2.1 Global Fourier–Jacobi periods

Let $(W_{2},q_{2})$ be a $2m$ -dimensional symplectic space over $F$ . We choose a basis $\{e_{m}^{\ast },\ldots ,e_{1}^{\ast },e_{1},\ldots ,e_{m}\}$ of $W_{2}$ so that $q_{2}(e_{i}^{\ast },e_{j})=\unicode[STIX]{x1D6FF}_{ij}$ . For $1\leqslant i\leqslant m$ , let $R_{i}=\langle e_{m-i+1},\ldots ,e_{m}\rangle$ and $R_{i}^{\ast }=\langle e_{m}^{\ast },\ldots ,e_{m-i+1}^{\ast }\rangle$ be isotropic subspaces of $W_{2}$ . Put $R_{0}=R_{0}^{\ast }=\{0\}$ . Let $0\leqslant r\leqslant m$ be an integer and put $n=m-r$ and $(W_{0},q_{0})$ the orthogonal complement of $R_{r}+R_{r}^{\ast }$ . We define $(W_{1},q_{1})=W_{0}+\langle e_{n+1},e_{n+1}^{\ast }\rangle$ . Let $G_{i}=\operatorname{Sp}(W_{i})$ and $\widetilde{G_{i}}=\operatorname{Mp}(W_{i})$ .

Let $0\leqslant i\leqslant n$ be an integer. Let $P_{i}$ be the parabolic subgroup of $G_{2}$ stabilizing the flag

with the Levi decomposition $P_{i}=M_{i}N_{i}$ . Here and below in this article, the notation $P=MN$ signifies that $M$ is the Levi subgroup and $N$ is the unipotent radical of $P$ . We denote by $W^{i}$ the orthogonal complement of $R_{i}+R_{i}^{\ast }$ and $G^{i}=\operatorname{Sp}(W^{i})$ . Then $M_{i}=G^{i}\times \operatorname{GL}_{1}^{i}$ . Let $\unicode[STIX]{x1D713}_{m}$ be the character of $N_{m}$ defined by

Let $\unicode[STIX]{x1D713}_{i}$ be the restriction of $\unicode[STIX]{x1D713}_{m}$ to $N_{i}$ .

Let $H=H(W_{0})$ be the Heisenberg group attached to the symplectic space $W_{0}$ . By definition, $H=W_{0}\ltimes F$ and the group law is given by

The group $H$ embeds in $G_{2}$ as a subgroup of $G_{1}$ and $H=G_{1}\cap N_{r}$ , $N_{r}=N_{r-1}H$ . Let $L=\langle e_{1},\ldots ,e_{n}\rangle$ and $L^{\ast }=\langle e_{n}^{\ast },\ldots ,e_{1}^{\ast }\rangle$ . Then $W_{0}=L+L^{\ast }$ is a complete polarization. We sometimes write an element $h\in H$ as $h(l+l^{\ast },t)$ where $l\in L$ , $l^{\ast }\in L^{\ast }$ and $t\in F$ . Let $v$ be a place of $F$ and $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}$ be the Weil representation of $H(F_{v})$ which is realized on ${\mathcal{S}}(L^{\ast }(F_{v}))$ . It is defined by

This is the unique irreducible infinite-dimensional representation of $H(F_{v})$ whose central character is $\unicode[STIX]{x1D713}_{v}$ . It induces an action of $\widetilde{G_{0}}(F_{v})$ on ${\mathcal{S}}(L^{\ast }(F_{v}))$ . We denote the joint action of $H(F_{v})\rtimes \widetilde{G_{0}}(F_{v})$ on ${\mathcal{S}}(L^{\ast }(F_{v}))$ again by $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}$ . We take the convention that if $W_{0}=\{0\}$ , then $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}=\unicode[STIX]{x1D713}_{v}$ .

Taking restricted tensor product of the Weil representations $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}$ , we obtain a global Weil representation $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$ of $H(\mathbb{A}_{F})\rtimes \widetilde{G_{0}}(\mathbb{A}_{F})$ which is realized on ${\mathcal{S}}(L^{\ast }(\mathbb{A}_{F}))$ . We define the theta series

We now talk about automorphic representations. There are two cases.

Case  $\operatorname{Mp}$ . Let $\unicode[STIX]{x1D70B}_{2}=\bigotimes \unicode[STIX]{x1D70B}_{2,v}$ be an irreducible cuspidal genuine automorphic representation of $\widetilde{G_{2}}(\mathbb{A}_{F})$ and $\unicode[STIX]{x1D70B}_{0}=\bigotimes \unicode[STIX]{x1D70B}_{0,v}$ be an irreducible cuspidal automorphic representation of $G_{0}(\mathbb{A}_{F})$ .

Case  $\operatorname{Sp}$ . Let $\unicode[STIX]{x1D70B}_{2}=\bigotimes \unicode[STIX]{x1D70B}_{2,v}$ be an irreducible cuspidal automorphic representation of $G_{2}(\mathbb{A}_{F})$ and $\unicode[STIX]{x1D70B}_{0}=\bigotimes \unicode[STIX]{x1D70B}_{0,v}$ be an irreducible cuspidal genuine automorphic representation of $\widetilde{G_{0}}(\mathbb{A}_{F})$ .

Let $S$ be a sufficiently large finite set of places of $F$ containing all archimedean places and finite places whose residue characteristic is two, such that $\unicode[STIX]{x1D70B}_{2,v}$ and $\unicode[STIX]{x1D70B}_{0,v}$ are both unramified and the conductor of $\unicode[STIX]{x1D713}_{v}$ is $\mathfrak{o}_{F,v}$ if $v\not \in S$ . Let $(\unicode[STIX]{x1D6FC}_{1,v},\ldots ,\unicode[STIX]{x1D6FC}_{m,v})$ and $(\unicode[STIX]{x1D6FD}_{1,v},\ldots ,\unicode[STIX]{x1D6FD}_{n,v})$ be the Satake parameters of $\unicode[STIX]{x1D70B}_{2,v}$ and $\unicode[STIX]{x1D70B}_{0,v}$ , respectively. Put

and

We then define the tensor product $L$ -function

The partial $L$ -function is convergent for $\Re s\gg 0$ . We denote by $L_{\unicode[STIX]{x1D713}_{v}}(s,\unicode[STIX]{x1D70B}_{i,v},\operatorname{Ad})$ and $L_{\unicode[STIX]{x1D713}}^{S}(s,\unicode[STIX]{x1D70B}_{i},\operatorname{Ad})=\prod _{v\not \in S}L_{\unicode[STIX]{x1D713}_{v}}(s,\unicode[STIX]{x1D70B}_{i,v},\operatorname{Ad})$ the (local and partial) adjoint $L$ -functions of $\unicode[STIX]{x1D70B}_{i}$ . If $\unicode[STIX]{x1D70B}_{i}$ is an automorphic representation of the metaplectic group (respectively symplectic group), then they depend (respectively do not depend) on $\unicode[STIX]{x1D713}$ . We include the subscript $\unicode[STIX]{x1D713}$ in both cases to unify notation. We assume that these $L$ -functions can be meromorphically continued to the whole complex plane.

Let $\unicode[STIX]{x1D711}_{2}\in \unicode[STIX]{x1D70B}_{2}$ , $\unicode[STIX]{x1D711}_{0}\in \unicode[STIX]{x1D70B}_{0}$ and $\unicode[STIX]{x1D719}\in {\mathcal{S}}(L^{\ast }(\mathbb{A}_{F}))$ . Define

The measures $du$ and $dh$ are the self-dual measures on $N_{r-1}(\mathbb{A}_{F})$ and $H(\mathbb{A}_{F})$ , respectively. The measure $dg_{0}$ is the Tamagawa measures on $G_{0}(\mathbb{A}_{F})$ .

2.2 Local Fourier–Jacobi periods

We fix a Haar measure $dg_{0,v}$ on $G_{0}(F_{v})$ for each $v$ such that the volume of $G_{0}(\mathfrak{o}_{v})$ equals one for almost all $v$ . Then there is a constant $C_{0}$ such that $dg_{0}=C_{0}\prod _{v}\,dg_{0,v}$ . Following [Reference Ichino and IkedaII10], we call $C_{0}$ the measure constant.

Let ${\mathcal{B}}_{\unicode[STIX]{x1D70B}_{i}}$ ( $i=0,2$ ) be the canonical bilinear pairing between $\unicode[STIX]{x1D70B}_{i}$ and $\unicode[STIX]{x1D70B}_{i}^{\vee }$ defined by

We fix a bilinear pairing ${\mathcal{B}}_{\unicode[STIX]{x1D70B}_{i,v}}$ between $\unicode[STIX]{x1D70B}_{i,v}$ and $\unicode[STIX]{x1D70B}_{i,v}^{\vee }$ for each place $v$ such that ${\mathcal{B}}_{\unicode[STIX]{x1D70B}_{i}}=\prod _{v}{\mathcal{B}}_{\unicode[STIX]{x1D70B}_{i,v}}$ . Put $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{i,v},\unicode[STIX]{x1D711}_{i,v}^{\vee }}(g)={\mathcal{B}}_{\unicode[STIX]{x1D70B}_{i,v}}(\unicode[STIX]{x1D70B}_{i,v}(g)\unicode[STIX]{x1D711}_{i,v},\unicode[STIX]{x1D711}_{i,v}^{\vee })$ if $\unicode[STIX]{x1D711}_{i,v}\in \unicode[STIX]{x1D70B}_{i,v}$ and $\unicode[STIX]{x1D711}_{i,v}^{\vee }\in \unicode[STIX]{x1D70B}_{i,v}^{\vee }$ .

The contragredient representation of $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$ is $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}^{-1}}$ (again realized on ${\mathcal{S}}(L^{\ast }(\mathbb{A}_{F}))$ ) and there is a canonical pairing between $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}$ and $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}^{-1}}$ given by

where the measure $dl^{\ast }$ is the self-dual measure on $L^{\ast }(\mathbb{A}_{F})$ . Similarly, for any place $v$ , there is a canonical pairing between $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}$ and $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}^{-1}}$ given by

where the measure $dl^{\ast }$ is the self-dual measure on $L^{\ast }(F_{v})$ . Then ${\mathcal{B}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}}=\prod _{v}{\mathcal{B}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}}\!.$ Put $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D719}_{v},\unicode[STIX]{x1D719}_{v}^{\vee }}(g)={\mathcal{B}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}}(\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}_{v}}(g)\unicode[STIX]{x1D719}_{v},\unicode[STIX]{x1D719}_{v}^{\vee })$ .

We now fix a place $v$ of $F$ . Recall that the group $P_{m}$ of $G_{2}$ is a minimal parabolic subgroup which is contained in $P_{r-1}$ . For any real number $\unicode[STIX]{x1D6FE}$ or $\unicode[STIX]{x1D6FE}=-\infty$ , define

For any $\unicode[STIX]{x1D6FE}\geqslant -\infty$ , we define $N_{i,\unicode[STIX]{x1D6FE}}=N_{i}(F_{v})\cap N_{m,\unicode[STIX]{x1D6FE}}$ . Define

where $h\in H(F_{v})$ and $g_{0}\in G_{0}(F_{v})$ in the case $\operatorname{Sp}$ (respectively $g_{0}\in \widetilde{G_{0}}(F_{v})$ in the case $\operatorname{Mp}$ ). Define

for $\unicode[STIX]{x1D711}_{i,v}\in \unicode[STIX]{x1D70B}_{i,v},\unicode[STIX]{x1D711}_{i,v}^{\vee }\in \unicode[STIX]{x1D70B}_{i,v}^{\vee }$ , $\unicode[STIX]{x1D719}_{v},\unicode[STIX]{x1D719}_{v}^{\vee }\in {\mathcal{S}}(L^{\ast }(F_{v}))$ . If $r\leqslant 1$ , then it is to be understood that ${\mathcal{F}}_{\unicode[STIX]{x1D713}}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{2,v},\unicode[STIX]{x1D711}_{2,v}^{\vee }}=\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{2,v},\unicode[STIX]{x1D711}_{2,v}^{\vee }}$ . Moreover, if $r=0$ , then it is to be understood that the integral over $H(F_{v})$ is void.

If $\unicode[STIX]{x1D70B}_{i,v}$ is unitary, then we may identify $\unicode[STIX]{x1D70B}_{i,v}^{\vee }$ with $\overline{\unicode[STIX]{x1D70B}_{i,v}}$ . We then define

These two propositions will be proved in § 3.

We now consider the unramified situation. Note first that the symplectic spaces $W_{i}$ , the isotropic subspaces $R_{i}$ and hence the groups $G_{i}$ are naturally defined over $\mathfrak{o}_{F}$ . Let $S$ be a sufficiently large finite set of places of $F$ containing all archimedean places and finite places whose residue characteristic is two, such that if $v\not \in S$ , then the following conditions hold.

1. (i) The conductor of $\unicode[STIX]{x1D713}_{v}$ is $\mathfrak{o}_{F,v}$ .

2. (ii) We have $\unicode[STIX]{x1D719}_{v}=\unicode[STIX]{x1D719}_{v}^{\vee }=\operatorname{1}_{L^{\ast }(\mathfrak{o}_{F,v})}$ .

3. (iii) For $i=0,2$ , $\unicode[STIX]{x1D711}_{i,v}$ and $\unicode[STIX]{x1D711}_{i,v}^{\vee }$ are fixed by $G_{i}(\mathfrak{o}_{F,v})$ and satisfy ${\mathcal{B}}_{\unicode[STIX]{x1D70B}_{i,v}}(\unicode[STIX]{x1D711}_{i,v},\unicode[STIX]{x1D711}_{i,v}^{\vee })=1$ . In particular, the representations $\unicode[STIX]{x1D70B}_{i,v}$ and $\unicode[STIX]{x1D70B}_{i,v}^{\vee }$ are unramified.

4. (iv) We have $\int _{G_{0}(\mathfrak{o}_{F,v})}\,dg_{0,v}=1$ .

We will prove this proposition in § 4. Note that in this proposition, we do not assume that the representations $\unicode[STIX]{x1D70B}_{2,v}$ and $\unicode[STIX]{x1D70B}_{0,v}$ are tempered.

2.3 Conjectures

Following [Reference Ichino and IkedaII10] and [Reference LiuLiu16], we say that the representations $\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{0}$ are almost locally generic if for almost all places $v$ of $F$ , the local components $\unicode[STIX]{x1D70B}_{2,v}$ and $\unicode[STIX]{x1D70B}_{0,v}$ are generic. Suppose that we are in the case of $\operatorname{Mp}$ . As explained in [Reference Ichino and IkedaII10], the automorphic representations $\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{0}$ should come from some elliptic Arthur parameters

where $L_{F}$ is the (hypothetical) Langlands group of $F$ . If $\unicode[STIX]{x1D70B}_{i}$ is tempered, then $\unicode[STIX]{x1D6F9}_{i}$ is trivial on $\operatorname{SL}_{2}(\mathbb{C})$ . It is believed (Ramanujan conjecture) that almost locally generic representations are tempered. We define $S_{\unicode[STIX]{x1D70B}_{2}}$ (respectively $S_{\unicode[STIX]{x1D70B}_{0}}$ ) to be the centralizer of the image of $\unicode[STIX]{x1D6F9}_{2}$ in $\widehat{\widetilde{G_{2}}}$ (respectively $\widehat{G_{0}}$ ). They are finite abelian 2-groups. In the case $\operatorname{Sp}$ , we have the same discussion, except that we replace $\widetilde{G_{2}}$ by $G_{2}$ and replace $G_{0}$ by $\widetilde{G_{0}}$ .

We end this section by writing Conjecture 2.3.1(3) in an equivalent form which does not involve the finite set $S$ . We may define the completed $L$ -functions

The actual definition of the local Euler factor of these $L$ -functions is not essential to us since Conjecture 2.3.1 does not depend on the definition of these Euler factors. Put

and let ${\mathcal{L}}_{v}$ be its local Euler factor evaluated at $s=\frac{1}{2}$ at the place $v$ . Define

Then the identity (2.3.1) can be rewritten as

The product is convergent since there are only finitely many terms which do not equal to one. This is an equality of elements in

Note that by [Reference SunSun12, Reference Sun and ZhuSZ12, Reference Liu and SunLS13], this space is at most one dimensional. So we know a priori that there is a constant $C$ such that

The point of Conjecture 2.3.1 is thus to compute the constant $C$ .

3.1 Preliminaries

We recall some basic estimates in this subsection. We follow [Reference Ichino and IkedaII10, § 4] rather closely.

Let $G$ be a reductive group over $F$ . Let $A_{G}$ be a maximal split subtorus of $G$ , $M_{0}$ the centralizer of $A_{G}$ in $G$ . We fix a minimal parabolic subgroup $P_{0}$ of $G$ with the Levi decomposition $P_{0}=M_{0}N_{0}$ . Let $\unicode[STIX]{x1D6E5}$ be the set of simple roots of $(P_{0},A_{G})$ . Let $\unicode[STIX]{x1D6FF}_{P_{0}}$ be the modulus character of $P_{0}$ . Let

We fix a special maximal compact subgroup $K$ of $G$ . Then we have a Cartan decomposition $G=KA_{G}^{+}K$ . We also have the Iwasawa decomposition

Let $f$ and $f^{\prime }$ be two nonnegative functions on $G$ . We say that $f\ll f^{\prime }$ if there is a constant $C$ such that $f(g)\leqslant Cf^{\prime }(g)$ for all $g\in G$ . We say that $f\sim f^{\prime }$ if $f\ll f^{\prime }$ and $f^{\prime }\ll f$ . In this case we say that $f$ and $f^{\prime }$ are equivalent.

For any function $f\in L^{1}(G)$ ,

where $\unicode[STIX]{x1D708}(m)$ is a positive function on $A_{G}^{+}$ such that

Let $\operatorname{1}$ be the trivial representation of $M_{0}$ and let $e(g)=\unicode[STIX]{x1D6FF}_{P_{0}}(m_{0}(g))^{1/2}$ be an element in $\operatorname{Ind}_{P_{0}}^{G}\operatorname{1}$ . Let $dk$ be the measure on $K$ such that $\operatorname{vol}K=1$ . We define the Harish-Chandra function

This function is bi- $K$ -invariant. This function depends on the choice of $K$ . However, different choices of $K$ give equivalent functions on $G$ . So this choice will not affect our estimates.

We define a height function on $G$ . We fix an embedding $\unicode[STIX]{x1D70F}:G\rightarrow \operatorname{GL}_{n}$ . Write $\unicode[STIX]{x1D70F}(g)=(a_{ij})$ and $\unicode[STIX]{x1D70F}(g^{-1})=(b_{ij})$ . Define

There is a positive real number $d$ such that

Now let $\unicode[STIX]{x1D70B}$ be an irreducible admissible tempered representation of $G$ . Let $\unicode[STIX]{x1D6F7}$ be a smooth matrix coefficient of $G$ . Then there is a constant $B$ such that

This is classical and is called the weak inequality when $\unicode[STIX]{x1D6F7}$ is $K$ -finite and due to [Reference SunSun09] when $\unicode[STIX]{x1D6F7}$ is smooth.

We finally assume that $G=\operatorname{Mp}(2n)$ . This is not an algebraic group, but it behaves in many ways like an algebraic group. In particular, we have a Cartan decomposition for $G$ , i.e.  $G=KA_{G}^{+}K$ where $K$ is the inverse image of a special maximal compact subgroup of $\operatorname{Sp}(2n)$ (e.g.  $\operatorname{Sp}(2n)(\mathfrak{o}_{F})$ if $F$ is non-archimedean and $\operatorname{U}(n)$ is $F$ is archimedean) and $A_{G}^{+}$ is the inverse image of $A_{\operatorname{Sp}(2n)}^{+}$ in $G$ . We define $\unicode[STIX]{x1D6EF}_{G}=\unicode[STIX]{x1D6EF}_{\operatorname{Sp}(2n)}\circ p$ where $p:G\rightarrow \operatorname{Sp}(2n)$ is the canonical projection. Then the weak inequality holds for tempered representations of $G$ .

3.2 Some estimates

3.3 Proof of Proposition 2.2.1

The case $r=0$ is rather straightforward. Indeed, in this case $G_{0}=G_{1}=G_{2}$ . By the weak inequality, we only need to prove that

is absolutely convergent. By the Cartan decomposition and the estimates (3.1.2) and (3.1.4), the convergence is reduced to the convergence of

This is clear. Proposition 2.2.1 is thus proved when $r=0$ .

The case $r\geqslant 2$ follows from the case $r=1$ by Lemma 3.2.6.

We now treat the case $r=1$ . In this case $G_{2}=G_{1}$ . The defining integral of $\unicode[STIX]{x1D6FC}$ reduces to

Since $\unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D70B}_{0}$ are both tempered, we need to prove that

is convergent.

Let $g_{0}=k_{1}d(a)k_{2}$ be the Cartan decomposition of $g_{0}$ where $a=(a_{1},\ldots ,a_{n})\in (F^{\times })^{n}$ with $|a_{s}|\leqslant \cdots \leqslant |a_{1}|\leqslant 1$ . We first estimate $|\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}^{\vee }}(hd(a))|$ . We claim that there is a function $f$ on $L^{\ast }$ so that $f(l^{\ast })P(l^{\ast })$ is bounded for any polynomial function $P$ on $L^{\ast }$ , such that

Indeed

Thus, to prove (3.3.1), it is enough to prove that for any polynomial function $P$ on $L^{\ast }$ ,

We have

Since $P$ is a polynomial function, we may choose a sufficiently large $N$ , such that

where $x=(x_{1},\ldots ,x_{n})\in L^{\ast }$ . We have the second inequality because $|a_{i}|\leqslant 1$ for all $i$ . Then

We have thus proved (3.3.1).

By (3.1.1), to prove the convergence of the defining integral of $\unicode[STIX]{x1D6FC}$ , it is enough to show the convergence of

Then Lemma 3.2.5 reduces the convergence of this integral to the case $r=0$ .

3.4 Proof of Proposition 2.2.2

We are going to use the notation in the proof of Lemma 3.2.6, one paragraph before (3.2.1). To simplify notation, we write $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{i}}=\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{i},\unicode[STIX]{x1D711}_{i}}$ , $i=0,2$ and $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D719}}=\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D719},\unicode[STIX]{x1D719}}$ .

To facilitate understanding, we divide the proof into several steps.

Step 1. The goal is to reduce the Proposition to the inequality (3.4.1).

In order to prove that $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D711}_{2},\unicode[STIX]{x1D711}_{0},\unicode[STIX]{x1D719})\geqslant 0$ , it is enough to show that for any function $f\in {\mathcal{C}}_{c}^{\infty }(T)$ , we have

We denote this expression by $I$ . Since $f(t)$ is compactly supported, by Fubini’s theorem, we have

We denote the integral in the parentheses by $\mathit{II}$ . It follows from Lemma 3.2.6 that

is bounded by a constant which depends continuously on $\unicode[STIX]{x1D713}_{r-1}$ . Since $f$ is compactly supported on $T$ , we can choose this constant to be independent of $t$ (but depends on $hg_{0}$ ). Then by the Lebesgue dominated convergence theorem, we have

Moreover, the double integral on the right-hand side is absolutely convergent. We can thus interchange the order of integration. Finally, we conclude that

Let $f_{1}(t)=f(t)|t_{1}\cdots t_{r-1}|^{-1/2}\in {\mathcal{C}}_{c}^{\infty }(T)$ . Recall that the map $t\mapsto \unicode[STIX]{x1D713}^{t}$ identifies $T$ with $\widehat{N_{\dagger }}^{\text{reg}}$ which is an open subset of $\widehat{N_{\dagger }}$ consisting of generic characters. The measure $|t_{1}\cdots t_{r-1}|\,dt$ is identified with the self-dual measure on $\widehat{N_{\dagger }}$ under this map. In this way, $f$ , as well as $f_{1}$ , are viewed as compactly supported functions on $\widehat{N_{\dagger }}$ and we may talk about their Fourier transform $\widehat{f}$ and $\widehat{f_{1}}$ which are functions on $N_{\dagger }$ . The Fourier transform of a product of two functions is the convolution of the Fourier transforms of these two functions. We conclude that

Therefore,

where $\unicode[STIX]{x1D70B}_{2}(\widehat{f_{1}})\unicode[STIX]{x1D711}_{2}=\int _{N_{\dagger }}\widehat{f_{1}}(n)\unicode[STIX]{x1D70B}_{2}(n)\unicode[STIX]{x1D711}_{2}\,dn$ . This expression makes sense since $\widehat{f_{1}}$ is a Schwartz function on $N_{\dagger }$ . Thus to show that $I\geqslant 0$ , it remains to show that

Actually, we will show that

for all smooth vectors $\unicode[STIX]{x1D711}_{2}\in \unicode[STIX]{x1D70B}_{2}$ and $\unicode[STIX]{x1D711}_{0}\in \unicode[STIX]{x1D70B}_{0}$ . Unlike the proof of [Reference Ichino and IkedaII10, Proposition 1.1] and [Reference LiuLiu16, Theorem 2.1(2)], we cannot apply [Reference HeHe03, Theorem 2.1] directly, as $G_{2}\times HG_{0}$ is not reductive. However, we are going to mimic the proof of [Reference HeHe03, Theorem 2.1] to prove (3.4.1).

Step 2. The goal is to reduce (3.4.1) to the case of $K$ -finite vectors.

We claim that it is enough to prove (3.4.1) for a $K_{2}$ -finite (respectively $K_{0}$ -finite) vector $\unicode[STIX]{x1D711}_{2}\in \unicode[STIX]{x1D70B}_{2}$ (respectively $\unicode[STIX]{x1D711}_{0}\in \unicode[STIX]{x1D70B}_{0}$ ). This is only an issue when $F$ is archimedean. So we assume temporarily that $F$ is archimedean. Since $K_{2}$ -finite vectors are dense in the space of smooth vectors in $\unicode[STIX]{x1D70B}_{2}$ , we may choose a sequence of $K_{2}$ -finite vectors $\unicode[STIX]{x1D711}_{2}^{(i)}$ which is convergent to $\unicode[STIX]{x1D711}_{2}$ . It follows that $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{2}}$ is approximated pointwisely by $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{2}^{(i)}}$ . Moreover, by [Reference SunSun09], there exists an element $X$ in the Lie algebra of $G_{2}$ , which depends on $K_{2}$ only, such that

Since $\unicode[STIX]{x1D711}_{2}^{(i)}$ is convergent to $\unicode[STIX]{x1D711}_{2}$ , we see that $|\unicode[STIX]{x1D70B}_{2}(X)\unicode[STIX]{x1D711}_{2}^{(i)}|^{2}$ is convergent to $|\unicode[STIX]{x1D70B}_{2}(X)\unicode[STIX]{x1D711}_{2}|^{2}$ . In particular, it is bounded by some constant which is independent of $\unicode[STIX]{x1D711}_{2}^{(i)}$ . Similarly we choose a sequence $\unicode[STIX]{x1D711}_{0}^{(i)}$ of $K_{0}$ -finite vectors in $\unicode[STIX]{x1D70B}_{0}$ which approximate $\unicode[STIX]{x1D711}_{0}$ . Since

is absolutely convergent, by the Lebesgue dominated convergence theorem

So the positivity in (3.4.1) for smooth vectors follows from the positivity for $K$ -finite vectors.

From now on, we assume that $\unicode[STIX]{x1D711}_{2}$ and $\unicode[STIX]{x1D711}_{0}$ in (3.4.1) are $K_{2}$ -finite and $K_{0}$ -finite, respectively. We come back to the situation $F$ being an arbitrary local field of characteristic zero.

Step 3. The goal is to reduce (3.4.1) to the inequality (3.4.2).

Since $\unicode[STIX]{x1D70B}_{2}$ is tempered, by (the proof of) [Reference HeHe03, Theorem 2.1] (which is also valid when $F$ is non-archimedean), one can find a sequence of compactly supported continuous functions $f_{2,j}^{(i)}$ on $G_{2}$ and a sequence of positive real numbers $a_{j}^{(i)}$ , $j=1,\ldots ,s_{i}$ , such that $\sum _{j=1}^{s_{i}}a_{j}^{(i)}=1$ and the functions

approximate $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{2}}(g_{2}^{\prime })$ pointwisely. Moreover, there is a constant $C_{2}$ , such that

Similarly, we can find a sequence of compactly supported continuous functions $f_{0,j}^{(i)}$ on $G_{0}$ and a sequence of positive real numbers $b_{j}^{(i)}$ , $j=1,\ldots ,k_{i}$ , such that $\sum _{j=1}^{k_{i}}b_{j}^{(i)}=1$ and the functions

approximate $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D711}_{0}}(g_{0}^{\prime })$ pointwisely. Moreover, there is a constant $C_{0}$ , such that

Since the integral

is absolutely convergent, by the Lebesgue dominated convergence theorem, to prove (3.4.1), it is enough to prove that for any $i,j$ ,

We denote the left-hand side by $Q$ . Note that this integral is absolutely convergent. To simplify notation, we write $f_{2}=f_{2,j}^{(i)}$ and $f_{0}=f_{0,j}^{(i)}$ .

Step 4. Proof of (3.4.2).

We can write the inner product on ${\mathcal{S}}(L^{\ast })$ as

Using this expression of the inner product, we have

Note that we have used the fact the pairing ${\mathcal{B}}_{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D713}}}$ is $G_{0}$ -invariant.

We make the following change of variables

Then

where $L+F$ embeds in $H\times H$ diagonally.

Finally we decompose the integral over $G_{2}$ as

We conclude that

We have thus proved (3.4.2) and, hence, Proposition 2.2.2.

3.5 Regularization via stable unipotent integral

In this subsection, we give an alternative but equivalent way to define the linear functional $\unicode[STIX]{x1D6FC}$ when $F$ is non-archimedean following [Reference Lapid and MaoLM15a, Reference LiuLiu16]. This definition is better for the unramified computation and is valid for nontempered representations. In this subsection, $F$ is always assumed to be non-archimedean.

Let $N$ be a unipotent group over $F$ and $f$ a smooth function on $N$ . We say that $f$ is compactly supported on average if there are compact subsets $U$ and $U^{\prime }$ of $N$ , such that $\text{L}(\unicode[STIX]{x1D6FF}_{U^{\prime }})\text{R}(\unicode[STIX]{x1D6FF}_{U})f$ is compactly supported. Here $\unicode[STIX]{x1D6FF}_{U}$ stands for the Dirac measure on $U$ , i.e.  $\unicode[STIX]{x1D6FF}_{U}=(\operatorname{vol}U)^{-1}\operatorname{1}_{U}$ , and

If $f$ is compactly supported on average, we then define

This is called the stable integral of $f$ on $N$ . The definition is independent of the choice of $U$ and  $U^{\prime }$ .

We denote temporarily by $G$ a reductive group over $F$ . Let $P_{\text{min}}=M_{\text{min}}N_{\text{min}}$ be a fixed minimal parabolic subgroup of $G$ . Let $P=MN\supset P_{\text{min}}$ be a parabolic subgroup of $G$ . Let $\unicode[STIX]{x1D6F9}$ be a generic character of $N$ , i.e. the stabilizer of $\unicode[STIX]{x1D6F9}$ in $M_{\text{min}}$ is the center of $M_{\text{min}}$ . Let $\unicode[STIX]{x1D70B}$ be an irreducible admissible representation of $G$ and $\unicode[STIX]{x1D6F7}$ a matrix coefficient of $\unicode[STIX]{x1D70B}$ . Then we have the following result.

Now let $G=\operatorname{Mp}(2n)$ . Then Proposition 3.5.1 still holds. The same proof as in [Reference LiuLiu16, Proposition 3.3] goes through as it uses only the Bruhat decomposition and Jacquet’s subrepresentation theorem, which are valid for $G$ .

Now we retain the notation $G_{0}$ , $G_{1}$ , $G_{2}$ , etc. Let $\unicode[STIX]{x1D6F7}$ be a matrix coefficient on $G_{2}$ (respectively $\widetilde{G_{2}}$ ). Define

which is a function on $G_{2}$ (respectively $\widetilde{G_{2}}$ ). This definition makes sense because of Proposition 3.5.1.

Thanks to Lemma 3.5.2, if $F$ is non-archimedean, then we may use ${\mathcal{F}}_{\unicode[STIX]{x1D713}}^{\text{st}}$ instead of ${\mathcal{F}}_{\unicode[STIX]{x1D713}}$ in the definition of the local linear form $\unicode[STIX]{x1D6FC}$ . We will not distinguish ${\mathcal{F}}_{\unicode[STIX]{x1D713}}^{\text{st}}$ and ${\mathcal{F}}_{\unicode[STIX]{x1D713}}$ from now on and write just ${\mathcal{F}}_{\unicode[STIX]{x1D713}}$ .

4.1 Setup

For $i=0,1,2$ , let $B_{i}=P_{m}\cap G_{i}=T_{i}U_{i}$ be the upper triangular Borel subgroup of $G_{i}$ where $T_{i}$ is the diagonal maximal torus of $G_{i}$ . We have a hyperspecial subgroup $K_{i}=\operatorname{Sp}(W_{i})(\mathfrak{o}_{F})$ of $\operatorname{Sp}(W_{i})$ . Recall that the twofold cover $\widetilde{G_{i}}\rightarrow G_{i}$ splits uniquely over $K_{i}$ . We can thus view $K_{i}$ as a subgroup of $\widetilde{G_{i}}$ . Let $\unicode[STIX]{x1D6EF}$ (respectively $\unicode[STIX]{x1D709}$ ) be an unramified character of $T_{2}$ (respectively $T_{0}$ ). In the case $\operatorname{Sp}$ , we consider the unramified principal series $\unicode[STIX]{x1D70B}_{2}=I(\unicode[STIX]{x1D6EF})$ of $G_{2}$ and $\unicode[STIX]{x1D70B}_{0}=I(\unicode[STIX]{x1D709})$ of $\widetilde{G_{0}}$ . In the case $\operatorname{Mp}$ , we consider the unramified principal series $\unicode[STIX]{x1D70B}_{2}=I(\unicode[STIX]{x1D6EF})$ of $\widetilde{G_{2}}$ and $\unicode[STIX]{x1D70B}_{0}=I(\unicode[STIX]{x1D709})$ of $G_{0}$ . Note that the unramified principal series representation of the metaplectic group depends on the additive character $\unicode[STIX]{x1D713}$ , even though this is not reflected in the notation. We frequently identify $\unicode[STIX]{x1D6EF}$ with an element in $\mathbb{C}^{m}$ which we also denote by $\unicode[STIX]{x1D6EF}=(\unicode[STIX]{x1D6EF}_{1},\ldots ,\unicode[STIX]{x1D6EF}_{m})$ , the correspondence being given by

Similarly we identify $\unicode[STIX]{x1D709}$ with an element in $\mathbb{C}^{n}$ . The contragredient of $\unicode[STIX]{x1D70B}_{2}$ (respectively $\unicode[STIX]{x1D70B}_{0}$ ) is $I(\unicode[STIX]{x1D6EF}^{-1})$ (respectively $I(\unicode[STIX]{x1D709}^{-1})$ ). Let $f_{\unicode[STIX]{x1D6EF}}\in I(\unicode[STIX]{x1D6EF})$ , $f_{\unicode[STIX]{x1D6EF}^{-1}}\in I(\unicode[STIX]{x1D6EF}^{-1})$ (respectively $f_{\unicode[STIX]{x1D709}}\in I(\unicode[STIX]{x1D709})$ , $f_{\unicode[STIX]{x1D709}^{-1}}\in I(\unicode[STIX]{x1D709}^{-1})$ ) be the $K_{2}$ -fixed (respectively $K_{0}$ -fixed) elements with $f_{\unicode[STIX]{x1D6EF}}(1)=f_{\unicode[STIX]{x1D6EF}^{-1}}(1)=1$ (respectively $f_{\unicode[STIX]{x1D709}}(1)=f_{\unicode[STIX]{x1D709}^{-1}}(1)=1$ ). Let

and

Then $\unicode[STIX]{x1D6FC}(f_{\unicode[STIX]{x1D6EF}},f_{\unicode[STIX]{x1D6EF}^{-1}},f_{\unicode[STIX]{x1D709}},f_{\unicode[STIX]{x1D709}^{-1}},\unicode[STIX]{x1D719},\unicode[STIX]{x1D719})=I(1,\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713})$ .

Let $J=H\rtimes G_{0}$ and $\widetilde{J}=H\rtimes \widetilde{G_{0}}$ . We define the Borel subgroup $B_{J}$ (respectively $B_{\widetilde{J}}$ ) of $J$ (respectively $\widetilde{J}$ ) as a subgroup of $J$ (respectively $\widetilde{J}$ ) consisting of elements of the form $hb_{0}$ where $b_{0}\in B_{0}$ (respectively $\widetilde{B_{0}}$ , the inverse image of $B_{0}$ in $\widetilde{G_{0}}$ ) and $h\in H$ is of the form $h(l,t)$ , $l\in L$ . We define the unramified principal series representation of $J$ (respectively $\widetilde{J}$ ) as

respectively

where $\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D713}}(b_{0})=\unicode[STIX]{x1D709}(\operatorname{diag}[t_{n},\ldots ,t_{1},t_{1}^{-1},\ldots ,t_{n}^{-1}])\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D713}}(t_{1}\cdots t_{n})$ and $t_{n},\ldots ,t_{1},t_{1}^{-1},\ldots ,t_{n}^{-1}$ are diagonal entries of $b_{0}$ .

The group $J$ (respectively $\widetilde{J}$ ) acts on $I^{J}(\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}})$ (respectively $I^{\widetilde{J}}(\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}})$ ) via the right translation. Let $K_{J}=J\cap K_{1}$ . There is a canonical $J$ (respectively $\widetilde{J}$ )-invariant pairing given by

where $f\in I^{J}(\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}}),f^{\vee }\in I^{J}(\unicode[STIX]{x1D709}^{-1},\unicode[STIX]{x1D713})$ (respectively $f\in I^{\widetilde{J}}(\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}}),f^{\vee }\in I^{\widetilde{J}}(\unicode[STIX]{x1D709}^{-1},\unicode[STIX]{x1D713})$ ).

In the case $\operatorname{Mp}$ , there is a canonical inner product preserving isomorphism

where $f_{\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}}}(hg_{0})=\unicode[STIX]{x1D714}_{\overline{\unicode[STIX]{x1D713}}}(hg_{0})\unicode[STIX]{x1D719}(0)f_{\unicode[STIX]{x1D709}}(g_{0})$ , $h\in H$ and $g_{0}\in \widetilde{G_{0}}$ . In the case $\operatorname{Sp}$ , there is a canonical inner product preserving isomorphism

where $f_{\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}}}(hg_{0})=\unicode[STIX]{x1D714}_{\overline{\unicode[STIX]{x1D713}}}(h\unicode[STIX]{x1D704}(g_{0}))\unicode[STIX]{x1D719}(0)f_{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D704}(g_{0}))$ . Analogous isomorphism also holds in the case $\operatorname{Mp}$ .

For the ease of the exposition, we slightly modify our notation in the case $\operatorname{Mp}$ for the rest of this section. For $i=0,1,2$ , we put $G_{i}=\operatorname{Mp}(W_{i})$ and $B_{i}$ the standard Borel subgroup of $G_{i}$ . Denote by $J=H\rtimes \operatorname{Mp}(W_{0})$ , which is a subgroup of $G_{1}$ , and $B_{J}$ its Borel subgroup. We denote by $K_{i}=\operatorname{Sp}(W_{i})(\mathfrak{o}_{F})$ a hyperspecial maximal subgroup of $\operatorname{Sp}(W_{i})$ . The metaplectic cover $\operatorname{Mp}(W_{i})\rightarrow \operatorname{Sp}(W_{i})$ splits canonically over $K_{i}$ , so we view $K_{i}$ as a compact (but not maximal) subgroup of $G_{i}$ and an element in $K_{i}$ is naturally viewed as an element in $G_{i}$ . Let $K_{J}=K_{1}\cap J$ . The subgroup $P_{i}=M_{i}N_{i}$ ( $i=1,\ldots ,r-1$ ) is a parabolic subgroup of $\operatorname{Sp}(W_{2})$ as before. The metaplectic double cover splits canonically over $N_{i}$ , so we consider $N_{i}$ as subgroups of $G_{2}$ . By the Weyl group of $\operatorname{Mp}(W_{i})$ , we mean the Weyl group of $\operatorname{Sp}(W_{i})$ . We let

be representatives of the longest elements in the Weyl groups $W_{G_{2}}$ , $W_{G_{1}}$ and $W_{G_{0}}$ , respectively. They are viewed as elements in $G_{2}$ , $G_{1}$ and $G_{0}$ , respectively.

For $(\unicode[STIX]{x1D6EF}_{1},\ldots ,\unicode[STIX]{x1D6EF}_{m})\in \mathbb{C}^{m}$ and $(\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n})\in \mathbb{C}^{n}$ , we denote by $\unicode[STIX]{x1D6EF}$ and $\unicode[STIX]{x1D709}$ the genuine character of $B_{2}$ and $B_{0}$ , respectively, defined by

We have the unramified principal series representation $I(\unicode[STIX]{x1D6EF})$ of $G_{2}$ and $I(\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}})$ of $J$ . We let $f_{\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}}}$ be the $K_{J}$ fixed element in $I(\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}})$ such that $f_{\unicode[STIX]{x1D709},\overline{\unicode[STIX]{x1D713}}}(1)=1$ . We will need to integrate over $\operatorname{Mp}(W_{0})$ . For this, we pick a measure $dx$ on $\operatorname{Mp}(W_{0})$ , such that for any $f\in {\mathcal{C}}_{c}^{\infty }(\operatorname{Sp}(W_{0}))$ , we have $\int _{\operatorname{Sp}(W_{0})}f(g)\,dg=\int _{\operatorname{Mp}(W_{0})}f(x)\,dx$ . When integrating over $K_{i}$ ’s or $K_{J}$ , we always use the measure so that the volume of the domain of the integration is one.

With this modification of notation, the integral $I(g_{2},\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713})$ in both cases $\operatorname{Mp}$ and $\operatorname{Sp}$ can be written as

4.2 Reduction steps: $r\geqslant 1$

We distinguish two cases: $r=0$ and $r\geqslant 1$ . We treat the case $r\geqslant 1$ first.

Let ${\dot{w}}=w_{1,\text{long}}^{-1}w_{2,\text{long}}$ be a representative of the longest element in $W_{G_{1}}\backslash W_{G_{2}}$ .

By Lemma 4.2.1, we have

Let

Thanks to this lemma, we can define a function $Y_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ on $G_{2}$ with the following properties:

1. (i) $Y_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}(b_{2}g_{2}h(l,t)b_{0}u)=(\unicode[STIX]{x1D6EF}^{-1}\unicode[STIX]{x1D6FF}_{B_{2}}^{1/2})(b_{2})(\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FF}_{B_{J}}^{-1/2})(b_{0})\overline{\unicode[STIX]{x1D713}(t)\unicode[STIX]{x1D713}_{r-1}(u)}Y_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}(g_{2})$ for any $b_{2}\in B_{2}$ , $b_{0}\in B_{0}$ , $l\in L$ and $u\in N_{r-1}$ ;

2. (ii) the support of $Y_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ is $B_{2}\unicode[STIX]{x1D702}(N_{r-1}\rtimes B_{J})$ ;

3. (iii) $Y_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}(\unicode[STIX]{x1D702})=1$ .

The space of functions that satisfy the first two conditions is one dimensional by Lemma 4.2.2. We have

for $b_{2}\in B_{2}$ , $b_{0}\in B_{0}$ , $l\in L$ and $u\in N_{r-1}$ . We define a function $T_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ on $G_{2}$ as

If the defining integral of $T_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ is convergent, then we have

We assume that the defining integral of $T_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ is convergent for the moment. This will be proved later. It follows that

Define

Then we have

4.3 Reduction steps: $r=0$

We now treat the case $r=0$ .

The integral we need to compute is

We define

Similar to Lemma 4.2.2, it is straightforward to prove the following lemma.

We define a function $Y_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ on $J$ which is supported on $B_{J}\unicode[STIX]{x1D702}B_{0}$ by

We define the function $T_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ on $J$ by

and the function $S_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ by

It follows that

We now prove the convergence of the defining integral of $T_{\unicode[STIX]{x1D6EF},\unicode[STIX]{x1D709},\unicode[STIX]{x1D713}}$ and $S_{\unicode[STIX]{x1D6EF}^{-1},\unicode[STIX]{x1D709}^{-1},\unicode[STIX]{x1D713}^{-1}}$ . Assume that $r\geqslant 0$ .