2.1.

We denote by $\mathbb {Z}$ , $\mathbb {Q}$ , $\mathbb {R,}$ and $\mathbb {C}$ the ring of rational integers and the fields of rational numbers, real numbers, and complex numbers, respectively. For a place v of $\mathbb {Q}$ , $\mathbb {Q}_v$ denotes the v-adic completion of $\mathbb {Q}$ . For a finite place p of $\mathbb {Q}$ , let $\mathbb {Z}_p$ be the ring of p-adic integers and $|x|_p$ the p-adic absolute value of $x\in \mathbb {Q}_p^{\times }$ . We denote by $\mathbb {A}$ the adele ring of $\mathbb {Q}$ .

For a ring R, denote by $R^{(m,n)}$ the set of $m\times n$ matrices whose components are in R. We also write $\mathrm {M}_m(R)$ for $R^{(m,m)}$ . Let X be an algebraic group X over $\mathbb {Q}$ . We denote by $X_v$ the group of $\mathbb {Q}_v$ -rational points of X for a place v of $\mathbb {Q}$ , and by $X_{\mathbb {A}}$ the adelization of X.

For $x\in \mathbb {C}$ , we put $\mathbf {e}(x)=\exp (2\pi \sqrt {-1}x)$ . Let $\psi $ be the additive character of $\mathbb {A}/\mathbb {Q}$ such that $\psi (x_{\infty })=\mathbf {e}(x_{\infty })$ for $x_{\infty }\in \mathbb {R}$ . For a place v of $\mathbb {Q}$ , the restriction of $\psi $ to $\mathbb {Q}_v$ is denoted by $\psi _v$ .

Denote by $X_{\mathrm {unr}}(\mathbb {Q}_p^{\times })$ the set of quasi-characters of $\mathbb {Q}_p^{\times }$ trivial on $\mathbb {Z}_p^{\times }$ . For a prime number p, we write $\mathbb {F}_p$ for the finite field $\mathbb {Z}/p\mathbb {Z}$ .

We put

$$\begin{align*}\delta(P)= \begin{cases} 1,&\text{if }P\text{ holds},\\ 0,&\text{if }P\text{ does not hold} \end{cases} \end{align*}$$

for a condition P and $\delta _{i,j}=\delta (i=j)$ for $i,j\in \mathbb {Z}$ .

2.2.

Throughout the article, we fix a definite quaternion algebra B over $\mathbb {Q}$ with discriminant $d_B$ . Let $x\mapsto \overline {x}$ be the main involution of B. For a subset X of B, put $X^-=\{\alpha \in X\mid \alpha +\overline {\alpha }=0\}$ .

We write $N_r$ and $\mathrm {tr}_r$ for the reduced norm and the reduced trace of $\mathrm {M}_r(B)$ to $\mathbb {Q}$ , respectively. We often suppress the subscript r if there is no fear of confusion.

We fix an isomorphism from $B\otimes _{\mathbb {Q}}\mathbb {R}$ to the Hamilton quaternion algebra

$$\begin{align*}\mathbb{H}=\mathbb{R}\cdot 1+\mathbb{R}\cdot i+\mathbb{R}\cdot j+\mathbb{R}\cdot k\qquad(i^2=j^2=k^2=-1,\,ij=-ji=k) \end{align*}$$

and identify $B\otimes _{\mathbb {Q}}\mathbb {R}$ with $\mathbb {H}$ . Throughout the article, we always distinguish the element i of $\mathbb {H}$ from the imaginary unit $\sqrt {-1}\in \mathbb {C}$ . We define an embedding $A\colon \mathbb {H}\to \mathrm {M}_2(\mathbb {C})$ by

(2) $$ \begin{align} A(a_0+a_1i+a_2j+a_3k) = \left( \begin{array}{cc} a_0+a_1\sqrt{-1}&a_2+a_3\sqrt{-1} \\ -a_2+a_3\sqrt{-1} & a_0- a_1\sqrt{-1} \end{array} \right) \end{align} $$

for $a_0,a_1,a_2,a_3\in \mathbb {R}$ . The map A extends to an isomorphism of $\mathbb {H}\otimes _{\mathbb {R}}\mathbb {C}$ to $\mathrm {M}_2(\mathbb {C})$ .

2.3.

The measures on $\mathbb {R}$ and $\mathbb {C}$ are taken to be the usual Lebesgue measures.

We define the measures $d_L\alpha $ on $\mathbb {H}$ and $d_L\beta $ on $\mathbb {H}^-$ by $d_L\alpha =da_0da_1da_2da_3\;\; (\alpha =a_0+a_1i+a_2j+a_3k, a_i\in \mathbb {R})$ and $d_L\beta =db_1db_2db_3\;\;(\beta =b_1i+b_2j+b_3k, b_i\in \mathbb {R})$ , respectively. The invariant measure $d^{\times } t$ on $\mathbb {H}^1=\{t\in \mathbb {H}\mid N(t)=1\}$ is normalized by $\mathrm {vol}(\mathbb {H}^1)=1$ .

2.4.

For a place v of $\mathbb {Q}$ , we define the measures $d\alpha _v$ on $B_v$ and $d\beta _v$ on $B_v^-$ so that they are self-dual with respect to the pairings $(\alpha _v,\alpha _v')\mapsto \psi _v(\mathrm {tr}(\overline {\alpha _v}\alpha _v'))$ and $(\beta _v,\beta _v')\mapsto \psi _v(\mathrm {tr}(\overline {\beta _v}\beta _v'))$ , respectively. Note that, for $v=\infty $ , $d\alpha _{\infty }=4d_L\alpha _{\infty }$ . We define the measure on $B_v^{(m,n)}$ by $dX=\prod _{1\le i\le m,\,1\le j\le n}dx_{ij}\;\;(X=(x_{ij}))$ , and the measures $d\alpha $ on $B_{\mathbb {A}}$ and $d\beta $ on $B^-_{\mathbb {A}}$ by the product measures $\prod _{v\le \infty }d\alpha _v$ and $\prod _{v\le \infty }d\beta _v$ , respectively.

2.5.

For a place v of $\mathbb {Q}$ , we denote by $\mathscr {S}(B_v^{(m,n)})$ the space of Schwartz–Bruhat functions on $B_v^{(m,n)}$ .

2.6.

For a positive integer $\kappa $ , let $V_{\kappa }$ be the space of homogeneous polynomials in the two variables X and Y of total degree $\kappa $ . Then $\{P_r\}_{r=0}^{\kappa }$ forms a basis of $V_{\kappa }$ and $\dim V_{\kappa }=\kappa +1$ , where

(3) $$ \begin{align} P_r(X,Y)=X^r\,Y^{\kappa-r}\in V_{\kappa}\qquad(0\le r\le \kappa). \end{align} $$

We define $\sigma _{\kappa }\colon \mathbb {H}\to \mathrm {End}(V_{\kappa })$ by

(4) $$ \begin{align} (\sigma_{\kappa} (\alpha)P)(X,Y)=P((X,Y)A(\alpha)) \qquad(\alpha\in\mathbb{H},P\in V_{\kappa}), \end{align} $$

where A is defined by (2). Then $\sigma _{\kappa }$ defines an irreducible representation of $\mathbb {H}$ on $V_{\kappa }$ . The inner product of $V_{\kappa }$ defined by

$$\begin{align*}( P,P')_{\kappa}=\int_{\mathbb{C}^2}P(x,y)\,\overline{P'(x,y)}e^{-2\pi (|x|^2+|y|^2)}dxdy \qquad(P,P'\in V_{\kappa}) \end{align*}$$

is $A(\mathbb {H}^1)$ -invariant. Put $\|P\|_{\kappa }=\sqrt {(P,P)_{\kappa }}$ for $P\in V_{\kappa }$ . The following formula is often used in this article.

Lemma 2.6.1. For $\alpha ,\beta \in \mathbb {H}$ , we have

$$\begin{align*}\int_{\mathbb{H}}\mathbf{e}(\mathrm{tr}(\alpha y)+\sqrt{-1}\overline{y}y)\,\sigma_{\kappa}(y+\beta)dy =\mathbf{e}(\sqrt{-1}\overline{\alpha}\alpha)\,\sigma_{\kappa}(\sqrt{-1}\overline{\alpha}+\beta). \end{align*}$$

3.1.

In this article, we consider two quaternion unitary groups $G_1$ and $G_2$ over $\mathbb {Q}$ defined by

$$ \begin{align*} G_{i}(\mathbb{Q})=\left\{g\in\mathrm{GL}_2(B)\mid {}^t\!\overline{g}Q_ig=Q_i\right\}, \end{align*} $$

where

$$\begin{align*}Q_1=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}, \quad Q_2= \begin{pmatrix} 0&1\\1&0 \end{pmatrix}. \end{align*}$$

The groups $G_1$ and $G_2$ are denoted by $O^*(4)$ and $\mathrm {Sp}(1,1),$ respectively, and $(G_1,G_2)$ forms a dual pair.

Remark 3.1.1. Let $H'$ and $Z'$ be algebraic groups over $\mathbb {Q}$ defined by

$$ \begin{align*} H'(\mathbb{Q})&=\{(h,\alpha)\in\mathrm{GL}_2(\mathbb{Q})\times B^{\times} \mid \det h\cdot N(\alpha)=1\},\\ Z'(\mathbb{Q})&=\{(a1_2,a^{-1})\mid a\in\mathbb{Q}^{\times}\}\,\subset H'(\mathbb{Q}). \end{align*} $$

A homomorphism $H'\ni (h,\alpha )\mapsto \alpha h\in G_1$ defines an isomorphism $H'/Z'\cong G_1$ .

3.2.

Let

$$\begin{align*}X_i= \begin{cases} \{z\in\mathbb{C}\mid \mathrm{Im}(z)>0\} & (i=1),\\ \{z\in \mathbb{H}\mid \mathrm{tr}(z)>0\} & (i=2). \end{cases} \end{align*}$$

The group $G_{i,\infty }$ acts transitively on $X_i$ by

$$\begin{align*}g\cdot z=(az+b)(cz+d)^{-1}\qquad\left(g=\begin{pmatrix} a&b\\c&d \end{pmatrix}\in G_{i,\infty},z\in X_i\right). \end{align*}$$

Let

$$\begin{align*}K_{i,\infty}=\{k\in G_{i,\infty}\mid k\cdot z^0_{i}=z^0_{i}\}, \end{align*}$$

where

$$\begin{align*}z^0_i=\begin{cases} \sqrt{-1} & (i=1),\\ 1 & (i=2). \end{cases} \end{align*}$$

Then $K_{i,\infty }$ is a maximal compact subgroup of $G_{i,\infty }$ and

$$ \begin{align*} K_{1,\infty}&= \left\{t \begin{pmatrix} \cos \theta & \sin \theta\\-\sin \theta & \cos \theta \end{pmatrix} \mid t\in \mathbb{H}^1,\,0\le \theta <2\pi \right\},\\ K_{2,\infty}&=\left\{ \begin{pmatrix} a&b\\b&a \end{pmatrix} \mid a,b\in\mathbb{H},\,N(a)+N(b)=1,\,\mathrm{tr}(\overline{a}b)=0 \right\}. \end{align*} $$

Note that

$$\begin{align*}\begin{pmatrix} a&b\\b&a \end{pmatrix} \mapsto (a+b,a-b) \end{align*}$$

gives an isomorphism $K_{2,\infty } \cong \mathbb {H}^1\times \mathbb {H}^1$ . We normalize the Haar measure on $K_{i,\infty }$ by $\mathrm {vol}(K_{i,\infty })=1$ .

3.3.

For a positive integer $\kappa $ , we define irreducible representations $\tau _{1,\kappa }$ (resp., $\tau _{2,\kappa }$ and $\tau ^-_{2,\kappa }$ ) of $K_{1,\infty }$ (resp. $K_{2,\infty }$ ) on $V_{\kappa }$ as follows:

$$ \begin{align*} \tau_{1,\kappa}(k)&=e^{-\sqrt{-1}\kappa\theta}\,\sigma_{\kappa}(t) \qquad\left(k=t \begin{pmatrix} \cos \theta&\sin \theta\\-\sin \theta&\cos \theta \end{pmatrix}\in K_{1,\infty},\,t\in\mathbb{H}^1, 0\le \theta<2\pi \right),\\ \tau_{2,\kappa}(k)&=\sigma_{\kappa}(a+ b),\;\; \tau_{2,\kappa}^-(k)=\sigma_{\kappa}(a- b)\qquad \left(k= \begin{pmatrix} a&b\\b&a \end{pmatrix}\in K_{2,\infty} \right). \end{align*} $$

Note that $\tau _{2,\kappa }$ is not equivalent to $\tau _{2,\kappa }^-$ .

3.4.

We henceforth fix a maximal order $\mathcal {O}$ of B. For a finite place p of $\mathbb {Q}$ , let $\mathcal {O}_p=\mathcal {O}\otimes _{\mathbb {Z}}\mathbb {Z}_p$ . For $p|d_B$ , denote by $\mathfrak {P}_p$ the maximal ideal of $\mathcal {O}_p$ and fix an element $\Pi _p$ of $B_p$ such that $\mathfrak {P}_p=\Pi _p\mathcal {O}_p$ . We define a lattice $L_{1,p}$ (resp. $L_{2,p}$ ) of $B_p^{(2,1)}$ (resp. $B_p^{(1,2)}$ ) by

$$ \begin{align*} L_{1,p}&=\left( \begin{array}{c} \mathcal{O}_p\\ \mathcal{O}_p \end{array} \right),\\ L_{2,p} &= \begin{cases} (\mathcal{O}_p,\mathcal{O}_p)& (p\nmid d_B),\\ (\mathcal{O}_p,\mathfrak{P}_p^{-1})&(p|d_B). \end{cases} \end{align*} $$

Then $L_{1,p}$ and $L_{2,p}$ are maximal $\mathcal {O}_p$ -lattices of $B_p^{(2,1)}$ and $B_p^{(1,2)}$ , respectively (see [Reference Shimura17, Section 3.2]).

Set

$$ \begin{align*} K_{1,p}&=\{k\in G_{1,p}\mid kL_{1,p}=L_{1,p}\},\\ K_{2,p}&=\{k\in G_{2,p}\mid L_{2,p}k=L_{2,p}\}. \end{align*} $$

Then $K_{i,p}$ is a maximal open compact subgroup of $G_{i,p}$ .

Set

$$\begin{align*}K_{i,f}=\prod_{p<\infty}K_{i,p}. \end{align*}$$

3.5.

We normalize the Haar measures on $G_{1,\infty }$ and $G_{2,\infty }$ by

$$\begin{align*}\int_{G_{1,\infty}}\varphi(g)dg=\int_{\mathbb{R}}db\,\int_{0}^{\infty}a^{-2}d^{\times} a \int_{K_{1,\infty}}dk\,\varphi\left( \begin{pmatrix} 1&b\\0&1 \end{pmatrix} \begin{pmatrix} a&0\\0&a^{-1} \end{pmatrix} k \right) \quad(\varphi\in L^1(G_{1,\infty})) \end{align*}$$

and

$$\begin{align*}\int_{G_{2,\infty}}\varphi(g)dg=\int_{\mathbb{H}^-}d_L\beta\,\int_{0}^{\infty}a^{-6}d^{\times} a \int_{K_{2,\infty}}dk\,\varphi\left( \begin{pmatrix} 1&\beta \\0&1 \end{pmatrix} \begin{pmatrix} a&0\\0&a^{-1} \end{pmatrix} k \right) \quad(\varphi\in L^1(G_{2,\infty})), \end{align*}$$

where note that we have normalized the measures dk at the infinite place so that $\mathrm {vol}(K_{i,\infty })=1$ . In addition, the Haar measure on $G_{i,p}$ is normalized by $\mathrm {vol}(K_{i,p})=1$ .

4.1.

Let $\kappa $ be an integer greater than $4$ . We define an $\mathrm {End}(V_{\kappa })$ -valued spherical function $\omega _{i,\kappa }$ on $G_{i,\infty }$ by

$$\begin{align*}\omega_{i,\kappa}(g)= \begin{cases} \sigma_{\kappa}(\Delta_1(g))^{-1}&(i=1),\\ N(\Delta_2(g))^{-1}\sigma_{\kappa}(\Delta_2(g))^{-1} & (i=2) \end{cases} \end{align*}$$

for $g\in G_{i,\infty }$ , where

$$ \begin{align*} \Delta_1(g)&= \dfrac{1}{2}\,(-\sqrt{-1},1)g \begin{pmatrix} \sqrt{-1}\\1 \end{pmatrix}\in (\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C})^{\times} \qquad(g\in G_{1,\infty}),\\ \Delta_2(g)&= \dfrac{1}{2}\,(1,1)g \begin{pmatrix} 1\\1 \end{pmatrix}\in\mathbb{H}^{\times} \qquad(g\in G_{2,\infty}). \end{align*} $$

It is easily checked that

(5) $$ \begin{align} \omega_{i,\kappa}(kgk')=\tau_{i,\kappa}(k')^{-1} \omega_{i,\kappa}(g) \tau_{i,\kappa}(k)^{-1} \qquad(g\in G_{i,\infty}, k,k'\in K_{i,\infty}). \end{align} $$

Note that $\omega _{i,\kappa }$ is integrable on $G_{i,\infty }$ .

4.2.

Let $\mathcal {S}_{i,\kappa }$ be the space of smooth functions $f\colon G_i(\mathbb {Q})\backslash G_i(\mathbb {A})\to V_{\kappa }$ satisfying the following conditions:

1. (1) For $g\in G_i(\mathbb {A}),k_f\in K_{i,f}$ , and $k_{\infty }\in K_{i,\infty }$ , we have

   (6) $$ \begin{align} f(gk_fk_{\infty})=\tau_{i,\kappa}(k_{\infty})^{-1} f(g). \end{align} $$

2. (2) $\|f(g)\|_{\kappa }$ is bounded on $G_i(\mathbb {A})$ .

3. (3) For $g\in G_i(\mathbb {A})$ , we have

   (7) $$ \begin{align} f(g)=c_{i,\kappa}\,\int_{G_{i,\infty}}\, \omega_{i,\kappa}(x_{\infty})f(gx_{\infty}^{-1}) dx_{\infty}, \end{align} $$

where

(8) $$ \begin{align} c_{i,\kappa}= \begin{cases} \dfrac{\kappa-1}{2\pi}&(i=1),\\ & \\ \dfrac{\kappa(\kappa-1)}{8\pi^2}&(i=2). \end{cases} \end{align} $$

Note that $f\in \mathcal {S}_{i,\kappa }$ is cuspidal; namely,

$$\begin{align*}f_0(g_1)=\int_{\mathbb{Q}\backslash \mathbb{A}}f\left( \begin{pmatrix} 1&b\\0&1 \end{pmatrix}g_1 \right)db=0 \qquad(g_1\in G_1(\mathbb{A})) \end{align*}$$

if $i=1$ (this follows from the boundedness and the Fourier expansion of f) and

$$\begin{align*}f_0(g_2)=\int_{B^-\backslash B^-_{\mathbb{A}}}f\left( \begin{pmatrix} 1&\beta\\0&1 \end{pmatrix}g_2 \right)d\beta=0 \qquad(g_2\in G_2(\mathbb{A})) \end{align*}$$

if $i=2$ (see [Reference Arakawa2, Proposition 3.1]).

We define an inner product $\langle ,\rangle _{i}$ on $\mathcal {S}_{i,\kappa }$ by

$$\begin{align*}\langle f,f'\rangle_{i}= \int_{G_i(\mathbb{Q})\backslash G_i(\mathbb{A})}( f(g_i),f'(g_i))_{\kappa}dg_i \qquad(f,f'\in\mathcal{S}_{i,\kappa}), \end{align*}$$

where $dg_i$ is the right-invariant measure on $G_i(\mathbb {Q})\backslash G_i(\mathbb {A})$ induced by the product measure $\prod _v dg_v$ on $G_i(\mathbb {A})$ .

4.3.

For a finite place p of $\mathbb {Q}$ , let $\mathcal {H}(G_{i,p},K_{i,p})$ be the Hecke algebra: By definition, $\mathcal {H}(G_{i,p},K_{i,p})$ is the space of compactly supported bi- $K_{i,p}$ -invariant functions on $G_{i,p}$ and the product is defined by

$$\begin{align*}\left(\varphi_p*\varphi^{\prime}_p\right) (g_p)=\int_{G_{i,p}}\varphi_p(g_px_p^{-1})\varphi^{\prime}_p(x_p)dx_p \qquad(\varphi_p,\varphi^{\prime}_p\in \mathcal{H}(G_{i,p},K_{i,p}), g_p\in G_{i,p}). \end{align*}$$

The Hecke algebra $\mathcal {H}(G_{i,p},K_{i,p})$ acts on $\mathcal {S}_{i,\kappa }$ by

$$\begin{align*}\left(f*\varphi_p\right)(g)=\int_{G_{i,p}}f(gx_p^{-1})\varphi_p(x_p)dx_p \qquad(f\in\mathcal{S}_{i,\kappa},\varphi_p\in\mathcal{H}(G_{i,p},K_{i,p}),g\in G_{i}(\mathbb{A})). \end{align*}$$

We say that $f\in \mathcal {S}_{i,\kappa }$ is a Hecke eigenform if

(9) $$ \begin{align} f*\varphi_p=\lambda_{i,p}^f(\varphi_p)\,f \end{align} $$

holds for any finite place p of $\mathbb {Q}$ and any $\varphi _p\in \mathcal {H}(G_{i,p},K_{i,p})$ with a $\mathbb {C}$ -algebra homomorphism $\lambda _{i,p}^f$ from $\mathcal {H}(G_{i,p},K_{i,p})$ to $\mathbb {C}$ . The following result is a straightforward consequence of [Reference Satake16, Section 5.2].

4.4.

In this and the next sections, we assume that $f\in \mathcal {S}_{i,\kappa }$ is a Hecke eigenform and define the local factor $L_p(s,f,\mathrm {std})$ of the standard L-function attached to f at a finite place p of $\mathbb {Q}$ .

In this section, we suppose that B is unramified at p (namely, $p \nmid d_B$ ). We identify $B_p$ with $\mathrm {M}_2(\mathbb {Q}_p)$ by an isomorphism $B_p\simeq \mathrm {M}_2(\mathbb {Q}_p)$ such that $\mathcal {O}_{p}$ maps to $\mathrm {M}_2(\mathbb {Z}_p)$ and such that the main involution of $B_p$ corresponds to

$$\begin{align*}X=\left( \begin{array}{cc} a&b \\ c&d \end{array} \right)\mapsto J^{-1} {}^t\! XJ=\left( \begin{array}{cc} d&-b \\ -c&a \end{array} \right),\qquad J=\left( \begin{array}{cc} 0&1 \\ -1&0 \end{array} \right). \end{align*}$$

Then $\mathrm {tr}(X)$ and $N(X)$ are the trace and the determinant of $X\in \mathrm {M}_2(\mathbb {Q}_p)$ , respectively. We identify $G_{i,p}$ and $K_{i,p}$ with

$$\begin{align*}\left\{g\in \mathrm{GL}_4(\mathbb{Q}_p)\mid {}^t\! g\mathcal{Q}_ig =\mathcal{Q}_i\right\}, \end{align*}$$

and

$$\begin{align*}\left\{k\in \mathrm{GL}_4(\mathbb{Z}_p)\mid {}^t\! k\mathcal{Q}_ik =\mathcal{Q}_i\right\}, \end{align*}$$

respectively. Here,

$$\begin{align*}\mathcal{Q}_1= \begin{pmatrix} 0&J\\-J&0 \end{pmatrix},\quad \mathcal{Q}_2= \begin{pmatrix} 0&1_2\\-1_2&0 \end{pmatrix}. \end{align*}$$

For $\chi =(\chi _1,\chi _2)\in X_{\mathrm {unr}}(\mathbb {Q}_p^{\times })^2$ , we define the spherical function $\omega _{i,\chi }$ on $G_{i,p}$ by

$$\begin{align*}\omega_{i,\chi}(g)=\int_{K_{i,p}}\eta_{i,\chi}(kg)dk. \end{align*}$$

Here,

$$ \begin{align*} \eta_{1,\chi}(g)&= \chi_1(a_1)\chi_2(a_2)|a_1|_p\\ & \qquad\left(g= \begin{pmatrix} a_1 & * & * & *\\ 0 & a_2 & * & *\\ 0 & 0 & a_2^{-1} & *\\ 0 & 0 & 0 & a_1^{-1} \end{pmatrix}\,k\in G_{1,p},\ a_1,a_2\in \mathbb{Q}_p^{\times}, k\in K_{1,p} \right) \end{align*} $$

and

$$ \begin{align*} \eta_{2,\chi}(g)&= \chi_1(a_1)\chi_2(a_2)|a_1|_p^2|a_2|_p\\ & \qquad\left(g= \begin{pmatrix} a_1 & * & * & *\\ 0 & a_2 & * & *\\ 0 & 0 & a_2^{-1} & 0\\ 0 & 0 & * & a_1^{-1} \end{pmatrix}\,k\in G_{2,p},\ a_1,a_2\in \mathbb{Q}_p^{\times}, k\in K_{2,p} \right) \end{align*} $$

(for the definition of $X_{\mathrm {unr}}(\mathbb {Q}_p^{\times })$ , see Section 2.1).

We define the local factor of the standard L-function attached to f at p by

$$ \begin{align*} L_p(s, f,\mathrm{std})= \left\{ \begin{aligned} 1\quad\quad & (i=1)\\ \zeta_p(s)\,\quad & (i=2) \end{aligned} \right\} \cdot \prod_{j=1}^2\,L_p(s,\chi_j)L_p(s,\chi_j^{-1}), \end{align*} $$

where $\zeta _p(s)=(1-p^{-s})^{-1}$ and $L_p(s,\chi _j^{\pm 1})=(1-\chi _j^{\pm 1}(p)p^{-s})^{-1}$ . Note that the definition of $L_p(s,f,\mathrm {std})$ does not depend on the choice of $\chi $ .

4.5.

In this section, we suppose that B is ramified at p (namely, $p | d_B$ ).

Let $X_{\mathrm {unr}}(B_p^{\times })$ be the set of homomorphisms from $B_p^{\times }$ to $\mathbb {C}^{\times }$ trivial on $\mathcal {O}_p^{\times }$ . For $\chi \in X_{\mathrm {unr}}(B_p^{\times })$ , we define the spherical function $\omega _{i,\chi }$ on $G_{i,p}$ by

$$\begin{align*}\omega_{i,\chi}(g)=\int_{K_{i,p}}\,\eta_{i,\chi}(kg)dk. \end{align*}$$

Here,

$$ \begin{align*} \eta_{1,\chi}\left(\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right)\left( \begin{array}{cc} \alpha&0 \\ 0&\overline{\alpha}^{-1} \end{array} \right)k\right) &=\chi(\alpha)|N(\alpha)|_p^{1/2}\qquad (b\in \mathbb{Q}_p,\alpha\in B_p^{\times}, k\in K_{1,p}) \end{align*} $$

and

$$ \begin{align*} \eta_{2,\chi}\left(\left( \begin{array}{cc} 1&\beta \\ 0&1 \end{array} \right)\left( \begin{array}{cc} \alpha&0 \\ 0&\overline{\alpha}^{-1} \end{array} \right)k\right) &=\chi(\alpha)|N(\alpha)|_p^{3/2}\qquad (\beta\in B^-_p,\alpha\in B_p^{\times}, k\in K_{2,p}). \end{align*} $$

We define the local factor of the standard L-function attached to f at p by

$$ \begin{align*} L_p(s, f,\mathrm{std})= \left\{ \begin{aligned} 1\quad\quad & (i=1)\\ \zeta_p(s)\,\quad & (i=2) \end{aligned} \right\} \cdot \left(1-\chi(\Pi_p)p^{-s-1/2}\right)^{-1} \left(1-\chi^{-1}(\Pi_p)p^{-s-1/2}\right)^{-1}. \end{align*} $$

4.6.

We define the standard L-function attached to a Hecke eigenform $f\in \mathcal {S}_{i,\kappa }$ by

(10) $$ \begin{align} L(s,f,\mathrm{std}) =\prod_{p<\infty}\,L_p(s,f,\mathrm{std}). \end{align} $$

5.1.

Let v be a place of $\mathbb {Q}$ . We define the Weil representation $r_v$ of $G_{1,v}\times G_{2,v}$ on the space

$$\begin{align*}W_v= \begin{cases} \mathscr{S}(B_v^{(2,1)}) & (v<\infty),\\ \mathscr{S}(B_v^{(2,1)})\otimes \mathrm{End}(V_{\kappa}) & (v=\infty) \end{cases} \end{align*}$$

as follows: For $\varphi \in W_v$ and $X\in B_v^{(2,1)}$ ,

$$ \begin{align*} r_v(g,1_2)\varphi(X)&=\varphi({}^t\!\overline{g}X)\qquad (g\in G_{1,v}),\\ r_v\left(1_2,\left( \begin{array}{cc} \alpha&0 \\ 0&\overline{\alpha}^{-1} \end{array} \right)\right)\varphi(X) & = |N(\alpha)|_v^2\,\varphi(X\alpha)\qquad(\alpha\in B_v^{\times}),\\ r_v\left(1_2,\left( \begin{array}{cc} 1&\beta \\ 0&1 \end{array} \right)\right)\varphi(X) & = \psi_v\left(-\dfrac{1}{2}\mathrm{tr}(\beta{}^t\!\overline{X}JX)\right)\varphi(X)\qquad(\beta\in B_v^-),\\ r_v\left(1_2,\left( \begin{array}{cc} 0&1 \\ 1&0 \end{array} \right)\right)\varphi(X) & = \int_{B_v^{(2,1)}}\psi_v\left(-\mathrm{tr}({}^t\!\overline{Y}JX)\right)\varphi(Y)dY. \end{align*} $$

To give another realization of the Weil representation, we define an intertwining operator $I_v$ from $W_v$ to

$$\begin{align*}W_v'= \begin{cases} \mathscr{S}(B_v^{(1,2)}) & (v<\infty),\\ \mathscr{S}(B_v^{(1,2)})\otimes \mathrm{End}(V_{\kappa}) & (v=\infty) \end{cases} \end{align*}$$

by a partial Fourier transform

$$\begin{align*}I_v\varphi(x_1,x_2)=\int_{B_v}\psi_v\left(\mathrm{tr}(\overline{y}x_2)\right) \varphi\left( \begin{array}{c} x_1\\y \end{array} \right)dy \qquad (\varphi\in W_v, x_1,x_2\in B_v). \end{align*}$$

The inverse of $I_v$ is given by

$$\begin{align*}I_v^{-1}\phi\left( \begin{array}{c} x_1\\x_2 \end{array} \right)=\int_{B_v}\psi_v\left(-\mathrm{tr}(\overline{y}x_2)\right) \phi(x_1,y)dy \qquad (\phi\in W_v', x_1,x_2\in B_v). \end{align*}$$

We set

$$\begin{align*}r_v'(g_1,g_2)=I_v\circ r_v(g_1,g_2) \circ I_v^{-1}\qquad(g_i\in G_{i,v}), \end{align*}$$

which defines a smooth representation of $G_{1,v}\times G_{2,v}$ on $W_v'$ . A direct calculation shows the following.

Lemma 5.1.1. For $\phi \in W_v'$ and $X'\in B_v^{(1,2)}$ , we have

$$ \begin{align*} r_v'\left(\left( \begin{array}{cc} \alpha&0 \\ 0&\overline{\alpha}^{-1} \end{array} \right),1_2\right)\phi(X') & = |N(\alpha)|_v^2\,\phi(\overline{\alpha}X')\qquad(\alpha\in B_v^{\times}),\\ r_v'\left(\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right),1_2\right)\phi(X') & = \psi_v\left(-b\,\overline{X'}Q{}^t\! X'\right)\phi(X')\qquad(b\in \mathbb{Q}_v),\\ r_v'\left(\left( \begin{array}{cc} 0&1 \\ -1&0 \end{array} \right),1_2\right)\phi(X') & = \int_{B_v^{(1,2)}}\psi_v\left(-\mathrm{tr}(\overline{X'}Q{}^t\! Y)\right)\phi(Y)dY,\\ r_v'(1_2,g)\phi(X')&=\phi(X'g)\qquad (g\in G_{2,v}). \end{align*} $$

5.2.

Let $\mathscr {S}(B_{\mathbb {A}}^{(2,1)})$ and $\mathscr {S}(B_{\mathbb {A}}^{(1,2)})$ be the restricted tensor products of $\mathscr {S}(B_{v}^{(2,1)})$ and $\mathscr {S}(B_{v}^{(1,2)})$ over the places v of $\mathbb {Q}$ with respect to $\{\varphi _{0,p}\}_{p<\infty }$ and $\{\phi _{0,p}\}_{p<\infty }$ , respectively. Here, $\varphi _{0,p}$ and $\phi _{0,p}$ denote the characteristic functions of $L_{1,p}$ and $L_{2,p}$ , respectively (for the definitions of $L_{1,p}$ and $L_2,p$ , see Section 3.4). We define smooth representations r and $r'$ of $G_1(\mathbb {A})\times G_2(\mathbb {A})$ on $W=\mathscr {S}(B_{\mathbb {A}}^{(2,1)})\otimes \mathrm {End}(V_{\kappa })$ and $W'=\mathscr {S}(B_{\mathbb {A}}^{(1,2)})\otimes \mathrm {End}(V_{\kappa })$ as restricted tensor products of $r_v$ and $r_v'$ , respectively:

$$\begin{align*}r=\otimes_v\,r_v,\, r'=\otimes_v\,r^{\prime}_v. \end{align*}$$

Then $r'(g_1,g_2)=I\circ r(g_1,g_2)\circ I^{-1}\;\;(g_i\in G_i(\mathbb {A}))$ , where

$$\begin{align*}I\varphi(x_1,x_2)=\int_{B_{\mathbb{A}}}\psi\left(\mathrm{tr}(\overline{y}x_2)\right) \varphi\left( \begin{array}{c} x_1\\y \end{array} \right)dy \qquad (\varphi\in W, x_1,x_2\in B_{\mathbb{A}}). \end{align*}$$

We define test functions $\varphi _0=\otimes _v\varphi _{0,v}\in W$ and $\phi _0=\otimes _v\phi _{0,v}\in W'$ , where

$$ \begin{align*} \varphi_{0,\infty}(X)&=\mathbf{e}(\sqrt{-1}\,{}^t\!\overline{X}X)\,\sigma_{\kappa}((1,-\sqrt{-1})\overline{X})\qquad (X\in\mathbb{H}^{(2,1)}),\\ \phi_{0,\infty}(X')&=\mathbf{e}(\sqrt{-1}\,\overline{X'}\,\,{}^t\!\! X')\,\sigma_{\kappa}((1,1){}^t\!\overline{X'})\qquad (X\in\mathbb{H}^{(1,2)}). \end{align*} $$

The following results are proved in a straightforward manner.

Lemma 5.2.1. For a place v of $\mathbb {Q}$ , we have

$$\begin{align*}\phi_{0,v}=I_v\varphi_{0,v}. \end{align*}$$

5.3.

We define the theta kernel $\theta \colon G_1(\mathbb {A})\times G_2(\mathbb {A})\to \mathrm {End}(V_{\kappa })$ by

$$\begin{align*}\theta(g_1,g_2)=\sum_{X\in B^{(2,1)}}\,\left(r(g_1,g_2)\varphi_0\right)(X) \qquad(g_i\in G_i(\mathbb{A})). \end{align*}$$

By the Poisson summation formula, we have

$$\begin{align*}\theta(g_1,g_2)=\sum_{X'\in B^{(1,2)}}\,\left(r'(g_1,g_2)\phi_0\right) (X'). \end{align*}$$

Lemma 5.3.1. We have

$$\begin{align*}\theta\left(\gamma_1 g_1 k_{1,f} k_{1,\infty},\gamma_2 g_2 k_{2,f}k_{2,\infty}\right) =\tau_{2,\kappa}(k_{2,\infty})^{-1}\circ \theta(g_1,g_2)\circ \tau_{1,\kappa}(k_{1,\infty}) \end{align*}$$

for $\gamma _i\in G_i(\mathbb {Q}),g_i\in G_i(\mathbb {A}), k_{i,f}\in K_{i,f}$ , and $ k_{i,\infty }\in K_{i,\infty } $ .

5.4.

We define the theta lift $\mathcal {L}(f)$ of $f\in \mathcal {S}_{1,\kappa }$ by

(11) $$ \begin{align} \mathcal{L}(f)(g_2)=\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\theta(g_1,g_2)f(g_1)dg_1 \qquad (g_2\in G_2(\mathbb{A})). \end{align} $$

Here, the Haar measure $dg_1$ on $G_1(\mathbb {A})$ is normalized by $\mathrm {vol}(G_1(\mathbb {Q})\backslash G_1(\mathbb {A}))=1$ . The integral (11) is absolutely convergent.

The theta lifting $\mathcal {L}\colon \mathcal {S}_{1,\kappa }\,\to \,\mathcal {S}_{2,\kappa }$ was first introduced essentially by Arakawa in his unpublished notes. The second named author of this article, based on Arakawa’s notes, developed Arakawa’s work to the lifting from elliptic modular forms to automorphic forms on $\mathrm {Sp}(1,q)$ generating the quaternionic discrete series representations at the archimedean place [Reference Narita11].

Specifying some arithmetic group defined by a global maximal lattice, Arakawa and the second named author proved Theorem 5.4.1 by showing that the lifts of Poincaré series on $G_1$ satisfy (7) under the assumption $\kappa> 6$ . In Section 7, we give another proof of Theorem 5.4.1 by using an archimedean analog of Eichler commutation relations under the assumption $\kappa>8$ .

We call $\mathcal {L}(f)$ the Arakawa lift of f. The arithmetic properties of the Arakawa lifting $\mathcal {L}$ (in particular the Fourier expansion of $\mathcal {L}(f)$ ) have been studied in [Reference Murase and Narita5], [Reference Murase and Narita6], and [Reference Murase and Narita7]. In [Reference Narita12] and [Reference Narita13], the second named author treated the case where the involved representation at the archimedean place belongs to a more general discrete series. He also studied a relation between the theta lifting for $(GO^*(4),\mathrm {GSp}(1,1))$ (or $(GO^*(4),\mathrm {GSp}^*(2))$ ) and that for $(GO(2,2),\mathrm {GSp}(2))$ in terms of the Jacquet–Langlands–Shimizu correspondence.

6.1.

Define $\varphi ^*_0\in W$ and $\phi ^*_{0}\in W'$ by

$$ \begin{align*} \varphi^*_0 &=\varphi^*_{0,\infty}\otimes\bigotimes_{p<\infty}\varphi_{0,p}\\ \phi^*_0 &=\phi^*_{0,\infty}\otimes\bigotimes_{p<\infty}\phi_{0,p}, \end{align*} $$

where

$$ \begin{align*} \varphi^*_{0,\infty}(X) &=\mathbf{e}\left(\sqrt{-1}{}^t\!\overline{X}X\right) \sigma_{\kappa}((1,\sqrt{-1})X)\qquad(X\in\mathbb{H}^{(2,1)}),\\ \phi^*_{0,\infty}(X') &=\mathbf{e}\left(\sqrt{-1}X'\,{}^t\!\overline{X'}\right) \sigma_{\kappa}((1,-1){}^t\! X')\qquad(X'\in\mathbb{H}^{(1,2)}). \end{align*} $$

The proofs of the following results are straightforward and omitted.

Lemma 6.1.2. We have

$$\begin{align*}(r(g_1,g_2)\varphi_0(X)v,v')_{\kappa} =(v,r(g_1,g_2^-)\varphi^*_0(X)v')_{\kappa} \qquad(g_i\in G_i(\mathbb{A}), X\in B_{\mathbb{A}}^{(2,1)},v,v'\in V_{\kappa}), \end{align*}$$

where

$$\begin{align*}g_2^-=\left( \begin{array}{cc} a&-b \\ -c&d \end{array} \right)\qquad \text{ for }\ g_2=\left( \begin{array}{cc} a&b \\ c&d \end{array} \right)\in G_2(\mathbb{A}). \end{align*}$$

6.2.

Set

$$\begin{align*}\theta^*(g_1,g_2) =\sum_{X\in B^{(2,1)}}\,\left(r(g_1,g_2^-)\varphi^*_0\right)(X)\in \mathrm{End}(V_{\kappa})\qquad (g_i\in G_i(\mathbb{A})). \end{align*}$$

By the Poisson summation formula, we have

$$\begin{align*}\theta^*(g_1,g_2) =\sum_{X'\in B^{(1,2)}}\,\left(r'(g_1,g_2^-)\phi^*_0\right)(X'). \end{align*}$$

The following facts are direct consequences of Lemmas 6.1.1(3) and 6.1.2.

For $F\in \mathcal {S}_{2,\kappa }$ , we define

(12) $$ \begin{align} \mathcal{L}^*(F)(g_1)=\int_{G_2(\mathbb{Q})\backslash G_2(\mathbb{A})} \theta^*(g_1,g_2)\,F(g_2)dg_2 \qquad(g_1\in G_1(\mathbb{A})). \end{align} $$

Here, the Haar measure $dg_2$ on $G_2(\mathbb {A})$ is normalized by $\mathrm {vol}(G_2(\mathbb {Q})\backslash G_2(\mathbb {A}))=1$ .

The proof of (1) is given in the next section. The second assertion follows from Lemma 6.2.1. The third assertion is easily verified.

In a forthcoming paper, we will investigate $\mathcal {L}(\mathcal {L}^*(F))$ by using (13).

7.1.

The aim of this section is to prove Theorems 5.4.1 and 6.2.2(1) by using the following archimedean analog of Eichler commutation relations.

Theorem 7.1.1. Suppose that $\kappa>8$ . For $(g_1,g_2)\in G_1(\mathbb {A})\times G_2(\mathbb {A})$ , we have

(14) $$ \begin{align} c_{1,\kappa}\,\int_{G_{1,\infty}}\,\theta(g_1x,g_2)\omega_{1,\kappa}(x)dx = c_{2,\kappa}\,\int_{G_{2,\infty}}\,\omega_{2,\kappa}(y)\theta(g_1,g_2y^{-1})dy \end{align} $$

and

(15) $$ \begin{align} c_{1,\kappa}\,\int_{G_{1,\infty}}\,\omega_{1,\kappa}(x)\theta^*(g_1x^{-1},g_2)dx = c_{2,\kappa}\,\int_{G_{2,\infty}}\,\theta^*(g_1,g_2y)\omega_{2,\kappa}(y)dy, \end{align} $$

where the Haar measures $dx$ and $dy$ on $G_{1,\infty }$ and $G_{2,\infty }$ , respectively, are given in Section 3.5.

7.2.

We first prove Theorem 5.4.1 assuming (14). We can show Theorem 6.2.2(1) assuming (15) in a similar manner and omit its proof. Throughout this section, we fix an integer $\kappa>4$ . To simplify the notation, we write $\omega _1$ and $\omega _2$ for $\omega _{1,\kappa }$ and $\omega _{2,\kappa }$ , respectively.

Let $f\in S_{1,\kappa }$ and put $F=\mathcal {L}(f)$ . The condition (6) for $F=\mathcal {L}(f)$ follows from Lemma 5.3.1. We have

$$ \begin{align*} F(g_2) &= \int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})} \,\theta(g_1,g_2)f(g_1)dg_1\\ &=\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\,\theta(g_1,g_2) \left(c_{1,\kappa}\,\int_{G_{1,\infty}}\omega_1(x)f(g_1x^{-1})dx\right) dg_1\\ &=\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})} \left(c_{1,\kappa}\,\int_{G_{1,\infty}}\,\theta(g_1x,g_2)\omega_1(x)dx\right) f(g_1)dg_1\\ &=\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})} \left(c_{2,\kappa}\,\int_{G_{2,\infty}}\,\omega_2(y) \theta(g_1,g_2y^{-1})dy\right) f(g_1)dg_1\\ &=c_{2,\kappa}\,\int_{G_{2,\infty}}\omega_2(y) \left(\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\,\theta(g_1,g_2y^{-1})f(g_1) dg_1\right)dy\\ &=c_{2,\kappa}\,\int_{G_{2,\infty}}\,\omega_2(y)F(g_2y^{-1})dy, \end{align*} $$

proving (7) for $F=\mathcal {L}(f)$ .

It now remains to show that F is cuspidal. We have

$$ \begin{align*} F_0(g_2)& =\int_{B^-\backslash B^-_{\mathbb{A}}}\,F\left(\left( \begin{array}{cc} 1&\beta \\ 0&1 \end{array} \right)g_2\right) d\beta\\ &=\int_{B^-\backslash B^-_{\mathbb{A}}}\,\left( \int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\, \sum_{X\in B^{(2,1)}}\, r\left(g_1,\left( \begin{array}{cc} 1&\beta \\ 0&1 \end{array} \right)g_2\right)\varphi_0(X)f(g_1)dg_1 \right) d\beta\\ & = \int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\, \sum_{X\in B^{(2,1)}}\, \left(\int_{B^-\backslash B^-_{\mathbb{A}}}\, \psi\left(-\dfrac{1}{2}\mathrm{tr}(\beta {}^t\!\overline{X}JX)\right)d\beta\right) r(g_1,g_2)\varphi_0(X)f(g_1)dg_1\\ &=\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\, \sum_{X\in B^{(2,1)},\,{}^t\!\overline{X}JX=0}\, r(1,g_2)\varphi_0({}^t\! \overline{g_1}X)f(g_1)dh. \end{align*} $$

Observe that

$$\begin{align*}\left\{X\in B^{(2,1)}\mid {}^t\!\overline{X}JX=0\right\}= \left\{\left( \begin{array}{c} 0\\0 \end{array} \right)\right\}\cup \left\{{}^t\!\overline{\gamma}\left( \begin{array}{c} 0\\1 \end{array} \right)\mid \gamma\in N_1(\mathbb{Q}) \backslash G_1(\mathbb{Q})\right\}, \end{align*}$$

where

$$\begin{align*}N_1(\mathbb{Q}) =\left\{\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right)\mid b\in\mathbb{Q}\right\}. \end{align*}$$

It follows that

$$ \begin{align*} & F_0(g_2)\\ &=(r(1,g_2)\varphi_0)(0)\int_{G_1(\mathbb{Q})\backslash G_1(\mathbb{A})}\,\,f(g_1)dg_1\\ &\quad +\int_{N_1(\mathbb{A})\backslash G_1(\mathbb{A})}\, \left(r(1,g_2)\varphi_0\right)\left({}^t\!\overline{g_1}\left( \begin{array}{c} 0\\1 \end{array} \right)\right) \left(\int_{\mathbb{Q}\backslash \mathbb{A}}\,f\left(\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right)g_1\right)db \right)dg_1\\ & =0, \end{align*} $$

since f is cuspidal. This shows the cuspidality of F and Theorem 5.4.1 has been proved.

7.3.

To prove Theorem 7.1.1, we need some preparations. Define $\Phi _{\kappa }\colon \mathbb {H}^{\times }\,\to \,\mathrm {End}(V_{\kappa })$ by

$$\begin{align*}\Phi_{\kappa}(x)=N(x)^{-1} \sigma_{\kappa}(x)^{-1}\qquad(x\in\mathbb{H}^{\times}). \end{align*}$$

Note that $\omega _2(g)=\Phi _{\kappa }\left (2^{-1}(1,1)g\left ( \begin {array}{c} 1\\1 \end {array} \right )\right )$ for $g\in G_{2,\infty }$ . The following is easily verified.

We define a map $M_{\kappa }\colon \mathbb {H}\to \mathrm {End}(V_{\kappa })$ by

(16) $$ \begin{align} M_{\kappa}(\xi)=\int_{\mathbb{H}^-}\,\mathbf{e}(-\mathrm{tr}(\xi\beta))\Phi_{\kappa}(1+\beta) d_L\beta\qquad(\xi\in\mathbb{H}). \end{align} $$

It is easily verified that $M_{\kappa }(\xi )=M_{\kappa }(x)$ for $\xi =t+x\;\;(t\in \mathbb {R},x\in \mathbb {H}^-)$ and that

$$\begin{align*}M_{\kappa}(\alpha\xi\alpha^{-1})=\sigma_{\kappa}(\alpha) M_{\kappa}(\xi) \sigma_{\kappa}(\alpha)^{-1} \qquad (\alpha\in \mathbb{H}^{\times},\xi\in\mathbb{H}). \end{align*}$$

For $\xi \in \mathbb {H}$ , there exists a $u\in \mathbb {H}^1$ such that $\xi =u(s+it)u^{-1}\;\;(s,t\in \mathbb {R},t\ge 0)$ . Then $M_{\kappa }(\xi )= \sigma _{\kappa }(u) M_{\kappa }(it)\sigma _{\kappa }(u)^{-1} $ . The following fact is proved in [Reference Arakawa1, Lemma 1.2]. Recall that we have introduced $P_r(X,Y)=X^rY^{\kappa -r}\in V_{\kappa }$ at (3).

Proposition 7.3.2. For $t\ge 0$ , we have

$$\begin{align*}M_{\kappa}(it)P_r=\delta_{r\kappa}\,\lambda(\kappa)\,t^{\kappa-1} e^{-4\pi t}\,P_{\kappa}\qquad(0\le r\le \kappa), \end{align*}$$

where

$$\begin{align*}\lambda(\kappa)=\dfrac{2^{2\kappa-1}\pi^{\kappa+1}}{\kappa!}. \end{align*}$$

In particular, $M_{\kappa }(0)=0$ .

Lemma 7.3.3. For $a,b\in \mathbb {C}$ , we have

(17) $$ \begin{align} \sigma_{\kappa}(a+ib)\,P_r= (a+b\sqrt{-1})^{r}(a-b\sqrt{-1})^{\kappa-r}\,P_r \qquad(0\le r\le \kappa). \end{align} $$

Proof. This follows from the definition (4) and

$$\begin{align*}A(a+bi)=\left( \begin{array}{cc} a+b\sqrt{-1}&0 \\ 0&a-b\sqrt{-1} \end{array} \right) \end{align*}$$

(for the definition of A, see (2)).

The following formula is well-known.

Lemma 7.3.4. For $s,t\in \mathbb {R}$ with $t>0$ , we have

$$\begin{align*}\int_{\mathbb{R}}(t+\sqrt{-1}x)^{-\kappa}\,e(sx)dx= \begin{cases} \dfrac{(2\pi)^{\kappa}}{(\kappa-1)!}\,s^{\kappa-1}\mathbf{e}(\sqrt{-1}st) & (s>0),\\ 0& (s\le 0). \end{cases} \end{align*}$$

7.4.

Define $\phi _{\omega _1},\phi _{\omega _1}^*\colon \mathbb {H}^{(1,2)}\to \mathrm {End}(V_{\kappa })$ and $\varphi _{\omega _2},\varphi _{\omega _2}^*\colon \mathbb {H}^{(2,1)}\to \mathrm {End}(V_{\kappa })$ by

$$ \begin{align*} \phi_{\omega_1}(X')& =\int_{G_{1,\infty}}\,r'(g_1,1)\phi_{0,\infty}(X')\,\omega_1(g_1)dg_1 \\ \phi^*_{\omega_1}(X')& =\int_{G_{1,\infty}}\,\omega_1(g_1)\,r'(g_1^{-1},1)\phi^*_{0,\infty}(X')dg_1, \\ \varphi_{\omega_2}(X) &=\int_{G_{2,\infty}}\,\omega_2(g_2)\,r(1,g_2^{-1})\varphi_{0,\infty}(X)dg_2 \\ \varphi^*_{\omega_2}(X)& =\int_{G_{2,\infty}}\,r(1,g_2^-)\varphi^*_{0,\infty}(X)\,\omega_2(g_2)dg_2 \end{align*} $$

for $X'\in \mathbb {H}^{(1,2)}$ and $X\in \mathbb {H}^{(2,1)}$ . The following facts are easily verified.

One of the keys to the proof of Theorem 7.1.1 is the following results.

The proof of Proposition 7.4.2 is given in Sections 7.5–7.7. We remark that the assumption $\bar {x_1}x_2=s-it$ in the second assertion of both propositions is due to the fact that any element in pure quaternions is conjugate to an element in $\mathbb {R} i$ , as we have used before Proposition 7.3.2. We omit the proof of Proposition 7.4.3, since it is quite similar to that of Proposition 7.4.2 and slightly easier.

7.5.

In this section, we prove Proposition 7.4.2(1). Let $x_1,x_2\in \mathbb {H}$ . By (5) and Lemma 5.2.2, we have

$$ \begin{align*} & \phi_{\omega_1}(x_1,x_2)\\ &= \int_{G_{1,\infty}}\,r'(g_1,1)\phi_{0,\infty}(x_1,x_2)\omega_1(g_1)dg_1\\ & =\int_{\mathbb{R}}db\,\int_0^{\infty}a^{-2}d^{\times} a\, \int_{K_{1,\infty}}dk \,r'\left(\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right)\left( \begin{array}{cc} a&0 \\ 0&a^{-1} \end{array} \right)k,1\right)\phi_{0,\infty} (x_1,x_2)\\ &\qquad \omega_1\left(\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right)\left( \begin{array}{cc} a&0 \\ 0&a^{-1} \end{array} \right)k\right)\\ &= \int_{\mathbb{R}}db\,\int_0^{\infty}a^{-2}d^{\times} a\, \mathbf{e}(-b\, \mathrm{tr}(\overline{x_1}x_2))\,a^4\,\phi_{0,\infty}(ax_1,ax_2)\, \left(\dfrac{a+a^{-1}-\sqrt{-1}a^{-1} b}{2}\right)^{-\kappa}\\ &=2^{\kappa} \sigma_{\kappa}(\overline{x_1+x_2})\, \int_0^{\infty}\,a^{2\kappa+2}\mathbf{e}(\sqrt{-1}a^2(\overline{x_1}x_1+\overline{x_2}x_2)) d^{\times} a \int_{\mathbb{R}}\,\left(a^2+1-\sqrt{-1}b\right)^{-\kappa} \mathbf{e}(-b\,\mathrm{tr}(\overline{x_1}x_2))db. \end{align*} $$

By Lemma 7.3.4, we see that the integral over $\mathbb {R}$ is equal to

$$\begin{align*}\dfrac{(2\pi)^{\kappa}}{(\kappa-1)!}\,\delta(\mathrm{tr}(\overline{x_1}x_2)>0) \,(\mathrm{tr}(\overline{x_1}x_2))^{\kappa-1}\,\mathbf{e}(\sqrt{-1}(a^2+1)\mathrm{tr}(\overline{x_1}x_2)). \end{align*}$$

Thus, we obtain

$$ \begin{align*} \phi_{\omega_1}(x_1,x_2) & =\dfrac{(4\pi)^{\kappa}}{(\kappa-1)!}\,\delta(\mathrm{tr}(\overline{x_1}x_2)>0) \,\sigma_{\kappa}(\overline{x_1+x_2})\,(\mathrm{tr}(\overline{x_1}x_2))^{\kappa-1}\, \mathbf{e}(\sqrt{-1}\,\mathrm{tr}(\overline{x_1}x_2))\\ &\quad\,\int_0^{\infty}\,a^{2\kappa+2} \,\mathbf{e}(\sqrt{-1}a^2(\overline{x_1}x_1+\overline{x_2}x_2+\mathrm{tr}(\overline{x_1}x_2)))\,d^{\times} a. \end{align*} $$

Since the last integral is equal to

$$ \begin{align*} \dfrac{1}{2}\,\int_0^{\infty}\, a^{\kappa+1}\mathbf{e}(\sqrt{-1}N(x_1+x_2)a)d^{\times} a =\dfrac{\kappa!}{2}(2\pi N(x_1+x_2))^{-\kappa-1}, \end{align*} $$

we have

$$ \begin{align*} \phi_{\omega_1}(x_1,x_2) =2^{\kappa-2}\pi^{-1} \,\kappa\cdot\delta(\mathrm{tr}(\overline{x_1}x_2)>0) \,(\mathrm{tr}(\overline{x_1}x_2))^{\kappa-1}\, \mathbf{e}(\sqrt{-1}\mathrm{tr}(\overline{x_1}x_2)) \Phi_{\kappa}(x_1+x_2), \end{align*} $$

which completes the proof of Proposition 7.4.2(1). Here, we used Lemma 7.3.1(3).

7.6.

In this section, we prove Proposition 7.4.2(2).

Lemma 7.6.1. Let $x_1,x_2\in \mathbb {H}$ . We have

$$ \begin{align*} \varphi_{\omega_2}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)& =2^{\kappa+1}\int_0^{\infty}\,a^{\kappa}(a+1)^{1-\kappa} \mathbf{e}(\sqrt{-1}(N(x_1)+N(x_2))a)\,M_{\kappa}(-(a+1)\overline{x_1}x_2) d^{\times} a\\ &\qquad \cdot\sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2}). \end{align*} $$

Proof. By (5) and Lemma 5.2.2, we have

$$ \begin{align*} \varphi_{\omega_2}\left( \begin{array}{c} x_1\\x_2 \end{array} \right) &=\int_{K_{2,\infty}}dk\,\int_0^{\infty}a^6d^{\times} a\,\int_{\mathbb{H}^-}d_L\beta\, \; \omega_2\left(k\left( \begin{array}{cc} a&0 \\ 0&a^{-1} \end{array} \right)\left( \begin{array}{cc} 1&\beta \\ 0&1 \end{array} \right)\right)\\ &\qquad r\left(1,\left( \begin{array}{cc} 1&-\beta \\ 0&1 \end{array} \right)\left( \begin{array}{cc} a^{-1}&0 \\ 0&a \end{array} \right)k^{-1}\right) \,\varphi_{0,\infty}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\\ &=\int_0^{\infty}a^6d^{\times} a\,\int_{\mathbb{H}^-}d_L\beta\, \Phi_{\kappa}\left(\dfrac{a}{2}(1+a^{-2}+\beta)\right)\, \mathbf{e}(\mathrm{tr}(\beta\overline{x_1}x_2))a^{-4}\,\varphi_{0,\infty}\left( \begin{array}{c} a^{-1} x_1\\a^{-1} x_2 \end{array} \right)\\ &=\int_0^{\infty}a^{-2}d^{\times} a\,\int_{\mathbb{H}^-}d_L\beta\, \left(\dfrac{a^{-1}}{2}\right)^{-\kappa-2} \Phi_{\kappa}\left(1+a^2+\beta\right)\, \mathbf{e}(\mathrm{tr}(\beta\overline{x_1}x_2))\,\varphi_{0,\infty}\left( \begin{array}{c} a x_1\\a x_2 \end{array} \right)\\ &=2^{\kappa+2}\int_0^{\infty}a^{2\kappa}d^{\times} a\,\int_{\mathbb{H}^-}d_L\beta\, \Phi_{\kappa}\left(1+a^2+\beta\right)\\ &\qquad \mathbf{e}\left(\mathrm{tr}(\beta\overline{x_1}x_2)+\sqrt{-1}a^2(N(x_1)+N(x_2))\right) \,\sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2}). \end{align*} $$

Changing the variable $\beta $ into $(1+a^2)\beta $ , we obtain

$$ \begin{align*} \varphi_{\omega_2}\left( \begin{array}{c} x_1\\x_2 \end{array} \right) &=2^{\kappa+2} \int_0^{\infty} a^{2\kappa}(a^2+1)^{1-\kappa} \mathbf{e}\left(\sqrt{-1}a^2(N(x_1)+N(x_2))\right) d^{\times} a\,\\ &\qquad \int_{\mathbb{H}^-}\,\mathbf{e}(\mathrm{tr}((a^2+1)\beta\overline{x_1}x_2))) \Phi_{\kappa}(1+\beta)d_L\beta\, \cdot\sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2})\\ &=2^{\kappa+1} \int_0^{\infty} a^{\kappa}(a+1)^{1-\kappa} \mathbf{e}\left(\sqrt{-1}(N(x_1)+N(x_2))a\right) M_{\kappa}(-(a+1) \overline{x_1}x_2) d^{\times} a\\ &\qquad \cdot\sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2}) \end{align*} $$

and the lemma has been proved.

We now suppose that $\overline {x_1}x_2=s-it\;\;(s,t\in \mathbb {R},t\ge 0)$ . Since

$$ \begin{align*} (\overline{x_1}-\sqrt{-1}\overline{x_2})(x_1+\sqrt{-1}x_2)& =N(x_1)+N(x_2)+\sqrt{-1}(\overline{x_1}x_2-\overline{x_2}x_1)\\ &=N(x_1)+N(x_2)-2\sqrt{-1}it, \end{align*} $$

we have

$$\begin{align*}\sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2}) \,\sigma_{\kappa}(x_1+\sqrt{-1}x_2)\,P_r =(N(x_1)+N(x_2)+2t)^r\,(N(x_1)+N(x_2)-2t)^{\kappa-r}\,P_r \end{align*}$$

by Lemma 7.3.3. We also have

$$\begin{align*}M_{\kappa}(-(a+1)\overline{x_1}x_2)P_r= M_{\kappa}(i(a+1)t)P_r=\delta_{r\kappa}\,\lambda(\kappa)(a+1)^{\kappa-1} t^{\kappa-1}e^{-4\pi(a+1)t}P_{\kappa} \qquad(0\le r\le \kappa) \end{align*}$$

by Proposition 7.3.2. We thus have

$$ \begin{align*} & \varphi_{\omega_2}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\, \sigma_{\kappa}(x_1+\sqrt{-1}x_2)\,P_r\\ &= \delta_{r\kappa}\, 2^{\kappa+1}\lambda(\kappa)\, (N(x_1)+N(x_2)+2t)^{\kappa} \, t^{\kappa-1}e^{-4\pi t}\, \int_0^{\infty}\,a^{\kappa}e^{-2\pi(N(x_1)+N(x_2)+2t)a}d^{\times} a \,P_{\kappa}\\ &=\delta_{r\kappa}\, 2^{2\kappa}\pi\kappa^{-1}\, t^{\kappa-1}e^{-4\pi t}\, P_{\kappa}, \end{align*} $$

which completes the proof of Proposition 7.4.2(2).

7.7.

In this section, we prove Proposition 7.4.2(3). Observe that $\varphi _{\omega _2}\left ( \begin {array}{c} 0\\x_2 \end {array} \right )=0$ and $I_{\infty }^{-1}\phi _{\omega _1}\left ( \begin {array}{c} 0\\x_2 \end {array} \right )=0$ in view of Propositions 7.4.2(1) and (2) and Lemma 7.4.1(1). We hereafter assume that $x_1\ne 0$ .

Lemma 7.7.1. We have

$$ \begin{align*} & I_{\infty}^{-1}\phi_{\omega_1}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\\ &=2^{2\kappa-1}\pi^{-1}\kappa\, \int_0^{\infty}\,a^{\kappa}(a+1)^{1-\kappa} \,\mathbf{e}(\mathrm{tr}(-\overline{x_1}x_2+\sqrt{-1}N(x_1))a)\, M_{\kappa}((a+1)\overline{x_2}x_1)\,d^{\times} a\,\cdot\, \sigma_{\kappa}(\overline{x_1}). \end{align*} $$

Proof. We put $c(\kappa )=2^{\kappa }\pi ^{-1} \kappa $ . By Proposition 7.4.2(1), we have

$$ \begin{align*} & I_{\infty}^{-1}\phi_{\omega_1}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\\ &=4\int_{\mathbb{H}}\,\mathbf{e}(-\mathrm{tr}(\overline{y}x_2)) \phi_{\omega,\infty}(x_1,y)d_Ly\\ & =c(\kappa)\,\int_{\mathbb{H}}\,\delta(\mathrm{tr}(\overline{x_1}y)>0)\,(\mathrm{tr}(\overline{x_1}y))^{\kappa-1} \,\mathbf{e}(-\mathrm{tr}(\overline{x_2}y)+\sqrt{-1}\,\mathrm{tr}(\overline{x_1}y))\Phi_{\kappa}(x_1+y)d_Ly. \end{align*} $$

Changing the variable y into $x_1y$ and using Lemma 7.3.1(2), we obtain

$$ \begin{align*} & I_{\infty}^{-1}\phi_{\omega_1}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\\ & =c(\kappa)\,\int_{\mathbb{H}}\,\delta(\mathrm{tr}(y)>0)\, (\mathrm{tr}(y))^{\kappa-1}\,\mathbf{e}(\mathrm{tr}(-\overline{x_2}x_1y+\sqrt{-1}N(x_1)y)) \Phi_{\kappa}(1+y)d_Ly\cdot N(x_1)^{\kappa+1}\Phi_{\kappa}(x_1)\\ &=c(\kappa)\, \int_{\mathbb{R}}da\,\int_{\mathbb{H}^-}d_L\beta\, \delta(\mathrm{tr}(a+\beta)>0)\, (\mathrm{tr}(a+\beta))^{\kappa-1}\, \mathbf{e}(\mathrm{tr}((-\overline{x_2}x_1+\sqrt{-1}N(x_1))(a+\beta)))\\ &\qquad \Phi_{\kappa}(1+a+\beta)\, \sigma_{\kappa}(\overline{x_1})\\ &=c(\kappa)\, \int_{0}^{\infty} (2a)^{\kappa-1}\,\mathbf{e}(\mathrm{tr}((-\overline{x_2}x_1+\sqrt{-1}N(x_1)))a)da\, \int_{\mathbb{H}^-}\,\mathbf{e}(-\mathrm{tr}(\overline{x_2}x_1\beta))\,\Phi_{\kappa}(1+a+\beta)d_L\beta \cdot \sigma_{\kappa}(\overline{x_1}). \end{align*} $$

Changing the variable $\beta $ into $(1+a)\beta $ , we obtain

$$ \begin{align*} & I_{\infty}^{-1}\phi_{\omega_1}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\\ &=c(\kappa)\, 2^{\kappa-1}\int_0^{\infty}\,a^{\kappa-1}(a+1)^{1-\kappa} \mathbf{e}(\mathrm{tr}(-\overline{x_2}x_1+\sqrt{-1}N(x_1))a)da\,\\ & \qquad \int_{\mathbb{H}^-}\,\mathbf{e}(-\mathrm{tr}((a+1)\overline{x_2}x_1\beta))\,\Phi_{\kappa}(1+\beta)d_L\beta\, \cdot\,\sigma_{\kappa}(\overline{x_1})\\ &=2^{2\kappa-1}\pi^{-1}\kappa \int_0^{\infty}\,a^{\kappa}(a+1)^{1-\kappa} \,\mathbf{e}(\mathrm{tr}(-\overline{x_1}x_2+\sqrt{-1}N(x_1))a)\, M_{\kappa}((a+1)\overline{x_2}x_1)\,d^{\times} a\,\cdot\,\sigma_{\kappa}(\overline{x_1}) \end{align*} $$

and we are done.

In view of Lemma 7.4.1(2) and Propositions 7.4.2(1) and (2), it suffices to show the following lemma to complete the proof of Proposition 7.4.2(3).

Lemma 7.7.2. Suppose that $\overline {x_1}x_2=s-it\;\;(s,t\in \mathbb {R},\,t\ge 0)$ . Then

$$ \begin{align*} I_{\infty}^{-1} \phi_{\omega_1}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\, \sigma_{\kappa}(x_1+\sqrt{-1}x_2)\, P_r = \delta_{r\kappa}\cdot 2^{2\kappa-2}t^{\kappa-1}e^{-4\pi t} \, P_{\kappa} \quad(r=0,\ldots,\kappa). \end{align*} $$

Proof. Put $c'(\kappa )=2^{2\kappa -1}\pi ^{-1}\kappa $ . By Lemmas 7.7.1 and 7.3.3 and Proposition 7.3.2, we have

$$ \begin{align*} & I_{\infty}^{-1} \phi_{\omega_1}\left( \begin{array}{c} x_1\\x_2 \end{array} \right)\, \sigma_{\kappa}(x_1+\sqrt{-1}x_2)\, P_r\\ &=c'(\kappa) \int_0^{\infty}\,a^{\kappa}(a+1)^{1-\kappa} \,\mathbf{e}(\mathrm{tr}(-\overline{x_1}x_2+\sqrt{-1}N(x_1)a)\, M_{\kappa}((a+1)\overline{x_2}x_1)\,d^{\times} a\\ &\qquad\cdot \,\sigma_{\kappa}(N(x_1)+\sqrt{-1}\overline{x_1}x_2)\,P_r\\ &=c'(\kappa) \int_0^{\infty}\,a^{\kappa}(a+1)^{1-\kappa} \,\mathbf{e}(2(-s+\sqrt{-1}N(x_1))a)\, M_{\kappa}(i(a+1)t)\,d^{\times} a\\ &\qquad\cdot \,(N(x_1)+t+\sqrt{-1}s)^{r}(N(x_1)-t+\sqrt{-1}s)^{\kappa-r} P_r\\ &=\delta_{r\kappa}\cdot c'(\kappa)\,\lambda(\kappa)\cdot (N(x_1)+t+\sqrt{-1}s)^{\kappa}\\ &\qquad \int_{0}^{\infty}\,a^{\kappa}(a+1)^{1-\kappa} ((a+1)t)^{\kappa-1}\,e^{4\pi\sqrt{-1}(-sa+\sqrt{-1}N(x_1)a)-4\pi (a+1)t} d^{\times} a\cdot P_{\kappa}\\ &=\delta_{r\kappa}\cdot\dfrac{2^{4\kappa-2}\pi^{\kappa}}{(\kappa-1)!} t^{\kappa-1}e^{-4\pi t} (N(x_1)+t+\sqrt{-1}s)^{\kappa} \int_0^{\infty}\,a^{\kappa}e^{-4\pi(N(x_1)+t+\sqrt{-1}s)a}d^{\times} a\cdot P_{\kappa}\\ &=\delta_{r\kappa}\cdot 2^{2\kappa-2}t^{\kappa-1}e^{-4\pi t}\cdot P_{\kappa}. \end{align*} $$

7.8.

We define the norm of $X=(x_{ij})\in \mathbb {H}^{(m,n)}$ by

$$\begin{align*}\|X\|=\sqrt{\sum_{i=1}^m\sum_{j=1}^n\,N(x_{ij})}. \end{align*}$$

The aim of this section is to prove the following estimates, which are needed in the proof of Theorem 7.1.1.

To prove the proposition, we need some preparations.

Proof. The first assertion is an immediate consequence of the Cauchy–Schwarz inequality and the inequality of arithmetic and geometric means. The second one follows from the first one and the fact that $t=\frac {1}{2}\mathrm {tr}(\overline {x_1}x_2i)$ . The third one follows from

$$\begin{align*}(\overline{x_1}-\sqrt{-1}\overline{x_2})(x_1+\sqrt{-1}x_2)=N(x_1)+N(x_2)-2\sqrt{-1}it \end{align*}$$

and

$$\begin{align*}A(N(x_1)+N(x_2)-2\sqrt{-1}it)=\left( \begin{array}{cc} N(x_1)+N(x_2)+2t&0 \\ 0&N(x_1)+N(x_2)-2t \end{array} \right), \end{align*}$$

which is invertible in $\mathbb {H}\otimes _{\mathbb {R}}\mathbb {C}=\mathrm {M}_2(\mathbb {C})$ if and only if $x_2\not =- x_1i$ in view of the second assertion (for the definition of A, see (2)).

Proof. First suppose that $x_2\ne -x_1i$ . By Lemma 7.8.2, $\overline {x_1}-\sqrt {-1}\overline {x_2}$ is invertible in $\mathbb {H}\otimes _{\mathbb {R}}\mathbb {C}$ . By Lemmas 7.3.3 and 7.8.2(3) and Proposition 7.4.2(2), we have

$$ \begin{align*} &\varphi_{\omega_2}(X) \sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2})^{-1} P_r\\ &=\varphi_{\omega_2}(X) \sigma_{\kappa}(x_1+\sqrt{-1}x_2) \sigma_{\kappa}(\|X\|^2-2\sqrt{-1}it)^{-1} P_r\\ &=(\|X\|^2+2t)^{-r}(\|X\|^2-2t)^{-\kappa+r} \varphi_{\omega_2}(X) \sigma_{\kappa}(x_1+\sqrt{-1}x_2)\,P_r\\ &= \delta_{r\kappa}\cdot 2^{2\kappa}\pi\kappa^{-1}\cdot t^{\kappa-1} e^{-4\pi t}(\|X\|^2+2t)^{-\kappa}\,P_{\kappa}, \end{align*} $$

which proves (20).

Next suppose that $x_2=-x_1i\ne 0$ . Then $\overline {x_1}x_2=-N(x_1)i=-\frac {1}{2}\|X\|^2\,i$ . Note that, if $r\ne \kappa $ , we have

$$\begin{align*}\sigma_{\kappa}(\overline{x_1}-\sqrt{-1}\overline{x_2})\sigma_{\kappa}(x_1)\,P_r =N(x_1)^{\kappa}\,\sigma_{\kappa}(1-\sqrt{-1}\,i)\,P_r=0 \end{align*}$$

by Lemma 7.3.3. It follows from Lemma 7.6.1 that

$$\begin{align*}\varphi_{\omega_2}(X)\sigma_{\kappa}(x_1)\,P_r =0 \qquad(0\le r\le \kappa-1). \end{align*}$$

Since $\sigma _{\kappa }(x_1+\sqrt {-1}x_2)\,P_{\kappa } = \sigma _{\kappa }(x_1)\,\sigma _{\kappa }(1-\sqrt {-1}\,i)\,P_{\kappa } =2^{\kappa }\, \sigma _{\kappa }(x_1)\,P_{\kappa }$ by Lemma 7.3.3, we have

$$ \begin{align*} \varphi_{\omega_2}(X) \sigma_{\kappa}(x_1)\,P_{\kappa} =2\pi\kappa^{-1}\cdot \|X\|^{2\kappa-2} e^{-2\pi \|X\|^2}\, P_{\kappa} \end{align*} $$

by Proposition 7.4.2, which proves (21).

For $z\in \mathbb {H}\otimes _{\mathbb {R}}\mathbb {C}$ , define $(s_{a,b}(z))_{a,b=0}^{\kappa }\in \mathbb {C}^{(\kappa +1,\kappa +1)}$ by

$$\begin{align*}\sigma_{\kappa}(z)P_b=\sum_{a=0}^{\kappa}\,P_a\,s_{a,b}(z) \qquad(0\le b\le\kappa). \end{align*}$$

Then

(22) $$ \begin{align} \sum_{j=0}^{\kappa}\,s_{a,j}(z)s_{j,b}(w)=s_{a,b}(zw) \qquad (z,w\in \mathbb{H}\otimes_{\mathbb{R}}\mathbb{C},\,0\le a,b\le \kappa). \end{align} $$

We see that $x\mapsto N(x)^{-\kappa /2}\left |s_{a,b}(x)\right |$ is bounded on $\mathbb {H}^{\times }$ , from which the following two lemmas are deduced.

Lemma 7.8.4. There exists a positive constant c such that

(23) $$ \begin{align} |(\Phi_{\kappa}(x)P_a,P_b)_{\kappa}|\le c\,N(x)^{-\kappa/2-1} \end{align} $$

for $x\in \mathbb {H}^{\times }$ and $0\le a,b\le \kappa $ .

Lemma 7.8.5. There exists a positive constant $c'$ such that

(24) $$ \begin{align} |s_{a,b}(x+\sqrt{-1}y)|\le c'\,(N(x)+N(y))^{\kappa/2} \end{align} $$

holds for any $x,y\in \mathbb {H}$ and $0\le a,b\le \kappa $ .

We now prove Proposition 7.8.1. It suffices to show the following estimates: There exist positive constants $C_1,C_2$ such that the inequalities

(25) $$ \begin{align} |(\phi_{\omega_1}(X')P_a,P_b)_{\kappa}|&\le\,C_1\,(1+\|X\|)^{-\kappa-2}, \end{align} $$

(26) $$ \begin{align} |(\varphi_{\omega_2}(X)P_a,P_b)_{\kappa}|&\le\,C_2\,(1+\|X\|)^{-\kappa} \end{align} $$

hold for any $X'\in \mathbb {H}^{(1,2)}$ and $X\in \mathbb {H}^{(1,2)}$ .

We first show (25). Let $X'=(x_1,x_2)\in \mathbb {H}^{(1,2)}$ and $0\le a,b\le \kappa $ . By Proposition 7.4.2 and Lemma 7.8.4, we have

$$ \begin{align*} \left|( \phi_{\omega_1}(X')P_a,P_b)_{\kappa}\right| \le c_1\cdot \delta(\mathrm{tr}(\overline{x_1}x_2)>0)\,(\mathrm{tr}(\overline{x_1}x_2))^{\kappa-1} \,e^{-2\pi \mathrm{tr}(\overline{x_1}x_2)}\,N(x_1+x_2)^{-\kappa/2-1} \end{align*} $$

with a positive constant $c_1$ . Observe that, if $\mathrm {tr}(\overline {x_1}x_2)>0$ , we have $N(x_1+x_2)>\|X'\|^2$ and $\mathrm {tr}(\overline {x_1}x_2)\le \|X'\|^2$ by Lemma 7.8.2. It follows that

$$\begin{align*}\left|( \phi_{\omega_1}(X')P_a,P_b)_{\kappa}\right| \le c_1\cdot\,\delta(\mathrm{tr}(\overline{x_1}x_2)>0)\|X'\|^{\kappa-4} \end{align*}$$

and hence

(27) $$ \begin{align} \mathrm{Sup}_{\|X'\|\le 1}\, \left|( \phi_{\omega_1}(X')P_a,P_b)_{\kappa}\right| <\infty \end{align} $$

under the assumption $\kappa>4$ . On the other hand, since $t\mapsto t^{\kappa -1}e^{-2\pi t}$ is bounded on $\mathbb {R}_{>0}$ , we have

(28) $$ \begin{align} \left|( \phi_{\omega_1}(X')P_a,P_b)_{\kappa}\right| \le c_2\cdot \delta(\mathrm{tr}(\overline{x_1}x_2)>0)\|X'\|^{-\kappa-2} \end{align} $$

with a positive constant $c_2$ . The estimate (25) follows from (27) and (28).

We next show (26). Let $X={}^t\!(x_1,x_2)\in \mathbb {H}^{(2,1)}$ and $0\le a,b\le \kappa $ . In view of Lemma 7.4.1(2), we may (and do) suppose that $\overline {x_1}x_2=s-it\;\;(s,t\in \mathbb {R},t\ge 0)$ .

We first consider the case $x_2\ne -x_1i$ . Then $\xi =\overline {x_1}-\sqrt {-1}\overline {x_2}$ is invertible in $\mathbb {H}\otimes _{\mathbb {R}}\mathbb {C}$ by Lemma 7.8.2. By (22) and Lemma 7.8.3(1), we have

$$ \begin{align*} \varphi_{\omega_2}(X)P_a& =\sum_{j=0}^{\kappa}\,s_{j,a}(1)\varphi_{\omega_2}(X)P_j\\ & =\sum_{j=0}^{\kappa} \left(\sum_{r=0}^{\kappa}s_{j,r}(\xi^{-1})s_{r,a}(\xi) \right)\,\varphi_{\omega_2}(X)P_j\\ & =\sum_{r=0}^{\kappa}\,s_{r,a}(\xi)\,\varphi_{\omega_2}(X) \sigma_{\kappa}(\xi)^{-1} P_r\\ &=2^{2\kappa}\pi\kappa^{-1}\cdot s_{\kappa,a}(\xi)\,t^{\kappa-1} e^{-4\pi t}\,(\|X\|^2+2t)^{-\kappa}\,P_{\kappa}. \end{align*} $$

By Lemma 7.8.5, there exists a positive constant $c_3$ such that

(29) $$ \begin{align} | (\varphi_{\omega_2}(X)P_a,P_b)_{\kappa}| \le c_3\,t^{\kappa-1}e^{-4\pi t}\,\|X\|^{\kappa}\,(\|X\|^2+2t)^{-\kappa} \end{align} $$

holds for any $X\in \mathbb {H}^{(2,1)}$ . Since

$$\begin{align*}t^{\kappa-1}e^{-4\pi t}\,\|X\|^{\kappa}\,(\|X\|^2+2t)^{-\kappa} \le \left(\dfrac{t}{\|X\|^2+2t}\right)^{\kappa-1} \dfrac{\|X\|^{\kappa}}{\|X\|^2+2t} \le 2^{-\kappa+1}\|X\|^{\kappa-2}, \end{align*}$$

we have

(30) $$ \begin{align} \mathrm{Sup}_{\|X\|\le 1}|(\varphi_{\omega_2}(X)P_a,P_b)_{\kappa}|<\infty. \end{align} $$

We also have

(31) $$ \begin{align} | (\varphi_{\omega_2}(X)P_a,P_b)_{\kappa}| \le c_3c_4\,\|X\|^{-\kappa}, \end{align} $$

where $c_4=\mathrm {Sup}_{t>0}\,t^{\kappa -1}e^{-4\pi t}$ . The estimate (26) in the case $x_2\ne -x_1i$ follows from (30) and (31).

We next consider the remaining case where $x_2=-x_1i$ and $x_1\ne 0$ . A similar argument as above and Lemma 7.8.3(2) show that

$$ \begin{align*} (\varphi_{\omega_2}(X)P_a,P_b)_{\kappa} &=2\pi\kappa^{-1}\cdot s_{\kappa,a}(x_1^{-1}) \,\|X\|^{2\kappa-2}\,e^{-2\pi\|X\|^2} (P_{\kappa},P_b)_{\kappa}. \end{align*} $$

Since $|s_{\kappa ,a}(x_1^{-1})|\,\|X\|^{\kappa }$ is bounded, we have

$$\begin{align*}|(\varphi_{\omega_2}(X)P_a,P_b)_{\kappa}|\le c_5\cdot \|X\|^{\kappa-2}e^{-2\pi\|X\|^2} \end{align*}$$

with a positive constant $c_5$ , which implies (26) in this case.

7.9.

In this section, we complete the proof of (14).

Let

$$ \begin{align*} \Psi_{\omega_1}(X') &=\phi_{\omega_1}(X^{\prime}_{\infty})\prod_{p<\infty}\phi_{0,p}(X^{\prime}_p) \qquad(X'=(X^{\prime}_v)_{v\le\infty}\in B_{\mathbb{A}}^{(1,2)}),\\ \Phi_{\omega_2}(X) &=\varphi_{\omega_2}(X_{\infty})\prod_{p<\infty}\varphi_{0,p}(X_p) \qquad(X=(X_v)_{v\le\infty}\in B_{\mathbb{A}}^{(2,1)}). \end{align*} $$

Proposition 7.8.1 immediately implies the following.

We now complete the proof of (14). In view of Corollary 2.6 in Chapter VII of [Reference Stein and Weiss18] and Proposition 7.8.1, under the assumption that $\kappa>\dim _{\mathbb {Q}}B^{(1,2)}=8$ , we can apply the Poisson summation formula to the sum (32) and obtain

$$ \begin{align*} \int_{G_{1,\infty}}\,\theta(g_1x,g_2)\omega_1(x)dx &=\sum_{X'\in B^{(1,2)}}\,r'(g_1,g_2)\Psi_{\omega_1}(X')\\ &=\sum_{X\in B^{(2,1)}}\,I^{-1} (r'(g_1,g_2)\Psi_{\omega_1})(X). \end{align*} $$

By Lemma 5.2.1 and Proposition 7.4.2, we have $I^{-1} (r'(g_1,g_2)\Psi _{\omega _1})= \frac {c_{2,\kappa }}{c_{1,\kappa }}\cdot r(g_1,g_2)\Phi _{\omega _2}$ . We thus have

$$ \begin{align*} c_{1,\kappa}\,\int_{G_{1,\infty}}\,\theta(g_1x,g_2)\omega_1(x)dx &=c_{2,\kappa}\,\sum_{X\in B^{(2,1)}}\,r(g_1,g_2)\Phi_{\omega_2}(X)\\ &=c_{2,\kappa}\, \int_{G_{2,\infty}}\,\omega_2(y)\theta(g_1,g_2y^{-1})dy, \end{align*} $$

proving (14).

9.1.

In this section, we consider the unramified case; namely, we assume that p does not divide the discriminant $d_B$ of B. Throughout this section, we often drop the subscript p. As in Section 4.4, we let

$$ \begin{align*} G_1 & =\left\{g\in \mathrm{GL}_4(\mathbb{Q}_p)\mid {}^t\! g\left( \begin{array}{cc} 0&J \\ -J&0 \end{array} \right)g= \left( \begin{array}{cc} 0&J \\ -J&0 \end{array} \right)\right\}, \qquad J=\left( \begin{array}{cc} 0&1 \\ -1&0 \end{array} \right),\\ K_1 &=G_1 \cap \mathrm{GL}_4(\mathbb{Z}_p),\\ G_2 & =\left\{g\in \mathrm{GL}_4(\mathbb{Q}_p)\mid {}^t\! g\left( \begin{array}{cc} 0&1_2 \\ -1_2&0 \end{array} \right)g= \left( \begin{array}{cc} 0&1_2 \\ -1_2&0 \end{array} \right)\right\},\\ K_2 &=G_2 \cap \mathrm{GL}_4(\mathbb{Z}_p). \end{align*} $$

Define subgroups $N_1$ and $M_1$ of $G_1$ by

$$ \begin{align*} N_1&= \left\{ n_1(b):=\left( \begin{array}{cc} 1_2&b1_2 \\ 0&1_2 \end{array} \right) \mid b\in\mathbb{Q}_p \right\},\\ M_1&= \left\{ d_1(A):=\left( \begin{array}{cc} A&0 \\ 0&J^{-1} {}^t\! A^{-1} J \end{array} \right) \mid A\in\mathrm{GL}_2(\mathbb{Q}_p) \right\}. \end{align*} $$

Then $G_1=N_1M_1K_1$ . We also define subgroups $N_2$ and $M_2$ of $G_2$ by

$$ \begin{align*} N_2 &= \left\{ n_2(b,c,d):= \left( \begin{array}{c|c} {\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}} & {\begin{array}{cc} b & c \\ c & d \end{array}} \\ \hline 0 & {\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}} \end{array} \right) \mid b,c,d\in\mathbb{Q}_p \right\},\\ M_2&= \left\{ d_2(A):=\left( \begin{array}{cc} A&0 \\ 0& {}^t\! A^{-1} \end{array} \right) \mid A\in\mathrm{GL}_2(\mathbb{Q}_p) \right\}. \end{align*} $$

Then $G_2=N_2M_2K_2$ .

The Weil representation $r'$ given in Lemma 5.1.1 is reformulated as follows: For $\phi \in \mathcal {S}(\mathbb {Q}_p^{(2,4)})$ and $x,y\in \mathbb {Q}_p^{(2,2)}$ ,

$$ \begin{align*} r'(1,g)\phi(x,y)&=\phi((x,y)g) \qquad(g\in G_2),\\ r'(d_1(A),1)\phi(x,y)&=|\det A|^2\,\phi({}^t\! Ax,{}^t\! Ay) \qquad(A\in\mathrm{GL}_2(\mathbb{Q}_p)), \\ r'(n_1(b),1)\phi(x,y)& = \psi(-b\,\mathrm{Tr}(J^{-1}{}^t\! xJy))\phi(x,y) \qquad(b\in \mathbb{Q}_p). \end{align*} $$

9.2.

In this section, we prepare several facts on Hecke algebras.

For $i=1,2$ , let $\mathcal {T}_{G_1}^i$ and $\mathcal {T}_{G_2}^i$ be the characteristic functions of

$$\begin{align*}T_{G_1}^i =\{g_1\in G_1\mid pg_1\in \mathrm{M}_4(\mathbb{Z}_p),\, \mathrm{rank}_{\mathbb{F}_p}(pg_1)=i\} \end{align*}$$

and

$$\begin{align*}T_{G_2}^i =\{g_2\in G_2\mid pg_2\in \mathrm{M}_4(\mathbb{Z}_p),\, \mathrm{rank}_{\mathbb{F}_p}(pg_2)=i\}, \end{align*}$$

respectively. Here, for $X\in \mathrm {M}_r(\mathbb {Z}_p)$ , we write $\mathrm {rank}_{\mathbb {F}_p}(X)$ for the rank of $(X \bmod {p\,\mathrm {M}_r(\mathbb {Z}_p)})$ in $\mathrm {M}_r(\mathbb {F}_p)$ . Then $\mathcal {T}_{G_1}^i\in \mathcal {H}(G_1,K_1)$ and $\mathcal {T}_{G_2}^i\in \mathcal {H}(G_2,K_2)$ .

Lemma 9.2.1. We have

$$ \begin{align*} \mathcal{H}(G_1,K_1)&=\mathbb{C}[\mathcal{T}_{G_1}^1,\mathcal{T}_{G_1}^2],\\ \mathcal{H}(G_2,K_2)&=\mathbb{C}[\mathcal{T}_{G_2}^1,\mathcal{T}_{G_2}^2]. \end{align*} $$

For $a\in \mathbb {Q}_p$ , put

$$ \begin{align*} \nu_1(a)&=d_1\left( \begin{array}{cc} 1&a \\ 0&1 \end{array} \right)= \begin{pmatrix} 1&a&0&0\\ 0&1 &0 &0 \\ 0&0 &1 &a \\ 0&0 &0 &1 \end{pmatrix}\in G_1,\\ \nu_2(a)&=d_2\left( \begin{array}{cc} 1&a \\ 0&1 \end{array} \right)= \begin{pmatrix} 1&a&0&0\\ 0&1 &0 &0 \\ 0&0 &1 &0 \\ 0&0 &-a &1 \end{pmatrix}\in G_2. \end{align*} $$

We put

$$ \begin{align*} \Lambda_i&=\{(b,c,d)\in(\mathbb{Z}_p/p\mathbb{Z}_p)^3\mid \mathrm{rank}_{ \mathbb{F}_p}\left( \begin{array}{cc} b&c \\ c&d \end{array} \right)=i\} \end{align*} $$

for $i=1,2$ . Note that $|\Lambda _1|=p^2-1$ and $|\Lambda _2|=p^3-p^2$ .

The proofs of the following facts are straightforward and we omit it.

Lemma 9.2.2. We have

$$ \begin{align*} T_{G_1}^1 &= \bigsqcup_{a,\,b\in \mathbb{Z}_p/p\mathbb{Z}_p}\,n_1(b)\nu_1(a)d_1\left( \begin{array}{cc} p&0 \\ 0&1 \end{array} \right)K_1 \,\sqcup\, \bigsqcup_{b\in \mathbb{Z}_p/p\mathbb{Z}_p}\,n_1(b)d_1\left( \begin{array}{cc} 1&0 \\ 0&p \end{array} \right)K_1 \\ & \qquad \,\sqcup\, \bigsqcup_{a\in \mathbb{Z}_p/p\mathbb{Z}_p}\, \nu_1(a)d_1\left( \begin{array}{cc} 1&0 \\ 0&p^{-1} \end{array} \right)K_1 \,\sqcup\, d_1\left( \begin{array}{cc} p^{-1}&0 \\ 0&1 \end{array} \right)K_1 \end{align*} $$

and

$$ \begin{align*} T_{G_1}^2 &= \bigsqcup_{a\in\mathbb{Z}_p/p^2\mathbb{Z}_p}\,\nu_1(a)d_1\left( \begin{array}{cc} p&0 \\ 0&p^{-1} \end{array} \right)K_1 \,\sqcup\, d_1\left( \begin{array}{cc} p^{-1}&0 \\ 0&p \end{array} \right)K_1 \,\sqcup\, \bigsqcup_{b\in \mathbb{Z}_p/p^2\mathbb{Z}_p}\,n_1(b)d_1\left( \begin{array}{cc} p&0 \\ 0&p \end{array} \right)K_1 \\ &\qquad \,\sqcup\, d_1\left( \begin{array}{cc} p^{-1}&0 \\ 0&p^{-1} \end{array} \right)K_1 \,\sqcup\, \bigsqcup_{a\in \mathbb{Z}_p^{\times}/p\mathbb{Z}_p}\,\nu_1(p^{-1} a)\,K_1 \,\sqcup\, \bigsqcup_{b\in \mathbb{Z}_p^{\times}/p\mathbb{Z}_p} n_1(p^{-1} b)\,K_1, \end{align*} $$

where $\mathbb {Z}_p^{\times }/p\mathbb {Z}_p$ means $(\mathbb {Z}_p- p\mathbb {Z}_p)/p\mathbb {Z}_p$ .

Lemma 9.2.3. We have

$$ \begin{align*} T_{G_2}^1 &=\bigsqcup_{a,\,c\,\in\mathbb{Z}_p/p\mathbb{Z}_p,\,b\in\mathbb{Z}_p/p^2\mathbb{Z}_p} n_2(b,c,0)\nu_2(a)d_2\left( \begin{array}{cc} p&0 \\ 0&1 \end{array} \right)K_2\\ & \qquad \,\sqcup\, \bigsqcup_{c\,\in\mathbb{Z}_p/p\mathbb{Z}_p,\,d\in\mathbb{Z}_p/p^2\mathbb{Z}_p} n_2(0,c,d)d_2\left( \begin{array}{cc} 1&0 \\ 0&p \end{array} \right)K_2 \,\sqcup\, \bigsqcup_{(b,c,d)\in\Lambda_1}n_2(p^{-1} b,p^{-1} c,p^{-1} d)K_2\\ &\qquad \, \sqcup\, \bigsqcup_{a\in\mathbb{Z}_p/p\mathbb{Z}_p}\nu_2(a)d_2\left( \begin{array}{cc} 1&0 \\ 0&p^{-1} \end{array} \right)K_2 \,\sqcup\, d_2\left( \begin{array}{cc} p^{-1}&0 \\ 0&1 \end{array} \right)K_2. \end{align*} $$

and

$$ \begin{align*} T_{G_2}^2 &= \bigsqcup_{b,\,c,\,d\in\mathbb{Z}_p/p^2\mathbb{Z}_p}\,n_2(b,c,d)d_2\left( \begin{array}{cc} p&0 \\ 0&p \end{array} \right)K_2 \\&\quad\sqcup\, \bigsqcup_{a,\,b\in\mathbb{Z}_p/p^2\mathbb{Z}_p}\,n_2(b,0,0) \nu_2(a)d_2\left( \begin{array}{cc} p&0 \\ 0&p^{-1} \end{array} \right)K_2\\ &\quad \,\sqcup\, \bigsqcup_{d\in\mathbb{Z}_p/p^2\mathbb{Z}_p}\, n_2(0,0,d)d_2\left( \begin{array}{cc} p^{-1}&0 \\ 0&p \end{array} \right)K_2 \,\sqcup\,d_2\left( \begin{array}{cc} p^{-1}&0 \\ 0&p^{-1} \end{array} \right)K_2\\ &\quad\,\sqcup\,\bigsqcup_{a,\,m\in \mathbb{Z}_p/p\mathbb{Z}_p,\,d\in\mathbb{Z}_p^{\times}/p\mathbb{Z}_p,\, l\in\mathbb{Z}_p/p^2\mathbb{Z}_p}\,n_2(p^{-1} a^2d+l,\,p^{-1} ad+m,\,p^{-1} d) \nu_2(a)d_2\left( \begin{array}{cc} p&0 \\ 0&1 \end{array} \right)K_2\\ &\quad\,\sqcup\,\bigsqcup_{b\in\mathbb{Z}_p^{\times}/p\mathbb{Z}_p,\,c\in\mathbb{Z}_p/p\mathbb{Z}_p,\,d\in\mathbb{Z}_p/p^2\mathbb{Z}_p}\,n_2(p^{-1} b,c,d)d_2\left( \begin{array}{cc} 1&0 \\ 0&p \end{array} \right)K_2\\ &\quad\,\sqcup\,\bigsqcup_{a\in\mathbb{Z}_p/p\mathbb{Z}_p,\,b\in\mathbb{Z}_p^{\times}/p\mathbb{Z}_p} \,n_2(p^{-1} b,0,0)\nu_2(a)d_2\left( \begin{array}{cc} 1&0 \\ 0&p^{-1} \end{array} \right)K_2\\ &\quad \sqcup\,\bigsqcup_{d\in\mathbb{Z}_p^{\times}/p\mathbb{Z}_p}\,n_2(0,0,p^{-1} d) d_2\left( \begin{array}{cc} p^{-1}&0 \\ 0&1 \end{array} \right)K_2\\ &\quad \,\sqcup\,\bigsqcup_{a\in\mathbb{Z}_p^{\times}/p\mathbb{Z}_p,\,c,\,m\in\mathbb{Z}_p/p\mathbb{Z}_p} n_2(p^{-2}(ac+pm),p^{-1} c,0)\nu_2(p^{-1} a)K_2\\&\quad\sqcup\, \bigsqcup_{(b,c,d)\in\Lambda_2}\,n_2(p^{-1} b,p^{-1} c,p^{-1} d)K_2. \end{align*} $$

Recall that the spherical functions $\omega _{1,\chi }\colon G_1\to \mathbb {C}$ and $\omega _{2,\Xi }\colon G_2\to \mathbb {C}$ for $\chi =(\chi _1,\chi _2),\Xi =(\Xi _1,\Xi _2)\in X_{\mathrm {unr}}(\mathbb {Q}^{\times }_p)^2$ are defined in Section 4.4. We have

$$ \begin{align*} (\omega_{1,\chi}*\varphi)(g_1)&:=\int_{G_1}\omega_{1,\chi}(g_1x^{-1}) \varphi(x)dx =\lambda_{1,\chi}(\varphi)\omega_{1,\chi}(g_1) \qquad (g_1\in G_1,\varphi\in\mathcal{H}(G_1,K_1)),\\ (\omega_{2,\Xi}*\Phi)(g_2)&:=\int_{G_2}\omega_{2,\Xi}(g_2y^{-1})\Phi(y)dy =\lambda_{2,\Xi}(\Phi)\omega_{2,\Xi}(g_2) \qquad (g_2\in G_2,\Phi\in\mathcal{H}(G_2,K_2)) \end{align*} $$

with $\lambda _{1,\chi }\in \mathrm {Hom}_{\mathbb {C}\text {-alg}}(\mathcal {H}(G_1,K_1),\mathbb {C})$ and $\lambda _{2,\Xi }\in \mathrm {Hom}_{\mathbb {C}\text {-alg}}(\mathcal {H}(G_2,K_2),\mathbb {C})$ (cf. [Reference Satake16, Section 5.3]). Using Lemmas 9.2.2 and 9.2.3, we get the following.

Lemma 9.2.4. We have

$$ \begin{align*} \lambda_{1,\chi}(\mathcal{T}_{G_1}^1)&=p(\chi_1+\chi_2+\chi_1^{-1}+\chi_2^{-1}),\\ \lambda_{1,\chi}(\mathcal{T}_{G_1}^2)&=p(\chi_1\chi_2+\chi_1\chi_2^{-1}+ \chi_1^{-1} \chi_2+\chi_1^{-1} \chi_2^{-1})+2p-2,\\ \lambda_{2,\Xi}(\mathcal{T}_{G_2}^1)&=p^2(\Xi_1+\Xi_2+\Xi_1^{-1}+\Xi_2^{-1})+p^2-1,\\ \lambda_{2,\Xi}(\mathcal{T}_{G_2}^2)&=p^3(\Xi_1\Xi_2+\Xi_1\Xi_2^{-1}+\Xi_1^{-1} \Xi_2 +\Xi_1^{-1} \Xi_2^{-1})+ p^2(p-1)(\Xi_1+\Xi_2+\Xi_1^{-1}+\Xi_2^{-1})+2p^3-2p^2, \end{align*} $$

where we write, for $i=1,2$ , $\chi _i$ and $\Xi _i$ for $\chi _i(p)$ and $\Xi _i(p)$ , respectively.

9.3.

Denote by $\phi _0\in \mathscr {S}(\mathbb {Q}_p^{(2,4)})$ the characteristic function of $\mathbb {Z}_p^{(2,4)}$ and set

$$ \begin{align*} J_{G_1}^i(x,y) &=\int_{T_{G_1}^i}\,r'(g_1,1)\phi_0(x,y)dg_1,\\ J_{G_2}^i(x,y) &=\int_{T_{G_2}^i}\,r'(1,g_2)\phi_0(x,y)dg_2 \end{align*} $$

for $x,y\in \mathbb {Q}_p^{(2,2)}$ and $i=1,2$ .

We now state the main result of this section. A sketch of its proof is given in Section 9.5.

Proposition 9.3.1. For $x,y\in \mathbb {Q}_p^{(2,2)}$ , we have

(38) $$ \begin{align} J_{G_2}^1(x,y)=(p^2-1)\phi_0(x,y)+pJ_{G_1}^1(x,y) \end{align} $$

and

(39) $$ \begin{align} J_{G_2}^2(x,y)=(p^2-p)J_{G_1}^1(x,y)+p^2J_{G_1}^2(x,y). \end{align} $$

This proposition immediately implies the following result.

Corollary 9.3.2. The algebra homomorphism $\upsilon \colon \mathcal {H}(G_2,K_2)\,\to \, \mathcal {H}(G_1,K_1)$ given by

$$ \begin{align*} \upsilon(\mathcal{T}_{G_2}^1)&=(p^2-1)\,\mathcal{T}_{G_1}^0+p\,\mathcal{T}_{G_1}^1,\\ \upsilon(\mathcal{T}_{G_2}^2)&=(p^2-p)\,\mathcal{T}_{G_1}^1+p^2\mathcal{T}_{G_1}^2 \end{align*} $$

satisfies (36) (and hence (37)), where $\mathcal {T}_{G_1}^0$ denotes the characteristic function of $K_1$ .

9.4.

In this section, assuming Corollary 9.3.2, we prove the following correspondence of the local factors of the standard L-functions.

Proof. The first assertion is a direct consequence of Lemma 9.2.1 and Corollary 9.3.2. To show the second, we take $\chi =(\chi _1,\chi _2)\in X_{\mathrm {unr}}(\mathbb {Q}_p^{\times })^2$ such that $\omega _{1,p}^f=\omega _{1,\chi }$ and $\Xi =(\Xi _1,\Xi _2)\in X_{\mathrm {unr}}(\mathbb {Q}_p^{\times })^2$ such that $\omega _{2,p}^{\mathcal {L}(f)}=\omega _{2,\Xi }$ (see Lemma 4.4.1). By abuse of notation, we write $\chi _i$ and $\Xi _i$ for $\chi _i(p)$ and $\Xi _i(p)$ ( $i=1,2$ ), respectively. In view of Lemma 9.2.4, we have

(40) $$ \begin{align} \lambda_{1,p}^f(\mathcal{T}_{G_1}^1)&=p\left(\chi_1+\chi_2 +\chi_1^{-1}+\chi_2^{-1}\right),\end{align} $$

(41) $$ \begin{align} \lambda_{1,p}^f(\mathcal{T}_{G_1}^2)&=p\left(\chi_1\chi_2+\chi_1\chi_2^{-1}+\chi_1^{-1} \chi_2+\chi_1^{-1}\chi_2^{-1}\right)+2p-2, \end{align} $$

(42) $$ \begin{align} \lambda_{2,p}^{\mathcal{L}(f)}(\mathcal{T}_{G_2}^1)& =p^2\left(\Xi_1+\Xi_2+\Xi_1^{-1}+\Xi_2^{-1}\right)+p^2-1 , \end{align} $$

(43) $$ \begin{align} \lambda_{2,p}^{\mathcal{L}(f)}(\mathcal{T}_{G_2}^2)& =p^3\left(\Xi_1\Xi_2+\Xi_1\Xi_2^{-1}+\Xi_1^{-1} \Xi_2+\Xi_1^{-1}\Xi_2^{-1}\right)\nonumber\\ \notag &\quad +p^2(p-1)\left(\Xi_1+\Xi_2+\Xi_1^{-1}+\Xi_2^{-1}\right) +2p^3-2p^2. \end{align} $$

By (37), Corollary 9.3.2, (40), and (41), we have

$$ \begin{align*} \lambda_{2,p}^{\mathcal{L}(f)}(\mathcal{T}_{G_2}^1)&=\lambda_{1,p}^f(\upsilon(\mathcal{T}_{G_2}^1))\\ &=(p^2-1)\,\lambda_{1,p}^f(\mathcal{T}_{G_1}^0)+p\,\lambda_{1,p}^f(\mathcal{T}_{G_1}^1) \\ & =p^2\,\left(\chi_1+\chi_2+\chi_1^{-1}+\chi_2^{-1}\right)+p^2-1 \end{align*} $$

and

$$ \begin{align*} \lambda_{2,p}^{\mathcal{L}(f)}(\mathcal{T}_{G_2}^2)&=\lambda_{1,p}^f(\upsilon(\mathcal{T}_{G_2}^2))\\ &=(p^2-p)\,\lambda_{1,p}^f(\mathcal{T}_{G_1}^1) +p^2\,\lambda_{1,p}^f(\mathcal{T}_{G_1}^2) \\ & =(p^2-p)\,\left(\chi_1+\chi_2+\chi_1^{-1}+\chi_2^{-1}\right)\\ &\quad+p^3\,\left(\chi_1\chi_2+\chi_1\chi_2^{-1}+\chi_1^{-1} \chi_2+\chi_1^{-1}\chi_2^{-1}\right) +2p^3-2p^2. \end{align*} $$

Comparing the above formulas with (42) and (43), we obtain

$$ \begin{align*} \Xi_1+\Xi_2+\Xi_1^{-1}+\Xi_2^{-1}&=\chi_1+\chi_2+\chi_1^{-1}+\chi_2^{-1},\\ \Xi_1\Xi_2+\Xi_1\Xi_2^{-1}+\Xi_1^{-1} \Xi_2+\Xi_1^{-1}\Xi_2^{-1} &=\chi_1\chi_2+\chi_1\chi_2^{-1}+\chi_1^{-1} \chi_2+\chi_1^{-1}\chi_2^{-1}. \end{align*} $$

Hence, we have

$$\begin{align*}\prod_{i=1}^2\,(1-\Xi_i p^{-s})^{-1}\,(1-\Xi_i^{-1} p^{-s})^{-1} =\prod_{i=1}^2\,(1-\chi_i p^{-s})^{-1}\,(1-\chi_i^{-1} p^{-s})^{-1}, \end{align*}$$

which immediately implies the proposition.

9.5.

In this section, we give a sketch of the proof of Proposition 9.3.1. The detail of the proof is given in [Reference Murase and Narita8, Section 12]. The following fact is easily verified.

Lemma 9.5.1. For $u_1,u_2\in \mathrm {GL}_2(\mathbb {Z}_p)$ and $x,y\in \mathbb {Q}_p^{(2,2)}$ , we have

$$ \begin{align*} J_{G_1}^i(u_1xu_2,y)&=J_{G_1}^i(x,u_1^{-1} y\,{}^t\! u_2),\\ J_{G_2}^i(u_1xu_2,y)&=J_{G_2}^i(x,u_1^{-1} y\,{}^t\! u_2). \end{align*} $$

Thus, we may (and do) assume $x=\left ( \begin {array}{cc} p^{\alpha }&0 \\ 0&p^{\beta } \end {array} \right )\;\; (\alpha ,\beta \in \mathbb {Z},\,\alpha \le \beta )$ and put

$$ \begin{align*} J_{G_1}^i(y;\alpha,\beta)& =J_{G_1}^i\left(\left( \begin{array}{cc} p^{\alpha}&0 \\ 0&p^{\beta} \end{array} \right),y\right),\\ J_{G_2}^i(y; \alpha,\beta)& =J_{G_2}^i\left(\left( \begin{array}{cc} p^{\alpha}&0 \\ 0&p^{\beta} \end{array} \right),y\right) \end{align*} $$

to simplify the notation. We have $J_{G_1}^i(y; \alpha ,\beta ) =J_{G_2}^i(y; \alpha ,\beta )=0$ if $\beta \le -2$ . Henceforth, we suppose that $\beta \ge -1$ .

We denote by $\sigma $ the characteristic function of $\mathbb {Z}_p$ and set

$$\begin{align*}[i_1,i_2,i_3,i_4](y)=\prod_{k=1}^4\,\sigma(p^{-i_k}y_k) \qquad \left(y =\begin{pmatrix}y_1&y_2\\y_3&y_4\end{pmatrix} \in\mathbb{Q}_p^{(2,2)}\right). \end{align*}$$

Proposition 9.3.1 is a straightforward consequence of the following four lemmas, whose proofs are technical and omitted (for the full proofs, see [Reference Murase and Narita8, Section 12]).

10.1.

In this section, we consider the ramified case; namely, we assume that p divides $d_B$ . Throughout this section, we often drop the subscript p. Recall that $\mathcal {O}$ is the maximal order of B and $\mathfrak {P}$ is the maximal ideal of $\mathcal {O}$ . We fix an element $\Pi $ of $\mathcal {O}$ such that $\mathfrak {P}=\Pi \mathcal {O}$ . Define $\mathrm {ord}_{\Pi }\colon B^{\times }\to \mathbb {Z}$ such that $\mathrm {ord}_{\Pi }(\alpha )=n$ for $\alpha \in \Pi ^n\mathcal {O}^{\times }$ . We fix a prime element $\pi $ of $\mathbb {Q}_p$ .

Let

$$ \begin{align*} G_1&=\left\{g\in\mathrm{GL}_2(B)\mid {}^t\!\overline{g}\left( \begin{array}{cc} 0&1 \\ -1&0 \end{array} \right)g =\left( \begin{array}{cc} 0&1 \\ -1&0 \end{array} \right)\right\},\\ K_1 &=G_1\,\cap \mathrm{GL}_2(\mathcal{O})\\ G_2 &=\left\{g\in\mathrm{GL}_2(B)\mid {}^t\!\overline{g}\left( \begin{array}{cc} 0&1 \\ 1&0 \end{array} \right)g =\left( \begin{array}{cc} 0&1 \\ 1&0 \end{array} \right)\right\},\\ K_2 &=G_2\,\cap \left( \begin{array}{cc} \mathcal{O}&\mathfrak{P}^{-1} \\ \mathfrak{P}&\mathcal{O} \end{array} \right). \end{align*} $$

Define subgroups of $G_1$ by

$$ \begin{align*} N_1 &= \left\{ n_1(b):=\left( \begin{array}{cc} 1&b \\ 0&1 \end{array} \right) \mid b\in\mathbb{Q}_p \right\},\\ M_1&= \left\{ d_1(\alpha):=\left( \begin{array}{cc} \alpha&0 \\ 0& \overline{\alpha}^{-1} \end{array} \right) \mid \alpha\in B^{\times} \right\}. \end{align*} $$

Then $G_1=N_1M_1K_1$ .

We also define subgroups of $G_2$ by

$$ \begin{align*} N_2 &= \left\{n_2(\beta):= \left( \begin{array}{cc} 1&\beta \\ 0&1 \end{array} \right) \mid \beta\in B^-\right\},\\ M_2&=\left\{ d_2(\alpha):=\left( \begin{array}{cc} \alpha&0 \\ 0&\overline{\alpha}^{-1} \end{array} \right) \mid \alpha\in B^{\times} \right\}. \end{align*} $$

Then $G_2=N_2M_2K_2$ .

The Weil representation $r'$ given in Lemma 5.1.1 is reformulated as follows: For $\phi \in \mathcal {S}(B^{(1,2)})$ and $X\in B^{(1,2)}$ ,

$$ \begin{align*} r'(1,g)\phi(X)&=\phi(Xg) \qquad(g\in G_2),\\ r'(d_1(\alpha),1)\phi(X)&=|N(\alpha)|^2\,\phi(\overline{\alpha}X) \qquad(\alpha\in B^{\times}), \\ r'(n_1(b),1)\phi(x,y)& = \psi(-b\,\overline{X}Q\,{}^t\! X)\phi(X) \qquad(b\in \mathbb{Q}_p). \end{align*} $$

10.2.

In this section, we prepare several facts on Hecke algebras.

For $n\in \mathbb {Z}$ , we put

$$ \begin{align*} X_n&=\{\beta\in B^-\mid \mathrm{ord}_{\Pi}(\beta)\ge n\},\\ X_n^0&=\{\beta\in B^-\mid \mathrm{ord}_{\Pi}(\beta)= n\}=X_n\setminus X_{n+1},\\ \mathbf{d}_{1}(n)&=d_1(\Pi^n),\\ \mathbf{d}_{2}(n)&=d_2(\Pi^n). \end{align*} $$

Note that

$$ \begin{align*} |X_n/\pi X_n|&=p^3,\\ |X_n/X_{n+1}|& = \begin{cases} p & (n\,\text{ is even}),\\ p^2 & (n\,\text{ is odd}). \end{cases} \end{align*} $$

Let $\mathcal {T}_{G_1}^1\in \mathcal {H}(G_1,K_1)$ (resp. $\mathcal {T}_{G_2}^1\in \mathcal {H}(G_2,K_2)$ ) be the characteristic function of $T_{G_1}^1=K_1\mathbf {d}_1(1)K_1$ (resp. $T_{G_2}^1=K_2\mathbf {d}_2(1)K_2$ ).

Lemma 10.2.1. We have

$$ \begin{align*} \mathcal{H}(G_1,K_1)&=\mathbb{C}[\mathcal{T}_{G_1}^1],\\ \mathcal{H}(G_2,K_2)&=\mathbb{C}[\mathcal{T}_{G_2}^1]. \end{align*} $$

The following facts are easily verified.

Lemma 10.2.2. We have

$$ \begin{align*} T_{G_1}^1&=\bigsqcup_{b\in \mathbb{Z}_p/\pi\mathbb{Z}_p} \,n_1(b)\mathbf{d}_{1}(1)K_1 \,\sqcup\, \mathbf{d}_{1}(-1)K_1. \end{align*} $$

Lemma 10.2.3. We have

$$ \begin{align*} T_{G_2}^1&=\bigsqcup_{\beta\in X_{-1}/\pi X_{-1}} \,n_2(\beta)\mathbf{d}_{2}(1)K_2 \, \sqcup\, \bigsqcup_{\beta\in X_{-2}^0/X_{-1}}\,n_G(\beta)K\,\sqcup\, \mathbf{d}_{2}(-1)K_2. \end{align*} $$

Recall that the spherical functions $\omega _{1,\chi }\colon G_1\to \mathbb {C}$ and $\omega _{2,\Xi }\colon G_2\to \mathbb {C}$ for $\chi ,\Xi \in X_{\mathrm {unr}}(B^{\times })$ are defined in Section 4.5. Then we have

$$ \begin{align*} (\omega_{1,\chi}*\varphi)(g_1)& :=\int_{G_1}\omega_{1,\chi}(hx^{-1})\varphi(x)dx =\lambda_{1,\chi}(\varphi)\omega_{1,\chi}(g_1)\qquad (g_1\in G_1,\varphi\in\mathcal{H}(G_1,K_1)),\\ (\omega_{2,\Xi}*\Phi)(g_2)&:=\int_{G_2}\omega_{2,\Xi}(g_2y^{-1})\Phi(y)dy =\lambda_{2,\Xi}(\Phi)\omega_{2,\Xi}(g_2)\qquad (g_2\in G_2,\Phi\in\mathcal{H}(G_2,K_2)) \end{align*} $$

with $\lambda _{1,\chi }\in \mathrm {Hom}_{\mathbb {C}\text {-alg}}(\mathcal {H}(G_1,K_1),\mathbb {C})$ and $\lambda _{2,\Xi }\in \mathrm {Hom}_{\mathbb {C}\text {-alg}}(\mathcal {H}(G_2,K_2),\mathbb {C})$ (cf. [Reference Satake16, Section 5.4]). Using Lemmas 10.2.2 and 10.2.3, we get the following.

Lemma 10.2.4. We have

$$ \begin{align*} \lambda_{1,\chi}(\mathcal{T}_{G_1}^1)&=p^{1/2}(\chi(\Pi)+\chi^{-1}(\Pi)),\\ \lambda_{2,\Xi}(\mathcal{T}_{G_2}^1)&=p^{3/2}(\Xi(\Pi)+\Xi^{-1}(\Pi))+p-1. \end{align*} $$

10.3.

Let $\phi _0\in W'$ be the characteristic function of $(\mathcal {O},\mathfrak {P}^{-1})$ and set

$$ \begin{align*} J_{G_1}(X)&= \int_{T_{G_1}^1}\,r'(g_1,1)\phi_0(X)dg_1,\\ J_{G_2}(X)&=\int_{T_{G_2}^1}\,r'(1,g_2)\phi_0(X)dg_2 \end{align*} $$

for $X\in B^{(1,2)}$ . We now state the main results of this section, whose proof is given in Sections 10.4–10.6.

Proposition 10.3.1. We have

(44) $$ \begin{align} J_{G_2}(X)=(p-1)\phi_0(X)+pJ_{G_1}(X)\qquad(X\in B^{(1,2)}). \end{align} $$

Corollary 10.3.2. The $\mathbb {C}$ -algebra homomorphism $\upsilon \colon \mathcal {H}(G_2,K_2)\,\to \,\mathcal {H}(G_1,K_1)$ given by

$$\begin{align*}\upsilon(\mathcal{T}_{G_2}^1)=(p-1)\mathcal{T}_{G_1}^0+p\,\mathcal{T}_{G_1}^1 \qquad(X\in B^{(1,2)}) \end{align*}$$

satisfies (36) (and hence (37)), where $\mathcal {T}_{G_1}^0$ denotes the characteristic function of $K_1$ .

We now prove the following result assuming Corollary 10.3.2.

Proposition 10.3.3. We have

(45) $$ \begin{align} L_p(s,\mathcal{L}(f),\mathrm{std})=\zeta_p(s)L_p(s,f,\mathrm{std}). \end{align} $$

Proof. We take $\chi \in X_{\mathrm {unr}}(B^{\times })$ such that $\omega _{1,p}^f=\omega _{1,\chi }$ and $\Xi \in X_{\mathrm {unr}}(B^{\times })$ such that $\omega _{2,p}^{\mathcal {L}(f)}=\omega _{2,\Xi }$ (see Lemma 4.5.1). An argument similar to the proof of Proposition 9.4.1 shows

$$\begin{align*}\Xi(\Pi)+\Xi^{-1}(\Pi)= \chi(\Pi)+\chi^{-1}(\Pi) \end{align*}$$

and hence

$$\begin{align*}(1-\Xi(\Pi)p^{-s})^{-1}\,(1-\Xi^{-1}(\Pi)p^{-s})^{-1} = (1-\chi(\Pi)p^{-s})^{-1}\,(1-\chi^{-1}(\Pi)p^{-s})^{-1}. \end{align*}$$

The proposition immediately follows from this.

10.4.

The proof of Proposition 10.3.1 is reduced to the following explicit formulas for $J_{G_1}(X)$ and $J_{G_2}(X)$ . We denote by $\sigma $ the characteristic function of $\mathcal {O}$ .

Lemma 10.4.1. For $x,y\in B$ , we have

$$ \begin{align*} & J_{G_1}(x,y)\\ &= \begin{cases} 0 & (\mathrm{ord}_{\Pi}(x)\le -2),\\ p^{-1} \delta(\mathrm{tr}(\overline{x}y)\in\mathbb{Z}_p)\,\sigma(\pi y) & (\mathrm{ord}_{\Pi}(x)=0,-1),\\ p^2\,\sigma(y)+p^{-1} \sigma(\pi y) & (\mathrm{ord}_{\Pi}x\ge 1). \end{cases} \end{align*} $$

Lemma 10.4.2. For $x,y\in B$ , we have

$$ \begin{align*} & J_{G_2}(x,y)\\ &= \begin{cases} 0 & (\mathrm{ord}_{\Pi}(x)\le -2),\\ \delta(\mathrm{tr}(\overline{x}y)\in\mathbb{Z}_p)\,\sigma(\pi y) & (\mathrm{ord}_{\Pi}(x)=-1),\\ \delta(\mathrm{tr}(\overline{x}y)\in\mathbb{Z}_p)\,\sigma(\pi y)+(p-1)\sigma(\Pi y) & (\mathrm{ord}_{\Pi}(x)=0),\\ p^3\,\sigma(y)+(p-1)\sigma(\Pi y)+\sigma(\pi y) & (\mathrm{ord}_{\Pi}x\ge 1). \end{cases} \end{align*} $$

To prove these results, we need the following facts.

Lemma 10.4.3 [Reference Narita12, Lemmas A.2 and A.3].

We have

$$ \begin{align*} [\mathcal{O}\colon \pi\mathcal{O}] &=p^4,\\ [\mathcal{O}\colon \Pi\mathcal{O}] &=p^2,\\ [\mathcal{O}^-\colon \pi\mathcal{O}^-] &=p^3,\\ [\mathcal{O}^-\colon (\Pi\mathcal{O})^-] &=p,\\ [(\Pi\mathcal{O})^-\colon \pi\mathcal{O}^-] &=p^2. \end{align*} $$

Lemma 10.4.4 [Reference Murase and Narita5, Lemma 8.2].

For $x\in B$ , we have

$$ \begin{align*} \sum_{\beta\in X_{-1}/X_0}\,\sigma(x+\beta)&=\sigma(\Pi x),\\ \sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(\Pi^{-1}(x+\beta))& =\delta(\mathrm{tr}(x)\in p\mathbb{Z}_p)\sigma(\Pi x),\\ \sum_{\beta\in X_{-2}/X_{-1}}\,\sigma(\Pi(x+\beta))& =\delta(\mathrm{tr}(x)\in \mathbb{Z}_p)\sigma(\pi x). \end{align*} $$

10.5. The proof of Lemma 10.4.1

Let $x,y\in B$ . By Lemma 10.2.2, we have

$$ \begin{align*} J_{G_1}(x,y)&= \sum_{b\in \mathbb{Z}_p/p\mathbb{Z}_p}\, r'\left(n_1(b)\mathbf{d}_{1}(1),1\right)\,\phi_0(x,y) +r'\left(\mathbf{d}_{1}(-1),1\right)\,\phi_0(x,y)\\ &=p^{-2} \,\sum_{b\in \mathbb{Z}_p/p\mathbb{Z}_p}\,\psi(-b\,\mathrm{tr}(\overline{x}y))\, \phi_0(\overline{\Pi}x,\overline{\Pi}y) +p^2\,\phi_0(\overline{\Pi}^{-1} x,\overline{\Pi}^{-1} y)\\ &= p^{-1}\,\delta(\mathrm{tr}(\overline{x}y)\in\mathbb{Z}_p)\,\sigma(\Pi x)\,\sigma(\pi y) +p^2\,\sigma(\Pi^{-1} x)\,\sigma(y). \end{align*} $$

We have $J_{G_1}(x,y)=0$ if $\mathrm {ord}_{\Pi }(x)\le -2$ . If $\mathrm {ord}_{\Pi }(x)=-1 \text { or }0$ , we have

$$\begin{align*}J_{G_1}(x,y)=p^{-1}\,\delta(\mathrm{tr}(\overline{x}y)\in\mathbb{Z}_p)\,\sigma(\pi y). \end{align*}$$

If $\mathrm {ord}_{\Pi }(x)\ge 1$ , we have

$$\begin{align*}J_{G_1}(x,y)=p^{-1}\,\sigma(\pi y) +p^2\,\sigma(y), \end{align*}$$

which completes the proof of Lemma 10.4.1.

10.6. The proof of Lemma 10.4.2

Lemma 10.6.1. For $x,y\in B$ , we have

$$ \begin{align*} J_{G_2}(x,y)&=J_1(x,y)+J_2(x,y)+J_3(x,y), \end{align*} $$

where

$$ \begin{align*} J_1(x,y)&=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(x\Pi)\, \sigma(y+x\beta),\\ J_2(x,y)&=\sum_{\beta\in X^0_{-2}/X_{-1}}\,\sigma(x)\,\sigma(\Pi(y+x\beta)),\\ J_3(x,y)&=\sigma(\Pi^{-1} x)\,\sigma(\pi y). \end{align*} $$

Proof. By Lemma 10.2.3, we have

$$ \begin{align*} & J_{G_2}(x,y)\\ &= \sum_{\beta\in X_{-1}/\pi X_{-1}}\,r'(1,n_2(\beta)\mathbf{d}_2(1))\phi_0(x,y)\\ &\quad +\sum_{\beta\in X^0_{-2}/X_{-1}}\,r'(1,n_2(\beta))\phi_0(x,y) +r'(1,\mathbf{d}_2(-1))\phi_0(x,y)\\ &= \sum_{\beta\in X_{-1}/\pi X_{-1}}\,\phi_0((x,y)n_2(\beta)\mathbf{d}_2(1)) +\sum_{\beta\in X^0_{-2}/X_{-1}}\,\phi_0((x,y)n_2(\beta)) +\phi_0((x,y) \,\mathbf{d}_2(-1))\\[-1pt] & = \sum_{\beta\in X_{-1}/\pi X_{-1}}\,\phi_0(x\Pi,(y+x\beta)\overline{\Pi}^{-1}) +\sum_{\beta\in X^0_{-2}/X_{-1}}\,\phi_0(x,y+x\beta) +\phi_0(x\Pi^{-1},y\overline{\Pi})\\[-1pt] &=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(x\Pi)\, \sigma(\Pi (y+x\beta)\overline{\Pi}^{-1}) ) +\sum_{\beta\in X^0_{-2}/X_{-1}}\,\sigma(x)\,\sigma(\Pi(y+x\beta)) \\[-1pt] &\quad +\sigma(x\Pi^{-1})\,\sigma(\Pi y\overline{\Pi})\\[-1pt] &=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(x\Pi)\,\sigma(y+x\beta)+ \sum_{\beta\in X^0_{-2}/X_{-1}}\,\sigma(x)\,\sigma(\Pi(y+x\beta)) +\sigma(\Pi^{-1} x)\,\sigma(\pi y). \end{align*} $$

We now prove Lemma 10.4.2.

It is easy to see that $J_{G_2}(x,y)=0$ if $\mathrm {ord}_{\Pi }(x)\le -2$ .

Suppose that $\mathrm {ord}_{\Pi }(x)=-1$ . Then $J_2(x,y)=J_3(x,y)=0$ and

$$ \begin{align*} J_1(x,y)&=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(y+x\beta)\\[-1pt] &=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(\Pi^{-1}(x^{-1} y+\beta))\\[-1pt] &=\delta(\mathrm{tr}(x^{-1} y)\in p\mathbb{Z}_p)\,\sigma(\Pi x^{-1} y) \\[-1pt] & =\delta(\mathrm{tr}(\overline{x} y)\in \mathbb{Z}_p)\,\sigma(\pi y). \end{align*} $$

Suppose that $\mathrm {ord}_{\Pi }(x)=0$ . Then $J_3(x,y)=0$ . We have

$$ \begin{align*} J_1(x,y)&=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(y+x\beta)\\[-1pt] &=\sum_{\beta\in X_{-1}/\pi X_{-1}}\,\sigma(x^{-1} y+\beta)\\[-1pt] &=\sum_{\beta'\in X_{-1}/\pi X_{0}}\,\sum_{\beta"\in X_{0}/\pi X_{1}}\, \sigma(x^{-1} y+\beta'+\beta")\\[-1pt] &=|X_0/X_1|\,\sum_{\beta'\in X_{-1}/\pi X_{0}}\, \sigma(x^{-1} y+\beta')\\[-1pt] &=p\,\sigma(\Pi x^{-1} y) \\ &=p\,\sigma(\Pi y) \end{align*} $$

and

$$ \begin{align*} J_2(x,y)&= \sum_{\beta\in X^0_{-2}/X_{-1}}\,\sigma(\Pi(x^{-1} y+\beta))\\[-1pt] &=\delta(\mathrm{tr}(x^{-1} y)\in\mathbb{Z}_p) \,\sigma(\pi x^{-1} y)-\sigma(\Pi x^{-1} y)\\[-1pt] & =\delta(\mathrm{tr}(\overline{x} y)\in\mathbb{Z}_p) \,\sigma(\pi y)-\sigma(\Pi y). \end{align*} $$

Suppose that $\mathrm {ord}_{\Pi }(x)\ge 1$ . We have

$$ \begin{align*} J_1(x,y)&=|X_{-1}/\pi X_{-1}|\,\sigma(y)=p^3\,\sigma(y), \\[-1pt] J_2(x,y)&=|X^0_{-}/X_{-1}|\,\sigma(\Pi y)=(p-1)\,\sigma(\Pi y), \\[-1pt] J_3(x,y)&=\sigma(\pi y), \end{align*} $$

which completes the proof of Lemma 10.4.2.