}$

For a “good” prime $p\nmid N$ we have

$\mathcal{Z}_{p}(s,f)=\sum_{(k,l)\in\Lambda_{+}}\chi_{D}(p^{k})\lambda_{f}(m(p^{l-k},1))p^{-s(k+l)}.$

Let $\chi$ be either of the characters $\chi_{D}$, $\chi_{D_{p}^{\perp}}$.
Then we can write for either of the above sums over $(k,l)\in\Lambda_{+}$:

$\displaystyle\sum_{(k,l)}\chi(p^{k})p^{-2ks}\lambda_{f}(m(p^{l-k},1))p^{-s(l-k)}.$

For any fixed $k\in\mathbb{N}_{0}$ the index $l$ runs through the set
$\\{2n+k\;|\;n\in\mathbb{N}_{0}\\}$ to satisfy the conditions $l+k\in
2\mathbb{Z}$ and $l\geq k$. Therefore, the index $l-k$ runs through

$\\{2(n-k)\;|\;n\in\mathbb{N}_{0}\text{ with }n\geq k\\}=2\mathbb{N}_{0}.$

Thus, we may rewrite the latter series as

$\displaystyle\sum_{k}\chi(p^{k})p^{-2ks}\sum_{n}\lambda_{f}(m(p^{2n},1))p^{-2ns}.$

Globally, we then have

(3.24)
$\mathcal{Z}(s,f)=\prod_{p\mid|D|}\left(\left(\frac{e(\operatorname{sig}(D_{p})/8)}{|D_{p}|^{1/2}}-1\right)+L_{p}(2s,\chi_{D_{p}^{\perp}})\right)L(2s,\chi_{D})Z(s,f),$

where

  1. i)

$L_{p}(s,\chi_{D_{p}^{\perp}})=(1-\chi_{D_{p}^{\perp}}(p)p^{-s})^{-1},$

  2. ii)

$L(s,\chi_{D})$ is the Dirichlet $L$-series associated to $\chi_{D}$.

We can now state the following theorem regarding the analytic properties of
$\mathcal{Z}(s,f)$.

######  Theorem 3.3.

Let $\kappa\in 2\mathbb{Z},\;\kappa\geq 3,$ satisfy
$2\kappa+\operatorname{sig}(L)\equiv 0\bmod{4}$ and $f\in
S_{\kappa}(\rho_{L})$ a common eigenform of all Hecke operators
$T(m(k^{2},1))$, $k\in\mathbb{N}$. Then the zeta function $\mathcal{Z}(s,f)$
has a meromorphic continuation to the whole $s$-plane.

###### Proof.

This results from (3.24): Theorem 6.6 of [St1] provides the desired property
for $Z(s,f)$. For $L(s,\chi_{D})$ this is well known and clear for the
remaining factor anyway. ∎

##  4\. The Weil representation on $\operatorname{GL}_{2}(\mathbb{A})$

Let $(L,(\cdot,\cdot))$ and $D=L^{\prime}/L$ be defined as in Section 3 We
further define $V=L\otimes\mathbb{Q}$ and let $H=O(V)$ be the orthogonal group
over $\mathbb{Q}$ attached to $(V,(\cdot,\cdot))$. In this section we collect
some well known facts on the Weil representation of
$\operatorname{SL}_{2}(\mathbb{A})\times H(\mathbb{A})$, which is suited for
our purposes in this paper. Here, we consider the Schrödinger model of the
Weil repesentation $\omega=\prod_{p\leq\infty}\omega_{p}$ on the space
$S(V(\mathbb{A}))$ of Schwartz-Bruhat functions associated to the character

(4.1)
$\psi=\prod_{p\leq\infty}\psi_{p}:\mathbb{A}/\mathbb{Q}\rightarrow\mathbb{C}^{\times},\;x=(x_{p})\mapsto\psi(x)=e^{2\pi
i(-x_{\infty}+\sum_{p<\infty}x_{p}^{\prime})},$

where $x_{p}^{\prime}\in\mathbb{Q}/\mathbb{Z}$ is the principal part of
$x_{p}$ and $V(\mathbb{A})=V\otimes\mathbb{A}$. Note that this character is
the complex conjugate of the standard additive character (see e. g. [St], [BY]
and [KL], Chapter 8).

A second goal of present section is the extension of $\omega$ to a subgroup of
$\operatorname{GL}_{2}(\mathbb{A})$ in the spirit of the extension of the
“finite” Weil representation $\rho_{L}$ in [BS].

For $\mu\in D$ we define $\varphi_{\mu}\in S(V(\mathbb{A}_{f}))$ with

(4.2)
$\displaystyle\varphi_{\mu}=\mathbbm{1}_{\mu+\hat{L}}=\prod_{p<\infty}\varphi^{(\mu)}_{p}=\prod_{p<\infty}\mathbbm{1}_{\mu+L_{p}}.$

Here $L_{p}=L\otimes\mathbb{Z}_{p}$, which is the $p$-part of
$\hat{L}=L\otimes\hat{\mathbb{Z}}$ with
$\hat{\mathbb{Z}}=\prod_{p<\infty}\mathbb{Z}_{p}$, and
$\mathbbm{1}_{\mu+L_{p}}$ is the characteristic function of $\mu+L_{p}$. Note
that there is a close relation between the finite groups
$L_{p}^{\prime}/L_{p}$ and the $p$-groups $D_{p}$. In fact, these groups are
isomorphic. This isomorphism additionally respects the quadratic forms, which
endow both groups, see e. g. [Ze], Section 3, [St], Remark 3.2 or [We1],
Theorem 4.30. In the following, we identify these groups and use them
interchangeably. As in [BY] we consider the $|D|$\- dimensional subspace

(4.3) $S_{L}=\bigoplus_{\mu\in D}\mathbb{C}\varphi_{\mu}\subset
S(V(\mathbb{A}_{f})).$

It is known that the space (4.3) is stable under the action of the group
$\operatorname{SL}(2,\hat{\mathbb{Z}})$ via the Weil representation
$\omega_{f}$ (see e. g. [BY], chapter 2 or [Ku]). Also, the $L^{2}$ scalar
product $\langle\cdot,\cdot\rangle$ on $S_{L}\subset S(V(\mathbb{A}_{f}))$
simplifies to

(4.4) $\langle\sum_{\mu\in D}F_{\mu}\varphi_{\mu},\sum_{\mu\in
D}F_{\mu}\varphi_{\mu}\rangle=\sum_{\mu\in D}|F_{\mu}|^{2}.$

Note that $D$ can be decomposed into $p$-groups
$D=\bigoplus_{p\mid|D|}D_{p}\cong\bigoplus_{p\mid|D|}L_{p}^{\prime}/L_{p}$.
For almost all primes $p$ \- those coprime to $|D|$ \- $L_{p}$ is unimodular
and thus $L_{p}^{\prime}/L_{p}=0+L_{p}$. Therefore, we can write
$D\cong\bigoplus_{p<\infty}L_{p}^{\prime}/L_{p}$. On the level of the space
$S_{L}$, this decomposition translates to the isomorphism

(4.5)
$S_{L}\cong\bigotimes_{p<\infty}S_{L_{p}},\quad\varphi_{\mu}\mapsto\bigotimes_{p<\infty}\varphi_{p}^{(\mu_{p})},$

where $\mu=\sum_{p\mid|D|}\mu_{p}$ and
$\varphi_{p}^{(\mu_{p})}=\varphi_{p}^{(0)}$ for all primes $p$ coprime to
$|D|$. The local Weil representation $\omega_{p}$ acts on the $p$-part
$S_{L_{p}}$ of $S_{L}$, where

(4.6) $S_{L_{p}}=\begin{cases}\bigoplus_{\mu\in
L_{p}^{\prime}/L_{p}}\mathbb{C}\varphi_{p}^{(\mu)},&p\mid|D|,\\\
\mathbb{C}\varphi_{p}^{(0)},&p\nmid|D|.\end{cases}$

We then have

$\omega_{f}(\gamma_{f})\varphi_{\mu}=\bigotimes_{p<\infty}\omega_{p}(\gamma_{p})\varphi_{p}^{(\mu_{p})}.$

According to [St], Lemma 3.4 and [BY], Proposition 2.5, the Weil
representation $\omega_{p}$ can be described explicitly on the generators of
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$ by

(4.7)
$\begin{split}&\omega_{p}(n(b))\varphi_{p}^{(\mu)}=\psi_{p}(bq(\mu))\varphi_{p}^{(\mu)}\\\
&\omega_{p}(w)\varphi_{p}^{(\mu)}=\frac{\gamma_{p}(L_{p}^{\prime}/L_{p})}{|L_{p}^{\prime}/L_{p}|^{1/2}}\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\psi_{p}((\mu_{p},\nu_{p}))\varphi_{p}^{(\nu_{p})}\\\
&\omega_{p}(m(a))\varphi_{p}^{(\mu_{p})}=\chi_{V,p}(a)\varphi_{p}^{(a^{-1}\mu_{p})},\end{split}$

where $\gamma(L_{p}^{\prime}/L_{p})$ is the local Weil index and
$\chi_{V,p}(a)=(a,(-1)^{m/2}|D_{p}|)_{p}$ is the local Hilbert symbol.
Evaluating the local Hilbert symbol gives

(4.8)
$\chi_{V,p}(a)=\left(\frac{a}{|L_{p}^{\prime}/L_{p}|}\right)=\chi_{D_{p}}(a),$

see e. g. [Se], Chapter III. These formulas imply that (see the proof Lemma
3.4 in [St])

  1. i)

the local Weil representations $\omega_{p}$ is trivial if $p$ is coprime to
$|D|$,

  2. ii)

if we identify $\mathbb{C}[D]$ with $S_{L}$ via
$\mathfrak{e}_{\mu}\mapsto\varphi_{\mu}$, then $\omega_{f}$ coincides with the
finite Weil representation $\rho_{L.1}$ in the following way

(4.9) $\rho_{L}(\gamma)=\omega_{f}(\gamma_{f}),$

where $\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$ and
$\gamma_{f}\in\operatorname{SL}_{2}(\widehat{Z})$ is the projection of
$\gamma$ into $\operatorname{SL}_{2}(\widehat{\mathbb{Z}})$. Note that by our
choice of the character $\psi$, the relation (4.9) differs from the one in
[BY], (2.7), by conjugation.

The following lemma provides the action of $\omega_{p}$ for the lower
triangular matrix $n\\_(c)\in\operatorname{SL}_{2}(\mathbb{Z}_{p})$:

######  Lemma 4.1.

Let $c\in\mathbb{Z}_{p}^{\times}$. Then

(4.10)
$\omega_{p}(n\\_(c))\varphi_{p}^{(\mu_{p})}=\frac{\gamma_{p}(L_{p}^{\prime}/L_{p})}{|L_{p}^{\prime}/L_{p}|^{1/2}}\chi_{V,p}(-c)\psi_{p}(c^{-1}q(\mu_{p}))\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\psi_{p}(c^{-1}q(\nu_{p}))\psi_{p}(-c(\mu_{p},\nu_{p}))\varphi_{p}^{(\nu_{p})}.$

###### Proof.

By the Bruhat decomposition (see e. g. [KL], p. 69),

$n\\_(c)=n(c^{-1})wn(c)m(-c).$

From this we infer that by means of (4.7)

$\omega_{p}(n\\_(c))\varphi_{p}^{(\mu_{p})}=\frac{\gamma_{p}(L_{p}^{\prime}/L_{p})}{|L_{p}^{\prime}/L_{p}|^{1/2}}\chi_{V,p}(-c)\psi_{p}(c^{-1}q(\mu_{p}))\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\psi_{p}(c^{-1}q(\nu_{p}))\psi_{p}(-c(\mu_{p},\nu_{p}))\varphi_{p}^{(\nu_{p})}.$

∎

Via the extension of $\rho_{L}$ to a subgroup of $\mathcal{G}(\mathbb{Q})$
(see Section 3) it is possible to extend $\omega_{f}$ (cf. [We], Def. 46) to
the group $\mathcal{K}=\prod_{p<\infty}\mathcal{K}_{p}$ as follows:

######  Definition 4.2.

Let $N=\prod_{i=1}^{r}p^{e_{i}}$ be the level of $D$,

$Q(N)=\\{M\in\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})\;|\;\det(M)\text{ a
square in }(\mathbb{Z}/N\mathbb{Z})^{\times}\\}$

(cf. [BS], (3.2)) and

$\displaystyle\mathcal{K}\xrightarrow{\pi}\prod_{p|N}\mathcal{K}_{p}\xrightarrow{\pi_{N}}Q(N),$

where $\pi$ is the projection onto the places $p\mid N$ and $\pi_{N}$ is the
composition of the canonical projection of $\mathbb{Z}_{p_{i}}$ to
$\mathbb{Z}/p^{e_{i}}\mathbb{Z}$ and the application of the Chinese remainder
theorem, applied to each component of the matrices. For $k\in\mathcal{K}$ we
then define

(4.11) $\begin{split}\omega_{f}(k)&=\bigotimes_{p<\infty}\omega_{p}(k_{p})\\\
&=\bigotimes_{p\nmid
N}\omega_{p}(k_{p})\bigotimes_{p|N}\omega_{p}(k_{p})\end{split}$

with $\omega_{p}(k_{p})=\operatorname{id}_{S_{L_{p}}}$ for all primes $p\nmid
N$ and

(4.12) $\begin{split}\bigotimes_{p\mid
N}\omega_{p}(k_{p})=\rho_{L}(\pi_{N}((k_{p})_{p\mid N})).\end{split}$

Here by $(k_{p})_{p\mid N}$ we mean the tuple of all components
$k_{p}\in\mathcal{K}_{p}$ of $k$ belonging to the primes $p$ dividing $N$.
Combining (4.11) and (4.12) we can write

(4.13) $\omega_{f}(k)=\rho_{L}((\pi\circ\pi_{N})(k)).$

Note that Definition 4.2 is compatible with (4.9). For, if we take
$k\in\operatorname{SL}_{2}(\widehat{\mathbb{Z}})$ as the projection of some
$\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$, we find
$(\pi_{N}\circ\pi)(k)=\pi_{N}(\gamma)\in\operatorname{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$
and $\omega_{f}(k)=\rho_{L}(\gamma)$ since $\rho_{L}$ factors through
$\operatorname{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$.

We can imitate the extension process of $\rho_{L}$ from $\Gamma$ to
$\mathcal{Q}(N)$ to extend the local Weil representation $\omega_{p}$ from
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$ to $\mathcal{K}_{p}$.

######  Definition 4.3.

Let $k_{p}\in\mathcal{K}_{p}$ with
$\det(k_{p})=t^{2}\in(\mathbb{Z}_{p})^{\times}$. Then we define in accordance
with [BS], (3.5),

(4.14) $\begin{split}\omega_{p}(\left(\begin{smallmatrix}t&0\\\
0&t\end{smallmatrix}\right))\varphi_{p}^{(\lambda_{p})}&=\frac{g(D_{p})}{g_{t}(D_{p})}\varphi_{p}^{(\lambda_{p})}\\\
&=\chi_{D_{p}}(t)\varphi_{p}^{(\lambda_{p})}\end{split}$

and

(4.15) $\omega_{p}(k_{p})=\omega_{p}(\left(\begin{smallmatrix}t&0\\\
0&t\end{smallmatrix}\right))\omega_{p}(\left(\begin{smallmatrix}t^{-1}&0\\\
0&t^{-1}\end{smallmatrix}\right)k_{p}),$

where $\left(\begin{smallmatrix}t^{-1}&0\\\
0&t^{-1}\end{smallmatrix}\right)k_{p}$ is an element of
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$.

If the level $N$ is equal to a prime $p$, we have $\omega_{f}=\omega_{p}$ and
Definition 4.2 specialises to an extension of the local Weil representation
$\omega_{p}$ to $\mathcal{K}_{p}$. Thus,
$\omega_{p}(k_{p})=\rho_{L}(\pi_{p}(k_{p}))$, which is compatible with
Definition 4.3.

Finally, we need to define the local Weil representation $\omega_{p}$ on
double cosets of the form $\mathcal{K}_{p}m(p^{-k},p^{-l})\mathcal{K}_{p}$.
Here $m(p^{-k},p^{-l})\in\mathcal{G}(\mathbb{Q})$ and $k\leq l$ (cf. Theorem
6.9). Again, we mirror the corresponding process in [BS], Chapter 5, and make
use of results in Section 3 of the present paper.

######  Definition 4.4.

Let $m(p^{-k},p^{-l})\in\mathcal{G}(\mathbb{Q})$ with $k\leq l$.

  1. i)

Then we put

(4.16)
$\begin{split}\omega_{p}^{-1}(m(p^{-k},p^{-l}))\varphi_{p}^{(\lambda_{p})}&=\omega_{p}^{-1}(m(p^{-l},p^{-l}))\omega_{p}^{-1}(m(p^{l-k},1)\varphi_{p}^{(\lambda_{p})}\\\
&=\frac{g(D_{p})}{g_{p^{l}}(D_{p})}\varphi_{p}^{(p^{(l-k)/2}\lambda_{p})}.\end{split}$

  2. ii)

For $\delta=\gamma
m(p^{-k},p^{-l})\gamma^{\prime}\in\mathcal{K}_{p}m(p^{-k},p^{-l})\mathcal{K}_{p}$
we define

(4.17)
$\omega_{p}^{-1}(\delta)\varphi_{p}^{(\lambda_{p})}=\omega_{p}^{-1}(\gamma^{\prime})\omega_{p}^{-1}(m(p^{-k},p^{-l}))\omega_{p}^{-1}(\gamma)\varphi_{p}^{(\lambda_{p})}.$

As already noted before, if the level $N$ is equal to a prime $p$, we have
$\omega_{p}(\gamma_{p})=\omega_{f}(\gamma_{p})=\rho_{L}(\pi_{p}(\gamma_{p}))$
for $\gamma_{p}\in\operatorname{SL}_{2}(\mathbb{Z}_{p})$. In view of (3.13),
we conclude $\omega_{p}^{-1}(\delta)=\rho_{L}^{-1}(\delta)$ for
$\delta\in\operatorname{SL}_{2}(\mathbb{Z}_{p})m(p^{-k},p^{-l})\operatorname{SL}_{2}(\mathbb{Z}_{p})$.
Proposition 5.1 in [BS] then shows that (4.17) is independent of the choice of
$\gamma,\gamma^{\prime}\in\operatorname{SL}_{2}(\mathbb{Z}_{p})$. Since the
action of $\omega_{p}$ on $\mathcal{K}_{p}$ differs from that on
$\operatorname{SL}_{2}(\mathbb{Z}_{p})$ only by constant factor (see
Definition 4.3), we obtain the same result for
$\gamma,\gamma^{\prime}\in\mathcal{K}_{p}$.

##  5\. The Hecke algebra
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$

In this section we will describe the structure of the local vector valued
spherical Hecke algebra
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ associated to the
pair of groups $(\mathcal{Q}_{p},\mathcal{K}_{p})$ and the local Weil
representation $\omega_{p}$. For each prime $p$ we will introduce a Satake
map, which allows us to understand the structure of the corresponding Hecke
algebra. For primes $p\nmid|D|$ the Hecke algebras are isomorphic to the
scalar valued algebras defined by the same groups. These are well understood
thanks to the classical Satake map. If $p$ divides $|D|$, the algebras
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ are considerably
more complicated because $\omega_{p}$ is non-trivial. However, under certain
restrictions for $D_{p}$, we will define a modified Satake map, which maps
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ to a simpler
algebra, whose structure can be easier determined.

The following general facts about spherical Hecke algebras can be found in
many places among them [BK], chapter 4, [Ho] and [Mu].

######  Definition 5.1.

Let $G$ be a locally compact group $G$, $K$ an open compact subgroup and
$\rho:K\rightarrow\operatorname{GL}(V)$ a representation of $K$. The Hecke
algebra $\mathcal{H}(G//K,\rho)$ of $\rho$-spherical functions is the set of
functions $f:G\rightarrow\text{End}(V)$ which are

  1. i)

compactly supported modulo $K$, i. e. each $f$ vanishes outside finitely many
double cosets $KgK$ and satisfy

  2. ii)

$f(k_{1}gk_{2})=\rho(k_{1})\circ f(g)\circ\rho(k_{2})\text{ for all
}k_{1},k_{2}\in K\text{ and all }g\in G.$

Since each element $f$ of $\mathcal{H}(G//K,\rho)$ is of the form

$f(g)=\sum_{i=1}^{n}a_{i}(f_{i})(g),$

where $f_{i}$ is an element of the subspace of functions of
$\mathcal{H}(G//K,\rho)$, which vanish outside $Kg_{i}K$, the whole algebra is
generated by the functions $f_{i}$. Similarly, we denote by
$\mathcal{H}(G//K)$ the set of functions $f:G\rightarrow\mathbb{C}$, which are
compactly supported modulo $K$ and $K$-bi-invariant, i. e.
$f(k_{1}gk_{2})=f(g)$ for all $k_{1},k_{2}\in K$ and all $g\in G$. We call
$\mathcal{H}(G//K)$ also a Hecke algebra.

It is well known that $\mathcal{H}(G//K,\rho)$ is an associative
$\mathbb{C}$-algebra with respect to convolution

(5.1) $(f_{1}\ast f_{2})(g)=\int_{G}f_{1}(h)\circ f_{2}(h^{-1}g)dh=\sum_{h\in
G/K}f_{1}(h)\circ f_{2}(h^{-1}g),$

where $dh$ is the standard Haar measure on $G$ normalized by $\int_{K}dh=1$

In order determine the structure of $\mathcal{H}(G//K,\rho)$, in view of the
remarks before, it is useful to study the space of functions in this Hecke
algebra, which vanish outside a single double coset $KgK$. It can be described
in terms of intertwining operators of $\rho$ associated with $g$. To state the
corresponding result, we fix some notation. For $g\in G$ we mean by $K^{g}$
the group $gKg^{-1}$ and write $\rho_{g}$ for the representation
$h\mapsto\;\rho_{g}(h)=\rho(g^{-1}hg)$ of $K^{g}$. As usual,

$\displaystyle\operatorname{Hom}_{K\cap
K^{g}}(\rho,\rho_{g})=\\{F:V\rightarrow V\;|\;F\text{ is linear and
}F\circ\rho_{g}(h)=\rho(h)\circ F\text{ for all }h\in K\cap K^{g}\\}.$

Then we have

######  Lemma 5.2.

Let $g\in G$. The subspace of $\mathcal{H}(G//K,\rho)$ consisting of functions
supported on $KgK$, is isomorphic to $\operatorname{Hom}_{K\cap
K^{g}}(\rho,\;\rho_{g})$.

###### Proof.

The assertion is well known (see e. g. [BK], Chapter 4). Nevertheless, for
later purposes, we indicate a proof by giving the maps of the claimed
isomorphism (without further explanation).

If $f\in\mathcal{H}(G//K,\rho)$ with $f(g)\not=0$ supported on $KgK$, then it
easily checked that $f(g)\in\operatorname{Hom}_{K\cap K^{g}}(\rho,\;\rho_{g})$
(non-zero). On the other hand, if $0\not=F\in\operatorname{Hom}_{K\cap
K^{g}}(\rho,\;\rho_{g})$, we put $f(g)=F$ and $f(k_{1}gk_{2})=\rho(k_{1})\circ
f\circ\rho(k_{2})$ and obtain thereby an element of the above stated subspace
of $\mathcal{H}(G//K,\rho)$. ∎

The following Lemma ensures that the groups $\mathcal{Q}_{p}$ and
$\mathcal{K}_{p}$ meet the conditions of Definition 5.1. It might be known.
Since I have not found it in the literature, I state it here and add a short
proof.

######  Lemma 5.3.

  1. i)

The group $\mathcal{K}_{p}$ is an open compact subgroup of $\mathcal{Q}_{p}$.

  2. ii)

The group $\mathcal{Q}_{p}$ is a locally compact subgroup of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

###### Proof.

It is well known that $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ is locally
compact. By Lemma 8, I.3, of [Ch], it follows that $\mathcal{Q}_{p}$ is also
locally compact. By [Ka], Thm. 2.15, we know that
$(\mathbb{Z}_{p}^{\times})^{2}$ is an open subgroup in
$\mathbb{Z}_{p}^{\times}$ Therefore, $\mathcal{K}_{p}$ is an open subgroup in
$\operatorname{GL}_{2}(\mathbb{Z}_{p})$, which implies that is is also closed
(see [HW], Thm. 5.5). As
$\mathcal{Q}_{p}\cap\operatorname{GL}_{2}(\mathbb{Z}_{p})=\mathcal{K}_{p}$, we
find that $\mathcal{K}_{p}$ is a compact subgroup of $\mathcal{Q}_{p}$.

∎

Note that there is analogue of the Cartan decomposition for the pair
$(\mathcal{Q}_{p},\mathcal{K}_{p})$.

######  Lemma 5.4.

The group $\mathcal{Q}_{p}$ can be written as a disjoint union of
$\mathcal{K}_{p}$-double cosets:

$\mathcal{Q}_{p}=\bigcup_{\begin{subarray}{c}k\leq l\\\ k+l\in
2\mathbb{Z}\end{subarray}}\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}.$

###### Proof.

The proof is the same as for the Cartan decomposition for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$, see e. g. [Mu], p. 17. In the quoted
proof the matrix $g=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{GL}_{2}(\mathbb{Q}_{p})$ with
$a\not=0$ and $|a|_{p}\geq\max\\{|b|_{p},|c|_{p},|d|_{p}\\}$ is transformed
into

$m(a,d-a^{-1}bc)=k_{3}gk_{4},$

where $k_{3}=n\\_(-a^{-1}c),k_{4}=n(-a^{-1}b)\in\mathcal{K}_{p}$. If we assume
that $\det(g)=p^{2x}y^{2}$ and $a=p^{k}s$, then
$d-a^{-1}bc=p^{2x-k}y^{2}s^{-1},y,s\in\mathbb{Z}_{p}^{\times},$ and

$m(p^{k},p^{2x-k})=n\\_(-a^{-1}c)g\;n(-a^{-1}b)m(s)m(1,y^{2}).$

Therefore, all used transformation matrices are contained in
$\mathcal{K}_{p}$. We have a similar decomposition if $d\not=0$, see [KL], p.
208.

Also, note that any two double cosets $\mathcal{K}_{p}g_{1}\mathcal{K}_{p}$,
$\mathcal{K}_{p}g_{2}\mathcal{K}_{p}$ are disjoint since otherwise the double
cosets
$\operatorname{GL}_{2}(\mathbb{Z}_{p})g_{1}\operatorname{GL}_{2}(\mathbb{Z}_{p})$,
$\operatorname{GL}_{2}(\mathbb{Z}_{p})g_{2}\operatorname{GL}_{2}(\mathbb{Z}_{p})$
would not be disjoint, contradicting the Cartan decomposition for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$. ∎

We also have an analogue of the Iwasawa decomposition of
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$.

######  Lemma 5.5.

(5.2) $\mathcal{Q}_{p}=(B(\mathbb{Q}_{p})\cap\mathcal{Q}_{p})\mathcal{K}_{p}.$

###### Proof.

This follows immediately from the Iwasawa decomposition for
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$ by the intersection with
$\mathcal{Q}_{p}$ on both sides. ∎

As already noted, there are two cases to consider regarding the structure of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$. It depends on
whether $p$ divides $|D|$ or not. In both cases we will determine a set of
generators with the help of Lemma 5.2. Afterwards, we will define a Satake
map. If $p$ divides $|D|$, we will show - under some restrictions on
$L_{p}^{\prime}/L_{p}$ \- that
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is isomorphic to a
subalgebra of the spherical Hecke algebra
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$,
where

$S_{L_{p}}^{N(\mathbb{Z}_{p})}=\\{\varphi\in
S_{L_{p}}\;|\;\omega_{p}(n)(\varphi)=\varphi\text{ for all }n\in
N(\mathbb{Z}_{p})\\}.$

If $(p,|D|)=1$, $\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is
isomorphic to the classical spherical Hecke algebra
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p})$, whose structure is well
known. We start with the discussion of the first mentioned case and consider
the latter case subsequently.

###  5.1. The case of primes $p$ dividing $|D|$

######  Lemma 5.6.

Let $D_{p}$ be an anisotropic discriminant form and
$g=m(p^{k},p^{l})\in\mathcal{Q}_{p}$. Put $\rho_{g}=(\omega_{p})_{g}$.

  1. i)

If $k<l$, then the space
$\operatorname{Hom}_{\mathcal{K}_{p}\cap\mathcal{K}_{p}^{g}}(\omega_{p},\rho_{g})$
is generated by the map

(5.3) $T(k,l):S_{L_{p}}\rightarrow
S_{L_{p}}^{N(\mathbb{Z}_{p})},\quad\varphi_{p}^{(\mu)}\mapsto
T(k,l)(\varphi_{p}^{(\mu)})=\frac{g(D_{p})}{g_{p^{l}}(D_{p})}\varphi_{p}^{(p^{(l-k)/2}\mu_{p})}=\frac{g(D_{p})}{g_{p^{l}}(D_{p})}\varphi_{p}^{(0)}.$

  2. ii)

If $D_{p}\cong\mathcal{A}_{p}^{t}\oplus\mathcal{A}_{p}^{1}$, then $S_{L_{p}}$
decomposes into the irreducible submodules

$\displaystyle
S_{\mathcal{A}_{p}^{t}}^{+}=\text{span}\left\\{\varphi_{p}^{(\mu)}+\varphi_{p}^{(-\mu)}\;|\;\mu\in\mathcal{A}_{p}^{t}\right\\},\quad
S_{\mathcal{A}_{p}^{t}}^{-}=\text{span}\left\\{\varphi_{p}^{(\mu)}-\varphi_{p}^{(-\mu)}\;|\;\mu\in\mathcal{A}_{p}^{t}\right\\}\text{
and }$ $\displaystyle
S_{\mathcal{A}_{p}^{1}}^{+}=\text{span}\left\\{\varphi_{p}^{(\mu)}+\varphi_{p}^{(-\mu)}\;|\;\mu\in\mathcal{A}_{p}^{1}\right\\},\quad
S_{\mathcal{A}_{p}^{1}}^{-}=\text{span}\left\\{\varphi_{p}^{(\mu)}-\varphi_{p}^{(-\mu)}\;|\;\mu\in\mathcal{A}_{p}^{1}\right\\}.$

For $k=l$ the space
$\operatorname{Hom}_{\mathcal{K}_{p}\cap\mathcal{K}_{p}^{g}}(\omega_{p},\rho_{g})$
is then generated by the four maps $T(k,k)^{(i)}$, where

(5.4)
$T(k,k)^{(i)}(\gamma_{p}^{(1)}+\gamma_{p}^{(2)}+\gamma_{p}^{(3)}+\gamma_{p}^{(4)})=\frac{g(D_{p})}{g_{p^{k}}(D_{p})}\gamma_{p}^{(i)},\;i=1,\dots,4,$

is the projection onto the $i$-th irreducible submodule. For later purposes we
introduce the operator

(5.5) $T(k)=T(k,k)^{(1)}+T(k,k)^{(2)}+T(k,k)^{(3)}+T(k,k)^{(4)},$

which is equal to
$\frac{g(D_{p})}{g_{p^{k}}(D_{p})}\operatorname{id}_{S_{L_{p}}}$.

  3. iii)

If $D_{p}\cong\mathcal{A}_{p}^{t}$, then $S_{L_{p}}$ decomposes into the
irreducible submodules

$S_{\mathcal{A}_{p}^{t}}^{+}=\text{span}\left\\{\varphi_{p}^{(\mu)}+\varphi_{p}^{-(\mu)}\;|\;\mu\in\mathcal{A}_{p}^{t}\right\\},\quad
S_{\mathcal{A}_{p}^{t}}^{-}=\text{span}\left\\{\varphi_{p}^{(\mu)}-\varphi_{p}^{-(\mu)}\;|\;\mu\in\mathcal{A}_{p}^{t}\right\\}.$

For $k=l$ the space
$\operatorname{Hom}_{\mathcal{K}_{p}\cap\mathcal{K}_{p}^{g}}(\omega_{p},\rho_{g})$
is then generated by the two maps $T(k,k)^{(i)}$, where

(5.6)
$T(k,k)^{(i)}(\gamma_{p}^{(1)}+\gamma_{p}^{(2)})=\frac{g(D_{p})}{g_{p^{k}}(D_{p})}\gamma_{p}^{(i)},\;i=1,2,$

is the projection onto the $i$-th irreducible submodules. For later purposes
we introduce the operator

(5.7) $T(k)=T(k,k)^{(1)}+T(k,k)^{(2)},$

which is equal to
$\frac{g(D_{p})}{g_{p^{k}}(D_{p})}\operatorname{id}_{S_{L_{p}}}$.

###### Proof.

In light of Lemma 5.4, it clearly suffices to choose $g=m(p^{k},p^{l})$ with
$k\leq l$.

i) First, note that

$m(p^{k},p^{l})\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)m(p^{k},p^{l})^{-1}=\left(\begin{smallmatrix}a&bp^{k-l}\\\
cp^{l-k}&d\end{smallmatrix}\right).$

In particular, for $\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)=n(p^{l-k})$ we find that $n(1)$ is an element of
$\mathcal{K}_{p}\cap\mathcal{K}_{p}^{g}$. Thus, for any
$F\in\operatorname{Hom}_{\mathcal{K}_{p}\cap\mathcal{K}_{p}^{g}}(\omega_{p},\rho_{g})$
the equation

$\displaystyle
F\circ\omega_{p}(m(p^{k},p^{l})^{-1}n(1)m(p^{k},p^{l}))=\omega_{p}(n(1))\circ
F$ $\displaystyle\Longleftrightarrow\omega_{p}(n(1))\circ
F=F\circ\omega_{p}(n(p^{l-k}))$
$\displaystyle\Longleftrightarrow\omega_{p}(n(1))\circ F=F$

must hold. For the last equivalence we have used that the level of
$L_{p}^{\prime}/L_{p}$ is $p$. It follows that the image of $F$ is a subset of
$S_{L_{p}}^{N(\mathbb{Z}_{p})}$. Since the identity
$\omega_{p}(n(b))\varphi_{p}^{(\gamma)}=\varphi_{p}^{(\gamma)}$ holds for all
$b\in\mathbb{Z}_{p}^{\times}$ if and only if $\gamma$ is isotropic and
$L_{p}^{\prime}/L_{p}$ anisotropic, we can conclude that
$S_{L_{p}}^{N(\mathbb{Z}_{p})}=\mathbb{C}\varphi_{p}^{(0)}$. Therefore, $F$ is
a scalar multiple of the map
$\varphi_{p}^{(\lambda_{p})}\mapsto\varphi_{p}^{(0)}$ and has the claimed
form.

ii) The decomposition of $S_{L_{p}}$ and the irreducibility of $\omega_{p}$ on
$S_{\mathcal{A}_{p}^{t}}^{\pm}$ and $S_{\mathcal{A}_{p}^{1}}^{\pm}$ is well
known, see for example [NW], Theorem 4. In the case $k=l$ the equation
$F\circ\rho_{g}(h)=\omega_{p}(h)\circ F$ simplifies to
$F\circ\omega_{p}(h)=\omega_{p}(h)\circ F$ for all $h\in\mathcal{K}_{p}$.
Thus, $F$ is an intertwining operator for $\omega_{p}$. The structure of the
space of intertwining operators can be found in books about representation
theory, cf. [JL], Chapter 11. ∎

######  Remark 5.7.

The factor $\frac{g(D_{p})}{g_{p^{l}}(D_{p})}$ is not necessary. We introduced
it for reasons of compatibility (see Theorem 6.9 for that matter).

######  Corollary 5.8.

Let $p$ be a prime dividing $|D|$ and $D_{p}$ anisotropic. Then the Hecke
algebra $\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is
generated by the following elements:

  1. i)

For $k<l$

(5.8) $T_{k,l}(k_{1}m(p^{k},p^{l})k_{2})=\omega_{p}(k_{1})\circ
T(k,l)\circ\omega_{p}(k_{2}),$

where $T_{k,l}$ is only supported on
$\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}$. Here $T(k,l)$ is the
intertwining operator specified in Lemma 5.6, i).

  2. ii)

For $k=l$

(5.9) $T_{k,k}^{(i)}(k_{1}m(p^{k},p^{k})k_{2})=\omega_{p}(k_{1})\circ
T(k,k)^{(i)}\circ\omega_{p}(k_{2}),$

where
$\operatorname{supp}(T_{k,k}^{(i)})=\mathcal{K}_{p}m(p^{k},p^{k})\mathcal{K}_{p}$.
Here $T(k,k)^{(i)}$ is given in Lemma 5.6, ii). By the operator $T_{k}$ we
mean

$T_{k}(k_{1}m(p^{k},p^{l})k_{2})=\omega_{p}(k_{1})\circ
T(k)\circ\omega_{p}(k_{2}),$

where $T(k)$ is also given in Lemma 5.6, ii).

The following theorem investigates the structure of the Hecke algebra
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
assuming $D_{p}$ is anisotropic.

######  Theorem 5.9.

Let $p$ be an odd prime dividing $|D|$ and $D_{p}$ anisotropic.

  1. i)

Then

(5.10)
$\begin{split}\omega_{p}(m(t_{1},t_{2}))\varphi_{p}^{(0)}&=\begin{cases}\left(\frac{t_{2}}{p}\right)\varphi_{p}^{(0)},&|D_{p}|=p,\\\
\varphi_{p}^{(0)},&|D_{p}|=p^{2}\end{cases}\\\
&=\chi_{D_{p}}(t_{2})\varphi_{p}^{(0)}\end{split}$

for all $m(t_{1},t_{2})\in\mathcal{D}_{p}$.

  2. ii)

Then $S_{L_{p}}^{N(\mathbb{Z}_{p})}$ is equal to $\mathbb{C}\varphi_{p}^{(0)}$
and the Hecke algebra
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
isomorphic to the scalar valued Hecke algebra
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})$.

###### Proof.

As already mentioned in Section 3, the order of an isotropic quadratic module
$D_{p}$ is either $p^{2}$ or $p$.

i) Let $m(t_{1},t_{2})\in\mathcal{D}_{p}$ with
$\det(m(t_{1},t_{2}))=t^{2}\in(\mathbb{Z}_{p}^{\times})^{2}$. Then by (4.15)

$\displaystyle\omega_{p}(m(t_{1},t_{2}))\varphi_{p}^{(0)}$
$\displaystyle=\omega_{p}(m(t,t))\omega_{p}(m(t^{-1}t_{1},t^{-1}t_{2}))\varphi_{p}^{(0)}$
$\displaystyle=\left(\frac{t}{|L_{p}^{\prime}/L_{p}|}\right)\left(\frac{t^{-1}t_{2}}{|L_{p}^{\prime}/L_{p}|}\right)\varphi_{p}^{(0)},$

where we have used (4.8) and (4.14). The claimed result now follows.

For ii) we first note that if $L_{p}^{\prime}/L_{p}$ is anisotropic, the space
$S_{L_{p}}^{N(\mathbb{Z}_{p})}$ is equal to $\mathbb{C}\varphi_{p}^{(0)}$ and
thus one-dimensional. From i) we know that $\mathcal{D}_{p}$ acts via
$\omega_{p}$ on $S_{L_{p}}^{N(\mathbb{Z}_{p})}$ by multiplication with the
quadratic character $\chi_{D_{p}}$. Consequently,

$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})=\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\chi_{D_{p}}),$

where the latter Hecke algebra is meant in the sense of Definition 5.1 with
the one-dimensional representation $\rho=\chi_{D_{p}}$. The structure of the
latter algebra was discussed in [Ho], Remark 5.1. It was stated there that
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\chi_{D_{p}})$ is isomorphic to
the usual spherical algebra $\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})$.
To state this isomorphism explicitly, we specify a set of generators for the
former algebra. It is generated by elements of the form

$T_{k,l}(m(t_{1},t_{2})m(p^{k},p^{l})m(s_{1},s_{2}))=\chi_{D_{p}}(m(t_{1},t_{2}))\circ
T(k,l)\circ\chi_{D_{p}}(m(s_{1},s_{2}))$

with

$T(k,l)=\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{l})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}}.$

The isomorphism is then given by

(5.11)
$T_{k,l}=\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{l})\mathcal{D}_{p}}\cdot\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}}\mapsto\chi_{D_{p}}T_{k,l},$

where have extended $\chi_{D_{p}}$ trivially to a quasi-character on the whole
group $\mathcal{M}_{p}$ (see also [Ho]). ∎

We now define the before mentioned Satake map to further clarify the structure
of $\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and to connect
it to the algebra $\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})$.

(5.12)
$\begin{split}&\mathcal{S}:\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})\rightarrow\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p}{{{{}_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}}}}),\\\
&T\mapsto\left(m\mapsto\delta(m)^{1/2}\sum_{n\in
N(\mathbb{Q}_{p})/N(\mathbb{Z}_{p})}T(mn)_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}\right).\end{split}$

######  Remark 5.10.

  1. i)

Note that this definition is analogous to the one given by Herzig ([He]) over
a field in characteristic $p$. The modulus character
$\delta\left(\begin{smallmatrix}m_{1}&0\\\
0&m_{2}\end{smallmatrix}\right)=\left|\frac{m_{1}}{m_{2}}\right|_{p}$ is also
part of the classical Satake map (see e. g. [De], Chap. 8), where it ensures
that the image of the Satake map is invariant under the natural action of the
Weyl group. Herzig omitted the modulus character in his definition of the
Satake map as it does not produce the invariance under the action of the Weyl
group. Nevertheless, we keep it in the definition of $\mathcal{S}$ since it
indeed does share the property of invariance under the Weyl group.

  2. ii)

With the same arguments as after the statement of Theorem 1.2 and as in its
proof (Step 0) in [He], it can be proved that $\mathcal{S}$ is a well defined
map. To prove that $\mathcal{S}$ is a $\mathbb{C}-$algebra homomorphism all
calculations of Step 2 in the proof of Theorem 1.2 in [He] remain valid in our
situation.

As is shown in Lemma 5.6 and Corollary 5.8, the space of maps in
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ with support equal
to $\mathcal{K}_{p}m(p^{k},p^{k})\mathcal{K}_{p}$ is two- or four-dimensional
if $\omega_{p}$ is considered on the whole space $S_{L_{p}}$. If we restrict
ourselves to an irreducible subspace of $S_{L_{p}}$, the before mentioned
space one-dimensional. On the one hand, this condition would guarantee that
the Satake map (5.12) is indeed an isomorphism (without it, (5.12) is not even
injective, as is easily checked). On the other hand, it is to restrictive for
our purposes. So, in order to obtain an isomorphism between
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and a subalgebra of
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
via (5.12), we restrict
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ to a subalgebra
where the space of maps supported on the double coset
$\mathcal{K}_{p}m(p^{k},p^{k})\mathcal{K}_{p}$ is replaced with the subspace
generated by $T_{k}$.

For the next theorem we fix some notation:
Let $N(\mathcal{M}_{p})$ be the normalizer of $\mathcal{M}_{p}$ in
$\mathcal{Q}_{p}$. Then the group $N(\mathcal{M}_{p})/\mathcal{M}_{p}$ is
called Weyl group. It is isomorphic to the symmetric group $S_{2}$ and acts on
$\mathcal{M}_{p}$ by changing the entries $t_{1},t_{2}$ of a matrix
$m(t_{1},t_{2})$.

Let $k,l\in\mathbb{Z}$ with $k\leq l$. By
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ we mean the
subalgebra of $\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$
generated by $T$ with

$T=\begin{cases}T_{k,l},&k<l,\\\ T_{k},&k=l\end{cases}$

as specified in Corollary 5.8. In order to state results for all generators of
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ in subsequent
sections we often write $T_{k,k}$ instead of $T_{k}$. Furthermore, let

$\displaystyle\tau_{k}=\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{k})\mathcal{D}_{p}}\cdot\operatorname{id}_{S_{L_{p}}},$
$\displaystyle\tau_{k,l}=\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{l})\mathcal{D}_{p}}+\mathbbm{1}_{\mathcal{D}_{p}m(p^{l},p^{k})\mathcal{D}_{p}}\cdot\operatorname{id}_{S_{L_{p}}}.$

Then we denote by
$\mathcal{H}^{W}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
the subalgebra of
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
generated by $\tau_{k,l}$ and $\tau_{k}$, which is nothing else but subalgebra
of all elements of
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
invariant under the Weyl group $W$.

######  Theorem 5.11.

Let $p$ be a prime dividing $|D|$ and $L_{p}^{\prime}/L_{p}$ anisotropic.

Then the Hecke algebras
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and
$\mathcal{H}^{W}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$
are isomorphic.

###### Proof.

In view of Remark 5.10, it suffices to prove that $\mathcal{S}$ is injective
and surjective. To this end, we compute $\mathcal{S}(T)$ for a non-zero
$T\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$. By Corollary
5.8, we may assume that $T$ is either $T_{k,l}$ or $T_{k}$.

We first consider the case $k<l$. Thus,
$T=T_{k,l}\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ with
$\operatorname{supp}(T_{k,l})=\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}$.
Let $m(p^{i},p^{j})\in\mathcal{M}_{p}$ for arbitrary $i,j\in\mathbb{Z}$ with
$i\leq j$ and $i+j$ a square. One can prove (see [De], Lemma 8.24) that
$m(p^{i},p^{j})N(\mathbb{Q}_{p})\cap\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}\not=\emptyset$
if and only if $i,j\geq k$ and $i+j=k+l$. Therefore,

(5.13)
$\operatorname{supp}(\mathcal{S}T_{k,l})\subset\\{\mathcal{D}_{p}m(p^{\nu},p^{k+l-\nu})\mathcal{D}_{p}\;|\;\nu=k,\dots,l\\}.$

For $x=0$ it is obvious that $m(p^{\nu},p^{k+l-\nu})n(0)$ lies in
$\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}$ if and only $\nu=k$ or $\nu=l$.
Let $0\not=x=p^{r}s$ with $r<0$ and $s\in\mathbb{Z}_{p}^{\times}$. Employing
the Cartan decomposition produces for all $\nu$

(5.14)
$m(p^{\nu},p^{k+l-\nu})n(p^{r}s)=n\\_(p^{k+l-2\nu-r}s^{-1})m(p^{\nu+r},p^{k+l-\nu-r})m(s)n(-p^{-r}s^{-1})w$

if $k+l-2\nu-r\geq 0$. It follows for $\nu=k,\dots,l$ that the matrix
$m(p^{\nu},p^{k+l-\nu})n(p^{r}s)$ lies in the double coset
$\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}$ if and only if $r=k-\nu$. Thus,
for $\nu=k+1,\dots,l-1$ the sum
$\sum_{x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}}T_{k,l}(m(p^{\nu},p^{k+l-\nu})n(x))$
runs over all elements of the form $x=p^{k-\nu}s$, $s$ traversing the set

$\mathcal{U}(\nu)=\left\\{\sum_{i=0}^{\nu-k-1}x_{i}p^{i}\;|\;x_{0}\in(\mathbb{Z}/p\mathbb{Z})^{\times}\text{
and }x_{i}\in\mathbb{Z}/p\mathbb{Z},\;i=1,\dots,\nu-k-1\right\\}.$

If $k+l-2\nu-r<0$, we find

(5.15)
$m(p^{\nu},p^{k+l-\nu})n(p^{r}s)=n(p^{2\nu+r-k-l}s)m(p^{\nu},p^{k+l-\nu}).$

Obviously, $m(p^{\nu},p^{k+l-\nu})n(p^{r}s)$ is contained in
$\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}$ if and only if $\nu=k$ or
$\nu=l$. In the latter case,

(5.16) $m(p^{\nu},p^{k+l-\nu})n(p^{r}s)=n(p^{l-k+r}s)w^{-1}m(p^{k},p^{l})w.$

As $k+l-2\nu-r<0$ is equivalent to $r>k+l-2\nu$, for $\nu=l$, the sum
$\sum_{x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}}T_{k,l}(m(p^{l},p^{k})n(x))$ runs
over all $x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}$ with $|x|_{p}<l-k$. Assuming a
representation of the form $x=p^{r}s$, we may put $r=k-l$ and write

$\sum_{x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}}T_{k,l}(m(p^{l},p^{k})n(x))=\sum_{s\in\mathcal{U}(l)^{0}}T_{k,l}(m(p^{l},p^{k})n(p^{k-l}s)),$

where

$\mathcal{U}(l)^{0}=\left\\{\sum_{i=0}^{l-k-1}x_{i}p^{i}\;|\;x_{0}=0\text{ and
}x_{i}\in\mathbb{Z}/p\mathbb{Z},\;i=1,\dots,l-k-1\right\\}.$

Note that $|s|_{p}<1$ for all $s\in\mathcal{U}(l)^{0}$.

Consequently, $\mathcal{U}(l)\cup\mathcal{U}(l)^{0}$ contains all principal
parts $x$ in $\mathbb{Q}_{p}/\mathbb{Z}_{p}$ with $\nu_{p}(x)\geq k-l$. As
already pointed out, these are all $x$, for which
$T_{k,l}(m(p^{l},p^{k})n(x))$ is non-zero.

By means of the decomposition (5.14), we are now able to compute
$(\mathcal{S}T_{k,l})(m(p^{\nu},p^{k+l-\nu}))$ explicitly for any
$\nu\in\\{k,\dots,l\\}$. Since the computations for $\nu=l$ are more
complicated, we treat them separately afterwards. Thus, let
$\nu\in\\{k,\dots,l-1\\}$. Then

(5.17)
$\begin{split}&\sum_{x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}}T_{k,l}\left(m(p^{\nu},p^{k+l-\nu})n(x)\right)_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}\\\
&=\begin{cases}\sum_{s\in\mathcal{U}(\nu)}T_{k,l}\left(m(p^{\nu},p^{k+l-\nu})n(p^{k-\nu}s)\right)_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}},&\nu\not=k,\\\
T_{k,l}(m(p^{k},p^{l}))_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}},&\nu=k\end{cases}\\\
&=\begin{cases}\sum_{s\in\mathcal{U}(\nu)}\omega_{p}(n\\_(p^{l-\nu}s^{-1}))\circ
T_{k,l}(m(p^{k},p^{l}))\circ\omega_{p}(m(s)n(-p^{\nu-k}s^{-1})w)_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}},&\nu\not=k,\\\
T_{k,l}(m(p^{k},p^{l}))_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}},&\nu=k.\end{cases}\end{split}$

Since the level of $L_{p}^{\prime}/L_{p}$ is $p$, the last expression in
(5.17) for $\nu\not=k,l$ simplifies to

$\displaystyle\sum_{s\in
U(\nu)}T_{k,l}(m(p^{k},p^{l}))\circ\omega_{p}\left(m(s)w\right)_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}$

With the help of the explicit formulas (4.7) of $\omega_{p}$ and Lemma 5.6, we
obtain

$\displaystyle(\mathcal{S}T_{k,l})(m(p^{\nu},p^{k+l-\nu}))\varphi_{p}^{(0)}$
$\displaystyle=\delta(m(p^{\nu},p^{k+l-\nu}))^{1/2}\frac{\gamma_{p}(L_{p}^{\prime}/L_{p})}{|L_{p}^{\prime}/L_{p}|^{1/2}}\sum_{\gamma\in
L_{p}^{\prime}/L_{p}}\sum_{s\in\mathcal{U}(\nu)}\left(\frac{s}{|L_{p}^{\prime}/L_{p}|}\right)T_{k,l}(m(p^{k},p^{l}))\varphi_{p}^{(s^{-1}\gamma)}$
$\displaystyle=\begin{cases}0,&\text{ if }|L_{p}^{\prime}/L_{p}|=p,\\\
\delta(m(p^{\nu},p^{k+l-\nu}))^{1/2}\gamma_{p}(L_{p}^{\prime}/L_{p})|L_{p}^{\prime}/L_{p}|^{1/2}|\mathcal{U}(\nu)|\varphi_{p}^{(0)},&\text{
if }|L_{p}^{\prime}/L_{p}|=p^{2}.\end{cases}$

In view of the discussion above, for $\nu=l$ we have

(5.18)
$\begin{split}&\sum_{x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}}T_{k,l}(m(p^{l},p^{k})n(x))\varphi_{p}^{(0)}=\\\
&\sum_{s\in\mathcal{U}(l)}\omega_{p}(n\\_(s^{-1}))T(m(p^{k},p^{l}))\omega_{p}(m(s)w)\varphi_{p}^{(0)}+\sum_{s\in\mathcal{U}(l)^{0}}\omega_{p}(w^{-1})T_{k,l}(m(p^{k},p^{l}))\omega_{p}(w)\varphi_{p}^{(0)}.\end{split}$

With the help of Lemma 4.1 and the calculations before, it can be verified
that the first summand of the above expression is equal to

(5.19)
$\begin{split}&\left(\frac{-1}{|L_{p}^{\prime}/L_{p}|}\right)\gamma_{p}(L_{p}^{\prime}/L_{p})^{2}\sum_{s\in\mathcal{U}(l)}\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\psi_{p}(sq(\nu_{p}))\varphi_{p}^{(\nu_{p})}\\\
&=\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\left(|\mathcal{U}(l)^{0}|\sum_{x_{0}\in(\mathbb{Z}/p\mathbb{Z})^{\times}}e(x_{0}q(\nu_{p}))\right)\varphi_{p}^{(\nu_{p})},\end{split}$

where we exploited for the last equation the fact that level of $L_{p}$ is $p$
and that
$\gamma_{p}(L_{p}^{\prime}/L_{p})^{2}=\left(\frac{-1}{|L_{p}^{\prime}/L_{p}|}\right)$
(see e. g. [Ze], p. 73). Similarly, the second summand can be evaluated to be

(5.20) $|\mathcal{U}(l)^{0}|\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\varphi_{p}^{(\nu_{p})}.$

Replacing the right-hand side of (5.18) with (5.19) and (5.20), yields

$\displaystyle\sum_{x\in\mathbb{Q}_{p}/\mathbb{Z}_{p}}T_{k,l}(m(p^{l},p^{k})n(x))\varphi_{p}^{(0)}$
$\displaystyle=|\mathcal{U}(l)^{0}|\sum_{\nu_{p}\in
L_{p}^{\prime}/L_{p}}\left(\sum_{x\in\mathbb{Z}/p\mathbb{Z}}e(xq(\nu_{p}))\right)\varphi_{p}^{(\nu_{p})}$
$\displaystyle=|\mathcal{U}(l)^{0}|p\varphi_{p}^{(0)}.$

Here we have used the standard formula for the Gauss sum
$\sum_{x\in\mathbb{Z}/p\mathbb{Z}}e(xq(\nu_{p}))$. This leads us finally to

(5.21)
$\begin{split}&(\mathcal{S}T_{k,l})(m(t_{1},t_{2}))=\frac{g(D_{p})}{g_{p^{l}}(D_{p})}\delta(m(t_{1},t_{2}))^{1/2}\times\\\
&\begin{cases}\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{l})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}},&m(t_{1},t_{2})=m(p^{k},p^{l}),\\\
\delta_{p}\gamma_{p}(L_{p}^{\prime}/L_{p})|L_{p}^{\prime}/L_{p}|^{1/2}|\mathcal{U}(\nu)|\mathbbm{1}_{\mathcal{D}_{p}m(p^{\nu},p^{k+l-\nu})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}},&m(t_{1},t_{2})=m(p^{\nu},p^{k+l-\nu}),\nu\not=k,l\\\
p^{l-k}\mathbbm{1}_{\mathcal{D}_{p}m(p^{l},p^{k})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}},&m(t_{1},t_{2})=m(p^{l},p^{k}),\\\
{\bf 0},&\text{ otherwise,}\end{cases}\\\
&=\frac{g(D_{p})}{g_{p^{l}}(D_{p})}p^{\frac{1}{2}(l-k)}\times\\\
&\begin{cases}\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{l})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}},&m(t_{1},t_{2})=m(p^{k},p^{l}),\\\
\delta_{p}\gamma_{p}(L_{p}^{\prime}/L_{p})\mathbbm{1}_{\mathcal{D}_{p}m(p^{\nu},p^{k+l-\nu})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}},&m(t_{1},t_{2})=m(p^{\nu},p^{k+l-\nu}),\nu\not=k,l\\\
\mathbbm{1}_{\mathcal{D}_{p}m(p^{l},p^{k})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}},&m(t_{1},t_{2})=m(p^{l},p^{k}),\\\
{\bf 0},&\text{ otherwise,}\end{cases}\end{split}$

where $\delta_{p}=1$ if $|L_{p}^{\prime}/L_{p}|=p^{2}$ and zero otherwise.

If $k=l$, it follows from (5.13) that
$\operatorname{supp}(\mathcal{S}T_{k})=\mathcal{D}_{p}m(p^{k},p^{k})\mathcal{D}_{p}$.
The same thoughts as for $k<l$ after equation (5.16) yield

$\displaystyle(\mathcal{S}T_{k}(m(p^{k},p^{k}))$
$\displaystyle=T_{k}(m(p^{k},p^{k}))_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}$
$\displaystyle=\frac{g(D_{p})}{g_{p^{k}}(D_{p})}\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{k})\mathcal{D}_{p}}\operatorname{id}_{S_{L_{p}}^{N(\mathbb{Z}_{p})}}.$

From the above follows immediately that $\mathcal{S}$ is injective. For the
surjectivity it suffices to proof that $\tau_{k}$ and $\tau_{k,l}$ are
contained in the image of $\mathcal{S}$. This can be done almost verbatim as
in [De], p. 212. ∎

Whenever $p$ divides $|D|$ and we deal with Elements $T_{k,l}$, $T_{k}$ or
$\tau_{k,l}$, $\tau_{k}$ of either of the Hecke algebras
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ or
$\mathcal{H}^{W}(\mathcal{M}_{p}//\mathcal{D}_{p},\omega_{p_{|S_{L_{p}}^{N(\mathbb{Z}_{p})}}})$,
we mean the above stated and assume that $L_{p}^{\prime}/L_{p}$ is
anisotropic.

###  5.2. The case of primes $p$ not dividing $|D|$

We denote with $\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})^{W}$ the
subalgebra of all elements of $\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})$
invariant under the Weyl group $W$. In this case it easily seen that
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is isomorphic to
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})^{W}$.

######  Theorem 5.12.

Let $p$ be a prime coprime to $|D|$. Then the Hecke algebras
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})^{W}$ are isomorphic as
algebras.

###### Proof.

By Lemma 3.4 in [St], we know that $L_{p}$ is unimodular and $\omega_{p}$ is
the trivial representation on the space
$S_{L_{p}}=\mathbb{C}\varphi_{p}^{(0)}$. A basis of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is then given by

$\left\\{\mathbbm{1}_{\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}}\cdot\operatorname{id}_{S_{L_{p}}}\;|\;k,l\in\mathbb{Z},k\leq
l\right\\},$

where $\mathbbm{1}_{\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}}$ is the
characteristic function of $\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}$. The
composition of

$F:\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})\rightarrow\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p}),\quad\mathbbm{1}_{\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}}\cdot\operatorname{id}_{S_{L_{p}}}\mapsto\mathbbm{1}_{\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}}$

with the classical Satake map (see e. g. [De] or [Ca]) gives the desired
isomorphism. ∎

For later purposes, we conclude this chapter with the observation that
$\mathcal{H}(\mathcal{M}_{p}//\mathcal{D}_{p})$ can be identified with the
group algebra $\mathbb{C}[\mathcal{M}_{p}/\mathcal{D}_{p}]$ via

$\mathbbm{1}_{\mathcal{D}_{p}m(p^{k},p^{l})\mathcal{D}_{p}}\mapsto\mathbbm{1}_{m(p^{k},p^{l})\mathcal{D}_{p}}.$

##  6\. Hecke operators and their relation to
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$

In his thesis [We], Werner introduced a generalized version of the Hecke
operators $T(m^{2})^{*}$ defined in [BS] (see Section 3). Most notably, an
extension of the Weil representation from
$\operatorname{SL}_{2}(\mathbb{Z}/N\mathbb{Z})$ to
$\operatorname{GL}_{2}(\mathbb{Z}/N\mathbb{Z})$ ($N$ is the level of $L$) was
specified without the condition that the parameter $m$ has to be a square.
Werner also laid the foundation of an adelic description of these generalized
Hecke operators. In particular, he assigned to each vector valued modular form
a vector valued automorphic form on $\operatorname{GL}_{2}(\mathbb{A})$. In
this section we continue this work and embed it into a more general framework
of vector valued automorphic forms. However, we stick to the Hecke operators
given in [BS]. As a consequence, we have to work with the extension of the
Weil representation as stated in Section 3 and its adelic counterpart in
Section 4.

###  6.1. Relation between vector valued automorphic forms and vector valued
modular forms

Instead of working with $\operatorname{GL}_{2}(\mathbb{A})$, we consider the
restricted product

(6.1)
$\mathcal{G}(\mathbb{A})=\sideset{}{{}^{\prime}}{\prod}_{p\leq\infty}\mathcal{Q}_{p}=\left\\{(g_{p})\in\prod_{p\leq\infty}\mathcal{Q}_{p}\;|\;g_{p}\in\mathcal{K}_{p}\text{
for almost all primes }p\right\\},$

where

$\mathcal{Q}_{\infty}=\\{M\in\operatorname{GL}_{2}(\mathbb{R})\;|\;\det(M)\in(\mathbb{R}^{\times})^{2}\\}.$

Note that $\mathcal{K}_{\infty}=\operatorname{SO}(2)$ is a subgroup of
$\mathcal{Q}_{\infty}$. The group $\mathcal{G}(\mathbb{Q})$ can be embedded
diagonally as a discrete subgroup of $\mathcal{G}(\mathbb{A})$. An important
decomposition for $\operatorname{GL}_{2}(\mathbb{A})$, which will be needed
for the definition of automorphic forms, is the strong approximation. An
analogous result holds for $\mathcal{G}(\mathbb{A})$.

######  Theorem 6.1.

Let
$\mathcal{K}=\prod_{p<\infty}\mathcal{K}_{p}\subset\operatorname{GL}_{2}(\mathbb{A}_{f})$.
Then

(6.2)
$\mathcal{G}(\mathbb{A})=\mathcal{G}(\mathbb{Q})(\mathcal{Q}_{\infty}\times\mathcal{K}).$

More generally, let $\mathcal{U}=\prod_{p<\infty}\mathcal{U}_{p}$ be any open
compact subgroup of $\mathcal{K}$ with the property that
$\det(\mathcal{U})=(\widehat{\mathbb{Z}}^{\times})^{2}$. Then

(6.3)
$\begin{split}&\mathcal{G}(\mathbb{A}_{f})=\mathcal{G}(\mathbb{Q})\cdot\mathcal{U}\text{
and }\\\
&\mathcal{G}(\mathbb{A})=\mathcal{G}(\mathbb{Q})(\mathcal{Q}_{\infty}\times\mathcal{U}).\end{split}$

###### Proof.

A proof for the classical result for $\operatorname{GL}_{2}(\mathbb{A})$ can
be found in many places among them in [KL], Section 5.2 and Section 6.3. One
can check that the the proofs of Proposition 5.10, Proposition 6.5 and Theorem
6.8 of [KL] carry over to the analogous statements in our setting. ∎

In [KL] and [Ge] functions
$f:\operatorname{GL}_{2}(\mathbb{Q})\setminus\operatorname{GL}_{2}(\mathbb{A})\rightarrow\mathbb{C}$
with certain properties were related to (scalar valued) modular forms. Here we
consider $\mathcal{G}(\mathbb{Q})$-invariant and $S_{L}$-valued functions

$F:\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A})\rightarrow S_{L}$

with a similar goal. With respect to the basis $(\varphi_{\mu})_{\mu\in D}$ of
$S_{L}$ such a function can be written in the form $F=\sum_{\mu\in
D}F_{\mu}\varphi_{\mu}$. In view of (4.5) and (4.6), we will consider only
factorizable functions, that is, only those $S_{L}$-valued functions $F$,
which possess a decompostion of the form

$F(\gamma(g_{\infty}\times
g_{f}))=\bigotimes_{p<\infty}F_{p}(g_{\infty},g_{p}),$

where

$F_{p}(g_{\infty},g_{p})=\begin{cases}\sum_{\lambda\in
L_{p}^{\prime}/L_{p}}F_{\lambda,\infty}(g_{\infty})F_{\lambda,p}(g_{p})\varphi_{p}^{(\lambda)},&p\mid|D|,\\\
\varphi_{p}^{(0)},&p\nmid|D|.\end{cases}$

Using the bilinearity of the tensor product, we have

$F(g)=\sum_{(\lambda_{p})_{p}\in\bigoplus_{p<\infty}L_{p}^{\prime}/L_{p}}F_{\lambda_{p}\infty}(g_{\infty})\prod_{p<\infty}F_{\lambda_{p},p}(g_{p})\bigotimes_{p<\infty}\varphi_{p}^{(\lambda_{p})}.$

Note that $F$ is well defined since any occurring sum, product or tensor
product is finite.

We denote the subspace of all these functions
$F:\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A})\rightarrow S_{L}$
with $\mathcal{F}_{L}$. Associated to the Weil representation $\omega_{f}$ of
$\mathcal{G}(\mathbb{A}_{f})$ on the space $S_{L}$ and analogous to the
corresponding scalar valued space in [KL], we define

(6.4)
$\begin{split}&L^{2}(\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A}),\omega_{f})=\left\\{F\in\mathcal{F}_{L}\;\left|\begin{array}[]{ll}\text{i)}&F_{\mu}\text{
is measurable }\text{ for all }\mu\in D\\\
\text{ii)}&F(zg)=\omega_{f}(z_{f})^{-1}F(g)\text{ for all }\\\
&z=z_{\mathbb{Q}}(z_{\infty}\times z_{f})\in\mathcal{Z}(\mathbb{A})\\\
\text{iii)}&\int_{\overline{\mathcal{G}}(\mathbb{Q})\setminus\overline{\mathcal{G}}(\mathbb{A})}\|F(g)\|^{2}dg<\infty\end{array}\right.\right\\}\\\
&\text{ and }\\\ &L^{2}_{0}(\omega_{f})\\\ &=\left\\{F\in
L^{2}(\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A}),\omega_{f})\;\left|\;\int_{N(\mathbb{Q})\setminus
N(\mathbb{A})}F_{\mu}(ng)dn=0\text{ for all }\mu\in D,\text{ a. e.
}g\in\mathcal{G}(\mathbb{A})\right.\right\\}.\end{split}$

Here by

  1. i)

$\|F(g)\|^{2}$ we mean $\langle F(g),F(g)\rangle$ as defined in (4.4),

  2. ii)

$\overline{\mathcal{G}}(R)=\mathcal{Z}(R)\setminus\mathcal{G}(R)$, where
$\mathcal{Z}(R)$ is the center of $\mathcal{G}(R)$,

  3. iii)

$dg$ and $dn$ we mean the Haar measure on
$\overline{\mathcal{G}}(\mathbb{Q})\setminus\overline{\mathcal{G}}(\mathbb{A})$
and $N(\mathbb{Q})\setminus N(\mathbb{A})$, respectively.

Measurability for each component function $F_{\mu}$ is meant in the sense of
Proposition 7.15 of [KL]: $F_{\mu}$ can be written as a product
$\prod_{p\leq\infty}F_{\mu,p}(g_{p})$, each component satisfying:

  1. i)

$F_{\mu,p}:\mathcal{Q}_{p}\rightarrow\mathbb{C}$ is measurable for all
$p\leq\infty$

  2. ii)

$F_{{\mu,p}_{|_{\mathcal{K}_{p}}}}=1$ for all $p\notin S,$ where $S$ is finite
set of places.

The above integrals over
$\overline{\mathcal{G}}(\mathbb{Q})\setminus\overline{\mathcal{G}}(\mathbb{A})$
and $N(\mathbb{Q})\setminus N(\mathbb{A})$ are explained in [KL], Proposition
7.43 and Proposition 12.2, and meant in the very same way. Also note that the
integral in iii) of
$L^{2}(\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A}),\omega_{f})$ is
well defined as $F$ satisfies ii) and the Weil representation $\omega_{f}$ is
unitary with respect to $\langle\cdot,\cdot\rangle$.

Werner assigned in [We], Def. 49, a $\mathbb{C}[D]$-valued Function $F_{f}$ on
$\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A})$ to a cusp form $f\in
S_{\kappa}(\rho_{L})$. We adopt his definition to our setting, which basically
means that we replace the group ring with the isomorphic space $S_{L}$.

######  Definition 6.2.

Let $f\in S_{\kappa}(\rho_{L})$ and $g\in\mathcal{G}(\mathbb{A})$ with
$g=\gamma(g_{\infty}\times k)$, where
$\gamma\in\mathcal{G}(\mathbb{Q}),g_{\infty}\in\mathcal{Q}_{\infty}$ and
$k\in\mathcal{K}$. Then in terms of this decomposition we define a map
$\mathscr{A}$

(6.5) $f\mapsto\mathscr{A}(f)=F_{f}\;\text{ with
}F_{f}(g)=\omega_{f}(k)^{-1}j(g_{\infty},i)^{-\kappa}f(g_{\infty}i).$

Lemma 50 in [We] shows that the definition of $F_{f}$ in (6.5) is independent
from the decomposition of $g$. Moreover, from its definition it follows
immediately that $F_{f}$ is $\mathcal{G}(\mathbb{Q})$-invariant.

######  Proposition 6.3.

Let $f\in S_{\kappa}(\rho_{L})$. Then the assigned function $F_{f}$ on
$\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A})$ lies in the space
$L^{2}(\mathcal{G}(\mathbb{Q})\setminus\mathcal{G}(\mathbb{A}),\omega_{f})$.

###### Proof.

  1. i)

By definition the $\mu$-th component of $F_{f}$ is given by

(6.6)
$\begin{split}(F_{f})_{\mu}(g)&=\langle\omega_{f}(k)^{-1}j(g_{\infty},i)^{-\kappa}f(g_{\infty}i),\varphi^{(\mu)}\rangle\\\
&=j(g_{\infty},i)^{-\kappa}\sum_{\lambda\in
D}f_{\lambda}(g_{\infty}i)\prod_{p<\infty}\langle\omega_{p}^{-1}(k_{p})\varphi_{p}^{(\lambda_{p})},\varphi_{p}^{(\mu_{p})}\rangle,\end{split}$

where $g=\gamma(g_{\infty}\times k)$. It is well known that
$j(g_{\infty},i)^{-\kappa}f_{\lambda}(g_{\infty}i)$ is measurable on
$\mathcal{Q}_{\infty}$ as $f_{\lambda}$ is a scalar valued cusp form for
$\Gamma(N)$ (cf. [Ge], $\S 2$, for this case). As a result of the discussion
in Chapter 4, we have that $\omega_{p}$ is trivial for all $p\nmid N$. For
$p\mid N$ we find by means of the explicit formulas of $\omega_{p}$ (see e. g.
[BY], p. 645, or [St], Lemma 3.4) that $\omega_{p}$ is trivial on the subgroup

$\mathcal{K}_{p}(p^{\operatorname{ord}_{p}(D)})=\left\\{\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\mathcal{K}_{p}\;|\;\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\equiv\left(\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\right)\bmod{p^{\operatorname{ord}_{p}(D)}}\mathbb{Z}_{p}\right\\}$

and factors thereby through
$\mathcal{K}_{p}/\mathcal{K}_{p}(p^{\operatorname{ord}_{p}(D)})$ for each $p$
dividing $N$. Since
$\mathcal{K}_{p}(p^{\operatorname{ord}_{p}(D)})$ has as compact subgroup a
finite measure,
$\langle\omega_{p}^{-1}(k_{p})\varphi_{p}^{\mu_{p}},\varphi_{p}^{\mu_{p}}\rangle$
is a measurable function for all primes $p$. We then obtain that
$(F_{f})_{\mu}$ is measurable in the above stated sense.

  2. ii)

Let $z=z_{\mathbb{Q}}(z_{\infty}\times z_{f})\in\mathcal{Z}(\mathbb{A})$. Then
it follows immediately from the definition of $F_{f}$ that
$F_{f}(zg)=\omega_{f}(z_{f})^{-1}F_{f}(g)$.

  3. iii)

It can be verified that Proposition 7.43 and the discussion before of [KL] is
also valid in our situation. We have to check that all steps of the proof are
still working if we replace the involved groups by the corresponding groups in
our setting. This is in fact the case, some steps are even easier since we
only have to deal with matrices whose determinant is a square. As a result, we
may replace the integral over
$\overline{\mathcal{G}}(\mathbb{Q})\setminus\overline{\mathcal{G}}(\mathbb{A})$
with the corresponding integral over $D\mathcal{K}_{\infty}\times\mathcal{K}$.
Here $D$ is a fundamental domain for $\Gamma(1)\setminus\mathbb{H}$
interpreted as subset of $\operatorname{SL}_{2}(\mathbb{R})$. Following the
proof of Proposition 12.15 in [KL], we find for $F_{f}$

(6.7)
$\begin{split}\int_{\overline{\mathcal{G}}(\mathbb{Q})\setminus\overline{\mathcal{G}}(\mathbb{A})}\|F_{f}(g)\|^{2}dg&=\int_{D\mathcal{K}_{\infty}}\int_{\mathcal{K}}\|F_{f}(g\times
k)\|^{2}dkdg\\\
&=\int_{D}\|j(g_{\infty},i)^{-\kappa}f(g_{\infty}i)\|^{2}dg,\end{split}$

where we have used that $\omega_{f}$ is unitary with respect to
$\langle\cdot,\cdot\rangle$ and that the Haar measure on $\mathcal{Q}_{p}$ is
normalized to be equal to one on $\mathcal{K}_{p}$ for $p\leq\infty$. If we
identify $g_{\infty}i$ with an element $\tau\in\Gamma(1)\setminus\mathbb{H}$,
the last integral in (6.7) becomes

$\int_{\Gamma(1)\setminus\mathbb{H}}\|f(\tau)\|^{2}\operatorname{Im}(\tau)^{\kappa}\frac{dxdy}{y^{2}}<\infty,$

which is the Petersson norm of $f\in S_{\kappa}(\rho_{L})$. Therefore, the
$L^{2}$-norm of $F_{f}$ is finite.

∎

######  Lemma 6.4.

Let $f\in S_{\kappa}(\rho_{L})$ and $F_{f}$ the assigned automorphic form
given by (6.5). Then

$\int_{N(\mathbb{Q})\setminus N(\mathbb{A})}F_{\mu}(ng)dn=0$

for almost every $g\in\mathcal{G}(\mathbb{A})$ and all $\mu\in D$.

###### Proof.

The proof proceeds along the lines of the one of Proposition 12.2 in [KL]. Let
$n=n(x_{\mathbb{Q}})(n(x_{\infty})\times n(x_{f}))\in N(\mathbb{A})$ and
$g=\gamma(g_{\infty}\times g_{f})\in\mathcal{G}(\mathbb{A})$. Then the
definition of $F_{f}$ and $\omega_{f}$ yields

$\displaystyle F_{\mu}(ng)$ $\displaystyle=\langle
j(g_{\infty},i)^{-\kappa}j(n(x_{\infty}),g_{\infty}i)^{-\kappa}\omega_{f}^{-1}(g_{f})\omega_{f}^{-1}(n(x_{f}))f(n(x_{\infty})(g_{\infty}i)),\varphi_{\mu}\rangle$
$\displaystyle=j(g_{\infty},i)^{-\kappa}j(n(x_{\infty}),g_{\infty}i)^{-\kappa}\sum_{\nu\in
D}\psi_{f}(-x_{f}q(\nu))f_{\nu}(n(x_{\infty})(g_{\infty}i))\langle\omega_{f}^{-1}(g_{f})\varphi_{\nu},\varphi_{\mu}\rangle.$

As suggested in [KL], Prop. 12.2., we calculate more generally for
$r\in\mathbb{Q}$

(6.8) $\begin{split}&\int_{N(\mathbb{Q})\setminus
N(\mathbb{A})}F_{\mu}(n(x)g)\psi(rx)dx\\\
&=j(g_{\infty},i)^{-\kappa}\sum_{\nu\in
D}\langle\omega_{f}^{-1}(g_{f})\varphi_{\nu},\varphi_{\mu}\rangle\times\\\
&\int_{\mathbb{N}(\mathbb{Z})\setminus(N(\mathbb{R})\times
N(\widehat{\mathbb{Z}}))}\psi_{f}(-x_{f}q(\nu))f_{\nu}(n(x_{\infty})(g_{\infty}i))\psi_{\infty}(rx_{\infty})\psi_{f}(rx_{f})dx_{f}dx_{\infty}.\end{split}$

We can write the integral in the last expression as

$\displaystyle\int_{0}^{1}f_{\nu}(n(x_{\infty})(g_{\infty}i))\psi_{\infty}(rx_{\infty})\int_{N(\widehat{\mathbb{Z}})}\psi_{f}((r-q(\nu))x_{f})dx_{f}dx_{\infty},$

where the integral over $N(\widehat{\mathbb{Z}})$ is one if and only if
$r\in\mathbb{Z}+q(\nu)$. For such $r$ (note that
$\psi_{\infty}(x_{\infty})=e(-x_{\infty})$), taking into account that
$\int_{0}^{1}f_{\nu}(x_{\infty}+\tau)e(-rx_{\infty})dx_{\infty}=e(r\operatorname{Re}(\tau))a(\nu,r)$,
where $a(\nu,r)$ is the Fourier coefficient of $f$ with respect to $(\nu,r)$,
we finally obtain

$\displaystyle\int_{N(\mathbb{Q})\setminus
N(\mathbb{A})}F_{\mu}(n(x)g)\psi(rx)dx=j(g_{\infty},i)^{-\kappa}\sum_{\nu\in
D}\langle\omega_{f}^{-1}(g_{f})\varphi_{\nu},\varphi_{\mu}\rangle
e(r\operatorname{Re}(\tau))a(\nu,r),$

where $\tau=g_{\infty}i$. Since $f$ is a cusp form, we have that for $r=0$ all
coefficients $a(\nu,r)$ vanish. This gives the desired result. ∎

The image of $S_{\kappa}(\rho_{L})$ under the map $\mathscr{A}$ in (6.5) can
be characterized more closely:

######  Theorem 6.5.

Let $A_{\kappa}(\omega_{f})$ be the space of functions $F\in
L^{2}_{0}(\omega)$ satisfying

  1. i)

$F(gk)=\omega_{f}(k)^{-1}F(g)$ for all $k\in\mathcal{K}$ and all
$g\in\mathcal{G}(\mathbb{A})$

  2. ii)

$F(g\left(\begin{smallmatrix}\cos(\theta)&\sin(\theta)\\\
-\sin(\theta)&\cos(\theta)\end{smallmatrix}\right))=e^{i\kappa\theta}F(g)$ for
all $\theta\in[0,2\pi)$ and all $g\in\mathcal{G}(\mathbb{A})$

  3. iii)

All the components $F_{\mu}$ of $F$, considered as a function of
$\mathcal{Q}_{\infty}$ alone, satisfy the differential equation $LF_{\mu}=0$.
Here $L$ is the differential operator given by

(6.9) $L=e^{-2i\theta}\left(-2iy\frac{\partial}{\partial
x}+2y\frac{\partial}{\partial y}+i\frac{\partial}{\partial\theta}\right)$

with respect to the coordinates referring to the decomposition

(6.10) $g_{\infty}=z_{\infty}\begin{pmatrix}1&x\\\
0&1\end{pmatrix}\left(\begin{smallmatrix}y^{\frac{1}{2}}&0\\\
0&y^{-\frac{1}{2}}\end{smallmatrix}\right)\left(\begin{smallmatrix}\cos(\theta)&\sin(\theta)\\\
-\sin(\theta)&\cos(\theta)\end{smallmatrix}\right)$

of $g_{\infty}\in\mathcal{Q}_{\infty}$.

Then the map $\mathscr{A}$ defines an isometry from $S_{\kappa}(\rho_{L})$
onto $A_{\kappa}(\omega_{f})$.

###### Proof.

This theorem is well known for scalar valued automorphic forms, see e. g. [Ge]
or [KL]. Most parts of its proof can be settled with reference to the proof of
its scalar valued analogue.

Let $f\in S_{\kappa}(\rho_{L})$. It follows from Proposition 6.3 and Lemma 6.4
that $F_{f}\in L^{2}_{0}(\omega_{f})$. The assertion in i) is proved in [We],
Theorem 51, the one in ii) results from a straightforward calculation
analogous to the scalar valued case (see [KL], Proposition 12.5). For iii)
note that $F_{\mu}(g_{\infty}\times 1_{f})=y^{k/2}e^{ik\theta}f_{\mu}(x+iy)$
if we decompose $g_{\infty}\in\mathcal{Q}_{\infty}$ according to (6.10). The
same proof as in [KL], applied to each component $F_{\mu}$, establishes the
result using the assumption $f\in S_{\kappa}(\rho_{L})$.

Kudla [Ku] defined a map that assigns to a vector valued function $F$ on
$\mathcal{G}(\mathbb{A})$ a vector valued function $f_{F}$ on $\mathbb{H}$:

(6.11) $F\mapsto f_{F},\;f_{F}(\tau)=j(g_{\tau},i)^{\kappa}F(g_{\tau}\times
1_{f}),$

where $g_{\tau}=\begin{pmatrix}1&x\\\
0&1\end{pmatrix}\left(\begin{smallmatrix}y^{\frac{1}{2}}&0\\\
0&y^{-\frac{1}{2}}\end{smallmatrix}\right)$ and
$\tau=g_{\tau}i=x+iy\in\mathbb{H}$. It is easily seen that this map is well-
defined and that it is the inverse map of $\mathscr{A}$ (see [KL], Prop. 12.5,
for the corresponding scalar valued result). It remains to show that $f_{F}$
is an element of $S_{\kappa}(\rho_{L})$ for any $F\in A_{\kappa}(\omega_{f})$.
Kudla proved that $f_{F}$ transforms like a vector valued modular form with
respect to $\omega_{f}$ if $F\in A_{\kappa}(\omega_{f})$ ([Ku], Lemma 1.1).
Since each component of $F$ satisfies the differential equation in iii), it
follows that each component of $f_{F}$ is holomorphic on the upper half plane
(see [KL], Prop. 12.5). In view of these two properties, $f_{F}$ possess a
Fourier expansion, see [Br1], p. 18. By Proposition 6.3 we know that the
Petersson norm of $f_{F}$ coincides with the $L^{2}$ norm of $F$, it is in
particular finite. One can prove in the same way as in Prop. 3.39 of [KL] that
$f_{F}$ is an element of $S_{\kappa}(\rho_{L})$. Thus, the map in (6.5) is
surjective and an isometry. ∎

###  6.2. The action of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ on
$A_{\kappa}(\omega_{f})$

The goal of this subsection is to define an action of
$\mathcal{G}(\mathbb{A})$ via the Hecke algebra
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ on the space
$A_{\kappa}(\omega_{f})$ of vector valued automorphic forms. Since
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ acts only on
the $p$-component of an element $F\in A_{\kappa}(\omega_{f})$, we need to
complement the contribution of
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ with suitable
operators on the other places. The envisaged action will be defined in such a
way that it is compatible with the action of Hecke operators on
$S_{\kappa}(\rho_{L})$. Werner proposed in [We], Chapter 6, the definition of
an adelic vector valued Hecke operator mimicking Gelbart’s approach of an
adelic scalar valued Hecke operator. Our approach is slightly more general and
transfers the action of the spherical Hecke algebra in [Mu1], $\S$ 6, to the
vector valued setting.

######  Definition 6.6.

Let $p\in\mathbb{Z}$ be a fixed prime, $g=\gamma(g_{\infty}\times
g_{f})\in\mathcal{G}(\mathbb{A})$ and
$T_{p}\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$. Then we
define for a fixed $h\in\mathcal{G}(\mathbb{A})$

(6.12) $R^{T_{p}}(h):\mathcal{F}_{L}\rightarrow\mathcal{F}_{L},\quad F\mapsto
R^{T_{p}}(h)F=\bigotimes_{q<\infty}R_{q}^{T_{p}}(h_{q})F_{q}$

with

(6.13)
$R_{q}^{T_{p}}(h_{q})F_{q}(g_{q})=\begin{cases}F_{q}(g_{\infty},g_{q}h_{q}),&q\not=p\\\
T_{p}(h_{p})(F_{p}(g_{\infty},g_{p})),&q=p.\end{cases}$

The operator

(6.14) $\mathcal{T}^{T_{p}}:A_{\kappa}(\omega_{f})\rightarrow
A_{\kappa}(\omega_{f}),\quad\mathcal{T}^{T_{p}}(F)(g)=\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}R^{T_{p}}(\iota_{p}(x_{p}))F(g\iota_{p}(x_{p}))$

can be interpreted as a vector valued analogue of the construction in [Mu1]
(see Section 2 for the definition of $\iota_{p}$). For the sake of better
readability, we omit the argument $g_{\infty}$ in the subsequent calculations
and assume tacitly that the local functions also depend on $g_{\infty}$.

######  Remark 6.7.

  1. i)

If we decompose $\mathcal{T}^{T_{p}}$ into into its components, we obtain

(6.15)
$\mathcal{T}^{T_{p}}(F)(g)=\bigotimes_{q\not=p}F_{q}(g_{g})\otimes\left(\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{p}(x_{p})F_{p}(g_{p}x_{p})\right).$

Since $T_{p}\in\mathcal{H}(\mathbb{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ has
compact support, the sum in (6.15) is finite. It can be verified by means of
Theorem 6.5, i), and Definition 5.1, ii), that (6.15) and therefore (6.14) is
independent of the representative $x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}$
and thus well defined. We will show later in the paper that
$\mathcal{T}^{T_{p}}(F)$ is indeed contained in $A_{\kappa}(\omega_{f})$.

  2. ii)

Let $p$ be a prime,
$T_{k,l},T_{r,s}\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$
as specified in Corollary 5.8 and Theorem 5.12. Then by a straightforward
calculation, using (6.15) and the bilinearity of the tensor product, we obtain

$\displaystyle\mathcal{T}^{xT_{k,l}+yT_{r,s}}(F)(g)=x\mathcal{T}^{T_{k,l}}(F)(g)+y\mathcal{T}^{T_{r,s}}(F)(g)$

for all $F\in A_{k}(\omega_{f})$, all $g\in\mathcal{G}(\mathbb{A})$ and all
$x,y\in\mathbb{C}$.

There is also a compatibility relation regarding convolution:

$\displaystyle\mathcal{T}^{T_{k,l}\ast
T_{r,s}}(F)(g)=\bigotimes_{q\not=p}F_{q}(g_{q})\otimes\left(\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}\left(\sum_{y_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(y_{p})\circ
T_{r,s}(y_{p}^{-1}x_{p})\right)(F_{p}(g_{p}x_{p}))\right).$

Here $T_{k,l}\ast T_{r,s}$ is the convolution of $T_{k,l}$ and $T_{r,s}$ (cf.
(5.1)). Since both sums over $\mathcal{Q}_{p}/\mathcal{K}_{p}$ are finite, we
can change their order and obtain

$\displaystyle\bigotimes_{q\not=p}F_{q}(g_{q})\otimes\left(\sum_{y_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}\left(\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(y_{p})\circ
T_{r,s}(y_{p}^{-1}x_{p})(F_{p}(g_{p}x_{p}))\right)\right)$
$\displaystyle=\bigotimes_{q\not=p}F_{q}(g_{q})\otimes\left(\sum_{z_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{r,s}(z_{p})\left(\sum_{y_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(y_{p})(F_{p}(g_{p}y_{p}z_{p}))\right)\right)$
$\displaystyle=\mathcal{T}^{T_{r,s}}\circ\mathcal{T}^{T_{k,l}}(F)(g),$

where we have made the substitution $z_{p}=y_{p}^{-1}x_{p}$ in the second last
equation and used the fact that $T_{k,l}\circ T_{r,s}=T_{r,s}\circ T_{k,l}$
for all
$T_{k,l},T_{r,s}\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$.

######  Lemma 6.8.

Let $p$ be a prime,
$T_{k,l}\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ as given
in Corollary 5.8 and Theorem 5.12 and $F\in A_{\kappa}(\omega_{f})$. Then
$\mathcal{T}^{T_{k,l}}$

  1. i)

is $\mathcal{G}(\mathbb{Q})$-invariant and

  2. ii)

fulfils

$\displaystyle\mathcal{T}^{T_{k,l}}(F)(gk)=\omega_{f}^{-1}(k)\mathcal{T}^{T_{k,l}}(F)(g)\text{
for all }k\in\mathcal{K}\text{ and all }g\in\mathcal{G}(\mathbb{A}),$
$\displaystyle\mathcal{T}^{T_{k,l}}(F)(zg)=\omega_{f}^{-1}(z_{f})\mathcal{T}^{T_{k,l}}(F)(g)\text{
for all }z\in\mathcal{Z}(\mathbb{A})\text{ and all
}g\in\mathcal{G}(\mathbb{A}).$

###### Proof.

  1. i)

Since $F\in A_{\kappa}(\omega_{f})$ is $\mathcal{G}(\mathbb{Q})$-invariant,
the same holds for $\mathcal{T}^{T_{k,l}}(F)$ as can be seen in (6.15).

  2. ii)

By Theorem 6.5, i), and Definition 5.1, ii) we have

$\displaystyle\mathcal{T}^{T_{k,l}}(F)(gk)$
$\displaystyle=\bigotimes_{q\not=p}F_{q}(g_{q}k_{q})\otimes\left(\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(x_{p})(F_{p}(g_{p}k_{p}x_{p}))\right)$
$\displaystyle=\bigotimes_{q\not=p}\omega_{q}^{-1}(k_{q})F_{q}(g_{q})\otimes\left(\sum_{y_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(k_{p}^{-1}y_{p})(F_{p}(g_{p}y_{p}))\right)$
$\displaystyle=\begin{cases}\bigotimes_{q\not=p}\omega_{q}^{-1}(k_{q})F_{q}(g_{q})\otimes\omega_{p}^{-1}(k_{p})\left(\sum_{y_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(y_{p})(F_{p}(g_{p}y_{p}))\right),&\text{
if }p||D|\\\
\bigotimes_{q\not=p}\omega_{q}^{-1}(k_{q})F_{q}(g_{q})\otimes\left(\sum_{y_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(y_{p})(F_{p}(g_{p}y_{p}))\right),&\text{
if }p\nmid|D|\end{cases}$
$\displaystyle=\omega_{f}^{-1}(k)\mathcal{T}^{T_{k,l}}(F)(g).$

For the second equation we used the substitution $y_{p}=k_{p}x_{p}$. This
settles the first claimed identity.

For the second identity we make use of the fact that $F\in
A_{\kappa}(\omega_{f})$ and that $\omega_{p}(z_{p})$ acts for
$z_{p}\in\mathcal{Z}(\mathcal{K}_{p})$ on $S_{L_{p}}$ by multiplication with a
scalar (cf. (4.14)), which commutes with the operator $T_{k,l}$. Let
$z=z_{\mathbb{Q}}(z_{\infty}\times z_{f})$ with
$z_{f}=(z_{q})_{q<\infty}\in\mathcal{K}$. Then

$\displaystyle\mathcal{T}^{T_{k,l}}(F)(zg)$
$\displaystyle=\bigotimes_{q\not=p}\omega_{q}^{-1}(z_{q})F_{q}(g_{q})\otimes\left(\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}T_{k,l}(x_{p})(\omega_{p}(z_{p})^{-1}F_{p}(g_{p}x_{p}))\right)$
$\displaystyle=\omega_{f}^{-1}(z_{f})\mathcal{T}^{T_{k,l}}(F)(g).$

∎

Let $p$ be a prime, $k,l$ integers with $k\leq l$ and $k+l\in 2\mathbb{Z}$,
$T_{k,l}\in\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ as in
Corollary 5.8 or Theorem 5.12 and $\mathcal{T}^{T_{k,l}}$ as in Definition
6.6. We now show that the map $\mathscr{A}$ commutes with the Hecke operators
$\mathcal{T}^{T_{k,l}}$ and $T(m(p^{-k},p^{-l}))$ on both sides and thereby
confirm that $\mathcal{T}^{T_{k,l}}$ indeed preserves
$A_{\kappa}(\omega_{f})$. For a prime $p\nmid|D|$ this result was in principle
proved by Werner (cf. [We], Theorem 53), but not in our framework and not for
a general Hecke operator $T(m(p^{-k},p^{-l}))$.

######  Theorem 6.9.

Let $p$ be a prime, $k,l\in\mathbb{Z}$ with $k\leq l$ and $k+l\in
2\mathbb{Z}$. If $p$ divides $|D|$, let
$T_{k,l}\in\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ as in
Corollary 5.8. If $(p,|D|)$, let
$T_{k,l}=\mathbbm{1}_{\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}}\operatorname{id}_{S_{L_{p}}}\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$
be as in Theorem 5.12. Further, let $\mathcal{T}^{T_{k,l}}$ be as in
Definition 6.6 and
$T(m(p^{-k},p^{-l}))$ the Hecke operator as defined in Section 3. Then for any
$f\in S_{\kappa}(\rho_{L})$ we have

(6.16)
$\mathcal{T}^{T_{k,l}}(F_{f})=F_{p^{(k+l)(1-\frac{\kappa}{2})}T(m(p^{-k},p^{-l}))f},$

where $F_{f}$ is the automorphic form related to $f$ via the map
$\mathscr{A}$.

###### Proof.

We know from Lemma 6.8 that for any $g=\gamma(g_{\infty}\times
k)\in\mathcal{G}(\mathbb{A})$ we have

$\displaystyle\mathcal{T}^{T_{k,l}}(F_{f})(\gamma(g_{\infty}\times k))$
$\displaystyle=\mathcal{T}^{T_{k,l}}(F_{f})((g_{\infty}\times 1_{f})(1\times
k))$
$\displaystyle=\omega_{f}^{-1}(k)\mathcal{T}^{T_{k,l}}(F_{f})(g_{\infty}\times
1_{f}).$

The same holds for $F_{p^{-(\frac{l}{2}-1)}T(m(p^{-k},p^{-l}))f}$ since it is
an element of $A_{\kappa}(\omega_{f})$. Hence, it suffices to prove (6.16) for
$g=g_{\infty}\times 1_{f}$.

The proof is an adaptation of the one of Lemma 3.7 in [Ge]. We have

(6.17)
$\mathcal{T}^{T_{k,l}}(F_{f})(g)=\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}R^{T_{k,l}}(\iota_{p}(x_{p}))F_{f}(g\iota_{p}(x_{p})).$

Following an idea of Gelbart we set

$\displaystyle\gamma=(x_{p},\dots,x_{p},\dots)\in\mathcal{G}(\mathbb{Q}),$
$\displaystyle
k(x_{p})=(x_{p}^{-1},\dots,x_{p}^{-1},1_{p},x_{p}^{-1},\dots)\in\mathcal{K},$
$\displaystyle x_{p}^{-1}\in\mathcal{Q}_{\infty},$

where the $1_{p}$ in $k(x_{p})$ is at $p$-th place. With these notations it is
easily verified that

$\iota_{p}(x_{p})=\gamma(x_{p}^{-1}\times k(x_{p})).$

Therefore, the right-hand side of (6.17) becomes

$\displaystyle\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}R^{T_{k,l}}(\iota_{p}(x_{p}))F_{f}(\gamma(x_{p}^{-1}g_{\infty}\times
k(x_{p}))).$

Using the fact that $F_{f}\in A_{\kappa}(\omega_{f})$ and equation (6.6)
subsequently, we find that the latter expression is equal to

$\displaystyle\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}R^{T_{k,l}}(\iota_{p}(x_{p}))\omega_{f}^{-1}(k(x_{p}))F_{f}(x_{p}^{-1}g_{\infty}\times
1_{f})$
$\displaystyle=\sum_{x_{p}\in\mathcal{Q}_{p}/\mathcal{K}_{p}}j(x_{p}^{-1}g_{\infty},i)^{-k}\sum_{\lambda\in
D}f_{\lambda}(x_{p}^{-1}g_{\infty}i)R^{T_{k,l}}(\iota_{p}(x_{p}))(\omega_{f}^{-1}(k(x_{p}))\varphi_{\lambda}).$

Decomposing $R^{T_{k,l}}$ and $\omega_{f}$ into its local factors, yields

(6.18)
$\begin{split}&\sum_{x_{p}\in\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}/\mathcal{K}_{p}}j(x_{p}^{-1}g_{\infty},i)^{-k}\times\\\
&\sum_{\lambda\in
D}f_{\lambda}(x_{p}^{-1}g_{\infty}i)\bigotimes_{q\not=p}\omega_{q}^{-1}(x_{p}^{-1})\varphi_{q}^{(\lambda_{p})}\otimes
T_{k,l}(x_{p})(\omega_{p}^{-1}(1_{p})\varphi_{p}^{(\lambda_{p})}).\\\
\end{split}$

To further simplify the right-hand side of (6.18), we evaluate
$\omega_{q}^{-1}(x_{p}^{-1})$ and $T_{k,l}(x_{p})$ on a concrete set of
representatives $x_{p}$. To this end, we first assume $k<l$. It is easily seen
that Lemma 13.4 of [KL] carries over to our situation. Keeping this in mind,
we can conclude that

(6.19)
$\begin{split}&\left\\{x_{s,b}=m(p^{k},p^{k})\left(\begin{smallmatrix}p^{s}&b\\\
0&p^{l-k-s}\end{smallmatrix}\right)\;|\;s=1,\dots
l-k-1,b\in(\mathbb{Z}/p^{s}\mathbb{Z})^{\times}\right\\}\\\
&\cup\left\\{x_{b}=m(p^{k},p^{k})\left(\begin{smallmatrix}p^{l-k}&b\\\
0&1\end{smallmatrix}\right)\;|\;b\in\mathbb{Z}/p^{l-k}\mathbb{Z}\right\\}\cup\left\\{m(p^{k},p^{k})m(1,p^{l-k})\right\\}\end{split}$

is a set of representatives of
$\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}/\mathcal{K}_{p}$ for any prime
$p$. We now distinguish the cases $p\mid|D|$ and $p\nmid|D|$. The latter is
easier and will be postponed to the end of the proof.

The decomposition

(6.20) $x_{s,b}^{-1}=\left(\begin{smallmatrix}r&-b\\\
t&p^{s}\end{smallmatrix}\right)m(p^{-k},p^{-l})n\\_(-p^{l-k-s}t)\in\Gamma
m(p^{-k},p^{-l})\Gamma$

with $rp^{s}+bt=1$ and

(6.21) $x_{b}^{-1}=wm(p^{-k},p^{-l})w^{-1}n(-b)\in\Gamma
m(p^{-k},p^{-l})\Gamma$

can easily be verified. Since $\Gamma\subset\mathcal{K}_{q}$ for all primes
$q$, these decomposition can also be interpreted as decomposition in
$\mathcal{K}_{q}m(p^{-k},p^{-l})\mathcal{K}_{q}$ for all primes $q$. If
$q\not=p$, we may utilize Definition 4.3 and obtain

(6.22)
$\begin{split}&\omega_{q}^{-1}(x_{s,b}^{-1})=\omega_{q}^{-1}(n\\_(-p^{l-k-s}t))\omega_{q}^{-1}(m(p^{-k},p^{-l}))\omega_{q}^{-1}(\left(\begin{smallmatrix}r&-b\\\
t&p^{s}\end{smallmatrix}\right)),\\\
&\omega_{q}^{-1}(x_{b}^{-1})=\omega_{q}^{-1}(w^{-1}n(-b))\omega_{q}^{-1}(m(p^{-k},p^{-l}))\omega_{q}^{-1}(w).\end{split}$

By Definition 5.1, we further have

(6.23) $\begin{split}&T_{k,l}(x_{s,b})=\omega_{p}(n\\_(-p^{l-k-s}t)^{-1})\circ
T_{k,l}(m(p^{k},p^{l}))\circ\omega_{p}(\left(\begin{smallmatrix}r&-b\\\
t&p^{s}\end{smallmatrix}\right)^{-1}),\\\
&T_{k,l}(x_{b})=\omega_{p}((w^{-1}n(-b))^{-1})\circ
T_{k,l}(m(p^{k},p^{l}))\circ\omega_{p}(w^{-1}).\end{split}$

Following the proof of Theorem 5.9, i), we obtain

$\omega_{q}^{-1}(m(p^{-k},p^{-l}))\varphi_{q}^{(\mu_{q})})=\frac{g(D_{q})}{g_{p^{l}}(D_{q})}\varphi_{q}^{(p^{(l-k)/2}\mu_{q})}.$

Moreover, comparing (5.3) with (4.16), it becomes apparent that the identity

$T_{k,l}(m(p^{k},p^{l}))\varphi_{p}^{(\mu_{p})}=\omega_{p}^{-1}(m(p^{-k},p^{-l}))\varphi_{p}^{(\mu_{p})}$

holds. Replacing $\omega_{q}^{-1}(x_{p})$ and $T_{k,l}(x_{p})$ in (6.18) with
the expressions calculated before and piecing together the local Weil
representations, we arrive at

(6.24)
$\omega_{f}^{-1}(m(p^{-k},p^{-l}))\varphi^{(\lambda)}=\frac{g(D)}{g_{p^{l}}(D)}\varphi^{(p^{(l-k)/2}\lambda)}$

and

(6.25) $\mathcal{T}^{T_{k,l}}(F)(g_{\infty}\times
1_{f})=\sum_{x_{p}\in\mathcal{K}_{p}m(p^{k},p^{l})\mathcal{K}_{p}/\mathcal{K}_{p}}j(x_{p}^{-1}g_{\infty},i)^{-k}\sum_{\lambda\in
D}f_{\lambda}(x_{p}^{-1}g_{\infty}i)\omega_{f}^{-1}(x_{p}^{-1})\varphi^{(\lambda)}.$

On the other hand, it is well known that

$\\{x_{s,b}^{-1}\;|\;s=1,\dots
l-k-1,b\in(\mathbb{Z}/p^{s}\mathbb{Z})^{\times}\\}\cup\\{x_{b}^{-1}\;|\;b\in\mathbb{Z}/p^{l-k}\mathbb{Z}\\}\cup\\{m(p^{k},p^{l})^{-1}\\}$

is a set of representatives of $\Gamma/\Gamma m(p^{-k},p^{-l})\Gamma$. In view
of (6.20) and (6.21) we find

$\displaystyle\rho_{L}^{-1}(x_{s,b}^{-1})=\rho_{L}^{-1}(n\\_(-p^{l-k-s}t))\rho_{L}^{-1}(m(p^{-k},p^{-l}))\rho_{L}^{-1}(\left(\begin{smallmatrix}r&-b\\\
t&p^{s}\end{smallmatrix}\right)),$
$\displaystyle\rho_{L}^{-1}(x_{b}^{-1})=\rho_{L}^{-1}(w^{-1}n(-b))\rho_{L}^{-1}(m(p^{-k},p^{-l}))\rho_{L}^{-1}(w),$

where
$\rho_{L}^{-1}(m(p^{-k},p^{-l}))\mathfrak{e}_{\lambda}=\frac{g(D)}{g_{p^{l}}(D)}\mathfrak{e}_{p^{(l-k)/2}\lambda}$.

Thus, taking (6.24) and (4.9) into account, we find that the right-hand side
of (6.25) equals

$\displaystyle j(g_{\infty},i)^{-k}\sum_{x\in\Gamma/\Gamma
m(p^{-k},p^{-l})\Gamma}j(x,g_{\infty}i)^{-k}\sum_{\lambda\in
D}f_{\lambda}(x(g_{\infty}i))\rho_{L}^{-1}(x)\mathfrak{e}_{\lambda}$
$\displaystyle=F_{(p^{2l})^{1-k/2}T(m(p^{-k},p^{-l}))f}(g_{\infty}\times
1_{f}).$

For $k=l$ the set
$\mathcal{K}_{p}m(p^{k},p^{k})\mathcal{K}_{p}/\mathcal{K}_{p}$ consists only
of the element $m(p^{k},p^{k})$. The Hecke operator $T(m(p^{-k},p^{-k}))$ acts
just by multiplication with $\frac{g(D)}{g_{p^{k}}(D)}$. On the other hand,

$T_{k,k}(m(p^{k},p^{k}))=T_{k}(m(p^{k},p^{k}))=\frac{g(D)}{g_{p^{k}}(D)}\operatorname{id}_{S_{L_{p}}}$

and the arguments as before show that (6.18) equals
$\frac{g(D)}{g_{p^{k}}(D)}F_{f}$.

The proof for $p\nmid|D|$ starts again with (6.18). It was already noted above
that if $x_{p}$ runs through the set of right cosets (6.19), then $x_{p}^{-1}$
runs through a set representatives of $\Gamma/\Gamma m(p^{-k},p^{-l})\Gamma$.
By Theorem 5.12 we know

$\displaystyle
T_{k,l}(x_{p})(\omega_{p}^{-1}(1_{p})\varphi_{p}^{(\lambda_{p})})$
$\displaystyle=\omega_{p}^{-1}(1_{p})\varphi_{p}^{(\lambda_{p})}$
$\displaystyle=\varphi_{p}^{(\lambda_{p})}.$

Therefore,

$\bigotimes_{q\not=p}\omega_{q}^{-1}(x_{p}^{-1})\varphi_{q}^{(\lambda_{p})}\otimes
T_{k,l}(x_{p})(\omega_{p}^{-1}(1_{p})\varphi_{p}^{(\lambda_{p})})=\omega_{f}^{-1}(x_{p}^{-1})\varphi^{(\lambda)}.$

The same thoughts as before then lead again to the claimed result. The proof
for $k=l$ works the same way as in the corresponding case for $p\mid|D|$. ∎

######  Remark 6.10.

  1. i)

The identity (6.16) can be rephrased with the help of the isomorphism
$\mathscr{A}$. Let $F\in A_{\kappa}(\omega_{f})$ with the associated modular
form $f_{F}\in S_{\kappa}(\rho_{L})$ and $f_{\mathcal{T}^{T_{k,l}}(F)}$ the
modular form corresponding to $\mathcal{T}_{T_{k,l}}(F)$. Then (6.16) is
equivalent to

(6.26)
$f_{\mathcal{T}^{T_{k,l}}(F)}=p^{(k+l)(1-\kappa/2)}T(m(p^{-k},p^{-l}))(f_{F}).$

  2. ii)

It is also an immediate but important consequence of Theorem 6.9 and (3.11)
that $f\in S_{\kappa}(\rho_{L})$ is a common eigenform for all Hecke operators
$T(m(p^{k},p^{l}))$ if and only if the associated automorphic form $F_{f}$ is
a common eigenform for all operators $\mathcal{T}^{T_{k,l}}$ for all primes
$p$ and all
$T_{k,l}\in\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$
($\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ if $p$ and $|D|$
are coprime). Remark 6.7, ii) allows us to extend this statement to the whole
Hecke algebra $\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$
(and $\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$). Thus,

$\mathcal{T}^{T}(F_{f})=\lambda_{F_{f},p}(T)F_{f}$

for all $T\in\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ if
and only if $f\in S_{\kappa}(\rho_{L})$ is a common Eigenform for all Hecke
operators $T(m(p^{-k},p^{-l}))$. As in the classical scalar valued theory, we
may then conclude that the map

$\lambda_{F,p}:\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})\rightarrow\mathbb{C},\quad
T\mapsto\lambda_{F,p}(T)$

associated to an eigenform $F\in A_{\kappa}(\omega_{f})$ defines an algebra
homomorphism.

##  7\. Standard $L$-function of a common Hecke eigenform

This chapter is concerned with several issues regarding a standard
$L$-function of a vector valued Hecke eigenform. First, we will motivate and
define a standard $L$-function $L(s,F)$ attached to a vector valued
automorphic form $F$. We follow the approach of Bouganis and Marzec ([BM]),
Böcherer and Schulze-Pillot ([BoSP]) and Shimura ([Sh]) adapted to our
situation. Via the correspondence in Theorem 6.9, it is then possible to
associate the same $L$-function to the corresponding Hecke eigenform $f_{F}$.
Along the way we establish most of the results to prove a relation between the
standard zeta function $\mathcal{Z}(s,f)$ and $L(s,f)$. Afterwards, based on
this relation, we prove that the introduced standard $L$-function can be
continued meromorphically to the whole $s$-plane.

We assume in the whole chapter that $L_{p}^{\prime}/L_{p}$ is anisotrophic. We
adopt the notation of Section 5.

###  7.1. Standard $L$-function of a vector valued automorphic form

This section provides the necessary theory to define a standard $L$-function
for a vector valued automorphic form as defined before. This is essentially
the well known theory of spherical functions as it appeared in many places
(see e. g. [Ca], [McD] or [Sa]). Here we largely follow [Ar], Chapter 5, and
translate several statements therein to our setting. For primes $p$ which are
coprime to $|D|$ the proofs carry over almost verbatim. In the case of primes
$p$ dividing $|D|$ the Hecke algebra
$\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is more
complicated due to the Weil representation, causing serious difficulties. The
established results then provide the means to associate an $L$-function
$L(s,F)$ to a vector valued automorphic form $F$ along the lines of [BM],
Chapter 7.2.

######  Lemma 7.1.

Let $p$ be a prime.

  1. i)

If $p$ is coprime to $|D|$, any $\mathbb{C}$-algebra homomorphism
$\xi:\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})\rightarrow\mathbb{C}$
is of the form

(7.1) $T\mapsto\xi(T)=\widehat{\chi}_{S}(T),$

Here $\chi$ is some uniquely determined unramified character of
$\mathcal{M}_{p}$ and

(7.2) $\widehat{\chi}_{S}(T)=\sum_{(k,l)\in\mathbb{Z}^{2}}S(\langle
T,\varphi_{p}^{(0)}\rangle)(m(p^{k},p^{l}))\chi(m(p^{k},p^{l})),$

$S$ being the classical Satake map.

  2. ii)

If $p$ divides $|D|$, any $\mathbb{C}$-algebra homomorphism
$\xi:\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})\rightarrow\mathbb{C}$
is of the form

(7.3)
$\begin{split}T\mapsto\xi(T)=\widehat{\chi}_{\mathcal{S}}(T)&=\sum_{(k,l)\in\mathbb{Z}^{2}}\langle(I_{\chi_{D_{p}}}\circ\mathcal{S}(T))(m(p^{k},p^{l})),\varphi_{p}^{(0)}\rangle)\chi(m(p^{k},p^{l}))\\\
&=\sum_{(k,l)\in\mathbb{Z}^{2}}\langle\mathcal{S}(T)(m(p^{k},p^{l})),\varphi_{p}^{(0)}\rangle)\chi(m(p^{k},p^{l})),\end{split}$

where $\mathcal{S}$ is the Satake map (5.12), $I_{\chi_{D_{p}}}$ is the
isomorphism in (5.11) and $\chi$ is again an unramified character of
$\mathcal{M}_{p}$.

###### Proof.

First, recall that $\widehat{\chi}_{S}$ and $\widehat{\chi}_{\mathcal{S}}$ are
well defined since $S(\langle T,\varphi_{p}^{(0)}\rangle)$ and
$\langle\mathcal{S}(T),\varphi_{p}^{(0)}\rangle$ have finite support on
$\mathcal{M}_{p}$.

i) As already noted in Theorem 5.12, $f\mapsto\langle
f,\varphi_{p}^{(0)}\rangle$ is an isomorphism of the Hecke algebras
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p})$. From [Ca], Corollary 4.2, we
know that any algebra homomorphism of
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p})$ is given by

$g\mapsto\int_{t\in\mathcal{M}_{p}}S(g)(t)\chi(t)dt,$

where $\chi$ is an unramified character of $\mathcal{M}_{p}$. Since $S(g)$ is
bi-invariant under $\mathcal{D}_{p}$ and $\chi$ unramified, we obtain the
above stated term. (note that we adopted the normalisation of the Haar
measures in [Ca], see Section 2).

ii) The proof in i) essentially relies on the fact that any
$\mathbb{C}$-algebra homomorphism $\xi$ of the group algbra
$\mathbb{C}[\mathcal{M}_{p}/\mathcal{D}_{p}]$ can be written in terms of a
uniquely determined unramified character $\chi$ of $\mathcal{M}_{p}$ by

$\xi(T)=\sum_{m\in\mathcal{M}_{p}/\mathcal{D}_{p}}T(m)\chi(m).$

By Theorems 5.11 and 5.9 we know that

$T\mapsto\langle I_{\chi_{D_{p}}}\circ\mathcal{S}(T),\varphi_{p}^{(0)}\rangle$

maps $\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$
isomorphically to a subalgebra of
$\mathbb{C}[\mathcal{M}_{p}/\mathcal{D}_{p}]$ inducing the claimed form of
$\xi$. ∎

Note that each unramified character $\chi$ of $\mathcal{M}_{p}$ is of the form
$\chi(m(t_{1},t_{2}))=\chi_{1}(t_{1})\chi_{2}(t_{2})$, where $\chi_{i}$ is an
umramified character of $\mathbb{Q}_{p}^{\times}$, that is, $\chi_{i}$ is
trivial on $\mathbb{Z}_{p}$.

Now we introduce for an unramified character $\chi$ of $\mathcal{M}_{p}$ the
zonal spherical function on $\mathcal{Q}_{p}$ (cf. e. g. [Ca], p. 150): Let

(7.4) $\phi_{\chi}:\mathcal{Q}_{p}\rightarrow\mathbb{C},\quad
g=ntk\mapsto\phi_{\chi}(mnk)=(\chi\delta^{\frac{1}{2}})(m),$

where $n\in N(\mathbb{Z}_{p})$, $m\in\mathcal{M}_{p}$ and
$k\in\mathcal{K}_{p}$. Here
$\delta(m(t_{1},t_{2}))=\left|\frac{t_{1}}{t_{2}}\right|_{p}$ is the modulus
character, which we already introduced in Remark 5.10. Then the zonal
spherical function $\omega_{\chi}$ on $\mathcal{Q}_{p}$ is defined by

(7.5) $\omega_{\chi}:\mathcal{Q}_{p}\rightarrow\mathbb{C},\quad
g\mapsto\omega_{\chi}(g)=\int_{\mathcal{K}_{p}}\phi_{\chi}(kg)dk,$

where $dk$ is the Haar measure on $\mathcal{K}_{p}$ normalized by
$\int_{\mathcal{K}_{p}}dk=1$. It follows from its definition (7.5) and (7.4)
that $\omega_{\chi}$ is bi-invariant under $\mathcal{K}_{p}$. The next fact is
in principle well known for primes $p$ coprime to $|D|$.

######  Lemma 7.2.

Let $p$ be a prime,
$T\in\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ and

$\kappa_{p}=\frac{pg(D_{p})^{-1}+1}{p+1}.$

Then the following identities hold:

(7.6) $\int_{\mathcal{Q}_{p}}\langle
T(g),\varphi_{p}^{(0)}\rangle\omega_{\chi}(g)dg=\widehat{\chi}_{S}(T)$

if $p$ is coprime to $|D|$.

(7.7) $\frac{1}{\kappa_{p}}\int_{\mathcal{Q}_{p}}\langle
T(g)_{|_{S^{N(\mathbb{Z}_{p})}}},\varphi_{p}^{(0)}\rangle\omega_{\chi}(g)dg=\widehat{\chi}_{\mathcal{S}}(T)$

if $p$ is a divisor of $|D|$.

###### Proof.

If $(p,|D|)=1$, the algebra
$\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is isomorphic to
the classical algebra $\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p})$. In
particular, $T$ is bi-invariant with respect to $\mathcal{K}_{p}$. Therefore,
the result can be proved as in [McD], p. 46, or [Ca], p. 150. Following the
proof of either of the cited sources, we end up with

(7.8)
$\int_{m\in\mathcal{M}_{p}}\chi(m)\delta(m)^{1/2}\int_{N(\mathbb{Q}_{p})}\langle
T(mn),\varphi_{p}^{(0)}\rangle dn.$

Since $S(\langle T,\varphi_{p}^{(0)}\rangle)$ is bi-invariant under
$\mathcal{D}_{p}$ and $\chi$ is unramified, this is equal to

$\sum_{(k,l)\in\mathbb{Z}^{2}}\chi(m(p^{k},p^{l}))S(\langle
T,\varphi_{p}^{(0)}\rangle)(m(p^{k},p^{l})).$

For $p\mid|D|$ the computations are more involved since
$T\in\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ is not bi-
invariant under $\mathcal{K}_{p}$. However, it is still possible to remedy the
absence of the bi-invariance. To this end, we use two facts. First, due to the
decomposition (7.10) and equation (7.11) below, we infer that the Weil
representation $\omega_{p}(k)$ acts on $S_{L_{p}}^{\mathbb{Z}_{p}}$ by
multiplication with the quadratic character $\chi_{D_{p}}$ for
$k\in\mathcal{K}_{0}(p)$. Secondly, by Lemma 13.1 of [KL], we know that
$\\{T^{-j}w\;|\;j\in\mathbb{Z}/p\mathbb{Z}\\}\cup\\{1_{2}\\}$ provides a set
of coset representatives of $\mathcal{K}_{p}/\mathcal{K}_{0}(p)$. Thus, taking
this and the Iwasawa decomposition into account, it follows

(7.9) $\begin{split}&\int_{\mathcal{Q}_{p}}\langle
T(g)_{|_{S^{N(\mathbb{Z}_{p})}}},\varphi_{p}^{(0)}\rangle\omega_{\chi}(g)dg\\\
&=\int_{\mathcal{M}_{p}}\int_{N(\mathbb{Q}_{p})}\int_{\mathcal{K}_{0}(p)}\sum_{j\in\mathbb{Z}/p\mathbb{Z}}\langle
T(mn)\omega_{p}(T^{-j}w)\omega_{p}(k)\varphi_{p}^{(0)},\varphi_{p}^{(0)}\rangle\delta(m)^{1/2}\chi(m)dm\;dn\;dk.\\\
\end{split}$

For further evaluation of the integral over $\mathcal{K}_{0}(p)$, we remark
that any matrix $k=\left(\begin{smallmatrix}a&b\\\
pc&d\end{smallmatrix}\right)\in\mathcal{K}_{0}(p)$ can be written as product

(7.10) $k=m(\det(k),1)m(d^{-1})n(b)n\\_(pcd^{-1}),$

which allows to conclude that $\mathcal{K}_{0}(p)$ is equal to the direct
product

$M(\mathbb{Z}_{p}^{\times})D((\mathbb{Z}_{p}^{\times})^{2})N(\mathbb{Z}_{p})U(p\mathbb{Z}_{p}).$

We may therefore replace

$\int_{\mathcal{K}_{0}(p)}\text{ with
}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{D(\mathbb{Z}_{p}^{2})}\int_{N(\mathbb{Z}_{p})}\int_{U(p\mathbb{Z}_{p})}$

by Fubini’s theorem. For the sake of clarity we write
$\int_{\mathcal{K}^{\prime}}$ for the integrals
$\int_{D(\mathbb{Z}_{p}^{2})}\int_{N(\mathbb{Z}_{p})}\int_{U(p\mathbb{Z}_{p})}$.
Since the level of $L_{p}^{\prime}/L_{p}$ is $p$, we have in view of (4.8) and
Theorem 5.9, i)

(7.11) $\omega_{p}(m(d^{-1})m(ad-
bpc,1)n(b)n\\_(pcd^{-1}))\varphi_{p}^{(0)}=\chi_{D_{p}}(d)\varphi_{p}^{(0)}$

and

(7.12)
$T(m(p^{k}s_{1},p^{l}s_{2})n)\varphi_{p}^{(0)}=\chi_{D_{p}}(s_{2})T(m(p^{k},p^{l})n)\varphi_{p}^{(0)}.$

Thus, taking (7.11) and (7.12) into account, the right-hand side of (7.9)
equals

$\displaystyle\sum_{(k,l)\in\mathbb{Z}^{2}}\int_{\mathcal{D}_{p}}\int_{N(\mathbb{Q}_{p})}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{\mathcal{K}^{\prime}}\times$
$\displaystyle\sum_{j\in\mathbb{Z}/p\mathbb{Z}}\chi_{D_{p}}(s_{2})\chi_{D_{p}}(r)\langle
T(m(p^{k},p^{l})n)\omega_{p}(T^{-j}w)\varphi_{p}^{(0)},\varphi_{p}^{(0)}\rangle\delta(m(p^{k},p^{l}))^{1/2}\chi(m(p^{k},p^{l}))ds\>dn\;dr\;dk$
$\displaystyle=\sum_{(k,l)\in\mathbb{Z}^{2}}\int_{\mathcal{D}_{p}}\int_{N(\mathbb{Q}_{p})}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{\mathcal{K}^{\prime}}\times$
$\displaystyle\sum_{j\in\mathbb{Z}/p\mathbb{Z}}\langle
T(m(p^{k},p^{l})n)\omega_{p}(T^{-j}w)\varphi_{p}^{(0)},\varphi_{p}^{(0)}\rangle\delta(m(p^{k},p^{l}))^{1/2}\chi(m(p^{k},p^{l}))ds\>dn\;dr\;dk.$

The last equation results from the transformation

$(m(s_{1},s_{2}),n,m(r^{-1}),k)\mapsto(m(s_{1},s_{2})m(r^{-1}),n,m(r^{-1})m(s_{2}^{-1}),k).$

In light of the assumption $\mu(\mathcal{D}_{p})=1$, we may write for the last
expression above

$\displaystyle\mu(\mathcal{K}_{0}(p))\left\\{\sum_{(k,l)\in\mathbb{Z}^{2}}\delta(m(p^{k},p^{l}))^{1/2}\chi(m(p^{k},p^{l}))\times\right.$
$\displaystyle\int_{N(\mathbb{Q}_{p})}\langle
T(m(p^{k},p^{l})n)\sum_{j\in\mathbb{Z}/p\mathbb{Z}}\omega_{p}(T^{-j}w)\varphi_{p}^{(0)},\varphi_{p}^{(0)}\rangle
dn$
$\displaystyle\left.+\sum_{(k,l)\in\mathbb{Z}^{2}}\delta(m(p^{k},p^{l}))^{1/2}\chi(m(p^{k},p^{l})))\int_{N(\mathbb{Q}_{p})}\langle
T(m(p^{k},p^{l})n)\varphi_{p}^{(0)},\varphi_{p}^{(0)}\rangle dn\right\\},$

where $\mu(\mathcal{K}_{0}(p))=\frac{1}{p+1}$ (see [KL], (13.3)). The formulas
(4.7) of $\omega_{p}$ allow us to evaluate the sum over $j$ explicitly

(7.13)
$\begin{split}\sum_{j\in\mathbb{Z}/p\mathbb{Z}}\omega_{p}(T^{-j}w)\varphi_{p}^{(0)}&=g(D_{p})^{-1}\sum_{\gamma_{p}\in
L_{p}^{\prime}/L_{p}}\sum_{j\in\mathbb{Z}/p\mathbb{Z}}e(-jq(\gamma_{p}))\varphi_{p}^{(\gamma_{p})}\\\
&=pg(D_{p})^{-1}\varphi_{p}^{(0)},\end{split}$

which gives

$\displaystyle\int_{\mathcal{Q}_{p}}\langle
T(g)_{|_{S^{N(\mathbb{Z}_{p})}}},\varphi_{p}^{(0)}\rangle\phi_{\chi}(g)dg$
$\displaystyle=\kappa_{p}\sum_{(k,l)\in\mathbb{Z}^{2}}\delta(m(p^{k},p^{l}))^{1/2}\chi(m(p^{k},p^{l})))\int_{N(\mathbb{Q}_{p})}\langle
T(m(p^{k},p^{l})n)_{|_{S^{N(\mathbb{Z}_{p})}}},\varphi_{p}^{(0)}\rangle dn$
$\displaystyle=\kappa_{p}\sum_{(k,l)\in\mathbb{Z}^{2}}\langle\mathcal{S}(T)(m(p^{k},p^{l}),\varphi_{p}^{(0)}\rangle\chi(m(p^{k},p^{l}))$

as $T_{|_{S^{N(\mathbb{Z}_{p})}}}$ is right-invariant under
$N(\mathbb{Z}_{p})$. ∎

For $s\in\mathbb{C}$ we define $\nu_{s}:\mathcal{Q}_{p}\rightarrow\mathbb{C}$
by

(7.14) $\begin{split}&\nu_{s}(k_{1}gk_{2})=\nu_{s}(g)\text{ for all
}k_{1},k_{2}\in\mathcal{K}_{p}\text{ and all }g\in\mathcal{Q}_{p},\\\
&\nu_{s}(m(p^{k},p^{l}))=\begin{cases}p^{-(k+l)s},&(k,l)\in\Lambda_{+},\\\
0,&\text{ otherwise. }\end{cases}\end{split}$

To relate $L(s,F)$ and $\mathcal{Z}(s,f_{F})$, we calculate the integral

$\int_{\mathcal{Q}_{p}}\nu_{s}(g)\omega_{\chi}(g)dg$

in two different ways. The first one is an analogue of Lemma 5.2 in [Ar].

######  Lemma 7.3.

Let $p$ be a prime, $\chi$ be an unramified character of $\mathcal{M}_{p}$,

$T_{k,l}\in\begin{cases}\mathcal{H}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p}),&p\nmid|D|\\\
\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p}),&p\mid|D|,\end{cases}$
$C(L_{p})=\frac{g_{p^{l}}(D_{p})}{\kappa_{p}g(D_{p})}$

and

$B_{S}(\chi,X)=\sum_{(k,l)\in\Lambda_{+}}\widehat{\chi_{S}}(T_{k,l})X^{k+l}\text{
and
}B_{\mathcal{S}}(\chi,X)=\sum_{(k,l)\in\Lambda_{+}}\widehat{\chi_{\mathcal{S}}}(T_{k,l})X^{k+l}$

where $\kappa_{p}$ is specified in Lemma 7.2 and $\widehat{\chi_{S}}$,
$\widehat{\chi_{\mathcal{S}}}$ are defined in Lemma 7.1. Then

  1. i)

(7.15) $\int_{\mathcal{Q}_{p}}\nu_{s}(g)\omega_{\chi}(g)dg=B(\chi,p^{-s})$

if $p\nmid|D|$.

  2. ii)

(7.16)
$\int_{\mathcal{Q}_{p}}\nu_{s}(g)\omega_{\chi}(g)dg=C(L_{p})B(\chi,p^{-s})$

if $p\mid|D|$.

###### Proof.

For $p$ coprime to $|D|$ the proof is essentially the same as the one of Lemma
5.2 in [Ar]. One has just to replace the term $\nu_{s}$ with our corresponding
$\nu_{s}$ and $\varphi_{\alpha}$ with $T_{k,l}$.

Again, the proof for $p\mid|D|$ is more complicated. It uses the same ideas as
the ones in the proof of Lemma 7.2 and proceeds similar to the proof of [Ar],
Lemma 5.2. In light of the Cartan decomposition, we have

(7.17)
$\begin{split}\int_{\mathcal{Q}_{p}}\nu_{s}(g)\omega_{\chi}(g)dg&=\sum_{(k,l)\in\mathbb{Z}^{2}}\int_{\mathcal{K}_{p}}\int_{\mathcal{K}_{p}}\nu_{s}(m(p^{k},p^{l}))\omega_{\chi}(m(p^{k},p^{l}))dk_{1}\;dk_{2}\\\
&=\sum_{(k,l)\in\Lambda_{+}}p^{-s(k+l)}\omega_{\chi}(m(p^{k},p^{l}))\end{split}$

On the other hand, for
$T_{k,l}\in\mathcal{H}^{+}(\mathcal{Q}_{p}//\mathcal{K}_{p},\omega_{p})$ (with
$T_{k,k}=T_{k}$),

$\displaystyle\sum_{(k,l)\in\Lambda_{+}}\int_{\mathcal{K}_{p}}\int_{\mathcal{K}_{p}}\langle
T_{k,l}(k_{2}m(p^{k},p^{l})k_{1})_{|_{S^{N(\mathbb{Z}_{p})}}},\varphi_{p}^{(0)}\rangle\omega_{\chi}(m(p^{k},p^{l}))dk_{1}\;dk_{2}$
$\displaystyle=\sum_{(k,l)\in\Lambda_{+}}\sum_{u\in\mathbb{Z}/p\mathbb{Z}}\sum_{v\in\mathbb{Z}/p\mathbb{Z}}\int_{\mathcal{K}_{0}(p)}\int_{\mathcal{K}_{0}(p)}\times$
$\displaystyle\langle
T_{k,l}(m(p^{k},p^{l}))\omega_{p}(T^{-u}wk_{1}^{\prime})\varphi_{p}^{(0)},\omega_{p}(T^{-v}wk_{2}^{\prime})\varphi_{p}^{(0)}\rangle\omega_{\chi}(m(p^{k},p^{l}))dk_{1}^{\prime}\;dk_{2}^{\prime}+$
$\displaystyle\sum_{(k,l)\in\Lambda_{+}}\int_{\mathcal{K}_{0}(p)}\int_{\mathcal{K}_{0}(p)}\langle
T_{k,l}(m(p^{k},p^{l}))\omega_{p}(k_{1}^{\prime})\varphi_{p}^{(0)},\omega_{p}(k_{2}^{\prime})\varphi_{p}^{(0)}\rangle\omega_{\chi}(m(p^{k},p^{l}))dk_{1}^{\prime}\;dk_{2}^{\prime}.$

By means of the same notations and techniques as in the proof of Lemma 7.2, we
may replace the last expression with

(7.18)
$\begin{split}&\sum_{(k,l)\in\Lambda_{+}}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{\mathcal{K}^{\prime}}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{\mathcal{K}^{\prime}}\chi_{D_{p}}(r_{1})\chi_{D_{p}}(r_{2})\times\\\
&\langle
T_{k,l}(m(p^{k},p^{l}))\sum_{u\in\mathbb{Z}/p\mathbb{Z}}\omega_{p}(T^{-u}w)\varphi_{p}^{(0)},\sum_{v\in\mathbb{Z}/p\mathbb{Z}}\omega_{p}(T^{-v}w)\varphi_{p}^{(0)}\rangle\omega_{\chi}(m(p^{k},p^{l}))dr_{1}\;dk_{1}\;dr_{2}\;dk_{2}+\\\
&\sum_{(k,l)\in\Lambda_{+}}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{\mathcal{K}^{\prime}}\int_{M(\mathbb{Z}_{p}^{\times})}\int_{\mathcal{K}^{\prime}}\chi_{D_{p}}(r_{1})\chi_{D_{p}}(r_{2})\times\\\
&\langle
T_{k,l}(m(p^{k},p^{l}))\varphi_{p}^{(0)},\varphi_{p}^{(0)}\rangle\omega_{\chi}(m(p^{k},p^{l}))dr_{1}\;dk_{1}\;dr_{2}\;dk_{2}.\end{split}$

We can get rid of the characters $\chi_{D_{p}}$ by employing the
transformation