# MACE: Multi-Agent Autonomous Collaborative Exploration of Unknown
Environments

Charbel Toumieh and Alain Lambert The two authors are with the Université
Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique,
91405, Orsay, France https://www.lisn.upsaclay.fr

###### Abstract

In this paper, we propose a new framework for multi-agent collaborative
exploration of unknown environments. The proposed method combines state-of-
the-art algorithms in mapping, safe corridor generation and multi-agent
planning. It first takes a volume that we want to explore, then proceeds to
give the multiple agents different goals in order to explore a voxel grid of
that volume. The exploration ends when all voxels are discovered as free or
occupied, or there is no path found for the remaining undiscovered voxels. The
state-of-the-art planning algorithm uses time-aware Safe Corridors to
guarantee intra-agent collision safety as well safety from static obstacles.
The presented approach is tested in a state of the art simulator for up to 4
agents.

video: https://youtu.be/v7P7HpBRY50

##  I INTRODUCTION

###  I-A Problem statement

Multi-agent exploration has numerous real world applications such as search
and rescue, infrastructure inspection and cave exploration. An exploration
framework that is computationally efficient and safe is thus tremendously
beneficial. It is the purpose of this paper to present a new method for multi-
agent exploration that is suitable for low compute embedded systems.

###  I-B Related work

####  I-B1 Single multirotor planning

Many works address single quadrotor planning. Recent state-of-the-art
contributions use Safe Corridors to account for unknown/unexplored part of the
environment and guarantee trajectory safety [1], [2]. Safe Corridors are a
series of overlapping convex polyhedra that cover only free space and are used
in many autonomous navigation methods [3] [4]. In [2], the authors use Safe
Corridors in a MPC/MIQP formulation that is computationally efficient and
accounts for drag forces. Our work is based on their approach with
modifications to account for other moving agents.

####  I-B2 Multi-agent planning

While some works have addressed multi-agent planning [5], [6], [7], [8], [9],
[10], [11], only few works address the problem of low compute and safe multi-
agent autonomous exploration while avoiding static obstacles and other
exploring agents. In [12] the authors present an autonomous multi-agent
planning solution using only on-board resources. The presented approach does
not address full volume exploration and focuses only on intra-agent collision
avoidance as well as static obstacle collision avoidance.

####  I-B3 Exploration

Single quadrotor exploration and frontier analysis/selection has been
extensively studied in the literature. Recent work [13] has shown significant
improvement over other methods such as [14], [15], [16]. The proposed method
uses a frontier information structure (FIS) that is maintained incrementally.
It is provided to the exploration planner that plans exploration motions in
three sequential steps: it first finds global tours that are efficient; then
it selects a local set of optimal viewpoints; finally it generates minimum-
time local trajectories. However, it is unclear how to scale this approach to
multi-agent planning and whether the computational efficiency would be
maintained. It is why we inspired our goal selection method from the Classical
method [15] for frontier/goal selection, which scales well for multiple
agents.

Figure 1: Schematics of the multirotor model with the used coordinate systems.

###  I-C Contribution

The main contribution of our paper is a novel exploration framework that is
based on a decentralized and synchronous planning method. The planning method
takes inspiration mainly from [2] for single quadrotor planning and static
obstacle avoidance, [10] for intra-agent collision avoidance, and [15] for
goal selection for each agent. The framework also uses recent advances in
state-of-the-art mapping [17] and Safe Corridor generation [18] [19] that
makes it low compute and suitable for embedded systems. The planning framework
is tested in simulation using the state-of-the-art Airsim [20] simulator.

###  I-D Assumptions

The following assumptions are made:

  * •

The position of each agent is known within a certain range of uncertainty. In
the simulation, the position of each agent is known perfectly, but our
framework can account for uncertainties by inflating each agents collision
radius.

  * •

All agents can communicate with each other within a certain delay. The
communication only needs to happen when agents are close to each other.

  * •

All agents can communicate with a central hub within a certain delay. The
central hub will be tasked with merging each agent’s map with a global map and
sending goals to each agent.

##  II Agent Model

Table I: Nomenclature $g$ | gravity
---|---
$m$ | multirotor mass
$\bm{p}$ | position vector $x,y,z$ in the world frame
$\bm{v}$ | velocity vector $v_{x},v_{y},v_{z}$ in the world frame
$\bm{a}$ | acceleration vector from thrust and gravity in the world frame
$\bm{j}$ | jerk vector $j_{x},j_{y},j_{z}$ in the world frame
$\bm{z}_{W}$ | world frame $z$
$\bm{z}_{B}$ | body frame $z$
$\bm{R}$ | rotation matrix from body to world frame
$\bm{D}$ | quadratic drag matrix
$\phi$ | roll angle
$\theta$ | pitch angle
$\psi$ | yaw angle
$c_{cmd}$ | total thrust command

Each agent is modeled after a multirotor whose control inputs are the attitude
and thrust. The equations of motion are the following (nomenclature Tab. I):

$\displaystyle\dot{\bm{p}}=\bm{v}$ (1)
$\displaystyle\dot{\bm{v}}=-g\bm{z}_{W}+\dfrac{c_{cmd}}{m}\bm{z}_{B}-\bm{R}\bm{D}\bm{R^{\prime}}\bm{v}||\bm{v}||_{2}$
(2) $\displaystyle\dot{\phi}=\dot{\phi}_{cmd}$ (3)
$\displaystyle\dot{\theta}=\dot{\theta}_{cmd}$ (4)
$\displaystyle\dot{\psi}=\dot{\psi}_{cmd}$ (5)

The above-mentioned equations are simplified into the following model:

$\displaystyle\dot{\bm{p}}=\bm{v}$ (6)
$\displaystyle\dot{\bm{v}}=\bm{a}-\bm{D}_{lin\\_max}\bm{v}$
$\displaystyle\dot{\bm{a}}=\bm{j}$

$\bm{D}_{lin\\_max}$ is a diagonal matrix that represents the maximum linear
drag coefficient in all directions. This matrix is identified offline as shown
in [2]. The quadratic drag model is replaced with a linear worst-case scenario
model. Many off-the-shelf solvers are very efficient for linear constraints,
which is why we transformed the model into a linear one.

##  III The Framework

The framework takes a volume that we want to explore, then proceeds to give
all the agents different goals until all the volume is explored. The volume is
represented as a voxel grid, and it is considered fully explored when all the
voxels are either free or occupied, or no agent can find a path to the
remaining unknown voxels. The framework is divided into 2 modules (Fig. 2):

  1. 1.

A local module that is run on each agent and that contains mapping, planning
and control submodules.

  2. 2.

A global module that merges all the measurements of all the agents into a
single global voxel grid. This module then sends goals to each agent to
explore the volume.

Each of these modules as well as each submodule will be detailed separately in
this section. We make the following assumptions:

  1. 1.

The positions of all the agents are known within a certain range of
uncertainty.

  2. 2.

When the agents are close to each other they can communicate their positions
and generated trajectories to each others within a certain delay.

  3. 3.

The agents can all communicate with a central hub (running the global module)
within a certain delay.

Figure 2: We show the global pipeline of the exploration framework with 3
agents. The broadcast network communicates the local maps of each agent to the
global module, the goals computed by the global module to the agents, and
generated trajectories of all the agents to each agent.

###  III-A Local module - run on each agent

The local module is run on each agent an consists of 3 submodules: mapping,
planning and control.

####  III-A1 Mapping

Each agent is equipped with an omnidirectional lidar that allows is to see in
all directions. The output of the lidar is a pointcloud that is transformed
into a voxel grid using a GPU accelerated method [17]. We remove the points
associated to other agents appearing in the lidar scan. This is done by
removing all the points of the scan within a certain distance from the center
of other agents (which we know due to the communication of trajectories
between agents). The local voxel grid moves along with the agent such that the
agent’s position is always in the center of the grid. Each voxel grid is sent
to the global module to be merged into the global voxel grid.

####  III-A2 Planning

Each agent plans locally to reach the goal sent to it by the global module.
The planning framework is the same as [2], with modifications made to the Safe
Corridor part to become time-aware as presented in [10]. This allows to avoid
collision with other agents as well as static obstacles.

The planning framework is divided into the following steps that are executed
sequentially and at a constant rate (Fig. 4):

  1. 1.

Generate a global path that avoids static obstacles.

  2. 2.

Generate a Safe Corridor.

  3. 3.

Generate a time-aware Safe Corridor to avoid collision with other agents.

  4. 4.

Generate a local reference trajectory using the global path and the the Safe
Corridor.

  5. 5.

Solve the MPC/MIQP problem to generate a trajectory close to the local
reference trajectory and within the constraints of the time-aware Safe
Corridor to guarantee safety.

First a global path is generated using Jump Point Search (JPS) [21] and
Distance Map Planner (DMP) [22]. JPS is a shortest path algorithm that
preserves A*’s optimality, while potentially lowering the computation time by
an order of magnitude. The Distance Map Planner (DMP) [22] is used to make the
generated path safer. It leverages the artificial potential field to push the
path away from the obstacles (Fig. 3). We use the local voxel grid to search
for the path. If the goal is outside the local voxel grid, an intermediate
goal inside it is chosen as shown in [2].

Figure 3: We show a planning iteration of our algorithm. The global path is
generated using JPS and DMP. DMP uses the artificial potential field (pink) to
push the path away from the obstacles. The convex polyhedra (blue) span only
the free space. A local reference trajectory is sampled from the global path
(green circles). We show the separating hyperplane (cyan) generated between
the planning agent and another moving agent using the third predicted
position. This hyperplane constraint is added to the convex polyhedron
constraints to create $SC_{1}$ which will be used as constraint for the second
and third positions in the next planning iteration. This procedure is done for
all positions to create all $SC_{i}$ (time aware SC). It is then given at the
next iteration with the local reference trajectory to the MIQP solver to
generate a feasible trajectory (yellow circles).

Then, we create a Safe Corridor around that path to avoid static obstacles
using the method described in [18]. The Safe Corridor covers only free voxels
(unknown voxels are considered as obstacles). We only generate $P_{hor}$
(polyhedra horizon used for optimization) overlapping polyhedra around the
global path. $P_{hor}$ is typically chosen equal to $2$ to avoid high solving
times.

Using the Safe Corridor (SC), we generate $N$ local Safe Corridors $SC_{i}$,
$i=1,2,...,N$ with $N$ the number of discretization steps of the MPC/MIQP.
Each $SC_{i}$ is generated by adding the separating hyperplanes between the
$i^{th}$ predicted position of the agent and the $i^{th}$ predicted position
of all the other agents. These positions correspond to the solution of the
MPC/MIQP at the last iteration. The generation of the $SC_{i}$ is detailed in
[10]. The series of $SC_{i}$ is called time-aware SC.

We then generate a local reference trajectory $\bm{x}_{ref}$ by sampling the
global path at a sampling velocity $v_{samp}$ and sampling acceleration
$a_{samp}$ starting from the current agent position. These values are a user
input but are usually chosen close to the agent dynamical limits. Any sampled
point that is outside the SC is replaced by the most recent sampled point that
is inside the SC. At each iteration, a new local reference trajectory is
generated only if the predicted agent position $\bm{x}_{N}$ is within
$thresh\\_dist$ of $\bm{x}_{N,ref}$. More details about this step can be found
in [2].

Finally, we use the time-aware SC and the local reference trajectory in a
MIQP/MPC formulation to generate a trajectory that is safe from static
obstacles and other moving agents. With $\bm{x}=[\bm{p}\ \bm{v}\ \bm{a}]^{T}$,
$\bm{u}=\bm{j}$, $f(\bm{x}(t),\bm{u}(t))$ defined by Eq. 6, the model is
discretized using Euler to obtain the discrete dynamics
$\bm{x}_{k+1}=f_{d}(\bm{x}_{k},\bm{u}_{k})$. The agent velocity is limited by
the drag forces. The maximum bounds on the acceleration and the jerk in each
direction are determined by the dynamics of the agent.

Collision avoidance is guaranteed by forcing every two consecutive discrete
points $k$ and $k+1$ (and thus the segment formed by them) to be in one of the
overlapping polyhedra of a given local SC ($SC_{k}$). Let’s assume we have $P$
overlapping polyhedra in $SC_{k}$. They are described by
$\\{(\bm{A}_{kp},\bm{c}_{kp})\\}$, $p=0:P-1$. The constraint that the discrete
position $\bm{p}_{k}$ is in a polyhedron $p$ is described by
$\bm{A}_{kp}.\bm{p}_{k}\leq\bm{c}_{kp}$. We introduce binary variables
$b_{kp}$ ($P$ variables for each $\bm{x}_{k}$, $k=0:N-1)$ that indicate that
$\bm{p}_{k}$ and $\bm{p}_{k+1}$ are in the polyhedron $p$. We force all the
segments to be in at least one of the polyhedra with the constraint
$\sum_{p=0}^{P-1}b_{kp}\geq 1$.

We formulate our MPC under the following Mixed-Integer Quadratic Program
(MIQP) formulation. $\bm{R}_{x}$, $\bm{R}_{N}$ and $\bm{R}_{u}$ are the weight
matrix for the discrete state errors without the final state, the weight
matrix for the final discrete state error (terminal state), and the weight
matrix for the input, respectively.

$\displaystyle\underset{\begin{subarray}{c}\bm{x}_{k},\bm{u}_{k}\end{subarray}}{\text{minimize}}$
$\displaystyle\sum_{k=0}^{N-1}(||\bm{x}_{k}-\bm{x}_{k,ref}||_{\bm{R}_{x}}^{2}+||\bm{u}_{k}||_{\bm{R}_{u}}^{2})$
$\displaystyle+||\bm{x}_{N}-\bm{x}_{N,ref}||_{\bm{R}_{N}}^{2}$ (7) subject to
$\displaystyle\bm{x}_{k+1}=f_{d}(\bm{x}_{k},\bm{u}_{k}),\quad k=0:N-1$ (8)
$\displaystyle\bm{x}_{0}=\bm{X}_{0}$ (9) $\displaystyle\bm{v}_{N}=\bm{0}$ (10)
$\displaystyle\bm{a}_{N}=\bm{0}$ (11) $\displaystyle|a_{x,k}|\leq a_{x,max}$
(12) $\displaystyle|a_{y,k}|\leq a_{y,max},\quad a_{z,k}\leq a_{z,max}$ (13)
$\displaystyle a_{z,k}\geq a_{z,min},\quad|j_{x,k}|\leq j_{x,max}$ (14)
$\displaystyle|j_{y,k}|\leq j_{y,min},\quad|j_{z,k}|\leq j_{z,max}$ (15)
$\displaystyle
b_{kp}=1\implies\begin{cases}\bm{A}_{kp}\bm{p}_{k}\leq\bm{c}_{kp}\\\
\bm{A}_{kp}\bm{p}_{k+1}\leq\bm{c}_{kp}\end{cases}$ (16)
$\displaystyle\sum_{p=0}^{P_{hor}-1}b_{kp}\geq 1$ (17) $\displaystyle
b_{kp}\in\\{0,1\\}$ (18)

The MIQP is solved using the Gurobi solver [23].

Figure 4: We show the global pipeline of the planning framework of a single
planning agent.

####  III-A3 Control

Each agent is controlled using a nonlinear MPC [24], with the acados toolkit
[25]. The MPC minimizes the cost function:

$\displaystyle J=$
$\displaystyle\int_{t=0}^{T}||\bm{x}(t)-\bm{x}_{ref}(t)||_{\bm{Q}_{x}}^{2}+||\bm{u}(t)-\bm{u}_{ref}(t)||_{\bm{R}_{u}}^{2}dt$
(19) $\displaystyle+||\bm{x}(T)-\bm{x}_{ref}(T)||_{\bm{P}}^{2}$

We use the model described in section II with $\bm{u}=[c_{cmd}\
\dot{\phi}_{cmd}\ \dot{\theta}_{cmd}\ \dot{\psi}_{cmd}]^{T}$ and
$\bm{x}=[\bm{p}\ \bm{v}\ \phi\ \theta\ \psi]^{T}$.

###  III-B Global module - run on a central hub

This module takes a volume that we want to explore, merges the
measurements/maps of all the agents, and sends a goal to each agent at a
constant rate until that volume is fully explored. The volume is represented
as a voxel grid and is considered fully explored when all the voxels are
either free or occupied, or no agent can find a path to the remaining unknown
voxels.

####  III-B1 Merging maps from all agents

Each agent has its local voxel grid that is used for local obstacle avoidance
and navigation. When this map is transmitted to the global module, we are able
to efficiently merge it into the global map if the voxels of the local and
global maps overlap with each others i.e. the vector formed by the origin of
the local map and the origin of the global map has its $x$, $y$ and $z$
components as multiples of the voxel size. This can be achieved during the
initialization phase, by setting the initial origin of all maps (local and
global) to $(0,0,0)$. As presented in [17], the origin of the local voxel grid
will be always offset in all directions by a multiple of the voxel size to
keep the robot at its center.

We only update a voxel of the global grid if its corresponding voxel in the
local voxel grid is free or occupied. In the case of static environments, we
can add an additional condition to only update a voxel grid in a global grid
if its not occupied (and its corresponding voxel in the local voxel grid is
free or occupied).

####  III-B2 Computing goals for agents

After updating the global map with the local voxel grids of all the agents, we
proceed to compute goals for each agent. We define a neighbour voxel as a
voxel that is reachable by moving at most 1 voxel unit in each direction from
the current voxel. The goals are computed by doing the following steps
sequentially (Fig. 5):

  1. 1.

Find the border voxels: this is done by going over all the voxels of the
global map. If a voxel is free and has a neighbour voxel that is unknown, then
it is designated as a border voxel.

  2. 2.

Cluster the border voxels: we form clusters out of the border voxels using the
following rule: if two border voxel are neighbors, they belong to the same
cluster.

  3. 3.

Compute cluster centroid: we compute the centroid of each cluster by averaging
the positions of all the voxels that belong to that cluster.

  4. 4.

Compute potential goals: we compute potential goals by finding the voxel
border in a cluster that is the closest to its centroid. If multiple voxels
have the same closest distance, we chose one randomly.

  5. 5.

Compute goals: the finals goals are computed by first choosing the closest
potential goal to the first agent, and removing it from the potential goals’
list so that two agents don’t get the same goal. From the remaining goals, we
chose the closest one to the second agent and remove it from the list. We
continue in this fashion until all the agents have goals, or no potential
goals exist anymore.

Figure 5: We show an example of the borders, clusters and computed potential goals of a voxel grid. Number of agents | Flight distance (m) | Flight velocity (m/s) | Exploration time (s) | Computation time (ms) | Safety ratio
---|---|---|---|---|---
1 | 239.7 / 263.3 / 20.5 | 1.74 / 3.4 / 0.7 | 139.92 / 158 / 15.6 | 9 / 45 / 3.7 | -
2 | 129.5 / 137.7 / 7.72 | 1.8 / 3.47 / 0.74 | 78.2 / 83.3 / 4.67 | 12 / 49.8 / 4.7 | 1.12
3 | 81.2 / 110.91 / 13.4 | 1.9 / 3.62 / 0.81 | 49.6 / 60.2 / 6.36 | 13 / 61 / 5.2 | 1.03
4 | 73 / 86.8 / 9.2 | 2.04 / 3.9 / 0.88 | 44.9 / 50 / 4.27 | 14.5 / 64 / 5.3 | 1.13
Table II: Comparison between 1,2,3 and 4 agents using our framework on 5
randomly generated maps of size $30\ m\times 30\ m\times 3\ m$ and with
obstacle density $0.1\ obst/m^{2}$. We show the mean / max / standard
deviation per agent of each metric, except for the Safety ratio.

(a) Exploration with 1 agent

(b) Exploration with 4 agents

Figure 6: The trajectories and velocity of 1 exploring agent and 4 exploring
agents on the same map. The initial position of agents are offset by $3\ m$
from each others in the $x$ direction in the case of 4 agents

##  IV Simulation

The simulation is done in a $30\ m\times 30\ m\times 3\ m$ environment that
contains 90 cylinder obstacles of radius $0.35\ m$ and height $3\ m$. Their
positions are generated randomly, following a uniform distribution. The
environment is generated in Airsim [20]. The Gurobi solver is set to use one
thread only as this resulted in faster computation times during our
simulations. All testing is done on the Intel Core i7-9750H up to 4.50 GHz CPU
and NVIDIA’s GeForce RTX 2060 up to 1.62 GHz. We simulate up to 4 agents due
to the limited computing power to run the Airsim simulation,
mapping/planning/control for each agent and the global module. The agents
communicate between themselves using a ROS [26] topic. They also communicate
with the global module using a ROS topic.

###  IV-A Mapping parameters

The local map of each agent is of size $20\ m\times 30\ m\times 3\ m$. The
global map size is $30\ m\times 30\ m\times 3\ m$. The voxel size used for
both maps is $0.3\ m$. The update rate for the local map is $10$ Hz (lidar
measurement frequency) and the update rate of the global map is $5$ Hz. Every
time the global map is updated, a new set of goals is computed and sent to
each agent. The local map update takes less than $1\ ms$ due to the GPU
accelerated method we use [17] and the global map update and goal computation
takes less than $3\ ms$.

###  IV-B Planner parameters

We choose the following parameters: $N=12$, $h=100\ ms$,
$a_{samp}=a_{x,max}=a_{y,max}=a_{z,max}=0.7*g$, $a_{z,min}=-g$,
$j_{x,max}=j_{y,max}=j_{z,max}=8\ m/s^{2}$, $v_{samp}=3.5\ m/s$,
$\bm{D}_{lin\\_max}=diag(1,1,1)$, $thresh\\_dist=0.4\ m$, $P_{hor}=2$. The
weight matrices are diagonal: $\bm{R}_{x}=diag(200,200,200,0,0,0,0,0,0)$,
$\bm{R}_{N}=diag(100,100,100,0,0,0,0,0,0)$ and
$\bm{R}_{u}=diag(0.01,0.01,0.01)$. The drone radius is $d_{rad}=0.3\ m$. The
DMP planner pushes the JPS path $0.6\ m$ (2 voxels) away from obstacles (when
possible). The planning frequency is $10$ Hz. The planning requires
synchronized clocks between the agents. This is satisfied as all the agents
use the same CPU clock during the simulation. In a real world situation, the
clocks can be synchronized at the beginning of the exploration since the drift
is minimal (1 microsecond every second) for the duration of an exploration
task.

###  IV-C Controller parameters

The frequency of the MPC controller is $20$ Hz. The weights of the controller
are:

$\displaystyle\bm{P}=\bm{Q}_{x}=diag(15,15,15,0.01,0.01,0.01,0,0,1)$ (20)
$\displaystyle\bm{R}_{u}=diag(0.05,0.1,0.1,0.1)$ (21)

All parameters are set by approximation/experimentation and may not be
optimal. We limit $|\phi|\leq 85\ deg$, $\theta\leq 85\ deg$,
$|\dot{\phi}_{cmd}|\leq 120\ deg/s$, $|\dot{\theta}_{cmd}|\leq 120\ deg/s$ and
$|\dot{\psi}_{cmd}|\leq 60\ deg/s$.

###  IV-D Simulation results

We show the results of doing a series of 5 explorations on 5 randomly
generated maps using 1,2,3 and 4 agents in Tab. II. We also show the
performance of 1 agent and 4 agents on the same map (Fig. 6). The starting
position of the first agent is $(0,0,0)\ m$. When using multiple agents, the
starting position of agents are offset from each others $3\ m$ in the $x$
direction (Fig. 6(b)). All agents stop moving when the exploration is done
i.e. all voxels are discovered as free or occupied, or there is no path found
for the remaining undiscovered voxels.

The average flight distance per agent decreases in a way that is approximately
inversely proportional to the number of agents used. The average flight
velocity slightly increases since an agent has less chances of being torn
between two close goals as other agents explore goals that are close to it.
The average exploration time which corresponds to the time required to explore
the full volume also decreases in an approximately inversely proportional way
to the number of agents used. The average computation time increases as we use
more agents due to the fact that there are more constraints that are added to
each agent’s MIQP/MPC that correspond to the separating hyperplanes with other
agents. Note that the computation time can be decreased by lowering the number
of discretization steps $N$. Since the maximum computation time is $64\ ms$,
this would leave $100-64=36\ ms$ to communicate the trajectory to other agents
using a broadcasting network (in our case a ROS topic). This means that we can
modify $N$ to lower the computation time enough to satisfy communication
constraints. Finally we show the safety ratio which corresponds to the ratio:
(minimum distance between any two agents at any time during the
exploration)/(minimum collision distance). The planner satisfies safety
constraints when using any number of agents.

##  V Conclusions and Future Works

In this paper, we presented a novel framework for multi-agent autonomous
collaborative exploration of unknown environments. The method uses recent
advances in Safe Corridor generation, high-speed mapping and multi-agent
planning to guarantee safety and computational efficiency during the
exploration task. The framework is tested in a state-of-the-art simulator
using up to 4 agents.

In the future, we plan to implement the exploration framework in a fully
autonomous setting. This requires to add one more submodule in the local
module for localization/odometry and account for localization uncertainty in
the planning submodule to guarantee safety.

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S}A^{\psi^{-1}}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\Pi_{v}\right)\tilde{W}^{\sharp}_{v},\Phi_{v}\right).$

Counting on the fact that $\Pi_{v}$ is good in Proposition 4.10, we employ
Proposition 4.9 to insert the identity (4.17) for all $v$ with $c_{\Pi_{v}}=1$
for almost all $v$. In doing so, we write

(4.21)
$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}({\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi}^{\sharp},\Phi))\\\
=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\left(\prod_{v\in S}c^{-1}_{\Pi_{v}}\int^{\rm
st}_{N^{\prime}(F_{v})}J\left(\frac{1}{2},\sigma_{v}(u_{v})W_{v},\tilde{W}_{v}^{\sharp},\Phi_{v}\right)\psi^{-1}_{N^{\prime}(F_{v})}(u_{v})\,du_{v}\right).$

After integrating (4.18) over $N^{\prime}(F_{S})$ for $v\in S$, a product of
regularized integrals becomes

(4.22) $\prod_{v\in S}\int^{\rm
st}_{N^{\prime}(F_{v})}J\left(\frac{1}{2},\sigma_{v}(u_{v})W_{v},\tilde{W}_{v}^{\sharp},\Phi_{v}\right)\psi^{-1}_{N^{\prime}(F_{v})}(u_{v})\,du_{v}\\\
=\frac{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash N^{\prime}(F_{S}))\cdot
n^{S}_{-k,1}}\int^{\rm st}_{N^{\prime}(F_{S})}\langle\sigma(u)\varphi,{\rm
FJ}_{\psi_{N_{\mathbb{M}}}}(\mathcal{E}_{-k}\tilde{\varphi}^{\sharp},\Phi)\rangle_{G^{\prime}}\psi^{-1}_{N^{\prime}}(u)\,du.$

The factorization
$L^{S}(s,\pi\times\pi)=L^{S}(s,\pi\times\pi^{\vee})=L^{S}(s,\pi,{\rm
Sym}^{2})L^{S}(s,\pi,\wedge^{2})$ allows us to simplify the fraction

(4.23)
$m^{S}_{-k,1}/n^{S}_{-k,1}=1/(L^{S}(1,\pi,\wedge^{2})L^{S}(1,\pi\times\omega_{\pi}))$

whose partial $L$-function occurring in the denominator is nothing but the
adjoint $L$-function $L^{S}(s,\sigma,{\rm Ad})$ for $\sigma$. Combining (4.21)
and (4.22) with (4.23) leads us to the result that we seek for. ∎

##  5\. The Case for Odd Special Orthogonal Groups

###  5.1. Groups, Embeddings, and Weyl Elements

Let $G={\rm SO}_{4n}$ be the even split special orthogonal group defined by

$G={\rm SO}_{4n}=\\{g\in{\rm
GL}_{4n}\,|\,\prescript{t}{}{g}w_{4n}g=w_{4n}\\}.$

It acts on the $4n$-dimensional space $F^{4n}$ with the standard basis
$e_{1},\dotsm,e_{2n},e_{-2n},\dotsm,e_{-1}$. We denote by
$\langle\cdot,\cdot\rangle$ the symmetric form defined on the space $F^{4n}$
by $\langle u,v\rangle=\prescript{t}{}{u}w_{4n}v$ for any $u,v\in F^{4n}$. We
define $G^{\prime}={\rm SO}_{2n+1}\subseteq G$ consisting of elements fixing
$e_{1},\dotsm,e_{n-1},e_{1-n},\dotsm,e_{-1}$ and $e_{2n}+e_{-2n}$. (Note that
we take $\alpha=-2$ in their notation [GRS11, (3.40),(9.8)]). Let
$\mathcal{V}$ be the orthogonal complement of the space spanned by
$e_{1},\dotsm,e_{n-1},e_{1-n},\dotsm,e_{-1}$ and $e_{2n}+e_{-2n}$. We choose
$\mathcal{B}_{G^{\prime}}=\\{e_{n},\dotsm,e_{2n-1},e_{2n}-e_{-2n},e_{1-2n},\dotsm,e_{-n}\\}$
as an ordered basis for $\mathcal{V}$. The group $G^{\prime}$ is given by

$G^{\prime}={\rm SO}_{2n+1}=\\{g\in{\rm
GL}_{2n+1}\,|\,\prescript{t}{}{g}\begin{pmatrix}&&w_{n}\\\ &-2&\\\
w_{n}&&\end{pmatrix}g=\begin{pmatrix}&&w_{n}\\\ &-2&\\\
w_{n}&&\end{pmatrix}\\}.$

Let
$\mathcal{B}^{\prime}_{G^{\prime}}=\\{e_{n},\dotsm,e_{2n-1},e_{2n}+e_{-2n},e_{2n}-e_{-2n},e_{1-2n},\dotsm,e_{-n}\\}$
be the basis of $\mathcal{V}\oplus\langle e_{2n}+e_{-2n}\rangle$ obtained by
adding $e_{2n}+e_{-2n}$ to $\mathcal{B}_{G^{\prime}}$ as the $n+1^{th}$
vector. In coordinates, the embedding $\eta:{\rm SO}_{2n+1}\rightarrow{\rm
SO}_{4n}$ is given by

$\eta:\begin{pmatrix}A_{1}&A_{2}&A_{3}\\\ A_{4}&A_{5}&A_{6}\\\
A_{7}&A_{8}&A_{9}\\\ \end{pmatrix}\mapsto{\rm
diag}(1_{n-1},\mathcal{M}\begin{pmatrix}A_{1}&&A_{2}&A_{3}\\\ &1&&\\\
A_{4}&&A_{5}&A_{6}\\\ A_{7}&&A_{8}&A_{9}\\\
\end{pmatrix}\mathcal{M}^{-1},1_{n-1}),$

where $A_{1}$ and $A_{9}$ are $n\times n$ matrices, $A_{5}$ is a $1\times 1$
matrix, and

$\mathcal{M}={\rm diag}(1_{n},\begin{pmatrix}1&\;\;\;1\\\
1&-1\end{pmatrix},1_{n})$

is the change of coordinate matrix from the standard basis to
$\mathcal{B}^{\prime}_{G^{\prime}}$. Specifically, the image can be explicitly
computed as follows,

$\eta(\begin{pmatrix}A_{1}&A_{2}&A_{3}\\\ A_{4}&A_{5}&A_{6}\\\
A_{7}&A_{8}&A_{9}\\\ \end{pmatrix})={\rm
diag}(1_{n-1},\begin{pmatrix}\;\;\;A_{1}&\frac{1}{2}A_{2}&-\frac{1}{2}A_{2}&\;\;\;A_{3}\\\
\;\;\;A_{4}&1/2\cdot(1+A_{5})&1/2\cdot(1-A_{5})&\;\;\;A_{6}\\\
-A_{4}&1/2\cdot(1-A_{5})&1/2\cdot(1+A_{5})&-A_{6}\\\
\;\;\;A_{7}&\frac{1}{2}A_{8}&-\frac{1}{2}A_{8}&\;\;\;A_{9}\\\
\end{pmatrix},1_{n-1}).$

For instance [GRS11, (9.8)], a standard unipotent subgroup $g$ in ${\rm
SO}_{2n+1}$ and the image $\eta(g)$ in ${\rm SO}_{4n}$, with respect to the
standard basis of $F^{4n}$, are

$g=\begin{pmatrix}1_{n}&2x&y\\\ &1&\check{x}\\\ &&1_{n}\end{pmatrix}\in{\rm
SO}_{2n+1},\quad\eta(g)={\rm diag}(1_{n-1},\begin{pmatrix}1_{n}&(x,-x)&y\\\
&1_{2}&\begin{pmatrix}[r]\check{x}\\\ -\check{x}\end{pmatrix}\\\
&&1_{n}\end{pmatrix},1_{n-1})\in{\rm SO}_{4n}.$

Let $P^{\prime}=M^{\prime}\ltimes U^{\prime}$ be the Seigel parabolic subgroup
of $G^{\prime}$ with the Levi part
$M^{\prime}=\varrho^{\prime}(\mathbb{M}^{\prime})$ and the Seigel unipotent
subgroup $U^{\prime}$, where $\varrho^{\prime}:m\mapsto{\rm
diag}(1_{n-1},m,1_{2},m^{\ast},1_{n-1})$. Let us set

$\mathfrak{a}_{n}=\\{x\in{\rm Mat}_{n,n}\,|\,\check{x}=-x\\}.$

Then $U^{\prime}=U^{\prime}_{1}\ltimes U^{\prime}_{0}$ with

$U^{\prime}_{0}=\\{{\rm diag}(1_{n-1},\begin{pmatrix}1_{n}&&u\\\ &1_{2}&\\\
&&1_{n}\end{pmatrix},1_{n-1})\,|\,u\in\mathfrak{a}_{n}\\}$

and

$U^{\prime}_{1}=\\{{\rm diag}(1_{n-1},\begin{pmatrix}1_{n}&(v,-v)&\\\
&1_{2}&\begin{pmatrix}[r]\check{v}\\\ -\check{v}\end{pmatrix}\\\
&&1_{n}\end{pmatrix},1_{n-1})\,|\,v\in F^{n}\\}.$

Let $N^{\prime}$ (resp. $N$) be the standard maximal unipotent subgroup of
$G^{\prime}$ (resp. $G$). Then $N^{\prime}=N^{\prime}_{M^{\prime}}\ltimes
U^{\prime}$ with
$N^{\prime}_{M^{\prime}}=\varrho^{\prime}(N^{\prime}_{\mathbb{M}^{\prime}})$.
Let $V$ be the unipotent radical in $G$ of the standard parabolic subgroup
with Levi ${\rm GL}_{1}^{n-1}\times{\rm SO}_{2n+2}$:

$V=\\{\begin{pmatrix}v&u&\;\;\;y\\\ &1_{2n+2}&-\check{u}\\\
&&\;\;\;v^{\ast}\end{pmatrix}\in{\rm SO}_{4n}\,|\,v\in Z_{n},u\in{\rm
Mat}_{n,2n+2}\\}.$

Let $P^{\diamond}=M^{\diamond}\ltimes U^{\diamond}$ (resp. $P=M\ltimes U$) be
the standard Seigel parabolic subgroup of ${\rm SO}_{2n+2}$ (resp. $G$) with
the Levi part $M^{\diamond}$ (resp. $M$) isomorphic to ${\rm GL}_{n+1}$ (resp.
${\rm GL}_{2n}$). We let $\mathbb{M}={\rm GL}_{2n}$ and use the isomorphism
$\varrho(h)={\rm diag}(h,h^{\ast})$ to identify $\mathbb{M}$ with $M$. Let

$w^{\prime}_{U^{\diamond}}={\rm diag}(1_{n-1},\begin{pmatrix}&1_{n+1}\\\
1_{n+1}&\end{pmatrix},1_{n-1})$

represent the longest $M^{\diamond}$-reduced Weyl element of ${\rm
SO}_{2n+2}$. Let $w_{U}=\begin{pmatrix}&1_{2n}\\\ 1_{2n}&\end{pmatrix}$
represent the longest $M$-reduced Weyl element of $G$. We write

$\gamma=w_{U}(w^{\prime}_{U^{\diamond}})^{-1}=\begin{pmatrix}&1_{n+1}&&\vspace{-.1cm}\\\
\vspace{-.1cm}&&&1_{n-1}\\\ 1_{n-1}&&\vspace{-.1cm}&\\\
&&1_{n+1}&\end{pmatrix}.$

Let us denote

$V_{\gamma}=V\cap\gamma^{-1}N\gamma=\\{\begin{pmatrix}v&&u&\vspace{-.1cm}\\\
&1_{n+1}&\vspace{-.1cm}&-\check{u}\\\ &&1_{n+1}&\vspace{-.1cm}\\\
&&&v^{\ast}\end{pmatrix}\,|\,v\in Z_{n},u\in{\rm Mat}_{n,n+1}\\}.$

Let $N_{M}$ (resp. $N_{\mathbb{M}}$) be the standard maximal unipotent
subgroup of $M$ (resp. $\mathbb{M}$). Let $K=K_{{\rm GL}_{4n}}\cap{\rm
SO}_{4n}$. Then $K$ is a maximal compact subgroup of ${\rm SO}_{4n}$.

###  5.2. Additive Characters

We fix a non-trivial additive character $\psi$ of $F$ and a non-degenerate
character $\psi_{N_{\mathbb{M}}}$ of $N_{\mathbb{M}}$. Let $\psi_{\circ}$ be
the non-trivial character of $F$ given by

$\psi_{\circ}(x)=\psi_{N_{\mathbb{M}}}(1_{2n}+x\epsilon_{n,n+1})$

for $x\in F$. As discussed in [LM17, Remark 6.4] (cf. [LM16, §4.1, §6.1]), the
statements in the sequel will not depend on the choice of
$\psi_{N_{\mathbb{M}}}$. For convenience, we set

$\psi_{N_{\mathbb{M}}}(u)=\psi(u_{1,2}+\dotsm+u_{2n-1,2n}).$

Therefore $\psi_{\circ}=\psi$. We define the characters of various unipotent
groups. We denote by $\psi_{N}$ the degenerate character on $N$ given by
$\psi_{N}(uv)=\psi_{N_{M}}(u)$ for any $u\in N_{M}$ and $v\in U$. Let
$\psi_{N_{M}}$ be the non-degenerate character of $N_{M}$ given by
$\psi_{N_{M}}(\varrho(u))=\psi_{N_{\mathbb{M}}}(u)$. Let
$\psi_{N^{\prime}_{M^{\prime}}}$ be the non-degenerate character of
$N^{\prime}_{M^{\prime}}$ such that
$\psi_{N^{\prime}_{M^{\prime}}}(\varrho(u))=\psi_{N^{\prime}_{\mathbb{M}^{\prime}}}(u)$.
With the choice of $\psi_{N_{\mathbb{M}}}$, we have

$\psi_{N^{\prime}_{\mathbb{M}^{\prime}}}(u^{\prime})=\psi(u^{\prime}_{1,2}+\dotsm+u^{\prime}_{n-1,n}).$

Let $\psi_{U^{\prime}}$ be the character on $U^{\prime}$ given by
$\psi_{U^{\prime}}(u)=\psi(u_{2n-1,2n})^{-1}$. Then the non-degenerate
character $\psi_{N^{\prime}}$ is given by (cf. [LM16, 7.1])

$\psi_{N^{\prime}}(uv)=\psi_{N^{\prime}_{M^{\prime}}}(u)\psi_{U^{\prime}}(v)=\psi(u_{n,n+1}+\dotsm+u_{2n-2,2n-1}-v_{2n-1,2n})$

with $u\in N^{\prime}_{M^{\prime}}$ and $v\in U^{\prime}$. Our choice is
consistent with the characters considered in [GRS11, (9.9)] as we have
specified $\lambda=-1$ in their notation (cf. [GRS11, Theorem 9.4]). An
element in $V$ can be written as $vu$ where $u$ fixes $e_{1},\dotsm,e_{n}$,
$v$ fixes $e_{n+1},e_{n+2},\dotsm,e_{-n-1}$ and we set [GRS11, (3.48)]

$\psi_{V}(uv)=\psi^{-1}(v_{1,2}+\dotsm+v_{n-1,n})\psi^{-1}(u_{n,n+1}+u_{n,n+2}).$

###  5.3. Whittaker Models and the Intertwining Operator

Let $\pi$ be an irreducible generic representation of $\mathbb{M}$ with its
Whittaker model $\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\pi)$ with respect to the
character $\psi_{N_{\mathbb{M}}}$. Similarly we use the notation
$\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}$, $\mathbb{W}^{\psi_{N_{M}}}$,
$\mathbb{W}^{\psi^{-1}_{N_{M}}}$.

Given $g\in G$, we define $\nu(g)$ by $\nu(u\varrho(m)k)=|\det m|$ for any
$u\in U$, $m\in\mathbb{M}$, and $k\in K$ via the Iwasawa decomposition. For
any $f\in C^{\infty}(G)$ and $s\in\mathbb{C}$, define
$f_{s}(g)=f(g)\nu(g)^{s}$, $g\in G$. Let ${\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ be the space of $G$-smooth left
$U$-invariant function $W:G\rightarrow\mathbb{C}$ such that for all $g\in G$,
the function $m\mapsto\delta^{\frac{1}{2}}_{P}(m)W(mg)$ on $M$ belongs to
$\mathbb{W}^{\psi_{N_{M}}}(\pi)$. Similarly we define ${\rm
Ind}(\mathbb{W}^{\psi^{-1}_{N_{M}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi_{N_{\mathbb{M}}}}(\pi))$, ${\rm
Ind}(\mathbb{W}^{\psi^{-1}_{N_{\mathbb{M}}}}(\pi))$. For any $s\in\mathbb{C}$,
we have a representation $I(s,g)$ on the space ${\rm
Ind}(s,\mathbb{W}^{\psi_{N_{M}}}(\pi))$ given by
$(I(s,g)W)_{s})(x)=W_{s}(xg)$, $x,g\in G$.

We define the intertwining operator [Kap15, §3.1]

$M(s,\pi):{\rm Ind}(s,\mathbb{W}^{\psi_{N_{M}}}(\pi))\rightarrow{\rm
Ind}(-s,\mathbb{W}^{\psi_{N_{M}}}(\pi^{\vee}))$

by (the analytic continuation of)

$M(s,\pi)W(g)=\nu(g)^{s}\int_{U}W_{s}(\varrho(\mathfrak{t})w_{U}ug)\,du,$

where $\mathfrak{t}\in{\rm GL}_{2n}$ is introduced in order to preserve the
character $\psi_{N_{M}}$.

In the case where $F$ is a $p$-adic and $\pi$ and $\psi$ are unramified, and
there exist (necessarily unique) $K$-fixed element $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $\check{W}^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi^{\vee}))$ such that
$W^{\circ}(e)=\check{W}^{\circ}(e)=1$, then we have [Sha88, §2] (assuming
${\rm vol}(U\cup K)=1$)

(5.1) $M(s,\pi)W^{\circ}={\rm vol}(U\cap
K)\frac{L(2s,\pi,\wedge^{2})}{L(2s+1,\pi,\wedge^{2})}\check{W}^{\circ}.$

The following result is an analogue of [LM16, Proposition 2.1] and [LM17,
Proposition 4.1].

######  Proposition 5.1.

Suppose that $\pi\in{\rm Irr}_{\rm gen}\,\mathbb{M}$ such that
$\pi\cong\pi^{\vee}$. Then $M(s,\pi)$ is holomorphic at $s=\frac{1}{2}$.

###### Proof.

We write $\pi=\sigma_{1}\boxplus\dotsm\boxplus\sigma_{k}$ where
$\sigma_{1},\dotsm,\sigma_{k}$ are essentially square-integrable. By
decomposing $M(s,\pi)$ into a product of co-rank one intertwining operators,
it is suffices to show that the following intertwining operators

  1. $(\mathrm{i})$

$\sigma_{i}|\cdot|^{s}\boxplus\sigma_{j}^{\vee}|\cdot|^{-s}\rightarrow\sigma_{j}^{\vee}|\cdot|^{-s}\boxplus\sigma_{i}|\cdot|^{s},\quad
i,j=1,2,\dotsm,k,i\neq j$,

  2. $(\mathrm{ii})$

$M(s,\sigma_{i}),\quad i=1,2,\dotsm,k$,

are holomorphic at $s=\frac{1}{2}$. The holomorphy of $(\mathrm{i})$ follows
from [LM17, Lemma 4.2] and the irreducibility of
$\sigma_{i}\boxplus\sigma_{j}^{\vee}$. If $F=\mathbb{R}$ or $\mathbb{C}$, the
holomorphy of $(\mathrm{ii})$ is an immediate consequence of [LM17, Lemma 4.3]
and the irreducibility of $\sigma_{i}\boxplus\sigma_{i}^{\vee}$. Under the
identical condition of [LM17, Lemma 4.3], Luo [Luo24, §2.4 Main Theorem 1]
recently resolves the conjecture which turns out to be the stronger conclusion
that $L(2s,\sigma_{i},\wedge^{2})^{-1}M(s,\sigma_{i})$ is holomorphic for the
$p$-adic case. ∎

###  5.4. Period and Distinction

####  5.4.1. Global period

Let $\mathfrak{S}$ be the Shalika subgroup

$\mathfrak{S}=\left\\{\begin{pmatrix}g&\\\
&g\end{pmatrix}\begin{pmatrix}1_{n}&X\\\
&1_{n}\end{pmatrix}\,\middle|\,g\in\mathbb{M}^{\prime},X\in{\rm
Mat}_{n,n}\right\\}$

of $\mathbb{M}$ with the Shalika character
$\psi_{\mathfrak{S}}\left(\begin{pmatrix}g&\\\
&g\end{pmatrix}\begin{pmatrix}1_{n}&X\\\ &1_{n}\end{pmatrix}\right)=\psi({\rm
tr}\,X)$. Let $F$ be a number field. We say that $\pi\in{\rm
Cusp}\,\mathbb{M}$ is of orthogonal type if

$\int_{\mathfrak{S}(F)\backslash\mathfrak{S}(\mathbb{A})}\varphi(s)\psi^{-1}_{\mathfrak{S}}(s)\,ds\neq
0$

for some $\varphi$ in the space of $\pi$. The following characterization is
due to Jacquet–Shalika .

######  Proposition 5.2.

Let $\pi\in{\rm Cusp}\,\mathbb{M}$. Then $\pi$ is of orthogonal type if and
only if $L^{S}(s,\pi,\wedge^{2})$ has a pole at $s=1$.

We let ${\rm Ocusp}_{k}\,\mathbb{M}$ be the set of automorphic representation
$\pi$ of the form $\pi_{1}\times\dotsm\pi_{k}$, where $\pi_{i}\in{\rm
Cusp}\,{\rm GL}_{2n_{i}}(F)$, $i=1,\dotsm,k$ are pairwise inequivalent (that
is, $\pi_{i}\not\cong\pi_{j}$ if $i\neq j$) and of orthogonal type with
$n_{1}+\dotsm+n_{k}=n$ and $n_{i}>1$ for all $i$. In particular
$\pi_{1}^{\vee}\cong\pi_{i}$ and the central character $\omega_{\pi}$ of $\pi$
is trivial on $\mathbb{A}^{\times}$. We let ${\rm
Ocusp}\,\mathbb{M}=\cup_{k}{\rm Ocusp}_{k}\,\mathbb{M}$.

####  5.4.2. Local distinction

Let $F$ be a local field. We say that $\pi\in{\rm Irr}\,\mathbb{M}$ is of
orthogonal type if it has a non-trivial
$(\mathfrak{S}(F),\psi_{\mathfrak{S}(F)})$-invariant linear form $\lambda$. In
particular, the central character $\omega_{\pi}$ of $\pi$ is trivial on
$F^{\times}$. We write ${\rm Irr}_{\rm orth}\,\mathbb{M}$ for the set of
irreducible representations of orthogonal type. Clearly, if $\pi$ is of
orthogonal type in the global setting, then all its local components $\pi_{v}$
are irreducible and of orthogonal type as well.

######  Lemma 5.3.

  1. $(\mathrm{1})$

([JR96]–$p$-adic case; [AGJ09]–archimedean case) Suppose that $\pi\in{\rm
Irr}_{\rm orth}\,{\rm GL}_{2n}(F)$. Then the space of
$(\mathfrak{S}(F),\psi_{\mathfrak{S}(F)})$-invariant linear forms on $\pi$ is
one-dimensional. Moreover $\pi^{\vee}\cong\pi$.

  2. $(\mathrm{2})$

([Jo20, Proposition 3.12]–$p$-adic case) Suppose that $\pi\in{\rm Irr}\,{\rm
GL}_{2n}(F)$ is square integrable and $F$ is p-adic. Then $\pi\in{\rm
Irr}_{\rm orth}(F)$ if and only if $L(s,\pi,\wedge^{2})$ has a pole at $s=0$.

  3. $(\mathrm{3})$

(cf. [Mat17, Theorem 1.1]–$p$-adic case) Suppose that $\pi_{i}\in{\rm
Irr}_{\rm orth}\,{\rm GL}_{2n_{i}}(F)$, $i=1,2$ and the parabolic induction
$\pi_{1}\boxplus\pi_{2}$ is irreducible. Then $\pi_{1}\boxplus\pi_{2}\in{\rm
Irr}_{\rm orth}\,{\rm GL}_{2n_{1}+2n_{2}}(F)$.

###### Proof.

Let $P^{\circ}=M^{\circ}\ltimes U^{\circ}$ be the parabolic subgroup of type
$(2n_{1},2n_{2})$ of $\mathbb{M}$ so that $\pi:=\pi_{1}\boxplus\pi_{2}$ is the
parabolic induction of $\pi_{1}\otimes\pi_{2}$. Let $\beta_{i}$ be a non-
trivial $(\mathfrak{S}(F),\psi_{\mathfrak{S}(F)})$-invariant linear
functionals on $\pi_{i}$ for $i=1,2$. Let $\beta$ be the functional
$\beta_{1}\otimes\beta_{2}$ on $\pi_{1}\otimes\pi_{2}$. We define a linear
form $\lambda$ on $\pi$ by

$\lambda(\phi)=\int_{{\rm Mat}_{n_{2}\times
n_{1}}}\beta(\phi(\begin{pmatrix}1_{n_{1}}&\vspace{-.5mm}&&\\\
&1_{n_{1}}&\vspace{-.5mm}&\\\ &x&1_{n_{2}}&\vspace{-.5mm}\\\
&&&1_{n_{2}}\end{pmatrix}))\,dx.$

Then $\lambda$ is a non-zero functionals on $\pi$ which is
$\psi_{\mathfrak{S}}(F)$-invariant under $\mathfrak{S}(F)$ (cf. [Mat17, Lemma
3.1]). Furthermore the integral converges absolutely. The absolute convergence
is asserted for archimedean cases in [FJ93, Lemma 3.1] and is settled for
$p$-adic case in [Mat17, Lemma 3.2]. ∎

On the purpose of completeness, we recall the following classification
theorem, due to Matringe, of the set ${\rm Irr}_{\rm gen,orth}\,{\rm
GL}_{n}(F)$ of generic representations of orthogonal types.

######  Proposition 5.4.

([Mat17, Corollary 1.1]) Assume that $F$ is a $p$-adic field. Then the set
${\rm Irr}_{\rm gen,orth}\,{\rm GL}_{2n}(F)$ consists of the irreducible
representations of the form

$\pi=(\sigma_{1}\boxplus\sigma^{\vee}_{1})\boxplus\dotsm\boxplus(\sigma_{k}\boxplus\sigma_{k}^{\vee})\boxplus\tau_{1}\boxplus\dotsm\boxplus\tau_{\ell}$

where $\sigma_{1},\dotsm,\sigma_{k}$ are essentially square-integrable and
$\tau_{1},\dotsm,\tau_{\ell}$ are square-integrable of orthogonal type (i.e.,
$L(0,\tau_{i},\wedge^{2})=\infty$ for all $i=1,\dotsm,\ell$).

###  5.5. Descent maps

####  5.5.1. Global descent

We now turn our sight to the global setting. For any automorphic form
$\varphi$ on $G(\mathbb{A})$, let ${\rm GG}(\varphi)$ be the Gelfand–Graev
coefficient (a function on $G^{\prime}(F)\backslash G^{\prime}(\mathbb{A})$)

${\rm GG}(\varphi)(g)=\int_{V(F)\backslash
V(\mathbb{A})}\varphi(vg)\psi^{-1}_{V}(v)\,dv,\quad g\in
G^{\prime}(\mathbb{A}).$

We study the automorphic representation $\pi$ of $\mathbb{M}(\mathbb{A})$ in
${\rm Ocusp}\,\mathbb{M}$ realized on the space of Eisenstein series induced
from $\pi_{1}\otimes\dotsm\otimes\pi_{k}$. We view $\pi$ as a representation
of $M(\mathbb{A})$ via $\varrho$. Let $\mathcal{A}(\pi)$ be the space of
functions $\varphi:M(F)U(\mathbb{A})\backslash
G(\mathbb{A})\rightarrow\mathbb{C}$ such that
$m\mapsto\delta^{\frac{1}{2}}_{P}(m)\varphi(mg)$, $m\in M(\mathbb{A})$ belongs
to the space of $\pi$ for all $g\in G(\mathbb{A})$. Let
$\varphi_{s}(g)=\nu(g)^{s}\varphi(g)$. We define Eisenstein series

$\mathcal{E}(s,\varphi)=\sum_{\gamma\in P(F)\backslash G(F)}\varphi_{s}(\gamma
g),\quad,\varphi\in\mathcal{A}(\pi).$

The space of $\pi$ is preserved under the conjugation by $w_{U}$. Then we
define intertwining operator
$M(s,\pi):\mathcal{A}(\pi)\rightarrow\mathcal{A}(\pi)$ given by

$M(s,\pi)\varphi(g)=\nu(g)^{s}\int_{U(\mathbb{A})}\varphi_{s}(w_{U}ug)\,du.$

Both $\mathcal{E}(s,\varphi)$ and $M(s,\pi)$ converge absolutely for ${\rm
Re}(s)\gg 0$ and extend meromorphically to $\mathbb{C}$. It follows from
[GRS11, Theorem 2.1] that both $\mathcal{E}(s,\varphi)$ and $M(s,\pi)$ have a
pole of order $k$ at $s=\frac{1}{2}$. Let

$\mathcal{E}_{-k}\varphi=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}\mathcal{E}(s,\varphi)$

and

$M_{-k}=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}M(s,\pi)$

be the leading coefficient in the Laurent series expansion at $s=\frac{1}{2}$.
The constant term

$\mathcal{E}^{U}_{-k}\varphi(g)=\int_{U(F)\backslash
U(\mathbb{A})}\mathcal{E}_{-k}\varphi(ug)\,du$

of $\mathcal{E}_{-k}\varphi$ along $U$ is given by

$\mathcal{E}^{U}_{-k}\varphi(g)=(M_{-k}(\varphi)(g))_{-\frac{1}{2}}.$

The descent of $\pi$ (with respect to $\psi_{N_{\mathbb{M}}}$) is the space
$\mathcal{D}_{\psi}(\pi)$ spanned by ${\rm GG}(\mathcal{E}_{-k}\varphi)$ with
$\varphi\in\mathcal{A}(\pi)$. It is known from [GRS11, Theorem 3.1] that
$\mathcal{D}_{\psi}(\pi)$ is multiplicity free. By [GRS11, Theorem 9.4],
$\mathcal{D}_{\psi}(\pi)$ is a non-trivial cuspidal (and globally generic)
automorphic representation of $G^{\prime}$. We expect it to be irreducible.
This result will be the analogue of [GJS12, Theorem 2.3] in metaplectic group
case and [Mor18, Theorem 1.1] in even unitary group case. It is likely that
the method of [GJS12, Theorem 2.3] and [Mor18, Theorem 1.1] is extended to the
case at hand. Since this is beyond the scope of the current paper we simply
make it a working hypothesis.

######  Working Hypothesis 5.5.

When $\pi\in{\rm Ocusp}\,\mathbb{M}$, $\mathcal{D}_{\psi}(\pi)$ is
irreducible.

The image of the weak lift is then a consequence of [GRS11, Theorem 11.2]. In
particular, [GRS11, Theorem 9.4 (3)] describes the image for unramified
places.

######  Proposition 5.6.

With the above Working Hypothesis 5.5,
$\pi\mapsto\sigma=\mathcal{D}_{\psi}(\pi)$ defines a bijection between ${\rm
Ocusp}\,\mathbb{M}$ and ${\rm Cusp}_{\psi_{N^{\prime}}}\,G^{\prime}$, the set
of $\psi_{N^{\prime}}$-generic cuspidal automorphic representation of
$G^{\prime}$. Furthermore $\sigma^{\vee}$ weekly lifts to $\pi$.

####  5.5.2. Local descent

We turn back to the local setup. Suppose now that $F$ is local fields. Let
$\pi\in{\rm Irr}_{\rm gen}\,M$. For any $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $s\in\mathbb{C}$, we define a
function on $G^{\prime}$:

$A^{\psi}(s,g,W)=\int_{V_{\gamma}\backslash V}W_{s}(\gamma
v\mathfrak{c}^{n+1}g)\psi^{-1}_{V}(v)\,dv,\quad g\in G^{\prime}.$

where

$\mathfrak{c}={\rm diag}(1_{n-1},-1_{n},\begin{pmatrix}0&1\\\
1&0\end{pmatrix},-1_{n},1_{n-1}).$

Let the sign $\pm$ denote $(-1)^{n+1}$. The (signed) outer automorphism
$\mathfrak{c}$ does not lie in ${\rm SO}_{4n}$. However $\mathfrak{c}{\rm
SO}_{4n}\mathfrak{c}^{-1}=\mathfrak{c}{\rm SO}_{4n}\mathfrak{c}={\rm
SO}_{4n}$. At first glance the presence of (signed) outer automorphism
$\mathfrak{c}$ is implicit in integrals of Ginzburg [Gin90] and Soudry
[Sou93], but it can be reconciled with $\beta_{n,2n}$ defined in [Kap15, p.
397] by

$\beta_{n,2n}\,\varrho(\begin{pmatrix}&1_{n+1}\\\
1_{n-1}&\end{pmatrix})=\gamma\mathfrak{c}^{n+1}\,{\rm diag}(1_{n-1},\pm
1_{n},1_{2},\pm 1_{n},1_{n-1}),$

while the Weyl element $\varrho(\begin{pmatrix}&1_{n+1}\\\
1_{n-1}&\end{pmatrix})$ accounts the discrepancy of our embedding $\eta$ with
$i_{n,2n}$ in [Kap15, §3.1]. The basic properties of $A^{\psi}$ are summarized
in the following proposition. Its proof is identical to [LM16, Lemmata 4.5 and
4.9] and [LM17, Lemmata 5.1 and 8.1], henceforth will be omitted.

######  Proposition 5.7.

Suppose that $\pi\in{\rm Irr}_{\rm gen}\,M$. Let $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $s\in\mathbb{C}$. Then

  1. $(\mathrm{1})$

The integral $A^{\psi}(s,g,W)$ is well-defined and absolutely convergent
uniformly for $s\in\mathbb{C}$. Thus $A^{\psi}(s,g,W)$ is entire in $s$ and
smooth in $g$. In the non-archimedean case the integrand is compactly
supported on $V_{\gamma}\backslash V$.

  2. $(\mathrm{2})$

For any $W\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ and $s\in\mathbb{C}$,
the function $g\mapsto A^{\psi}(s,g,W)$ is smooth and
$(N^{\prime},\psi_{N^{\prime}})$-equivariant.

  3. $(\mathrm{3})$

For an $g,x\in G^{\prime}$, we have

(5.2) $A^{\psi}(s,g,I(s,x)W)=A^{\psi}(s,gx,W).$

  4. $(\mathrm{4})$

Suppose that $\pi$ is unramified, $\psi$ is unramified, and $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ is the standard unramified vector. Then
$A^{\psi}(s,e,W^{\circ})\equiv 1$ (assuming ${\rm vol}(V\cap K)={\rm
vol}(V_{\gamma}\cap K)=1$).

Let $\pi\in{\rm Irr}_{\rm gen}\,\mathbb{M}$, regarded also as a representation
of $M$ via $\varrho$. By the same argument as in [GRS99, Theorem in §1.3]
together with [LM17, Remark 4.3], for any non-zero subrepresentation
$\pi^{\prime}$ of ${\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ there exists
$W^{\prime}\in\pi^{\prime}$ such that $A^{\psi}(s,\cdot,W^{\prime})\not\equiv
0$. Upon twisting $\pi$, we conclude that for any given $s_{0}\in\mathbb{C}$,
there exists $W^{\prime}\in\pi^{\prime}$ such that
$A^{\psi}(s_{0},\cdot,W^{\prime})\not\equiv 0$.

Assume now that $\pi\in{\rm Irr}_{\rm gen,orth}\,\mathbb{M}$. By Proposition
5.1, $M(s,\pi)$ is holomorphic at $s=\frac{1}{2}$. The local descent of $\pi$
is the space $\mathcal{D}_{\psi}(\pi)$ of Whittaker functions on $G^{\prime}$
generated by
$A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi\right)W\right)$ with
$W\in{\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$. By the above observation,
$\mathcal{D}_{\psi}(\pi)\neq 0$.

Let $\pi^{\prime}$ be the image of ${\rm Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$
under $M\left(\frac{1}{2},\pi\right)$. In virtue of (5.2), the space
$\mathcal{D}_{\psi}(\pi)$ is canonically a quotient of the $G^{\prime}$-module
$J_{V}(\pi^{\prime})$ of the $V$-coinvariant of $\pi^{\prime}$. In this
regard, we view $J_{V}(\pi^{\prime})$ as the abstract descent, whereas
$\mathcal{D}_{\psi}(\pi)$ as the explicit descent.

####  5.5.3. Whittaker function of descent

We set [GRS11, (8.14)]

$\psi_{N_{\mathbb{M}}}(u)=\psi(u_{1,2}+\dotsm+u_{n,n+1}-u_{n+1,n+2}-\dotsm-
u_{2n-1,2n}).$

The heart of the proof lies in unfolding integrals which appears in the
forthcoming work of Lapid and Mao [LM22, §4.3, Third paragraph].

######  Theorem 5.8.

$($Lapid and Mao$)$ Let $\pi\in{\rm Ocusp}\,\mathbb{M}$ and
$\varphi\in\mathcal{A}(\pi)$. Then for any $g\in G^{\prime}$ we have

$\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
GG}(\mathcal{E}_{-k}\varphi))=\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N}}(\gamma
v\mathfrak{c}^{n+1}g,\mathcal{E}_{-k}\varphi)\psi^{-1}_{V}(v)\,dv,$

and the integral is absolutely convergent.

###  5.6. Local Rankin–Selberg–Shimura integrals

Let $F$ be either $p$-adic or archimedean, $\pi\in{\rm Irr}_{\rm gen}\,M$ and
$\sigma\in{\rm Irr}_{{\rm gen},\psi^{-1}_{N^{\prime}}}\,G^{\prime}$ with
Whittaker model $\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$. For any
$W^{\prime}\in\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}})(\pi)$, we define the local
Rankin–Selberg–Shimura type integral

$J(s,W^{\prime},W):=\int_{N^{\prime}\backslash
G^{\prime}}W^{\prime}(g)A^{\psi}(s,g,W)\,dg.$

The analytic properties of the integral were initiated by Ginzburg [Gin90],
extended by Soudary [Sou93], and subsequently concluded by Kaplan [Kap15].

######  Proposition 5.9.

Suppose that $\pi\in{\rm Irr}_{\rm gen}\,M$ and $\sigma\in{\rm Irr}_{{\rm
gen},\psi^{-1}_{N^{\prime}}}\,G^{\prime}$. Let
$W^{\prime}\in\mathbb{W}^{\psi^{-1}_{N^{\prime}}}(\sigma)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$. Then

  1. $(\mathrm{1})$

$J(s,W^{\prime},W)$ converges in some right-half plane (depending only on
$\pi$ and $\sigma$) and admits a meromorphic continuation in $s$.

  2. $(\mathrm{2})$

For any $s\in\mathbb{C}$, we can choose $W^{\prime}$ and $W$ such that
$J(s,W^{\prime},W)\neq 0$.

  3. $(\mathrm{3})$

If $\pi$, $\sigma$ and $\psi$ are unramified, $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ is the standard unramified vector and
${W^{\prime}}^{\circ}$ is $K^{\prime}$-invariant with
${W^{\prime}}^{\circ}(e)=1$, then

$J(s,{W^{\prime}}^{\circ},W^{\circ})=\frac{{\rm vol}(K^{\prime})}{{\rm
vol}(N^{\prime}\cap
K^{\prime})}\frac{L(s+\frac{1}{2},\sigma\times\pi)}{L(2s+1,\pi,\wedge^{2})}$

assuming the Haar measure on $V$ and $V_{\gamma}$ are normalized so that ${\rm
vol}(V\cap K)$ and ${\rm vol}(V_{\gamma}\cap K)$ are all $1$.

  4. $(\mathrm{4})$

$J(s,W^{\prime},W)$ satisfies local functional equations:

$J(-s,W^{\prime},M(s,\pi)W)=\omega^{n}_{\pi}(-1)\frac{\gamma(s+\frac{1}{2},\sigma\times\pi,\psi)}{\gamma(2s+1,\pi,\wedge^{2},\psi)}J(s,W^{\prime},W)$

where $\gamma(s,\pi,\wedge^{2},\psi)$ is the gamma factor defined by Shahidi
(pertaining to the exterior square representation).

It is noteworthy that $\displaystyle{\rm
vol}(K^{\prime})=(\prod_{j=1}^{n}\zeta_{F}(2j))^{-1}.$

###  5.7. Factorization of Gelfand-Graev periods

Define $\varphi\in\mathcal{A}(\pi)$ an automorphic form on $G(\mathbb{A})$;

$A^{\psi}(s,g,\varphi):=\int_{V_{\gamma}(\mathbb{A})\backslash
V(\mathbb{A})}\mathcal{W}^{\psi_{N_{M}}}(\varphi_{s},\gamma
v\mathfrak{c}^{n+1}g)\psi^{-1}_{V}(g)\,dv.$

The global descent $\mathcal{D}_{\psi}(\Pi)$ and the local descent
$\mathcal{D}_{\psi}(\Pi_{v})$ are related in the following way;

######  Proposition 5.10.

Suppose that $\varphi\in\mathcal{A}(\pi)$ is factorizable vector and
$\mathcal{W}^{\psi_{N_{M}}}(\varphi,\cdot)=\prod_{v}W_{v}$ with $W_{v}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi_{v}))$. Then for any sufficiently large
finite set of places $S$, we have

(5.3) $\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
GG}(\mathcal{E}_{-k}\varphi))=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\prod_{v\in
S}A^{\psi}\left(-\frac{1}{2},g_{v},M\left(\frac{1}{2},\pi_{v}\right)W_{v}\right),$

where

$m^{S}_{-k,1}(\pi)=\frac{\lim_{s\rightarrow
1}(s-1)^{k}L^{S}(s,\pi,\wedge^{2})}{L^{S}(2,\pi,\wedge^{2})}.$

###### Proof.

We notice that

$\mathcal{W}^{\psi_{N}}(\mathcal{E}_{-k}\varphi)=\mathcal{W}^{\psi_{N_{M}}}(\mathcal{E}^{U}_{-k}\varphi)=\mathcal{W}^{\psi_{N_{M}}}((M_{-k}(\varphi))_{-\frac{1}{2}}).$

Keeping this in mind, Theorem 5.8 can be expressed in terms of the global
transform $A^{\psi}$ as

(5.4) $\mathcal{W}^{\psi_{N^{\prime}}}(g,{\rm
GG}(\mathcal{E}_{-k}\varphi))=A^{\psi}\left(-\frac{1}{2},g,M_{-k}(\varphi)\right).$

We can identify $\mathcal{A}(\pi)$ with ${\rm
Ind}^{G(\mathbb{A})}_{P(\mathbb{A})}\,\pi=\otimes_{v}{\rm
Ind}^{G(F_{v})}_{P(F_{v})}\,\pi_{v}$. For $\rm Re(s)\gg 0$, we have

$\mathcal{W}^{\psi_{N_{M}}}(g,M(s,\pi)\varphi)=\int_{N_{M}(F)\backslash
N_{M}(\mathbb{A})}M(s,\pi)\varphi(ng)\psi^{-1}_{N_{M}}(n)\,dn\\\
=\int_{N_{M}(F)\backslash
N_{M}(\mathbb{A})}\int_{U(\mathbb{A})}\varphi(\varrho(\mathfrak{t}w_{U}ung))\psi^{-1}_{N_{M}}(n)\,du\,dn=\int_{U(\mathbb{A})}\mathcal{W}^{\psi_{N_{M}}}(\varrho(\mathfrak{t})w_{U}ug,\varphi)\,du.$

For $S$ sufficiently large, we are led to

$\mathcal{W}^{\psi_{N_{M}}}(M(s,\pi)\varphi)=m^{S}(s,\pi)\left(\prod_{v\in
S}M(s,\pi_{v})W_{v}\right)\prod_{v\notin S}W_{v}$

as meromorphic functions in $s\in\mathbb{C}$ where

$m^{S}(s,\pi)=\frac{L^{S}(2s,\pi,\wedge^{2})}{L^{S}(2s+1,\pi,\wedge^{2})}.$

Taking the limit as $s\rightarrow\frac{1}{2}$, we get for $S$ large enough,

(5.5) $\mathcal{W}^{\psi_{N_{M}}}(M_{-k}\varphi)=m^{S}_{-k}(\pi)\prod_{v\in
S}M\left(\frac{1}{2},\pi_{v}\right)W_{v}\prod_{v\notin S}W_{v}$

for which

$m^{S}_{-k}(\pi)=\lim_{s\rightarrow\frac{1}{2}}\left(s-\frac{1}{2}\right)^{k}m^{S}(s,\pi)=2^{-k}\frac{\lim_{s\rightarrow
1}(s-1)^{k}L^{S}(s,\pi,\wedge^{2})}{L(2,\pi,\wedge^{2})}=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}.$

For any factorizable $\varphi\in\mathcal{A}(\pi)$, our desired result is
immediate from (5.4) and (5.5). ∎

We unfold Petersson inner products of automorphic forms against the
Gelfand–Graev coefficient of Eisenstein series [GRS11, Theorem 10.3, (10.4)]
and then perform unramified computations [GRS11, (10.59)] to obtain the
following Euler product.

######  Proposition 5.11.

Let $\pi\in{\rm Ocusp}_{k}\,\mathbb{M}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$. Assume our Working Hypothesis 5.5. Let
$\varphi^{\prime}\in\sigma$ be an irreducible $\psi^{-1}_{N^{\prime}}$-generic
cuspidal automorphic representation of $G^{\prime}$ and
$\varphi\in\mathcal{A}(\pi)$ Suppose that
$\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\cdot,\varphi^{\prime})=\prod_{v}W^{\prime}_{v}$
and $\mathcal{W}^{\psi_{N_{M}}}(\cdot,\varphi)=\prod_{v}W_{v}$. Then for any
sufficiently large finite set of places $S$ we have

(5.6) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}(s,\varphi))\rangle_{G^{\prime}}=\frac{1}{2}\cdot\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S}))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}\cdot\frac{L(s+\frac{1}{2},\sigma\times\pi)}{L(2s+1,\pi,\wedge^{2})}\prod_{v\in
S}J(s,W^{\prime}_{v},W_{v}),$

where on the right-hand side we take the unnormalized Tamagawa measure on
$G^{\prime}(F_{S})$, $N^{\prime}(F_{S})$, $V(F_{S})$ and $V_{\gamma}(F_{S})$
(which are independent of the choice of gauge forms when $S$ is sufficiently
large. Refer to [LM17, §6].)

###### Proof.

It is worthwhile to mention that the volume of $G^{\prime}(F)\backslash
G^{\prime}(\mathbb{A})$, appearing on the right-hand side of (2.1), is $2$
equal to the Tamagawa number when we use the Tamagawa measure as in [LM16,
LM17]. We apply [GRS11, Theorem 10.3, (10.4)] in order to unfold

(5.7) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}(s,\varphi))\rangle_{G^{\prime}}=\frac{1}{2}\cdot{\rm
vol}(N^{\prime}(F)\backslash
N^{\prime}(\mathbb{A}))\int_{N^{\prime}(\mathbb{A}\backslash
G^{\prime}(\mathbb{A}))}\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(g,\varphi^{\prime})A^{\psi}(s,g,\varphi)\,dg.$

Similar to [LM16, Propositions 5.7 and 8.4], our desired result is a
consequence of (5.7) in conjunction with the unramified computation [Gin90]
(cf. [GRS11, (10.59)]):

(5.8) $J(s,{W^{\prime}_{v}}^{\circ},W^{\circ}_{v})=\frac{{\rm
vol}(K^{\prime})}{{\rm vol}(N^{\prime}(F_{v})\cap
K^{\prime})}\frac{L(s+\frac{1}{2},\sigma_{v}\times\pi_{v})}{L(2s+1,\pi_{v},\wedge^{2})}$

with $K^{\prime}=K_{{\rm GL}_{2n+1}(F_{v})}\cap G^{\prime}(F_{v})$. Since we
use the Tamagawa measure, the factor $\Delta^{S}_{{\rm SO}_{2n+1}}(1)^{-1}$
arises, where $\Delta^{S}_{{\rm SO}_{2n+1}}(s)=L(M^{\vee}(s))$ is the partial
$L$-function of the dual motive $M^{\vee}$ of the motive $M$ associated to
${\rm SO}_{2n+1}$ by Gross [Gro97]. In light of [Xue17]*Notation and
Convention [II10], we have

$\Delta^{S}_{{\rm SO}_{2n+1}}(1)=\prod_{j=1}^{n}\zeta^{S}_{F}(2j)$

for any finite set of places $S$. ∎

###  5.8. Petersson inner products and Fourier coefficients

Let $F$ be a local field. Let $\pi\in{\rm Irr}_{\rm gen,orth}\,\mathbb{M}$. We
say that $\pi$ is good if the following conditions are satisfied for all
$\psi$;

  1. $(\mathrm{i})$

$\mathcal{D}_{\psi}(\pi)$ is irreducible.

  2. $(\mathrm{ii})$

$J(s,W^{\prime},W)$ is holomorphic at $s=\frac{1}{2}$ for any
$W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\pi)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$.

  3. $(\mathrm{iii})$

For any $W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\pi)$,
$J(\frac{1}{2},W^{\prime},W)$ factors through the map
$W\mapsto(A^{\psi}(-\frac{1}{2},\cdot,M(\frac{1}{2},\pi)W))$.

We verify from (5.2) that for $x\in G^{\prime}$ we have

$J(s,\sigma(x)W^{\prime},I(s,x)W)=J(s,W^{\prime},W),$

where $\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$. Therefore if $\pi$ is good, there
is a non-degenerate $G^{\prime}$-invariant pairing $[\cdot,\cdot]$ on
$\mathcal{D}_{\psi^{-1}}(\pi)\times\mathcal{D}_{\psi}(\pi)$ such that

$J\left(\frac{1}{2},W^{\prime},W\right)=\left[W^{\prime},A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi\right)W\right)\right]$

for any $W^{\prime}\in\mathcal{D}_{\psi^{-1}}(\pi)$ and $W\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$. By [LM15, §2], when $\pi$ is good,
there exists $c_{\pi}$ satisfying

(5.9) $\int^{\rm
st}_{N^{\prime}}\left(\frac{1}{2},\sigma(u)W^{\prime},W\right)\psi_{N^{\prime}}(u)\,du=c_{\pi}W^{\prime}(e)A^{\psi}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\pi\right)W\right).$

######  Proposition 5.12.

Suppose that $F$ is $p$-adic, $\pi\in{\rm Irr}_{\rm gen,orth}\,\mathbb{M}$ and
$\psi$ are unramified. Then $\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$ is
irreducible and unramified. Let $W^{\circ}\in{\rm
Ind}(\mathbb{W}^{\psi_{N_{M}}}(\pi))$ be the standard unramified vector and
let ${W^{\prime}}^{\circ}$ be $K^{\prime}$-invariant with
${W^{\prime}}^{\circ}(e)=1$. Then holds with $c_{\pi}=1$.

###### Proof.

With [GRS11, Theorem 5.6] in hand, the proof of [LM17, Lemma 5.4] can be
carried over verbatim to the first part of our current setting. Assuming ${\rm
vol}(U\cap K)=1$ as in (5.1), we know that

${\rm vol}(K^{\prime})=(\prod_{j=1}^{n}\zeta^{S}_{F}(2j))^{-1}$

Combining it and [LM15, Proposition 2.14] leads to

$\int^{\rm
st}_{N^{\prime}}J\left(\frac{1}{2},\sigma(u){W^{\prime}}^{\circ},W^{\circ}\right)\psi^{-1}_{N^{\prime}}(u)\,du=\frac{{\rm
vol}(N^{\prime}\cap K^{\prime})}{{\rm
vol}(K^{\prime})}\frac{J(\frac{1}{2},{W^{\prime}}^{\circ},W^{\circ})}{L(1,\sigma,{\rm
Ad})}.$

Using the description of $\sigma$ in this case [GRS11, Theorem 5.6] coupled
with [GGP12, §7], we infer that $L(1,\sigma,{\rm Ad})=L(1,\pi,{\rm Sym}^{2})$
and $L(s,\sigma\times\pi)=L(s,\pi\times\pi)$. We recall that $\pi=\pi^{\vee}$
in our case. Therefore (5.1) aligned with of (5.8) implies that

$\frac{1}{L(1,\pi,{\rm
Sym}^{2})}\frac{L(1,\pi\times\pi)}{L(2,\pi,\wedge^{2})}=\frac{L(1,\pi,\wedge^{2})}{L(2,\pi,\wedge^{2})}=M\left(\frac{1}{2},\pi\right)W^{\circ}(e)={W^{\prime}}^{\circ}(e)A^{\psi}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\pi\right)W^{\circ}\right)$

as requested. ∎

We note that a priori $c_{\pi}$ implicitly depends on the choice of Haar
measure on $G^{\prime}$ (the former used in the definition of
$J(s,W^{\prime},W)$) and $U$ (the latter used in the definition of the
intertwining operator), but not on any other groups (refer to [LM17, Remark
5.2] for further details). However the Lie algebras of $G^{\prime}$ and $U$
are both canonically isomorphic as vector spaces to

$\\{X\in{\rm Mat}_{2n,2n}(F)\,|\,{X^{t}}\,w_{2n}=-w_{2n}X\\}.$

We say that the Haar measures on $G^{\prime}$ and $U$ are compatible if they
are defined as unnormalized Tamagawa measure with respect to the same gauge
form via the above identification of Lie algebras. We can now see that
$c_{\pi}$ does not depend on any choice of measure provided that we choose
compatible measures on $G^{\prime}$ and $U$.

######  Proposition 5.13.

Let $\pi\in{\rm Ocusp}_{k}\,\mathbb{M}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$. Assume our Working Hypothesis 5.5. Let
$\varphi^{\prime}\in\sigma$ be an irreducible $\psi^{-1}_{N^{\prime}}$-generic
cuspidal automorphic representation of $G^{\prime}$ and
$\varphi\in\mathcal{A}(\pi)$. Suppose that
$\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\cdot,\varphi^{\prime})=\prod_{v}W^{\prime}_{v}$
and $\mathcal{W}^{\psi_{N_{M}}}(\cdot,\varphi)=\prod_{v}W_{v}$. Then for all
$v$ $\pi_{v}$ is good. Moreover for any sufficiently large finite set of
places $S$, we have

(5.10) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}_{-k}\varphi)\rangle_{G^{\prime}}=\frac{1}{2}\cdot\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S}\backslash
N^{\prime}(F_{S})))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}\cdot
n^{S}_{-k,1}\prod_{v\in S}J\left(\frac{1}{2},W^{\prime}_{v},W_{v}\right),$

where

$n^{S}_{-k,1}=\frac{\lim_{s\rightarrow
1}(s-1)^{k}L^{S}(s,\pi\times\pi)}{L^{S}(2,\pi,\wedge^{2})}.$

###### Proof.

We verify that $\pi_{v}$ is good for all $v$. To this end, it can be deduced
from (5.3) together with the irreducibility of the global descent that
$\mathcal{D}_{\psi^{-1}}(\pi_{v})$ is irreducible for all $v$.

When $\pi\in{\rm Ocusp}_{k}\,\mathbb{M}$ and
$\sigma=\mathcal{D}_{\psi^{-1}}(\pi)$, we obtain
$L^{S}(s,\sigma\times\pi)=L^{S}(s,\pi\times\pi)$, which has a pole of order
$k$ at $s=\frac{1}{2}$. On the one hand, $J(s,W^{\prime}_{v},W_{v})$ is non-
vanishing at $s=\frac{1}{2}$ for appropriate $W^{\prime}_{v}$ and $W_{v}$.
Therefore the right-hand side of (5.6) (when $\varphi^{\prime}\in\sigma$) has
a pole of order at least $k$ at $s=\frac{1}{2}$ for suitable
$\varphi^{\prime}$ and $\varphi$. On the other hand, the left-hand side of
(5.6) has a pole of order at most $k$ at $s=\frac{1}{2}$ because this is true
for $\mathcal{E}(s,\varphi)$ and $\varphi^{\prime}$ is rapidly decreasing.
Multiplying (5.6) by $(s-\frac{1}{2})^{k}$ and then taking the limit as
$s\rightarrow\frac{1}{2}$, we conclude that $J(s,W^{\prime}_{v},W_{v})$ is
holomorphic at $s=\frac{1}{2}$ for all $v$, and for
$\varphi^{\prime}\in\sigma$ and $\varphi\in\mathcal{A}(\pi)$ we get

(5.11) $\langle\varphi^{\prime},{\rm
GG}(\mathcal{E}_{-k}\varphi)\rangle_{G^{\prime}}=\frac{1}{2}\cdot\frac{{\rm
vol}(N^{\prime}(\mathcal{O}_{S}\backslash
N^{\prime}(F_{S})))}{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}\cdot
n^{S}_{-k,1}\prod_{v\in S}J\left(\frac{1}{2},W^{\prime}_{v},W_{v}\right).$

We fix a place $v_{0}$. We now assume that $W_{v_{0}}$ such that
$A^{\psi}\left(-\frac{1}{2},\cdot,M\left(\frac{1}{2},\pi_{v_{0}}\right)W_{v_{0}}\right)\equiv
0$. Then by (5.3), we see that $\mathcal{W^{\psi_{N^{\prime}}}}(\cdot,{\rm
GG}(\mathcal{E}_{-k}\varphi))\equiv 0$, which in turn implies that ${\rm
GG}(\mathcal{E}_{-k}\varphi)\equiv 0$ owing to the irreducibility and
genericity of the descent. We conclude from (5.11) that
$J(\frac{1}{2},W^{\prime}_{v_{0}},W_{v_{0}})=0$. Henceforth $\pi_{v_{0}}$ is
good. ∎

While we expect that any irreducible generic representation of orthogonal type
$\pi_{v}$ is good, the proof that we give under the assumption that $\pi_{v}$
is the local component of $\pi\in{\rm Scusp}_{k}\,\mathbb{M}$ is valid at the
very least for the case of our interest, which pertains to the global period
integrals. Our proof here requires a local to global argument. Whereas the
global approach is overkill and indirect, one may directly prove the desired
result by using a local functional equation [LM17, Proposition 5.3]. For the
sake of brevity, we only include this indirect mean in keeping with the spirit
of [LM16, Propositions 5.7 and 8.4].

######  Theorem 5.14.

Let $\pi\in{\rm Ocusp}_{k}\,\mathbb{M}$ and $\sigma=\mathcal{D}_{\psi}(\pi)$.
Assume our Working Hypothesis 5.5. Let $S$ be a finite set of places including
all the archimedean places such that $\pi$ and $\psi$ are all unramified
outside $S$. Then for any $\varphi\in\sigma$ and
$\varphi^{\vee}\in\sigma^{\vee}$ which are fixed under $K^{\prime}_{v}$ for
all $v\not\in S$, we have

(5.12)
$\quad\quad\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}(\varphi^{\vee})=2^{1-k}\left(\prod_{v\in
S}c^{-1}_{\pi_{v}}\right)\frac{\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}{L^{S}(1,\pi,{\rm
Sym}^{2})}\\\ \times({\rm vol}(N^{\prime}(\mathcal{O}_{S})\backslash
N^{\prime}(F_{S})))^{-1}\int_{N^{\prime}(F_{S})}^{\rm
st}\langle\sigma(u)\varphi,\varphi^{\vee}\rangle_{G^{\prime}}\psi^{-1}_{N^{\prime}}(u)\,du.\quad\quad$

###### Proof.

Since the descent of $\pi$ is the space
$\mathcal{D}_{\psi_{N_{\mathbb{M}}}}(\pi)$ spanned by ${\rm
GG}(\mathcal{E}_{-k}\varphi^{\flat})$ for
$\varphi^{\flat}\in\mathcal{A}(\pi)$, we initially tackle with this for the
case $\varphi^{\vee}={\rm GG}(\mathcal{E}_{-k}\varphi^{\sharp})$ with
$\varphi^{\sharp}\in\mathcal{A}(\pi)$, and this identity (5.12) then extends
via linearity to all $\varphi^{\vee}$. In view of Proposition 5.11, we find
that

$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}({\rm
GG}(\mathcal{E}_{-k}\varphi^{\sharp}))=\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\prod_{v\in
S}A^{\psi^{-1}}\left(-\frac{1}{2},e,M\left(\frac{1}{2},\pi_{v}\right),W^{\sharp}_{v}\right).$

Counting on the fact $\pi_{v}$ is good in Proposition 5.13, we exploit
Proposition 5.12 to insert the identity (5.9) for all $v$ with $c_{\pi_{v}}=1$
for almost all $v$. In doing so, the above can be read as

(5.13)
$\mathcal{W}^{\psi_{N^{\prime}}}(\varphi)\mathcal{W}^{\psi^{-1}_{N^{\prime}}}({\rm
GG}(\mathcal{E}_{-k}\varphi^{\sharp}))=\frac{m^{S}_{-k,1}(\pi)}{2^{k}}\left(\prod_{v\in
S}c^{-1}_{\pi_{v}}\int^{\rm
st}_{N^{\prime}(F_{v})}J\left(\frac{1}{2},\sigma_{v}(u_{v})W_{v},W^{\sharp}_{v}\right)\psi^{-1}_{N^{\prime}(F_{v})}(u_{v})\,du_{v}\right).$

Integrating (5.10) over $N^{\prime}(F_{S})$ for $v\in S$, a product of
regularized integrals is turned into

(5.14) $\prod_{v\in S}\int^{\rm
st}_{N^{\prime}(F_{v})}J\left(\frac{1}{2},\sigma_{v}(u_{v})W_{v},W^{\sharp}_{v}\right)\psi^{-1}_{N^{\prime}(F_{v})}(u_{v})\,du_{v}\\\
=\frac{2\prod_{j=1}^{n}\zeta^{S}_{F}(2j)}{{\rm
vol}(N^{\prime}(\mathcal{O}_{S})\backslash N^{\prime}(F_{S}))\cdot
n^{S}_{-k,1}}\int^{\rm st}_{N^{\prime}(F_{S})}\langle\sigma(u)\varphi,{\rm
GG}(\mathcal{E}_{-k}\varphi^{\sharp})\rangle_{G^{\prime}}\psi^{-1}_{N^{\prime}(F_{v})}(u)\,du.$

The factorization
$L^{S}(s,\pi\times\pi)=L(s,\pi\times\pi^{\vee})=L^{S}(s,\pi,{\rm
Sym}^{2})L^{S}(s,\pi,\wedge^{2})$ enables us to simplify the ratio

(5.15) $m^{S}_{-k,1}/n^{S}_{-k,1}=1/L^{S}(1,\pi,{\rm Sym}^{2})$

whose partial $L$-function appearing in the denominator is actually the
adjoint $L$-function $L^{S}(s,\sigma,{\rm Ad})$ for $\sigma$. Combining (5.13)
and (5.14) with (5.15), we are led to the conclusion. ∎

######  Acknowledgments.

This paper is inspired from [LM16] and owes much to [LM16], when the author
learned it in the workshop proceeding in honor of Prof. James Cogdell’s 60th
birthday. It is pleasure to revisit remaining cases and dedicate this paper to
Prof. James Cogdell on the occasion of his 70th birthday. We wish Prof. James
Cogdell many more happy and healthy years. We would like to thank to Prof. Sug
Woo Shin for drawing author’s attention to residues of Rankin–Selberg type
integrals. This work was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (No. RS-2023-00209992).

######  Conflicts of interest.

The authors state that there is no conflict of interest.

######  Data availability.

This manuscript has no associated data.

## References