1.1.1

In some sense, this paper is a sequel of [Reference Beuzart-Plessis, Chaudouard and ZydorBPCZ22], where we proved the global Gan-Gross-Prasad (see [Reference Gan, Gross and PrasadGGP12, section 24]) and the Ichino-Ikeda conjectures for a product of unitary groups $U(n)\times U(n+1)$ (see [Reference Ichino and IkedaII10] and [Reference HarrisHar14]). The goal of the present paper is two-fold: first, we state and prove an extension of these two conjectures to the case of some Eisenstein series. Second, we show that this extension, when applied to some specific Eisenstein series, implies the global Gan-Gross-Prasad conjecture and its refinement à la Ichino-Ikeda for general Bessel periods of unitary groups. To state our results, we first review the notion of Arthur parameter.

1.1.2 Hermitian Arthur parameter.

Let $E/F$ be a quadratic extension of number fields and c be the nontrivial element of the Galois group $\operatorname {\mathrm {Gal}}(E/F)$ . Let $\mathbb {A}$ be the ring of adèles of F. Let $n\geqslant 1$ be an integer. Let $G_n$ be the group of automorphims of the E-vector space $E^n$ . We view $G_n$ as an F-group by Weil restriction. For an automorphic representation $\Pi $ of $G_n(\mathbb {A})$ , we denote by $\Pi ^*$ its conjugate-dual. Let us introduce some definitions. A discrete Hermitian Arthur parameter of $G_n$ is an irreducible automorphic representation $\Pi $ of $G_n(\mathbb {A})$ such that

• • $\Pi $ is isomorphic to the full induced representation $\operatorname {\mathrm {Ind}}_Q^{G_n}(\Pi _1\boxtimes \ldots \boxtimes \Pi _r)$ , where Q is a parabolic subgroup of $G_n$ with Levi factor $G_{n_1}\times \ldots \times G_{n_r}$ where $n_1+\ldots +n_r=n$ ;

• • $\Pi _i$ is a conjugate self-dual cuspidal automorphic representation of $G_{n_i}(\mathbb {A})$ , and the Asai L-function $L(s,\Pi _i,\mathrm {As}^{(-1)^{n+1}})$ has a pole at $s=1$ for $1\leqslant i\leqslant r$ .

• • the representations $\Pi _i$ are mutually non-isomorphic for $1\leqslant i\leqslant r$ ;

The integer r and the representations $(\Pi _i)_{1\leqslant i \leqslant r}$ are unique (up to a permutation). We set $S_\Pi =(\mathbb {Z}/2\mathbb {Z})^{r}$ .

For our purpose, we need more general Arthur parameters of $G_n$ , which we call regular Hermitian Arthur parameters and which are by definition the automorphic representations $\Pi $ of $G_n$ such that

• • $\Pi $ is isomorphic to the full induced representation $\operatorname {\mathrm {Ind}}_Q^{G_n}(\Pi _1\boxtimes \ldots \boxtimes \Pi _r \boxtimes \Pi _0\boxtimes \Pi _r^* \boxtimes \ldots \boxtimes \Pi _1^*)$ , where Q is a parabolic subgroup of $G_n$ with Levi factor $M_Q=G_{n_1}\times \ldots \times G_{n_r}\times G_{n_0}\times G_{n_r}\times \ldots \times G_{n_1},$ where $n_0+2(n_1+\ldots +n_r)=n$ ;

• • $\Pi _0$ is a discrete Hermitian Arthur parameter of $G_{n_0}$ ;

• • $\Pi _i$ is a cuspidal automorphic representation of $G_{n_i}(\mathbb {A})$ (with character central trivial on $A_{G_{n_i}}^\infty $ ) for $1\leqslant i\leqslant r$ ;

• • the representations $\Pi _1,\ldots ,\Pi _r,\Pi _1^*,\ldots ,\Pi _r^*$ are mutually non-isomorphic .

The representation $\Pi _0$ is uniquely determined by $\Pi $ and is called the discrete component of $\Pi $ . We set

$$ \begin{align*} S_\Pi=S_{\Pi_0}. \end{align*} $$

The parabolic subgroup Q depends on the ordering on the representations $\Pi _1,\ldots ,\Pi _r,\Pi _1^*,\ldots ,\Pi _r^*$ : we fix one.

Let $\mathfrak {a}_{Q,\mathbb {C}}^*$ be the complex vector space of unramified characters of $Q(\mathbb {A})$ . We have the real subspaces $\mathfrak {a}_Q^*$ and $i\mathfrak {a}_Q^*$ , respectively, of real and unitary characters. Let w be the permutation that exchanges the two blocks $G_{n_i}$ corresponding to $\Pi _i$ and $\Pi _i^*$ for all $1\leqslant i\leqslant r$ . We set

$$ \begin{align*}\mathfrak{a}_{\Pi,\mathbb{C}}^*=\{\lambda\in \mathfrak{a}_{Q,\mathbb{C}}^*\mid w\lambda=-\lambda\}.\end{align*} $$

For any $\lambda \in \mathfrak {a}_{\Pi ,\mathbb {C}}^*$ , we define $\Pi _\lambda $ as the full induced representation

$$ \begin{align*} \operatorname{\mathrm{Ind}}_Q^{G_n}\left((\Pi_1\boxtimes \ldots \boxtimes \Pi_r \boxtimes \Pi_0\boxtimes \Pi_r^* \boxtimes \ldots \boxtimes \Pi_1^*)\otimes\lambda\right). \end{align*} $$

If $\lambda \in i\mathfrak {a}_\Pi ^*=\mathfrak {a}_{\Pi ,\mathbb {C}}^*\cap i \mathfrak {a}_Q^*$ , then $\Pi _\lambda $ is irreducible.

1.1.3 Unitary groups and (weak) base change

For any integer $n\geqslant 1$ , let $\mathcal {H}_n$ be the set of isomorphism classes of nondegenerate c-Hermitian spaces h over E of rank n. We identify any $h \in \mathcal {H}_n$ with a representative, and we denote by $U(h)$ its automorphism group. Let $h\in \mathcal {H}$ and $P\subset U(h)$ be a parabolic subgroup with Levi factor $M_P$ . There exist a decomposition $n_0+2(n_1+\ldots +n_r)=n$ and $h_{n_0} \in \mathcal {H}_{n_0}$ such that $M_P$ is identified with $G_{n_1}\times \ldots \times G_{n_r} \times U(h_{n_0})$ . Let $\sigma $ be a cuspidal automorphic subrepresentation of $M_P(\mathbb {A})$ (with central character trivial on the central subgroup $A_P^\infty $ defined in §2.1.6). Accordingly, we have $\sigma =\Pi _1\boxtimes \ldots \boxtimes \Pi _r\boxtimes \sigma _0$ with $\Pi _i$ a cuspidal automorphic representation of $G_{n_i}$ (with central character trivial on $A_{G_{n_i}}^\infty $ ).

We shall say that a regular Hermitian Arthur parameter $\Pi $ of $G_n$ is a weak base-change of $(P,\sigma )$ if there exist a parabolic subgroup Q of $G_n$ with Levi factor $M_Q=G_{n_1}\times \ldots \times G_{n_r}\times G_{n_0}\times G_{n_r}\times \ldots \times G_{n_1}$ and a discrete Hermitian Arthur parameter $\Pi _0$ of $G_{n_0}$ such that

1. 1. $\Pi $ is isomorphic to the full induced representation $\operatorname {\mathrm {Ind}}_Q^{G_n}(\Pi _1\boxtimes \ldots \boxtimes \Pi _r \boxtimes \Pi _0\boxtimes \Pi _r^* \boxtimes \ldots \boxtimes \Pi _1^*)$ ;

2. 2. for almost all places of F that split in E, the local component $\Pi _{0,v}$ is the split local base change of $\sigma _{0,v}$ .

Note that this implies that the representations $\Pi _1,\ldots ,\Pi _r,\Pi _1^*,\ldots ,\Pi _r^*$ are mutually non-isomorphic and that $\Pi _0$ is in fact the discrete component of $\Pi $ . If condition 2 above is satisfied, we shall also say that $\Pi _0$ is the weak base change of $\sigma _0$ .

If $\Pi $ is a weak base-change of $(P,\sigma )$ , we can naturally identify the space $\mathfrak {a}_{P,\mathbb {C}}^*$ of unramified characters of $P(\mathbb {A})$ with $\mathfrak {a}_{\Pi ,\mathbb {C}}^*$ , and so we will not distinguish between the two spaces. Thus, for $\lambda \in \mathfrak {a}_{\Pi ,\mathbb {C}}^*$ , we can consider the full induced representation $\Sigma _\lambda =\operatorname {\mathrm {Ind}}_P^{U_h}(\sigma \otimes \lambda )$ .

1.1.4

We can extend the notions above to the case of a product. Let $n,n'\geqslant 1$ be integers. A regular Hermitian Arthur parameter of $G_{n}\times G_{n'}$ is then an automorphic representation of the form $\Pi =\Pi _n\boxtimes \Pi _{n'}$ , where $\Pi _k$ is a regular Hermitian Arthur parameter of $G_k$ for $k=n,n'$ . Then we set $S_\Pi =S_{\Pi _n}\times S_{\Pi _{n'}}$ and $\mathfrak {a}_{\Pi ,\mathbb {C}}^*=\mathfrak {a}_{\Pi _n,\mathbb {C}}^*\times \mathfrak {a}_{\Pi _{n'},\mathbb {C}}^* $ etc. For $\lambda =(\lambda _n,\lambda _{n'})\in \mathfrak {a}_{\Pi ,\mathbb {C}}^*$ , we set $\Pi _\lambda =\Pi _{n,\lambda _n}\boxtimes \Pi _{n',\lambda _{n'}}$ . A parameter $\Pi $ is discrete if both $\Pi _n$ and $\Pi _{n'}$ are discrete.

Let $h\in \mathcal {H}_n$ and $h'\in \mathcal {H}_{n'}$ . Let $P=P_n\times P_{n'}$ be a parabolic subgroup of $U(h)\times U(h')$ . We say that a regular Hermitian Arthur parameter $\Pi =\Pi _n\boxtimes \Pi _{n'}$ of $G_n\times G_{n'}$ is a weak base-change of $(P,\sigma )$ if $\Pi _n$ and $\Pi _{n'}$ are, respectively, weak base-changes of $(P_n,\sigma _n)$ and $(P_{n'},\sigma _{n'})$ , where $\sigma =\sigma _n\boxtimes \sigma _{n'}$ .

1.2.1 Corank $1$ and regular Hermitian Arthur parameter

Let $n\geqslant 1$ . Consider the ‘corank $1$ ’ case $G=G_n\times G_{n+1}$ . Let $\Pi =\Pi _n\boxtimes \Pi _{n+1}$ be a regular Hermitian Arthur parameter of G. We can write $\Pi _k=\operatorname {\mathrm {Ind}}_{Q_k}^{G_k}(\Pi _{1,k}\boxtimes \ldots \boxtimes \Pi _{r_k,k})$ for some parabolic subgroup $Q_k\subset G_k$ for $k=n,n+1$ and some cuspidal automorphic representations of $G_{n_{i,k}}(\mathbb {A})$ with $n_{1,k}+\ldots +n_{r_k,k}=k$ . We shall say that the parameter $\Pi $ is H-regular if for all $1\leqslant i\leqslant r_n$ and $1\leqslant j\leqslant r_{n+1}$ , the representation $\Pi _{i,n}$ is not isomorphic to the contragredient of $\Pi _{j,n+1}$ .

On the unitary side, let $h_0\in \mathcal {H}_1$ be the element of rank $1$ given by the norm $N_{E/F}$ . Then we attach to any $h\in \mathcal {H}_n$ the following algebraic groups over F:

• • the product of unitary groups $U_h=U(h)\times U(h\oplus h_0)$ , where $h\oplus h_0$ denotes the orthogonal sum;

• • the unitary group $U^{\prime }_h$ of automorphisms of h viewed as a subgroup of $U_h$ by the obvious diagonal embedding.

1.2.2

Let $P=M_PN_P\subset U_h$ be a parabolic subgroup with Levi factor $M_P$ and unipotent radical $N_P$ . Let $\sigma $ be a cuspidal automorphic subrepresentation of $M_P(\mathbb {A})$ with central character trivial on $A_P^\infty $ . Let $\mathcal {A}_{P,\sigma }(U_h)$ be the space of automorphic forms on the quotient $A_P^\infty M_P(F) N_P(\mathbb {A})\backslash U_h(\mathbb {A})$ such that for all $g\in U_h(\mathbb {A})$ ,

$$ \begin{align*}m\in M_P(\mathbb{A})\mapsto \delta_P(m)^{-\frac12} \varphi(mg)\end{align*} $$

belongs to the space of $\sigma $ . Here, $N_P$ is the unipotent radical of P and $\delta _P$ is the modular character of $P(\mathbb {A})$ . The representation of $U_h(\mathbb {A})$ on $\mathcal {A}_{P,\sigma }(U_h)$ is isomorphic to the induced representation $\Sigma =\operatorname {\mathrm {Ind}}_P^{U_h}(\sigma )$ . Let $\varphi \in \mathcal {A}_{P,\sigma }(U_h)$ . For $\lambda \in \mathfrak {a}_{\Pi ,\mathbb {C}}^*$ , we introduce the Eisenstein series $E(\varphi ,\lambda )$ and the Ichino-Yamana regularized period

(1.2.2.1) $$ \begin{align} \mathcal{P}_{U^{\prime}_h}(\varphi,\lambda)=\int_{[U_h^{\prime}]} \Lambda_u^TE(x,\varphi,\lambda)\,dx, \end{align} $$

where $[U_h^{\prime }]=U_h^{\prime }(F)\backslash U_h^{\prime }(\mathbb {A})$ is equipped with the Tamagawa measure, $\Lambda _u^T$ is the truncation operator introduced by Ichino-Yamana in [Reference Ichino and YamanaIY19] depending on an auxiliary parameter T whose definition is recalled in §3.3.2. The integral is absolutely convergent. Moreover, if the base change of $\Sigma $ is a H-regular Arthur parameter (which will be our assumption), then the integral does not depend on T (see proposition 3.5.3.1 below). In this case, $\mathcal {P}_{U^{\prime }_h}(\varphi ,\lambda )$ is a meromorphic function, which is regular outside the singularities of the Eisenstein series. In particular, it is holomorphic on $i\mathfrak {a}_{\Pi }^*$ .

1.2.3 The Gan-Gross-Prasad conjecture for some Eisenstein series

1.2.4 Factorization of periods of some Eisenstein series à la Ichino-Ikeda

Let $h\in \mathcal {H}_n$ . Let P be a parabolic subgroup of $ U_h$ with Levi factor $M_P$ and let $\sigma $ be an irreducible cuspidal automorphic subrepresentation of $M_P(\mathbb {A})$ such that the weak base change of $(P,\sigma )$ is a regular Hermitian Arthur parameter $\Pi $ . We have a restricted tensor product decomposition $\sigma =\bigotimes _{v\in V_F}'\sigma _v$ over the set $V_F$ of places of F. We assume that $\sigma $ is tempered – that is, for every place v, the local representation $\sigma _v$ is tempered. Let $\lambda \in i\mathfrak {a}_\Pi ^*$ . We define $\Pi _\lambda $ and $\Sigma _\lambda $ as above. Let $\Sigma _{\lambda ,v}=\operatorname {\mathrm {Ind}}_P^{U_h}(\sigma _v\otimes \lambda )$ and $\Pi _{\lambda ,v}$ be their local components.

We set

$$ \begin{align*} \mathcal{L}(s,\Sigma_\lambda)= \left(s-\frac12\right)^{-\dim(\mathfrak{a}_{\Pi}^*)} \prod_{i=1}^{n+1}L(s+i-1/2,\eta^i)\frac{L(s,\Pi_\lambda)}{L(s+1/2,\Pi_\lambda,\mathrm{As}')}, \end{align*} $$

where $\eta $ denotes the quadratic idele class character associated to the extension $E/F$ , $L(s,\eta ^i)$ is the completed Hecke L-function associated to $\eta ^i$ and $L(s,\Pi _\lambda ,\mathrm {As}')$ is the L-function associated to $\mathrm {As}^{(-1)^n}\boxtimes \mathrm {As}^{(-1)^{n+1}}$ . Note that with our hypothesis, the function $L(s,\Pi _\lambda ,\mathrm {As}')$ has a pole of order $\dim (\mathfrak {a}_{\Pi }^*)$ at $s=1$ . Thus, the function $(s-1)^{-\dim (\mathfrak {a}_{\Pi }^*)} L(s,\Pi _\lambda ,\mathrm {As}')$ is holomorphic and nonvanishing at $s=1$ . In particular, the function $\mathcal {L}(s,\Sigma _\lambda )$ is holomorphic at $s=\frac 12$ .

We denote by $\mathcal {L}(s,\Sigma _{\lambda ,v})$ the corresponding quotient of local L-factors; namely, for s in some half-space, we have

$$ \begin{align*} \mathcal{L}(s,\Sigma_\lambda)= \left(s-\frac12\right)^{-\dim(\mathfrak{a}_{\Pi}^*)} \prod_{v\in V_F} \mathcal{L}(s,\Sigma_{\lambda,v}). \end{align*} $$

For each place v of F, we define a local normalized period $\mathcal {P}^\natural _{h,\sigma _v}: \Sigma _v\times \Sigma _v\to \mathbb {C}$ as follows:

$$ \begin{align*}\mathcal{P}^\natural_{h,\Sigma_{\lambda,v}}(\varphi_v,\varphi^{\prime}_v)=\mathcal{L}\left(\frac12,\Sigma_{\lambda,v}\right)^{-1}\int_{U^{\prime}_h(F_v)} (\Sigma_{\lambda,v}(h_v)\varphi_v,\varphi^{\prime}_v)_v dh_v,\;\; \varphi_v,\varphi^{\prime}_v\in \Sigma_v.\end{align*} $$

It depends on the choice of a Haar measure $dh_v$ on $U^{\prime }_h(F_v)$ as well as an invariant inner product on $\sigma _v$ which gives in the usual way an invariant product on $\Sigma _v$ denoted by $(\cdot ,\cdot )_v$ . By the temperedness assumption, the integral is absolutely convergent [Reference HarrisHar14, Proposition 2.1] and the local factor $\mathcal {L}(s,\Sigma _{\lambda ,v})$ has neither zero nor pole at $s=\frac 12$ .

We introduce on $\mathcal {A}_{P,\sigma }(U_h)$ the Petersson inner product given by

$$ \begin{align*}(\varphi,\varphi)_{\operatorname{\mathrm{Pet}}}=\int_{A_P^{\infty}M_P(F)N_P(\mathbb{A})\backslash U_h(\mathbb{A})} \lvert \varphi(g)\rvert^2 dg,\;\; \varphi\in \sigma.\end{align*} $$

Recall that we have normalized the period integral $\mathcal {P}_{U_h^{\prime }}(\lambda )$ by choosing the invariant Tamagawa measures on $[U_h^{\prime }]$ . We also normalize the Petersson product by using the quotient of Tamagawa measures. We assume that the local Haar measures $dh_v$ on $U^{\prime }_h(F_v)$ are such that the product $\prod _v dh_v$ gives the Tamagawa measure on $U_h^{\prime }(\mathbb {A})$ .

Theorem 1.2.4.1. Let $\Pi $ and $(P,\sigma )$ as above. For $\lambda \in i\mathfrak {a}_\Pi ^*$ and every nonzero factorizable vector $\varphi =\otimes _v^{\prime } \varphi _v\in \mathcal {A}_{P,\sigma }(U_h)\simeq \otimes ^{\prime }_{v\in V_F} \Sigma _v$ , we have

(1.2.4.1) $$ \begin{align} \frac{\lvert \mathcal{P}_{U_h^{\prime}}(\varphi,\lambda)\rvert^2}{(\varphi, \varphi)_{\operatorname{\mathrm{Pet}}}}=\lvert S_\Pi\rvert^{-1} \mathcal{L}\left(\frac12,\Sigma_\lambda\right)\prod_v \frac{\mathcal{P}^\natural_{h,\Sigma_{\lambda,v}}(\varphi_v,\varphi_v)}{(\varphi_v,\varphi_v)_v}. \end{align} $$

1.3.1

Let $n\geqslant m\geqslant 0$ be two integers of the same parity. We have $n=m+2r$ for some $r\geqslant 0$ . Recall that we denote by $h_0$ the $1$ -dimensional Hermitian space given by the norm $N_{E/F}$ . Let $h_s\in \mathcal {H}_2$ be the orthogonal sum of $h_0$ and $-h_0$ . For any $h\in \mathcal {H}_m$ , we define $\tilde h\in \mathcal {H}_n$ to be the orthogonal sum of h and r copies of $h_s$ denoted by $h_s^1,\ldots , h_s^r$ . For each $1\leqslant i\leqslant r$ , let $(x_i,y_i) $ be a hyperbolic basis of $h_s^i$ ; that is, we have $h_s^i(x_i,x_i)=h_s^i(y_i,y_i)=0$ and $h_s^i(x_i,y_i)=1$ . We consider also the orthogonal sum $h_{n+1}= \tilde h \oplus h_0 \in \mathcal {H}_{n+1}$ . We denote by $v_0$ the vector of $h_{0}$ corresponding to $1\in E$ . We have a diagonal embedding

$$ \begin{align*} U(h) \hookrightarrow \mathcal{G}_h=U(h)\times U(h_{n+1}) \end{align*} $$

for which the image of $U(h)$ in $U(h_{n+1})$ is the subgroup which acts by the identity on $h_0\oplus h_s^1\oplus \ldots \oplus h_s^r$ .

Let $B\subset U(h_{n+1})$ be the stabilizer of the isotropic flag

(1.3.1.1) $$ \begin{align} (0)\subsetneq \operatorname{\mathrm{vect}}(x_1)\subsetneq \operatorname{\mathrm{vect}}(x_1,x_2)\subsetneq \ldots \subsetneq \operatorname{\mathrm{vect}}(x_1,\ldots ,x_r). \end{align} $$

Let N be the unipotent radical of B. Then the group $U(h)$ normalizes N. Let $\mathcal {B}_h=U(h)\ltimes (\{1\}\times N)$ : this is the so-called Bessel subgroup of $\mathcal {G}_h$ .

1.3.2 Bessel periods

Let $\psi :\mathbb {A}/F\to \mathbb {C}^\times $ be a nontrivial continuous character. We define a character $\psi _N:[N]\to \mathbb {C}^\times $ by

$$ \begin{align*} \psi_N(u)=\psi\left(\sum_{i=1}^{r-1} h_{n+1}(ux_{i+1},y_i)+h_{n+1}(uv_0,y_r) \right),\;\;\; u\in [N]. \end{align*} $$

This character extends uniquely to a character $\psi _{\mathcal {B}_h}: [\mathcal {B}_h]\to \mathbb {C}^\times $ that coincides with $\psi _N$ on $[N]$ and is trivial on $[U(h)]$ . Let $\sigma $ be a cuspidal automorphic subrepresentation of $\mathcal {G}_h(\mathbb {A})$ . We define the global Bessel period for $\varphi \in \sigma $ by the absolute convergent integral

$$ \begin{align*} \mathcal{P}_{\mathcal{B}_h,\psi}(\varphi)=\int_{[\mathcal{B}_h]} \varphi(g) \psi_{\mathcal{B}_h}(g) dg. \end{align*} $$

1.3.3 The Gan-Gross-Prasad conjecture for Bessel periods

Let $G^\flat =G_{m}\times G_{2n+1}$ . We can now state our first theorem about Bessel periods.

1.3.4

In our approach, Theorem 1.3.3.1 is a consequence of Theorem 1.2.3.1. To explain this, we may and shall assume $r>0$ . We start with a discrete Hermitian Arthur parameter $\Pi $ of $G^\flat $ . It can be written $\Pi =\Pi _m\boxtimes \Pi _{n+1}$ , where $\Pi _m$ and $\Pi _{n+1}$ are respective discrete parameters of $G_m$ and $G_{n+1}$ . Let $\alpha _1,\ldots ,\alpha _r$ be r characters of $E^\times \backslash \mathbb {A}_E^1$ such that the characters $\alpha _1,\ldots ,\alpha _r,\alpha _1^*,\ldots ,\alpha _r^*$ are two by two distinct (we recall that $\alpha _i^*$ denotes the conjugate-dual of $\alpha _i$ ). Let $Q_n\subset G_n$ be a parabolic subgroup of Levi factor $G_1^{r}\times G_{m}\times G_1^r$ . Then

$$ \begin{align*} \tilde\Pi=\operatorname{\mathrm{Ind}} _{Q_n}^{G_n}(\alpha_1 \boxtimes \ldots \boxtimes\alpha_r \boxtimes \Pi_m\boxtimes \alpha_1^* \boxtimes \ldots \boxtimes\alpha_r^* ) \boxtimes \Pi_{n+1} \end{align*} $$

is a regular Hermitian Arthur parameter of $G=G_n\times G_{n+1}$ . Even if $\tilde \Pi $ is not discrete, it is at least H-regular in the sense of §1.2.1: this is an obvious consequence of remark 1.2.1.1 and the assumption on the characters $\alpha _i$ . We have an identification $\mathbb {C}^r \simeq \mathfrak {a}_{\tilde \Pi ,\mathbb {C}}^*$ such that if $\lambda _s\in \mathfrak {a}_{\tilde \Pi ,\mathbb {C}}^*$ is the image of $(s,\ldots ,s)$ with $s\in \mathbb {C}$ , we have

$$ \begin{align*} \tilde\Pi_{\lambda_s}=\operatorname{\mathrm{Ind}} _{Q_n}^{G_n}(\alpha_1 |\cdot|_E^s \boxtimes \ldots \boxtimes\alpha_r |\cdot|_E^s \boxtimes \Pi_m \boxtimes \alpha_1^* |\cdot|_E^{-s} \boxtimes \ldots \boxtimes\alpha_r^* |\cdot|_E^{-s}) \boxtimes \Pi_{n+1}. \end{align*} $$

For simplicity, we set $\tilde \Pi _{s}=\tilde \Pi _{\lambda _s}$ . By elementary properties of Rankin-Selberg L-function, it is clear that assertion 1 of 1.3.3.1 is equivalent to $1'$ :

1. 1^′. There exists $s\in i\mathbb {R}$ such that $L(\frac 12,\tilde \Pi _{s})\not =0$ .

Let $h\in \mathcal {H}_m$ and $\sigma $ be an automorphic cuspidal subrepresentation of $\mathcal {G}_h$ whose weak base to $G^\flat $ is $\Pi $ . Let $P_n\subset U(\tilde h)$ be the parabolic subgroup stabilizing the isotropic flag

$$ \begin{align*}0\subsetneq \operatorname{\mathrm{vect}}(x_r)\subsetneq \operatorname{\mathrm{vect}}(x_r,x_{r-1})\subsetneq \ldots \subsetneq \operatorname{\mathrm{vect}}(x_r,\ldots,x_1).\end{align*} $$

(Note that this flag is opposite position to (1.3.1.1).) and set $P=P_n\times U(h_{n+1})$ ; a parabolic subgroup of $U_{\tilde h}=U(\tilde h)\times U(h_{n+1})$ . Then $G_1^r \times \mathcal {G}_h$ is a Levi factor $M_P$ of P. Set $\tilde \sigma = \alpha _1\boxtimes \ldots \boxtimes \alpha _r \boxtimes \sigma $ . This is an automorphic cuspidal representation of $M_P(\mathbb {A})$ , and $\tilde \Pi $ is the weak base change of $(P,\sigma )$ . Let $\varphi \in \mathcal {A}_{P,\tilde \sigma }(U_{\tilde h})$ . As in subsection 1.2, we denote by $U^{\prime }_{\tilde h}$ the ‘diagonal’ subgroup of $U_{\tilde h}$ . In the case at hand, the restriction of the Eisenstein series $E(\varphi ,\lambda )$ to $[U^{\prime }_{\tilde h}]$ is rapidly decreasing for any $\lambda \in \mathfrak {a}_{\tilde \Pi ,\mathbb {C}}$ , where the Eisenstein series is regular, and for any such $\lambda $ , we have

$$ \begin{align*} \mathcal{P}_{U^{\prime}_{\tilde h}}(\varphi,\lambda)=\int_{[U_h^{\prime}]} E(x,\varphi,\lambda)\,dx \end{align*} $$

where the left-hand side is defined according to (1.2.2.1) and the right-hand side is absolutely convergent (see Proposition 3.5.3.1 assertion 3). Moreover, the map $s\mapsto \mathcal {P}_{U^{\prime }_{\tilde h}}(\varphi ,\lambda _s)$ is meromorphic and holomorphic on $i\mathbb {R}$ . We prove in Proposition 8.8.2.1 that the map $\varphi \in \mathcal {A}_{\sigma }(\mathcal {G}_h)\mapsto \mathcal {P}_{\mathcal {B}_h,\psi }(\varphi )$ does not vanish identically if and only if there is $s\in \mathbb {C}$ such that the map $\varphi \mapsto \mathcal {P}_{U^{\prime }_{\tilde h}}(\varphi ,\lambda _s)$ does not vanish identically on $\mathcal {A}_{P,\tilde \sigma }(U_{\tilde h})$ . This last fact is eventually a consequence of some unfolding identity that roughly takes the following form:

$$ \begin{align*}\mathcal{P}_{U^{\prime}_{\tilde h}}(\varphi,\lambda_s)=\int_{\mathcal{B}'(\mathbb{A})\backslash U^{\prime}_{\tilde h}(\mathbb{A})} \mathcal{P}_{\mathcal{B}_h,\psi}(\varphi_s(h))dh\end{align*} $$

for $\varphi \in \mathcal {A}_{P,\tilde \sigma }(U_{\tilde h})$ , where $\varphi _s$ stands for the corresponding element of $\operatorname {\mathrm {Ind}}_P^{U_{\tilde h}}(\tilde \sigma \otimes \lambda _s)$ (given through the choice of a suitable Iwasawa decomposition $U_{\tilde h}(\mathbb {A})=P(\mathbb {A})K$ that is implicit in the definition of the Eisenstein series $E(\varphi ,\lambda _s)$ ) and $\mathcal {B}'=U(h)\rtimes V$ with V the unipotent radical of the parabolic subgroup of $U(\tilde h)$ stabilizing the isotropic subspace $\operatorname {\mathrm {vect}}(x_1,\ldots ,x_r)$ . It should be emphasized however that this identity does not make sense per se, as the Eulerian integral on the right-hand side is not absolutely convergent in general. More precisely, it has to be ‘interpreted in the sense of L-functions’, which requires some nontrivial unramified computations of local integrals involving Bessel functions. We refer the reader to Section 8 and, more specifically, 8.7 and 8.8 for details.

It follows that condition 2 of Theorem 1.3.3.1 holds for $h\in \mathcal {H}_m$ and $\sigma $ if and only if the following assertion holds:

1. 2’. There exists $s\in i\mathbb {R}$ such that $\varphi \mapsto \mathcal {P}_{U^{\prime }_{\tilde h}}(\varphi ,\lambda _s)$ does not vanish identically on $\mathcal {A}_{P,\sigma }(U_{\tilde h})$ .

It is then straightforward to deduce Theorem 1.3.3.1 from Theorem 1.2.3.1.

1.3.5 Local Bessel periods

From now on, we fix $h\in \mathcal {H}_m$ and a decomposition of the character $\psi =\otimes _{v\in V_F} \psi _v$ from which we get a decomposition $\psi _{\mathcal {B}_h}= \otimes _{v\in V_F}\psi _{\mathcal {B}_h,v}$ , where $\psi _{\mathcal {B}_h,v}$ is a character of $\mathcal {B}_h(F_v)$ . Let v be a place of F. The integral

$$ \begin{align*} \int_{\mathcal{B}_h(F_v)} f_v(g_v) \psi_{\mathcal{B}_h,v}(g_v) dg_v \end{align*} $$

is well defined for a smooth and compactly supported function $f_v$ on $\mathcal {G}_h(F_v)$ and extends to a continuous linear form $ f_v\mapsto \mathcal {P}_{\mathcal {B}_h,\psi _v}(f_v)$ on the space of tempered functions; see subsection 8.4. It depends on the choice of a Haar measure on $\mathcal {B}_h(F_v)$ .

Let $\sigma _v$ be a tempered irreducible representation of $\mathcal {G}_h(F_v)$ equipped with an invariant inner product $(\cdot ,\cdot )_v$ . Let $\varphi _v$ and $\varphi _v^{\prime }$ be vectors of $\sigma _v$ . The associated matrix coefficient defined by $f_{ \varphi _v,\varphi _v^{\prime } }(g)= (\sigma _v(g) \varphi _v,\varphi ^{\prime }_v)_v$ for all $g \in \mathcal {G}_h(F_v) $ belongs to this space and we set

$$ \begin{align*} \mathcal{P}_{\mathcal{B}_h,\psi_v}( \varphi_v,\varphi^{\prime}_v)= \mathcal{P}_{\mathcal{B}_h,\psi_v}( f_{ \varphi_v,\varphi_v^{\prime} }). \end{align*} $$

1.3.6 The Ichino-Ikeda conjecture for Bessel periods

Let $\sigma $ be a tempered automorphic cuspidal subrepresentation $\sigma $ of $\mathcal {G}_h(\mathbb {A})$ . Tempered means that we have a decomposition $\sigma =\otimes _{v\in V_F}' \sigma _v$ with $\sigma _v$ tempered for all v. We also assume that the weak base change of $\sigma $ to $G^\flat $ is a discrete Hermitian parameter $\Pi $ . As in §1.2.4, we define the ratio of L-functions $\mathcal {L}(s,\sigma )$ and its local counterparts $\mathcal {L}(s,\sigma _v)$ for $s\in \mathbb {C}$ . Explicitly, we have

$$ \begin{align*} \mathcal{L}(s,\sigma_v)=\prod_{i=1}^{n+1}L(s+i-1/2,\eta^i_v)\frac{L(s,\Pi_v)}{L(s+1/2,\Pi_v,\mathrm{As}')}, \end{align*} $$

where $\mathrm {As}'=\mathrm {As}^{(-1)^m}\otimes \mathrm {As}^{(-1)^{n+1}}$ and $\mathcal {L}(s,\sigma )$ is the product of the local factors in some half-plane. We use the local factor to define the normalized local Bessel period

$$ \begin{align*} \mathcal{P}_{\mathcal{B}_h,\psi_v}^\natural( \varphi_v,\varphi^{\prime}_v)= \mathcal{L}\left(\frac12 ,\sigma_v\right)^{-1}\mathcal{P}_{\mathcal{B}_h,\psi,v}( f_{ \varphi_v,\varphi_v^{\prime} }). \end{align*} $$

We assume that the product of local measures on $\mathcal {B}_h(F_v)$ gives the Tamagawa measure on $\mathcal {B}_h(\mathbb {A})$ . On $\sigma $ , we use the Petersson inner product $(\cdot , \cdot )_{\operatorname {\mathrm {Pet}}}$ normalized by the Tamagawa measure on $\mathcal {G}_h(\mathbb {A})$ .

Theorem 1.3.6.1. Let $\sigma $ and $\Pi $ as above. For every nonzero factorizable vector $\varphi =\otimes _v^{\prime } \varphi _v\in \sigma $ , we have

(1.3.6.1) $$ \begin{align} \frac{\lvert \mathcal{P}_{\mathcal{B}_h,\psi}(\varphi)\rvert^2}{(\varphi, \varphi)_{\operatorname{\mathrm{Pet}}}}=\lvert S_\Pi\rvert^{-1} \mathcal{L}\left(\frac12,\sigma\right)\prod_v \frac{\mathcal{P}_{\mathcal{B}_h,\psi_v}^\natural(\varphi_v,\varphi_v)}{(\varphi_v,\varphi_v)_v}. \end{align} $$

1.4.1

In this subsection, we explain some new ingredients that play a role in the proof of Theorems 1.2.3.1 and 1.2.4.1. As many other contributions on the subject (among them, see [Reference ZhangZha14b], [Reference ZhangZha14a], [Reference XueXue19], [Reference Beuzart-PlessisBeu21], [Reference Beuzart-PlessisBP21], [Reference Beuzart-Plessis, Liu, Zhang and ZhuBLZZ21], [Reference Beuzart-Plessis, Chaudouard and ZydorBPCZ22]), we follow the strategy of the seminal paper [Reference Jacquet and RallisJR11] of Jacquet and Rallis. More precisely, besides the local harmonic analysis performed in the mentioned papers, our work is based on the geometric comparison, fully established in [Reference Chaudouard and ZydorCZ21], of the relative trace formulas constructed in [Reference ZydorZyd20] of the unitary groups $U_h$ for $h\in \mathcal {H}_n$ and the corresponding group G. However, to be able to exploit this comparison, we need to obtain more tractable expressions for the spectral contributions we are interested in.

1.4.2

Let us first explain our result in the unitary case namely for the group $U=U_h$ and its subgroup $U'=U_h^{\prime }$ . Let $\mathfrak {X}(U)$ be the set of cuspidal data of U. According to the work of Zydor (see [Reference ZydorZyd20, section 4]), the contribution of $\chi \in \mathfrak {X}(U)$ to the relative trace formula for the group U is built upon the absolutely convergent integral

$$ \begin{align*} \int_{[U']\times [U']} K^T_{f,\chi}(x,y) \, dxdy. \end{align*} $$

Here, $K^T_{f,\chi }$ is a suitably modified version à la Arthur of the $\chi $ -part $K_{f,\chi }$ of the automorphic kernel $K_f(x,y)=\sum _{\gamma \in U(F)}f(x^{-1}\gamma y)$ associated to a Schwartz function f on $U(\mathbb {A})$ ; see (3.2.2.1) below for the precise definition. It depends on a truncation paramerer T. It turns out that the integral above is an exponential-polynomial function in T whose purely polynomial part is constant and gives by definition the $\chi $ -contribution denoted by $J^U_\chi (f)$ of the relative trace formula; see Theorem 3.2.3.1 for this slight extension of Zydor’s work to Schwartz test functions. The problem, however, is to get an expression for $J_\chi (f)$ that reflects the Langlands spectral decomposition of $K_{f,\chi }$ and that is related to the periods (1.2.2.1) we are interested in. The starting point is the following new independent characterization of $J^U_\chi (f)$ : the integral

(1.4.2.1) $$ \begin{align} \int_{[U']\times [U']}( K_{f,\chi} \Lambda^T_u)(x,y) \, dxdy \end{align} $$

is absolutely convergent and is asymptotic to an polynomial exponential in the variable T whose purely polynomial term is constant and equal to $J_\chi ^U(f)$ ; see Corollary 3.3.5.2. Here, $K_{f,\chi } \Lambda ^T_u$ means that we have applied the Ichino-Yamana truncation operator $\Lambda ^T_u$ already mentioned in §1.2.2 to the right variable of the kernel $K_{f,\chi }$ . Let us now assume that the cuspidal datum $\chi $ is $(U,U')$ -regular in the sense of §3.5.2. Then the expression (1.4.2.1) does not depend on T and thus is equal to $J_\chi ^U(f)$ . To state our result, we fix a representative $(M_P,\sigma )$ where $M_P$ is a Levi factor of a parabolic subgroup $P=M_PN_P$ of U and $\sigma $ is a cuspidal automorphic representation of $M_P(\mathbb {A})$ . Let $\mathcal {A}_{P,\sigma ,\operatorname {\mathrm {cusp}}}(U_h)$ be the space of automorphic forms on the quotient $A_P^\infty M_P(F) N_P(\mathbb {A})\backslash U(\mathbb {A})$ such that for all $g\in U(\mathbb {A}),$

$$ \begin{align*}m\in M_P(\mathbb{A})\mapsto \delta_P(m)^{-\frac12} \varphi(mg)\end{align*} $$

belongs to the $\sigma $ -isotypic component of the space of cuspidal automorphic forms on the quotient $A_P^\infty M_P(F) \backslash M_P(\mathbb {A})$ . Working throughout Langlands spectral decomposition of $K_{f,\chi }$ , we get (see Theorem 3.5.7.1):

(1.4.2.2) $$ \begin{align} J^U_\chi(f)= \int_{i\mathfrak{a}_P^*} J_{P,\sigma}^{U}(\lambda, f)\,d\lambda, \end{align} $$

where the right-hand side is the absolutely convergent integral of the relative character defined by

$$ \begin{align*} J_{P,\sigma}^{U}(\lambda, f)=\sum_{\varphi\in \mathcal{B}_{P,\sigma}} \mathcal{P}_{U'}(I_P(\lambda,f)\varphi,\lambda)\overline{ \mathcal{P}_{U'}(\varphi,\lambda) }. \end{align*} $$

Here, the periods $\mathcal {P}_{U'}(\cdot ,\lambda )$ are those defined in (1.2.2.1), and $I_P(\lambda ,f)$ denotes the induced action of f twisted by $\lambda $ . The sum is over some orthonormal basis $\mathcal {B}_{P,\sigma }$ of $\mathcal {A}_{P,\sigma ,\operatorname {\mathrm {cusp}}}(U)$ ; see §3.5.5 for the Petersson inner product.

1.4.3

Let us turn to the linear case – namely, $G=G_n\times G_{n+1}$ . In this case, we have to consider two subgroups – namely, $H=G_n$ diagonally embedded in G and $G'=G_n^{\prime }\times G_{n+1}'$ , where $G^{\prime }_n=\mathrm {GL}(n,F)$ is naturally embedded in $G_n=\mathrm {GL}(n,E)$ . Let $\chi $ be a cuspidal datum of G and let f be a Schwartz function on $G(\mathbb {A})$ . As before, we denote by $K_{f,\chi }$ the $\chi $ -part of the automorphic kernel. According to [Reference Beuzart-Plessis, Chaudouard and ZydorBPCZ22, Theorem 1.2.4.1], the contribution $I_\chi (f)$ , as defined by Zydor in [Reference ZydorZyd20], is also the constant term of the polynomial exponential (in the variable T) which is asymptotic to the absolutely convergent integral

$$ \begin{align*} \int_{[H]} \int_{[G'] } \Lambda_r^TK_\chi (h,g)\,\eta_{G'}(g)dgdh. \end{align*} $$

Here, $ \eta _{G'}$ is the quadratic character defined in §4.1.1 and $\Lambda _r^T$ is a truncation operator (in the parameter T) introduced by Ichino-Yamana and well-suited for the study of Rankin-Selberg period. Assume that $\chi $ is represented by a pair $(M,\pi )$ , where M is the Levi factor of a parabolic subgroup P of G. We assume also that $\chi $ is $(G,H)$ -regular and Hermitian in the sense of §4.1.3. Then we have (see Theorem 4.1.8.1)

$$ \begin{align*} I_\chi(f)=2^{-\dim(\mathfrak{a}_L)}\int_{i\mathfrak{a}_M^{L,*}}I_{P,\pi}(\lambda,f)\,d\lambda. \end{align*} $$

Here, L is a Levi subgroup of G containing M and determined by $\pi $ (see §4.1.4). For $\lambda \in i\mathfrak {a}_M^{L,*}$ , the relative character $I_{P,\pi }(\lambda ,f)$ is given by one of the two expressions

$$ \begin{align*} I_{P,\pi}(\lambda,f)&=\sum_{\varphi\in \mathcal{B}_{P,\pi}} \mathbf{P}(E(I_P(\lambda,f)\varphi,\lambda))\cdot \overline{J(\xi, \varphi,\lambda)}\\ &=\sum_{\varphi\in \mathcal{B}_{P,\pi}} \frac{Z^{RS}(0, W(I_P(\lambda,f)\varphi,\lambda)) \overline{\beta_\eta(W(\varphi,\lambda))} }{ \langle W(\varphi,\lambda),W(\varphi,\lambda)\rangle_{\operatorname{\mathrm{Pet}}}}. \end{align*} $$

The sums are over some orthonormal bases $\mathcal {B}_{P,\pi }$ for the Petersson inner product of the space $\mathcal {A}_{P,\pi ,\operatorname {\mathrm {cusp}}}(G)$ (defined as above). The first expression is built upon $\mathbf {P}(E(\varphi ,\lambda ))$ and $J(\xi , \varphi ,\lambda )$ . The former is the regularized Rankin-Selberg period à la Ichino-Yamana (see [Reference Ichino and YamanaIY15]) of the Eisenstein series associated to the pair $(P,\pi )$ . The latter is the intertwining (Flicker-Rallis) period of Jacquet-Lapid-Rogawski [Reference Jacquet, Lapid and RogawskiJLR99]. The second expression uses Whittaker functionals $W(\varphi ,\lambda )$ associated to Eisenstein series and linear forms on the Whittaker models of $\operatorname {\mathrm {Ind}}_{P(\mathbb {A})}^{G(\mathbb {A})}(\pi \otimes \lambda )$ equipped with the Petersson product $\langle \cdot , \cdot \rangle _{\operatorname {\mathrm {Pet}}}$ . The linear forms $Z^{RS}$ and $\beta _\eta $ are counterparts of the period and the intertwining period. In the Rankin-Selberg case, the link is recalled in Proposition 4.2.5.1: it is based on [Reference Ichino and YamanaIY15] which generalizes the classical Rankin-Selberg theory. In the Flicker-Rallis case, the precise relation is given in Proposition 4.2.7.1, which generalizes the work of Flicker [Reference FlickerFli88]. Besides some reductions based on [Reference Beuzart-Plessis, Chaudouard and ZydorBPCZ22, section 9], the bulk of the proof of proposition is the object of Section 5. Note also that we prove in Section 5 a result that is of independent interest: we express a basic intertwining period of Jacquet-Lapid-Rogawski in terms of an integral of a Whittaker functional; see Theorem 5.5.1.1. The second expression of the relative character is better suited for the proof of Theorem 1.2.4.1. In Section 6, we give an alternative proof of this spectral expansion of $I_\chi (f)$ , for $(G,H)$ -regular $\chi $ , that is based on the theory of Zeta integrals.

Finally, let us remark that if $\chi $ is a $(U,U')$ -regular cuspidal datum of U attached to a cuspidal representation of $M_{P_n}(\mathbb {A})\times U(h\oplus h_0)(\mathbb {A})$ for some parabolic subgroup $P_n$ of $U(h)$ , then the modified kernel $K^T_{f,\chi }(x,y) $ coincides with the usual $\chi $ -part of the kernel. Then one can directly get the expression (1.4.2.2) in which the $U'$ -periods are absolutely convergent. This includes, in particular, the Eisenstein series needed for the deduction of the Gan-Gross-Prasad and Ichino-Ikeda conjectures for general Bessel periods, as outlined in subsection 1.3.

1.5.1

The reader will find the main notations and some prelimary results in Section 2. The Section 3 is devoted to the study of some spectral contributions of the Jacquet-Rallis trace formula for unitary groups. The main general result is Theorem 3.2.3.1, which gives a more tractable expression to compute spectral contributions. The proof of Theorem 3.2.3.1 is the bulk of subsections 3.3 and 3.4. Then in subsection 3.5, the spectral contribution for some cuspidal data are explicitly given in terms of relative characters (see Theorem 3.5.7.1). In Section 4, we turn to the Jacquet-Rallis trace formula for general linear groups. The main result is Theorem 4.1.8.1, which expresses the spectral contribution for some cuspidal data in terms of relative characters. In subsection 4.2, we show that these relative characters can expressed in terms of Whittaker functionals; see Theorem 4.2.8.1. The main result of Section 5 is Theorem 5.5.1.1, which relates some basic intertwining period of Jacquet-Lapid-Rogawski to some integral of a Whittaker functional: it is used in the proof of Theorem 4.2.8.1, but it is also of independent interest. Its proof occupies the whole part of Section 5. As explained above, we provide in Section 6 an alternative proof for the description of the spectral contributions to the Jacquet-Rallis trace formula for general linear groups that can be obtained by combining Theorem 4.1.8.1 with Theorem 4.2.8.1. In Section 7, we explain the proof of Theorems 1.2.3.1 and 1.2.4.1 based on the comparison of the Jacquet-Rallis trace formulas and the results obtained before. The aim of the final Section 8 is to establish the reduction of Theorems 1.3.3.1 and 1.3.6.1 to special cases of Theorems 1.2.3.1 and 1.2.4.1, respectively. Its most technical part is in subsection 8.7, where necessary unramified computation is performed. Finally, Appendix A presents a probably well-known extension of Weyl’s character formula to non-connected groups that is necessary for the unramified computation.