Spectral Reciprocity for the product of Rankin-Selberg $L$-functions

Let $F/\mathbb{Q}$ be a fixed number field with ring of intergers ${\mathcal{O}}_{F}$ and discriminant $\Delta_{F}$.

For a place $v$ of $F$, we let $F_{v}$ be a local field which is the completion of $F$ at the place $v$. If $v$ is non-Archimedean, we write ${\mathfrak{o}}_{v}$ for the ring of integers in $F_{v}$ with maximal ideal ${\mathfrak{m}}_{v}$ and uniformizer $\varpi_{v}$. The cardinality of the residue field is $p_{v}:=|{\mathfrak{o}}_{v}/{\mathfrak{m}}_{v}\rvert$. For $s\in{\mathbb{C}}$, we define the local zeta function $\zeta_{F_{v}}(s)$ to be $(1-p_{v}^{-s})^{-1}$ if $v$ is a finite place; $\zeta_{F_{v}}(s)=\pi^{-s/2}\Gamma(s/2)$ if $v$ is real and $\zeta_{F_{v}}(s)=2(2\pi)^{-s}\Gamma(s)$ if $v$ is complex.

The adele ring of $F$ is denoted by ${\mathbb{A}}_{F}$ and its unit group is given by ${\mathbb{A}}^{\times}_{F}$ (idele group). We also let ${\mathbb{A}}^{1}_{F}:=\{x\in{\mathbb{A}}_{F}^{\times}\ :\ |x|=1\}$, where $|\cdot|:{\mathbb{A}}_{F}^{\times}\rightarrow{\mathbb{R}}_{>0}$ is the adelic norm map. Note that ${\mathbb{A}}_{F}^{1}$ is exactly the kernel of the adelic norm map. We also call ${\mathbb{A}}^{1}_{F}$ norm one ideles.

We fix $\psi=\prod_{v}\psi_{v}$ be the additive character with the form as $\psi_{\mathbb{Q}}\circ\text{Tr}_{F/{\mathbb{Q}}},$ where $\text{Tr}_{F/{\mathbb{Q}}}$ is the trace map and $\psi_{\mathbb{Q}}$ is the standard additive character on ${\mathbb{Q}}\setminus{\mathbb{A}}_{\mathbb{Q}}$. For $v<\infty$, we let $d_{v}$ be the conductor of additive character $\psi_{v}$, which is the smallest non-negative integer such that $\psi_{v}$ is trivial on ${\mathfrak{m}}_{v}^{d_{v}}$. In this case, we will have $\Delta_{F}=\prod_{v<\infty}q_{v}^{d_{v}}$. We may also set $d_{v}=0$ when $v$ is Archimedean.

Now we consider some subgroups of ${\mathrm{GL}}(n)$.

Let ${\mathrm{G}}={\mathrm{GL}}(n)$. If $R$ is a field, by definition, ${\mathrm{G}}(R)$ is the group of $n\times n$ matrices with coefficients in $R$ and determinant in $R-\{0\}$. We also define the following standard and also important subgroups

and the Borel subgroup

We let $W$ be the Weyl group of ${\mathrm{G}}$. We write

be the longest Weyl element.

Moreover, for any place $v$, we let ${\mathrm{K}}_{v}$ be the maximal compact subgroup of ${\mathrm{G}}(F_{v})$ which is defined by

We also let ${\mathrm{K}}:=\prod_{v}{\mathrm{K}}_{v}$. If $v<\infty$ and $n\geqslant 0$, we define the congruence subgroup [JPS81] by (This will be used in the computation of local vectors and is useful in the local new-vector theory)

If ${\mathfrak{a}}$ is an ideal of ${\mathcal{O}}_{F}$ with prime decomposition ${\mathfrak{a}}=\prod_{v<\infty}{\mathfrak{p}}_{v}^{f_{v}({\mathfrak{a}})}$ where ${\mathfrak{p}}_{v}$ is the unique prime (also maximal) ideal corresponding to the finite place $v$, then we define

Now we have to normalize the measures we need.

At each place $v$, $dx_{v}$ denotes a self-dual measure on $F_{v}$ with respect to the standard additive character $\psi_{v}$. If $v<\infty$, $dx_{v}$ gives a Haar measure on $F_{v}$ which gives the volume $q_{v}^{-d_{v}/2}$ to the integer ring ${\mathfrak{o}}_{v}$. If $v$ is real, the measure $dx_{v}$ is the standard Lebesgue measure on ${\mathbb{R}}$. If $v$ is complex, the measure $dx_{v}$ is the multiplication of two and the standard Lebesgue measure on ${\mathbb{C}}$. We define $dx:=\prod_{v}dx_{v}$ on ${\mathbb{A}}_{F}$. Moreover, we will take $d^{\times}x_{v}:=\zeta_{F_{v}}(1)\frac{dx_{v}}{|x_{v}|}$ as the Haar measure on the multiplicative group $F_{v}^{\times}$ and we let $d^{\times}x:=\prod_{v}d^{\times}x_{v}$ be the Haar measure on the idele group ${\mathbb{A}}^{\times}_{F}$.

We equip ${\mathrm{K}}_{v}$ with the probability Haar measure $dk_{v}$. In other word, the volume of ${\mathrm{K}}_{v}$ equals to one.

Using the Iwasawa decomposition on ${\mathrm{G}}(F_{v})$ which gives ${\mathrm{G}}(F_{v})={\mathrm{Z}}_{n}(F_{v}){\mathrm{N}}_{n}(F_{v}){\mathrm{A}}_{n}(F_{v}){\mathrm{K}}_{v}$, a Haar measure on ${\mathrm{G}}(F_{v})$ can be given by

where $\delta(A_{n})=\delta(y_{1},y_{2},\cdots,y_{n-1})$ is the modular character defined on ${\mathrm{A}}_{n}(F_{v})$. The measure on the adelic points of the subgroups in ${\mathrm{GL}}(n)$ are just the product of the local measures defined above. We also denote by $dg$ the quotient measure on the space

with total volume $\mathrm{vol}({\mathrm{X}})<\infty$.

We recall some basic background of Whittaker functions.

We start from the generic representations. Let $\pi$ be a global genreic automorphic representation of ${\mathrm{GL}}_{n}({\mathbb{A}}_{F})$ and let $\phi\in\pi$ be a generic automorphic form. Let $W_{\phi}:{\mathrm{GL}}_{n}({\mathbb{A}}_{F})\rightarrow{\mathbb{C}}$ be the Whittaker function of $\phi$ which is given by

We will give the definition of the additive character $\psi_{1}$ later.

Since $\pi$ is generic, the Whittaker function does not vanish. The integral defined above (2.3) is absolute convergent since the integral domain ${\mathrm{N}}_{n}(F)\backslash{\mathrm{N}}_{n}({\mathbb{A}}_{F})$ is compact and the integral is moreover uniformly comvergent on any compact subsets in ${\mathrm{GL}}_{n}({\mathbb{A}}_{F})$.

By changing variables, we note that $W_{\phi}(ng)=\psi_{1}(n)W_{\phi}(g)$ for all $n\in{\mathrm{N}}_{n}({\mathbb{A}}_{F})$, where $\psi_{1}$ is a nontrivial standard additive character from ${\mathrm{N}}_{n}(F)\backslash{\mathrm{N}}_{n}({\mathbb{A}}_{F})$ to ${\mathbb{C}}$. In our paper, we will choose that $\psi_{1}(n):=\psi\left(\sum_{i=1}^{n-1}x_{i,i+1}\right)$, where $n=(x_{i,j})\in{\mathrm{N}}_{n}({\mathbb{A}}_{F})$. Here $\psi$ is a standard additive character from ${\mathbb{A}}_{F}/F$ to ${\mathbb{C}}$ which is defined in Section 2.1.

We also have the following Fourier series expansion (see [Cog07, Theorem 1.1]) when we further assume that $\phi$ is a cusp form. Therefore, a cusp form is automatically generic by the following equation.

We can also define local Whittaker functions for smooth generic admissible representations of ${\mathrm{GL}}_{n}$ over local fields $F_{v}$. We have the following well known decomposition theorem. If $\pi$ is a generic smooth irreducible admissible representation of ${\mathrm{GL}}_{n}({\mathbb{A}}_{F})$, then we know that $\pi$ factors as a restricted tensor product by $\pi\simeq\otimes^{\prime}_{v}\pi_{v}$. For each $v$, $\pi_{v}$ is a local generic smmoth irreducible admissible representation of ${\mathrm{GL}}_{n}(F_{v})$. For each local place $v$, we can define local Whittaker functions. For every $\phi\in\pi$, if we write $\phi=\otimes^{\prime}_{v}\phi_{v}$, then we have the decomposition

In fact, we know that the map $\phi\mapsto W_{\phi}$ intertwines the space of $\pi$ and the space

which is called the Whittaker model of $\pi$. We can similarly define the local Whittaker models ${\mathcal{W}}(\pi_{v},\psi_{1,v})$.

It is also important to consider Whittaker functions with respect to the character $\psi_{1}^{\prime}=\psi_{1}^{-1}$ since it will appear in the integral representations for ${\mathrm{GL}}(n+1)\times{\mathrm{GL}}(n)$ Rankin-Selberg $L$-functions (see Section 3.2). It also has close relation with the Whittaker model of the contragredient representation $\widetilde{\pi}$ when the representation $\pi$ is unitarizable (see [FLO12, Appendix A]). Locally, we can define $W^{\prime}_{\phi_{v}}$ by simply replacing $\psi_{1,v}$ by $\psi^{\prime}_{1,v}=\psi_{1,v}^{-1}$. Globally, we can define $W^{\prime}_{\phi}$ by replacing $\psi_{1}$ by $\psi_{1}^{\prime}$. Moreover, we have the following equation

For $W\in{\mathcal{W}}(\pi_{v},\psi_{1,v})$, the corresponding Whittaker model ${\mathcal{W}}(\widetilde{\pi}_{v},\psi_{1,v}^{-1})$ for contragredient representation is given by $\widetilde{W}(g):=W\left(w_{0}(g^{t})^{-1}\right)$, where $w_{0}$ is the longest Weyl element in ${\mathrm{GL}}(n)$ and $(g^{t})^{-1}$ means the transpose inverse, therefore it is an involution in ${\mathrm{GL}}(n)$. The global Whittaker model ${\mathcal{W}}(\widetilde{\pi},\psi_{1}^{-1})$ is defined in the same way. If a local generic smooth irreducible admissible representation $\pi_{v}$ is unramified, this is equivalent to say that there exists a vector which is right invariant by the action of the maximal compact subgroup of ${\mathrm{G}}(F_{v})$ in the space of $\pi$. We call such a vector spherical vector and spherical vectors are unique up to multiplication by scalars. We say that a vector $\phi_{v}\in\pi_{v}$ is normalized spherical if it is spherical. Moreover, its related Whittaker function $W_{\phi_{v}}$ satisfies $W_{\phi_{v}}(I_{n\times n})=1$.

Now we consider the $L^{2}$ space $L^{2}({\mathrm{X}})$ which is the Hilbert space of complex valued square integrable functions with the domain ${\mathrm{X}}$. The $L^{2}$-norm is defined by

For any $\varphi_{1},\varphi_{2}\in L^{2}({\mathrm{X}})$, we have

An important closed subspace of $L^{2}({\mathrm{X}})$ is the closed subspace of cusp forms. We let $L_{\mathrm{cusp}}^{2}({\mathrm{X}})$ be the closed subspace of cusp forms. A cusp form is the function $\varphi\in L^{2}({\mathrm{X}})$ with the following additional equation

for almost all $g\in{\mathrm{GL}}_{n}({\mathbb{A}}_{F})$.

The group ${\mathrm{G}}({\mathbb{A}}_{F})$ acts by right translations on both spaces $L^{2}({\mathrm{X}})$ and $L_{\mathrm{cusp}}^{2}({\mathrm{X}})$ and the corresponding representation is unitary with respect to the norm in (2.7). The decomposition theorem of automorphic representations is well known, which states that each irreducible component $\pi$ factors as a restricted tensor product $\pi=\otimes^{\prime}_{v}\pi_{v}$ for all places $k$, where $\pi_{v}$ are irreducible and unitary representations of the local groups ${\mathrm{G}}(F_{v})$. The spectral decomposition is established in 1970s by Gelbart, Jacquet and Langlands which gives the following orthogonal decomposition (see [MW95] for more details)

Here the closed subspace $L^{2}_{\mathrm{cusp}}({\mathrm{X}})$ decomposes as a direct sum of irreducible ${\mathrm{G}}({\mathbb{A}}_{F})$-representations which are called the cuspidal automorphic representations. $L^{2}_{\mathrm{res}}({\mathrm{X}})$ is called the residue spectrum which is the direct sum of all residue automorphic representations of $L^{2}({\mathrm{X}})$. Finally, the continuous part $L^{2}_{\mathrm{cont}}({\mathrm{X}})$ is a direct integral of irreducible ${\mathrm{G}}({\mathbb{A}}_{F})$-representations and it is expressed by the Eisenstein series.

For any ideal ${\mathfrak{a}}$ of ${\mathcal{O}}_{F}$, we let $L^{2}({\mathrm{X}},{\mathfrak{a}}):=L^{2}({\mathrm{X}})^{{\mathrm{K}}_{0}({\mathfrak{a}})}$ be the closed subspace of level ${\mathfrak{a}}$ automorphic forms. This is the closed subspace of $L^{2}$ functions that are invariant under the subgroup ${\mathrm{K}}_{0}({\mathfrak{a}})$.

We now have the following spectral orthogonal decomposition with the level restriction:

Now we consider automorphic representations. Let $\hat{{\mathrm{X}}}$ be the isomorphism class of unitary irreducible automorphic representations which will appear in the spectral decomposition of $L^{2}({\mathrm{X}})$. Since we will later use the integral representation of ${\mathrm{GL}}(n+1)\times{\mathrm{GL}}(n)$ Rankin-Selberg $L$-functions, we will only focus on the unitary irreducible generic automorphic representations. We will consider $\hat{{\mathrm{X}}}_{\text{gen}}$ be the subset of $\hat{{\mathrm{X}}}$ which is the isomorphism class of generic representations in $\hat{{\mathrm{X}}}$, which is the unitary irreducible automorphic representation class that have (unique) Whittaker models. We fix an automorphic Plancherel measure $d\mu_{\text{aut}}$ on $\hat{{\mathrm{X}}}$ which is compatible with the Haar measure on ${\mathrm{X}}$.

Fortunately, we have the Langlands classification for $\hat{{\mathrm{X}}}_{\text{gen}}$ (see [CPS94] and [MW95]). We take a partition $n=r_{1}+\dots+r_{k}$. Let $\pi_{j}$ be a unitary cuspidal automorphic representation for ${\mathrm{GL}}_{r_{j}}({\mathbb{A}}_{F})$ (If $r_{j}=1$, we simply take $\pi_{j}$ to be a unitary Hecke character). We now consider the unitary induced representation $\Pi$ from the Levi subgroup ${\mathrm{GL}}(r_{1})\times\dots\times{\mathrm{GL}}(r_{k})\subseteq{\mathrm{GL}}(n)$ to ${\mathrm{GL}}(n)$ with the tensor product representation $\pi_{1}\otimes\dots\otimes\pi_{k}$. There exists a unique irreducible constituent of $\Pi$ which we denote by the isobaric sum $\pi_{1}\boxplus\dots\boxplus\pi_{k}$. Then, Langlands classification says that every element in $\hat{{\mathrm{X}}}_{\text{gen}}$ is isomorphic to such an isobaric sum.

Moreover, we know that all residual automorphic representations in $L^{2}_{\mathrm{res}}({\mathrm{X}})$ is not generic (see [JL13] Proposition 2.1).

Now we note that all the unitary generic Eisenstein series will have the form of the isobaric sum $\pi_{1}|\cdot\rvert^{it_{1}}\boxplus\dots\boxplus\pi_{k}|\cdot\rvert^{it_{k}}$, where $t_{1},t_{2},\cdots,t_{k}\in{\mathbb{R}}$. We will write

with parameters $t_{1},\cdots,t_{k}$ ($k\geq 2$).

We have the following abstract spectral decomposition formula in ${\mathrm{GL}}(n)$ case (see [MV10, Section 2.2]). The discrete part is generated by cusp forms and residue representations. The continuous spectrum part which is expressed by Eisenstein series is complicated. However, we know that it depends on the partition of the positive integer $n$ (see Section 2.5).

We need to recall integral representations of ${\mathrm{GL}}(n+1)\times{\mathrm{GL}}(n)$ ($n\geq 3$) $L$-functions.

The theory is quite similar to the adelic Hecke-Jacquet-Langlands’ theory [God18] of twisted $L$-functions for ${\mathrm{GL}}_{2}\times{\mathrm{GL}}_{1}$. We let $\Pi$ be irreducible automorphic representations of ${\mathrm{GL}}_{n+1}({\mathbb{A}})$. Let $\Phi\in\Pi$ be an automorphic form. Let $\pi$ be irreducible automorphic representations of ${\mathrm{GL}}_{n}({\mathbb{A}})$ and let $\phi\in\pi$ be an automorphic form. We first assume that $\Phi$ is a cusp form and is of rapid decay. Therefore, for every $s\in{\mathbb{C}}$, we can consider the following period integral

which defines an entire function of $s$ and is bounded on vertical strips.

From the Whittaker-Fourier expansion of cusp forms (2.4) [Cog07, Theorem 1.1], if $\Phi$ is a cusp form, we will have (for ${\mathrm{Re}}(s)$ large enough)

where the global zeta integral is given by

The following result can be found in [Cog07] [JPS79] [JPS83] [JS90] [Jac09].

One of the key ingredients in our paper is the following generalization of integral representation on Rankin-Selberg $L$-functions for ${\mathrm{GL}}(n+1)\times{\mathrm{GL}}(n)$ given by Ichino and Yamana [IY15].

We can briefly summarize the proof of Theorem 3.7 now. The proof of Theorem 3.7 is a combination of above decomposition formula 3.1 and an identity between two periods. Later, we will relate the period to moments of certain $L$-functions. The proof of an identity between two periods is a rather simple matrix computation and is really important to abstract pre-spectral reciprocity formula.

Suppose that $\Phi\in C^{\infty}(Z_{n+1}({\mathbb{A}}){\mathrm{GL}}_{n+1}(F)\backslash{\mathrm{GL}}_{n+1}({\mathbb{A}}))$ is a cuspidal automorphic form. Therefore, it is of rapid decay.

Then, we can define the projection by

Since $\Phi$ is of rapid decay, the above average projection map over the center ${\mathcal{A}}_{s}\Phi$ is well-defined for every complex number $s$ and is again of rapid decay in terms of $h_{n}\in{\mathrm{GL}}(n)$. Moreover, we may easily check that ${\mathcal{A}}_{s}\Phi$ is invariant under the action of the center ${\mathrm{Z}}_{n}({\mathbb{A}})$.

We also give the following definition of the period

whenever it is converges. Here $\phi$ is an automorphic form defined on ${\mathrm{GL}}_{n}({\mathbb{A}})$ and $\varphi$ is a fixed everywhere unramified automorphic form for ${\mathrm{GL}}_{n-1}({\mathbb{A}})$. Moreover, we assume that $\varphi$ is invariant under the center $Z({\mathbb{A}})$, therefore its central character is trivial. If $n=2$, we see that $\varphi=1$ which is the trivial character. We can now state the abstract spectral reciprocity formula.