# Influence of spin-orbit coupling on chemical bonding

Andris Gulans1,2 Claudia Draxl2 1Department of Physics, University of Latvia,
Jelgavas iela 3, Riga, LV-1004 Latvia  2Physics Department and IRIS Adlershof,
Humboldt-Universität zu Berlin, Zum Großen Windkanal 2, 12489 Berlin, Germany

###### Abstract

The influence of spin-orbit interaction on chemical bonds in elemental solids
and homonuclear dimers is analyzed by means of density-functional-theory
calculations. Employing highly precise all-electron full-potential
methodology, our results represent benchmark quality. Comparison of the
scalar- and fully-relativistic approaches for elemental solids shows that the
spin-orbit interaction may contract or expand the volume of the considered
material. The largest variation of the volume is obtained for Au, Tl, I, Bi,
Po and Hg, exhibiting changes between 1.0–7.6%. Using the tight-binding model,
we show for diatomic molecules that the nature of this effect lies in the
angular rearrangement of bonding and antibonding orbitals introduced by spin-
orbit coupling. Such an angular rearrangement appears in partially filled
$p$\- or $d$-orbitals in heavy elements. Finally, we discuss the impact of the
relativistic effects on the chemical bonding in single-layer iodides and
transition metal dichalcogenides.

##  I Introduction

Relativistic quantum theory prescribes that the spin of an electron is coupled
with its spatial motion. This effect is known as the spin-orbit coupling
(SOC), and it gives rise to a number of physical and chemical properties, such
as the band gap in graphene and topological insulating phases. Previous
studies have shown that SOC not only impacts electronic bands but also
symmetry [1] and chemical bonding, particularly in solids with heavy elements
[2, 3, 4, 5]. For example, the volume of simple cubic Po expands by 5% due to
the spin-orbit interaction [2] demonstrating that there are materials where
this effect is sizeable and must be accounted for.

The vast majority of calculations performed using Kohn-Sham density-functional
theory (DFT) [6, 7] employ scalar-relativistic approximations, some even
disregard relativity completely. Such a choice does not affect bond lengths in
materials with light elements, but is questionable when it comes to materials
containing heavy atoms. A few previous studies [8, 3, 9] have compared
structural and elastic properties of elemental solids as calculated with the
scalar- and fully relativistic approximations, where e.g. Refs. [3] and [8]
were focusing on just a few specific groups of the Periodic Table of Elements
(PTE). Such selection exemplified a few chemical elements for which SOC plays
an important role but it does not allow one to make conclusions about the
trends in the PTE. In Ref. [9], the complete 5$d$ and almost the complete 6$p$
series are discussed. However, the presented data are not provided in detail
such to make them useful for quantitative analysis and, hence, benchmarking
purposes. None of these three studies provide a detailed reasoning why and how
SOC impacts the bond lengths. Such a discussion is available only in
exceptional cases, like for the expansion of the bond in Bi2 [10] using the
tight-binding framework. Similarly, Ref. [11] presents the mechanism how SOC
influences the chemical bond in Tl2 and TlH. A more general discussion on the
impact of SOC on chemical bonds is still missing.

The computational materials-science community performs scalar-relativistic
calculations on a daily basis, yet the numerical precision for capturing
subtle effects may not always be taken for granted. A recent study [12] has
shown that modern DFT codes show a sufficient level of reproducibility for
elemental solids, but this outcome was a result of a concerted effort. Here,
we go beyond by involving fully-relativistic calculations which are
substantially more demanding. First, the spin-orbit interaction requires a
spinor formalism, doubling the degrees of freedom. The frequently used second-
variational approach [13] allows one to work with just a small overhead
compared to the cost of scalar-relativistic calculations, but the numerical
uncertainties related to this approximation are often difficult to control.
Second, in contrast to scalar-relativistic theory, fully-relativistic
wavefunctions exhibit a different asymptotic power law in the vicinity of the
nuclei. As a consequence, one distinguishes between states with the same
angular momentum $l$, but different total momentum $j$. Overall, SOC effects
mainly originate from the spatial region close to the nuclei, and this is
where many DFT codes make use of approximations such as employing
pseudopotentials. Having these two points in mind, it is natural to expect
that including SOC presents an additional challenge concerning precision and
reproducibility of DFT data.

In this paper, we present our treatment of relativity in the all-electron
full-potential code exciting [14] which is suitable for reaching microhartree
precision as demonstrated in Ref. 15 for light atoms in the non-relativistic
case. Such a feature is vitally important for the quality control of
calculations and benchmarking, because only controlling the numerical
precision enables systematic improvements what concerns the accuracy of the
underlying methodology. We use this implementation for performing high-quality
DFT calculations of the equation of state employing scalar- and fully-
relativistic theory, where we consider 55 non-magnetic elemental solids of the
periods 4–6. We demonstrate the effect of SOC on the lattice parameters and
provide a qualitative explanation to it based on a tight-binding model applied
to dimers of $p$-elements. Finally, we discuss this effect in two-
dimensionsional transition-metal dichalcogenides (TMDCs) and group-IIB and
-IVA iodides.

##  II Method

###  II.1 Relativity in DFT calculations

The relativistic Kohn-Sham theory is based on approximations to the four-
component Dirac equation which, written for the large component $\Psi$ and the
small component $\Phi$ in matrix form, reads

$\left(\begin{array}[]{cc}V&c(\boldsymbol{\sigma p})\\\ c(\boldsymbol{\sigma
p})&V-2c^{2}\end{array}\right)\left(\begin{array}[]{c}\Psi\\\
\Phi\end{array}\right)=E\left(\begin{array}[]{c}\Psi\\\
\Phi\end{array}\right),$ (1)

where $V$, $\mathbf{p}$, and $\boldsymbol{\sigma}$ are the effective Kohn-Sham
potential, the momentum operator, and the vector of Pauli matrices,
respectively. Formal elimination of the small component leads to an equation
for the large component,

$\frac{1}{2}(\boldsymbol{\sigma
p})\left[1+(E-V)/2c^{2}\right]^{-1}(\boldsymbol{\sigma p})\Psi+V\Psi=E\Psi.$
(2)

The factor $[1-(E-V)/2c^{2}]^{-1}$ in the kinetic-energy term introduces a
non-trivial energy dependence which turns Eq. 2 into a non-linear eigenvalue
problem. The simplest approach to make this equation tractable is to introduce
the zero-order regular approximation (ZORA) [16, 17]. Unfortunately, ZORA is
not gauge invariant, namely, adding a constant to the effective potential $V$
results in a change of the total energy. Therefore, we apply the infinite-
order relativistic approximation (IORA) [18] that transforms Eq. 2 into

$\begin{array}[]{l}\frac{1}{2}(\boldsymbol{\sigma
p})\left(1-\frac{V}{2c^{2}}\right)^{-1}(\boldsymbol{\sigma p})\Psi+V\Psi=\\\
\frac{E}{4c^{2}}(\boldsymbol{\sigma
p})\left(1-\frac{V}{2c^{2}}\right)^{-2}(\boldsymbol{\sigma
p})\Psi+E\Psi.\end{array}$ (3)

This equation presents a generalized eigenvalue problem, and its solutions are
the large components that are not orthogonal among themselves. Orthogonality
is restored only for 4-spinors once the small component is considered too. In
practice, however, we apply these relativistic approximations not to the deep-
lying core states but to semicore, valence, and conduction bands, where the
small component contributes to the density and the total energy only
negligibly. Therefore, we restrict our calculations to considering only the
large component that we normalize to 1.

To proceed further with Eqs. 2–3, one uses the following identity:

$\frac{1}{2}(\boldsymbol{\sigma p})K(\boldsymbol{\sigma
p})=\frac{1}{2}\mathbf{p}K\mathbf{p}+\frac{i}{2}\boldsymbol{\sigma}(\nabla
K\times\mathbf{p}),$ (4)

where $K$ is some function of $\mathbf{r}$, for example,
$K=\left(1-\frac{V}{2c^{2}}\right)^{-1}$. The two terms on the right-hand
side, $\frac{1}{2}\mathbf{p}K\mathbf{p}$ and
$\frac{i}{2}\boldsymbol{\sigma}(\nabla K\times\mathbf{p})$, are known as the
scalar-relativistic and spin-orbit contributions to the kinetic energy,
respectively. Ignoring the latter, turns Eq. 3 into the scalar-relativistic
approximation:

$\begin{split}\frac{1}{2}\boldsymbol{p}\left(1-\frac{V}{2c^{2}}\right)^{-1}\boldsymbol{p}\psi+V\psi=\\\
\frac{E}{4c^{2}}\boldsymbol{p}\left(1-\frac{V}{2c^{2}}\right)^{-2}\boldsymbol{p}\psi+E\psi.\end{split}$
(5)

###  II.2  Scalar-relativistic LAPW+LO

In this work, we employ the basis consisting of linearized augmented
planewaves (LAPW) and local orbitals (LO). For all other LAPW variants the
procedure is analogous. This method exploits our physical intuition regarding
the shape of wave functions that have atomic character in the vicinity of
nuclei and vary slowly in regions far away from them. Therefore, the unit cell
is partitioned into non-overlapping atomic spheres (MTα, with $\alpha$
enumerating the atoms) and the interstitial region (I). We use this division
of space for defining LAPWs as atomic-like orbitals in the spheres and
planewaves in the interstitial as:

$\begin{array}[]{ll}\phi_{\mathbf{G+k},\sigma}(\mathbf{r})&=A^{\mathbf{G+k}}_{lm}u_{l}(r_{\alpha};\varepsilon_{l})Y_{lm}(\mathbf{\hat{r}_{\alpha}})\chi_{\sigma}\\\
\\\
&+B^{\mathbf{G+k}}_{lm}\dot{u}_{l}(r_{\alpha};\varepsilon_{l})Y_{lm}(\mathbf{\hat{r}_{\alpha}})\chi_{\sigma},\end{array}$
(6)

and

$\phi_{\mathbf{G+k},\sigma}(\mathbf{r})=\frac{1}{\sqrt{\Omega}}e^{i(\mathbf{G+k})\mathbf{r}}\chi_{\sigma},$
(7)

respectively. The coefficients $A^{\mathbf{G+k}}_{lm}$ and
$B^{\mathbf{G+k}}_{lm}$ are determined such that the LAPWs are continuous and
smooth at the spheres boundaries. The radial parts of the atomic-like orbitals
$u_{l}(r;\varepsilon_{l})$ and their energy derivatives
$\dot{u}_{l}(r;\varepsilon_{l})$ are obtained from the scalar-relativistic
radial Schrödinger equation using a predefined energy parameter
$\varepsilon_{l}$ for each $l$ value.

Local orbitals (LO), the second type of basis functions, are defined as
atomic-like functions that are non zero only in one atomic sphere:

$\phi_{\mu,\sigma}(\mathbf{r})=\left[a_{\mu}u_{l}(r_{\alpha};\varepsilon_{\mu})+b_{\mu}\tilde{u}_{l}(r_{\alpha};\varepsilon_{\mu}^{\prime})\right]Y_{lm}(\mathbf{\hat{r}})\chi_{\sigma}.$
(8)

The coefficients $a_{\mu}$ and $b_{\mu}$ ensure that
$\phi_{\mu,\sigma}(\mathbf{r})$ is normalized and turns to zero at the sphere
boundary. The radial functions $u_{l}(r_{\alpha};\varepsilon_{\mu})$ and
$\tilde{u}_{l}(r_{\alpha};\varepsilon_{\mu}^{\prime})$ are obtained from the
scalar relativistic radial Schrödinger equation similarly as above. Note that
we do not restrict the definition of local orbitals just to radial functions
and their derivatives ($u_{l}(r;\varepsilon_{l})$,
$\dot{u}_{l}(r;\varepsilon_{l})$). Instead, $\tilde{u}$ can be chosen quite
general, as an energy derivative of any order with a meaningful energy
parameter. Such flexibility is important for achieving the basis-set limit.

With the basis functions defined in Eqs. 6 and 8, we now discuss the
evaluation of matrix elements. The non-relativistic case has already been
described in detail in Ref. 14. The differences to the relativistic
Hamiltonians are solely within the kinetic energy operator $\hat{T}$. We
evaluate the corresponding matrix elements consistently as

$T_{\nu\nu^{\prime}}^{\sigma\sigma^{\prime}}=\frac{1}{2}\delta_{\sigma\sigma^{\prime}}\int\limits_{\Omega}[\boldsymbol{p}\phi^{\sigma}_{\nu}(\mathbf{r})]^{\ast}K[\boldsymbol{p}\phi^{\sigma^{\prime}}_{\nu^{\prime}}(\mathbf{r})]d^{3}r,$
(9)

where the integral is computed along similar lines as shown in Ref. [14].

###  II.3 Fully-relativistic LAPW+LO

The same basis, as defined in the previous section, can also be used in fully-
relativistic calculations. However, such an approach is limited in accuracy,
since the solutions of the atomic Dirac equation depend on the total angular
momentum $j$. In particular, the radial parts of atomic wave functions obey
different power laws depending on $j$ at small distances from a nucleus. Such
a behavior has to be accounted for also in calculations going beyond atoms.
The method described in Ref. 19 relies on relativistic augmented planewaves,
where the basis fulfills the requirements of a fully-relativistic calculation.
More specifically, the basis employs the radial Dirac equation for generating
the radial parts of atomic-like orbitals (spinors) and combines them with the
angular and spin degrees of freedom in terms of spin harmonics.

In the present work, we rather employ a strategy that requires relatively
small changes to the existing scalar-relativistic implementation of the
LAPW+LO method. We keep the definitions of LAPWs and LOs as in Eqs. 6 and 8,
except for the radial functions $u(r)$, $\dot{u}(r)$ and their further energy-
derivatives that are now solutions of the radial Dirac equation. Unlike in the
previous section, the radial functions depend on the relativistic quantum
number $\kappa$ which is related to the angular and total angular momenta $l$
and $j$, respectively. In case of LAPWs, we set $\kappa=-(l+1)$ corresponding
to $j=l+\frac{1}{2}$. In case of LOs, we choose $\kappa$ between the value
above and $\kappa=l$ corresponding to $j=l-\frac{1}{2}$. A basis defined in
such a way possesses all necessary degrees of freedom that are required for
reliably solving the two-component problem in Eq. 3.

Finally, we address another popular approximation. Typically, LAPW+LO
implementations of the fully-relativistic approach consider SOC only in the
atomic spheres while ignoring the interstitial contributions [13]. Using the
fact that the Kohn-Sham potential in the atomic spheres is approximately
spherically symmetric, the expression of the kinetic energy operator in Eq. 4
transforms into

$\hat{T}=-\frac{1}{2}\boldsymbol{p}K\boldsymbol{p}+\frac{K^{2}}{4c^{2}}\frac{1}{r}\frac{dV}{dr}\boldsymbol{\sigma
L},$ (10)

where $\boldsymbol{L}$ is the angular momentum operator. We go beyond this
approach, considering SOC also the interstitial region. Direct application of
Eq. 4 leads to a non-Hermitian Hamiltonian in the LAPW+LO basis. Instead, we
calculate the matrix elements of the kinetic energy operator following the
same spirit as in the scalar-relativistic case, i.e.,

$T_{\nu\nu^{\prime}}^{\sigma\sigma^{\prime}}=\frac{1}{2}\int\limits_{\Omega}[\boldsymbol{\sigma
p}\phi^{\sigma}_{\nu}(\mathbf{r})]^{\ast}K[\boldsymbol{\sigma
p}\phi^{\sigma^{\prime}}_{\nu^{\prime}}(\mathbf{r})]d^{3}r,$ (11)

and, therefore, hermiticity of the Hamiltonian is preserved.

Once the matrix elements are computed, we have to solve a full generalized
eigenvalue problem that now corresponds to a fully-relativistic Hamiltonian.
We note that the usual strategy for treating the spin-orbit coupling involves
the second-variational approach, which is a two-step procedure. First, one
considers only the scalar-relativistic part of the Hamiltonian. Then, a
limited number of the so-obtained solutions is used as the basis for the full
two-component problem [20, 21]. As it may be difficult to control the
precision in the latter approach, we do not employ it but solve the full
spinor problem.

##  III Computational details

To quantify differences between scalar- and fully-relativistic calculations of
elemental solids, we employ the protocol introduced in Refs. [9] and [12]. In
short, this protocol consists of the following steps: (i) The unit cell of a
chosen material is considered for 7 volumes ranging from 94% to 106% of the
equilibrium volume $V_{0}$. (ii) The Birch-Murnaghan equation of state (EOS)
is fitted to the obtained total energies and set to zero at the energy
minimum. (iii) The EOSs, $E_{1}(V)$ and $E_{2}(V)$, obtained for a given solid
with two different methods are compared. To this extent, the difference is
quantified by the so-called $\Delta$-factor defined as

$\Delta=\sqrt{\frac{\int[E_{1}(V)-E_{2}(V)]^{2}dV}{\Delta V}},$ (12)

where the integration is performed within the range of the volumes indicated
above.

The unit cells and atom positions are chosen like in Ref. [9]. Following Refs.
[9] and [12], we remain within the generalized gradient approximation (GGA) as
parametrized by Perdew, Burke, and Ernzernhof [22] for exchange-correlation
effects. The LAPW cutoff is set to $R_{\mathrm{MT}}G_{\mathrm{max}}=10$ and 12
for hydrogen and other chemical species, respectively. Such settings ensure
highly converged EOSs as verified in Ref. [23]. Further computational details,
such as the sets of LOs and $\mathbf{k}$-point grids, are available in Ref.
[24]

Figure 1:  $\Delta$ values (in meV/atom) for comparison between fully
relativistic and scalar-relativistic calculations. The intensity of the
background color of each cell is proportional to $\Delta$ obtained for the
respective element.  Figure 2:  Relative change of volume in elemental solids
due to spin-orbit coupling. Red and blue colors indicate expansion and
contraction of the elemental solid, respectively.

Since SOC is expected to impact only heavy atoms, we do not consider the
elements of the first three rows of the PTE. Furthermore, we exclude the
magnetic fourth-period metals and the rare-gas solids. The latter are very
poorly described by the PBE functional due to the lack of van der Waals
interaction. In particular, the weak spurious bonding obtained in PBE
calculations is a consequence of the overlapping density tails of atoms. A
fully-relativistic calculation of the magnetic fourth-period metals involves a
non-collinear treatment, which goes beyond the scope of this paper.

Calculations of dimers are performed in a unit cell with dimensions of
$15\times 15\times 20$ bohr3. The LAPW cutoff is set to
$R_{\mathrm{MT}}G_{\mathrm{max}}=10$.

Two-dimensional materials are modeled using the slab model with the thickness
of the vacuum layer set to $\sim$10 Å. The Brillouin zone is sampled by a grid
of $9\times 9\times 1$ $\mathbf{k}$-points. An LAPW cutoff of
$R_{\mathrm{MT}}G_{\mathrm{max}}=12$ is used in these cases.

##  IV Results

###  IV.1 Elemental solids

Figure 1 shows the impact of spin-orbit coupling in terms of $\Delta$ values,
i.e., differences between scalar- and fully-relativistic calculations.
$\Delta$ ranges from 0.01 meV/atom (for K) to 19.37 meV/atom (for Po). In
other words, some elements are barely affected by SOC, while some of them
experience a dramatic change. Overall, we observe two trends: First, the
effects increases with the period for elements within the same group. Second,
within the same period, $\Delta$ does not increase monotonously with the
nuclear charge, but rather follows a pattern. $\Delta$ is small for alkali and
alkali-earth metals, and it reaches its maximum for chalcogens and halogens.
Elemental solids of the other groups are affected by SOC to a variable degree,
and it is not straightforward to capture a pattern. However, we can already
conclude that a few elements from the fifth period and more than half of those
from the sixth period require a fully-relativistic treatment in order to
obtain accurate EOSs.

Figure 3:  $\Delta$ values (in meV/atom) evaluating fully-relativistic
calculations performed with $j$-unresolved and $j$-resolved bases. The
intensity of the color in the background of each cell scales proportionally to
$\Delta$ obtained for each element.  Figure 4:  Relative change of volume in
elemental solids due to spin-orbit coupling. Full bars correspond to the fcc
phase, whereas striped ones correspond to the structure given in Refs. [12]
and [9].

To get more insight into the observed trends, we analyze the calculated
equilibrium volumes. In Fig. 2, their variation upon inclusion of SOC is
illustrated. These results show that the equilibrium volume expands or shrinks
following a strict pattern within the PTE. More specifically, we observe a
volume reduction for elemental solids of groups IA–IIIA/VB and IB–IIIA/VIA and
an expansion for the rest. The largest effects (in ascending order) are
obtained for Au, Tl, I, Bi, Po and Hg, which either shrink or expand by
$-7.6$–6.2%.

A number of previous studies have addressed this issue [9, 8, 2, 3]. Philipsen
and Baerends have considered selected elemental solids, obtaining noticeable
variations of the lattice constant in Au and Bi [8]. Similarly, Lejaeghere and
coworkers [9] have found that SOC has the strongest impact on the equilibrium
volume of Re, Au, Po, Tl, Bi and Hg (in ascending order). This list contains
five of the six elements highlighted by us but misses iodine that was not
calculated fully relativistically in their work. A quantitative comparison
with their results shows a major discrepancy for Po. According to Ref. 9, its
volumes is only changed by $\approx 1\%$, in stark contrast to our value of
$6.1\%$. A different calculation of Po [2] predicts a 5%-volume increase, in
much better agreement with our result. Finally, the volume contraction of Pb
by as much as 8% reported in Ref. 3 is in striking disagreement with the
values of 0.1% and 0.0% obtained in this work and in Ref. 9, respectively.
These contradictions highlight the importance of benchmark data and show that
achieving high precision in fully-relativistic calculations appears to be
still challenging.

The above presented fully-relativistic data were obtained using the basis as
described in Sec. II.3 It is, however, quite common that in LAPW+LO
calculations, in practice not a $j$-resolved basis is employed. In other
words, the basis designed for scalar-relativistic calculations is often used
also in fully-relativistic ones. Therefore, we explore the corresponding
errors that are depicted in Fig. 3. Our calculations show that this error is
small for the elements of the fourth period. It then increases and becomes
significant in the second half of the fifth period, with a maximum of 0.71
meV/atom obtained for Sn. An error of this magnitude, i.e., the order of 1
meV/atom, is considered as a typical experimental uncertainty [12]. Finally,
in the sixth period, the error introduced by the $j$-unresolved basis
increases even further and goes beyond the 1 meV/atom threshold for elemental
solids from W to Pb as well as Po. The largest values of $\sim$5 meV/atom are
reached for Ir, Pt and Au. In case of Au, the obtained $\Delta$ is comparable
to the difference between the scalar- and fully-relativistic calculations
reported in Fig. 1. Here the different basis sets lead to SOC-induced volume
changes of about 0.2% and 1.1%, respectively.

Figure 5:  Rearrangement of electron density in fcc solids due to SOC in the
(100) plane for Au, Hg, and Tl (top) as well as Pb, Bi, and Po (bottom). The
contour lines correspond to 15 isovalues equally spaced between the minimum
and maximum values of the respective scale where blue (red) indicates charge
depletion (accumulation).

Since the considered elemental solids differ in terms of symmetry and atomic
coordination, analyzing the trends across the PTE is intricate. To overcome
this issue, we consider the $4p$-, $5p$\- and $6p$-elements assuming the face-
centered cubic (fcc) structure for all of them. Although there are numerical
differences between the results for different lattice types, the qualitative
picture is the same. For the SOC effect on their equilibrium volumes, we refer
to Fig. 4. The group-IIIA elemental solids shrink, while those from the group
IVA remain essentially unaffected. Finally, the elements from groups VA–VIIA
expand, where the maximum expansion within each period is obtained for
halogens, except for At in the sixth period.

###  IV.2 Dimers

The results described in the previous section clearly show that SOC may
introduce significant changes not only in the band structure, but also in the
geometry of a solid. In other words, there are cases where determining the
equilibrium structure in practice, requires a fully relativistic treatment.
Here, we analyze why this is so and whether there is a way of telling in
advance if it is necessary to include SOC.

According to the data presented in Fig. 2, it increases the equilibrium volume
of I by 1.7%, which is among the largest volume expansions for the elemental
solids. Interestingly, as we will discuss below, this does not mean that the
scalar-relativistic approximation is crude for all compounds containing this
element. To show an example, we calculate the equilibrium volume of CsI in the
rock-salt phase. The fully- and scalar-relativistic approaches yield values of
101.7 Å3 and 101.6 Å3 per formula unit, respectively. Thus, SOC expands the
volume by only 0.1%. Clearly, the difference between CsI and elemental I lies
in the chemistry of the materials.

Bulk I is a is molecular crystal consisting of I2 molecules, and a volume
change is basically attributed to a deformation of the I–I bond. To analyze
the influence of SOC, we adopt the tight-binding model [25] with a basis that
consists of atomic $5p$-orbitals. The corresponding Hamiltonian can be solved
analytically in both the scalar- and fully-relativistic cases. Following the
derivation presented in Ref. 10, we sketch a schematic of the orbital energies
in Fig. 6. The scalar-relativistic solution of the problem consists of three
bonding ($\sigma_{g}$, $\pi_{xg}$, $\pi_{yg}$) and three antibonding orbitals
($\sigma_{u}$, $\pi_{xu}$, $\pi_{yu}$). The $\pi_{xg}$ and $\pi_{yg}$ orbitals
are degenerate, and so are $\pi_{xu}$ and $\pi_{yu}$.

Figure 6:  Schematic of orbital energies in the I atom and the I2 molecule as
obtained from the tight-binding model. The labels SR and FR, correspond to the
scalar- and fully-relativistic results, respectively. The two outer (inner)
columns represent the atomic (molecular) energies. Red and blue lines
correspond to bonding and anti-bonding orbitals, respectively. For the states
with mixed character both colors are used.

The fully-relativistic solution of the problem differentiates itself from the
scalar-relativistic case in three qualitative points. First, the degeneracy of
the $\pi$-orbitals is lost. Second, only two orbitals keep their angular
character. The remaining four present a mixture of $\sigma$\- and
$\pi$-orbitals arising from the scalar-relativistic case, although in each
case some particular character dominates. In Fig. 6, this dominant
contribution is indicated in the labels as the first one. Third, only two
orbitals keep their symmetry while the others do not have a pure bonding or
antibonding character. Again, it is possible though to recognize in each case
whether an orbital is predominantly bonding or antibonding. The second and
third observations can be used for rationalizing the impact of SOC on the bond
lengths. As we show below, they can lead to contradictory conclusions in
certain examples.

We start our discussion with the reasoning based on the population of bonding
and antibonding orbitals [10]. An I2 molecule in its ground state has 10
electrons in the $5p$-shells. They occupy the lowest-energy orbitals that
include three bonding and two antibonding ones in the scalar-relativistic
case. If SOC is considered, the only unoccupied level has
$\sigma_{\mathrm{u}}$ character with a small contribution from
$\pi_{\mathrm{g}}$. This means that electrons effectively occupy slightly less
than three bonding and slightly more than two antibonding orbitals. Such a
rearrangement weakens and thus elongates the chemical bond. We note in passing
that these considerations not only apply to I2 but also to other halogen
dimers.

To generalize this result, we consider dimers of a few other elements.
Employing scalar-relativistic theory, dimers of the group-VA elements have
fully occupied bonding and entirely empty antibonding orbitals. Like in the
example above, inclusion of SOC changes this picture, and the effective
population of the bonding orbitals reduces, thereby expanding the bonds.

We also consider an alternative reasoning based on the redistribution of the
angular character of electrons upon SOC. Our results sketched in Fig. 6 show
that the fully-relativistic I2 has more (fewer) electrons with $\sigma$
($\pi$) character than the scalar-relativistic counterpart. As $\pi$ orbitals
are more compact in the direction of the bond than $\sigma$ orbitals it is
clear that the bond should expand upon spin-orbit interaction. Applying the
same analysis for group-VA dimers, the number of $\sigma$\- and
$\pi$-electrons changes by a small amount only. The $\sigma_{g}$ state
partially acquires $\pi$-character, forming a $\sigma_{g}+\pi_{u}$ state, but
one of the $\pi_{g}$ states partially loses $\pi$-character, forming a
$\pi_{g}+\sigma_{u}$ state. Thus, this model predicts a small change of bond
lengths in dimers such as Sb2 and Bi2.

Table 1: Bond lengths (in Å) of homonuclear dimers in scalar (SR) and fully-relativistic (FR) calculations.  | As2 | Sb2 | Bi2 | Br2 | I2 | At2
---|---|---|---|---|---|---
SR | 2.110 | 2.476 | 2.651 | 2.309 | 2.684 | 2.868
FR | 2.110 | 2.477 | 2.678 | 2.310 | 2.704 | 3.036

To test the two qualitative arguments, we perform Kohn-Sham calculations of
the group-V and group-VII dimers. The calculated bond lengths are given in
Table 1. As predicted by the tight-binding calculations, the internuclear
distances in the halogen molecules increase due to SOC. The variation is small
(0.04%) for Br2, substantial (0.7%) for I2, and enormous (5.5%) for At2. SOC
also increases the internuclear distances in the group-VA dimers, but just
slightly in comparison to the halogens of the same periods. This small
increase is in agreement with the argument based on the analysis of the
$\sigma$/$\pi$ population but contradicts the explanation in terms of the
(anti)bonding character of the orbitals.

Although the presented tight-binding model is, strictly speaking, not
justified for the group IIIA dimers, because of the triplet character of their
ground state [26, 27], we nevertheless apply it to facilitate the discussion.
We argue that the chemical bond is expected to shorten, since it partially
loses its $\sigma$ character and acquires some $\pi$ character in a fully-
relativistic formalism; and this is what we indeed observe. Thus, overall,
depending on the filling of the $p$-shells, SOC results in shrinking or
expanding chemical bonds.

Figure 7:  Rearrangement of electron density in homonuclear dimers due to
spin-orbit interaction. Color maps are shown for the planes through the
dimers. The contour lines correspond to 15 isovalues equally spaced between
the minimum and maximum values of the respective scale where blue (red)
indicates charge depletion (accumulation).

The impact of SOC on the bonding in dimers and solids shows similar
qualitative trends. Our results indicate that the key to a strong effect is
the partially occupied $p$-shell. If this shell is completely filled as in the
example of CsI above, SOC does not rearrange the angular character of the
chemical bond, and therefore the volume remains essentially unchanged.
Partially occupied $d$-shells play a role in the expansion or contraction of
solids with $d$-elements. Almost empty and almost full $d$-shells lead to a
contraction of a crystal, whereas half-filled $d$-shells give rise to an
expansion.

##  V 2D materials

Table 2: Calculated lattice constants, $a$, and bond lengths, $d$, (in Å), and Bader charges, $q$, (in $e$) of metal ions in the investigated 2D materials from scalar- (SR) and fully-relativistic (FR) calculations.  | $a_{\mathrm{SR}}$ | $a_{\mathrm{FR}}$ | $d_{\mathrm{SR}}$ | $d_{\mathrm{FR}}$ | $q_{\mathrm{SR}}$ | $q_{\mathrm{FR}}$
---|---|---|---|---|---|---
MoS2 | 3.1858 | 3.1858 | 2.4145 | 2.4145 | 1.215 | 1.215
MoSe2 | 3.3200 | 3.3202 | 2.5419 | 2.5421 | 0.962 | 0.962
MoTe2 | 3.5550 | 3.5565 | 2.7355 | 2.7363 | 0.591 | 0.590
WS2 | 3.1881 | 3.1883 | 2.4204 | 2.4213 | 1.355 | 1.353
WSe2 | 3.3209 | 3.3204 | 2.5486 | 2.5487 | 1.070 | 1.066
WTe2 | 3.5587 | 3.5595 | 2.7410 | 2.7419 | 0.642 | 0.635
ZnI2 | 4.0986 | 4.1058 | 2.8722 | 2.8761 | 0.787 | 0.776
CdI2 | 4.3284 | 4.3342 | 3.0334 | 3.0366 | 0.793 | 0.780
HgI2 | 4.3777 | 4.3795 | 3.0678 | 3.0667 | 0.451 | 0.414
GeI2 | 4.3008 | 4.3013 | 3.0343 | 3.0367 | 0.727 | 0.716
SnI2 | 4.5785 | 4.5743 | 3.2228 | 3.2251 | 0.924 | 0.907
PbI2 | 4.6714 | 4.6527 | 3.2768 | 3.2778 | 0.947 | 0.879

The impact of SOC on chemical bonds goes beyond elemental solids. Here, we
discuss this effect in single-layer 2D materials. Previous studies showed that
many of them exhibit strong SO effects in the electronic structure impacting
their response to light. It is natural then to ask whether SOC also has an
impact on their structure.

We focus on two classes of 2D materials, i.e., iodides (AI2, where A = Zn, Cd,
Hg, Ge, Sn, Pb) in the PbI2 structure and TMDCs (MX2, where M = Mo, W and X =
S, Se, Te) in the MoS2 structure. These geometries were observed
experimentally in all cases except for HgI2 [28], for which we nevertheless
use the PbI2 lattice for the sake of consistency.

Figure 8:  Structures of TMDCs and iodides shown with the examples of MoS2
(left) and PbI2 (right), respectively in top (top panels) and side views
(bottom panels), respectively. Thin full lines mark the boundaries of unit
cells; the dashed line indicates the plane in which electron-density
differences are shown in Fig. 9.

Both considered structures are depicted in Fig. 8. Owing to the high symmetry,
the crystal lattice of the materials is uniquely defined by two parameters,
i.e., the lattice constant $a$, and the bond length $d$. We relax the
structures with respect to both parameters. The results displayed in Table 2
show that the effect of SOC on the structure is generally weak for all
considered TMDCs. The largest differences in $a$ and $d$ are observed for
MoTe2, where the increase of both is still less than 0.002 Å. We perform a
Bader charge analysis [29, 30] which confirms that these materials are
predominantly covalently bound with an admixture of ionicity, what is
consistent with Ref. [31]. The charge of the metal ions ranges between 0.6 $e$
and 1.4 $e$. Since the difference between the scalar- and fully-relativistic
results is below 0.01 $e$ we conclude that SOC does not impact the nature of
bonding in these materials.

For the iodides, our calculations reveal a stronger effect of SOC. In
particular, we find increased lattice constants in group-IIB iodides.
Remarkably, it is the largest for ZnI2 despite Zn being the lightest cation of
the group. This trend indicates that the lattice expansion in ZnI2 is induced
by relativistic effects on the I side. Replacing Zn by Cd or Hg, their
presence counteracts the behavior of I, thus the chemical bonds are less
affected. To illustrate this argument, we analyze the charge densities. SOC
leads to a decrease of Bader charges of the metal ions by 0.012–0.035 $e$,
making the materials slightly less ionic. This effect is smallest in ZnI2, and
it increases for the heavier group-IIB elements. To gain further insight, we
plot charge-density differences, $\Delta q$, in (Fig. 9). Like in the Bader
charge analysis, we see that electrons are transferred from the I ions to the
metal ions, with the following trend: $\Delta q_{\mathrm{Zn}}<\Delta
q_{\mathrm{Cd}}<\Delta q_{\mathrm{Hg}}$. The transfer is accompanied by a
rearrangement of the charge density between orbitals of different angular
momentum around the nuclei. The strongest charge depletion happens right
between the metal and the iodine nuclei. The largest increase, in turn, is
visible in areas away from the direction of bonds.

Figure 9:  Rearrangement of the electron density in two-dimensional solids due
to the spin-orbit interaction. The color maps are shown for the bonding planes
passing through the I ion and cations. The contour lines correspond to 15
isovalues equally spaced between the minimum and maximum values of the scale.

The lattice constant in group-IVA iodides reduces as SOC is taken into
account. This reduction is negligibly small in GeI2, is larger in SnI2, and
reaches the maximum value of $\Delta a_{\mathrm{SO}}=0.0187$ Å for PbI2.
Curiously, the Pb–I distance increases by 0.0010 Å, while the reduction in $a$
is related to the change in the bond angle. Similarly to the other layered
materials considered here, SOC makes also group-IVA iodides less ionic. Also
this effect is most pronounced in PbI2 where the Bader charge of the Pb ions
reduces by 0.066 $e$.

The results presented in this section overall show that the SOC effect on the
geometries is negligibly in most cases. The lattice constant changes by 0.02 Å
only in PbI2, a change much smaller than, e.g., a difference caused by using
PBE and LDA exchange-correlation functionals. However, we expect that the
discussed relativistic effect may have a larger impact for the same materials
with defects.

##  VI Conclusions

We have presented and applied an implementation of fully-relativistic DFT
within the framework of the linearized augmented planewaves plus local
orbitals method. Our approach tackles the two-spinor problem directly without
applying the second-variational approximation. To ensure high precision, we
construct the basis by taking into account the $j$ dependence of the radial
functions for all angular momenta corresponding to semicore and valence
shells.

We have applied this method for calculating the equation-of-state parameters
for elemental solids of the rows 4–6 of the PTE. The obtained results show
that spin-orbit interaction influences equilibrium volumes and bulk moduli by
modifying the chemical bonds. The strongest impact is observed for heavy
elements with nearly empty or nearly complete $p$\- or $d$-shells. Based on a
tight-binding model, we trace this effect back to the angular redistribution
of density in occupied bonding and anti-bonding orbitals. This result,
obtained for elemental solids and homonuclear molecules, allows one to
anticipate scenarios where SOC can impact the geometry of more complex
materials in terms of chemical composition. We have illustrated this
conclusion with the example of single-layer iodides. For the latter, we have
found that the charge redistribution due to SOC introduces slight changes in
the nature of bonding, making these materials less ionic. The largest change
in the lattice constant among considered materials is about $0.02$ Å obtained
for PbI2.

###### Acknowledgements.

The work has received partial support from the European Union’s Horizon 2020
research and innovation programme, grant agreement No. 676580 through the
Center of Excellence NOMAD (Novel Materials Discovery Laboratory) and grant
agreement No. 951786 through the Center of Excellence NOMAD (NOMAD CoE). We
acknowledge also support from the German Research Foundation via SFB951,
project number 182087777.

## References

  * Pyykkö [2012] P. Pyykkö, Relativistic Effects in Chemistry: More Common Than You Thought, Annual Review of Physical Chemistry 63, 45 (2012), pMID: 22404585, https://doi.org/10.1146/annurev-physchem-032511-143755 .
  * Legut  _et al._ [2007] D. Legut, M. Friák, and M. Šob, Why is polonium simple cubic and so highly anisotropic?, Phys. Rev. Lett. 99, 016402 (2007).
  * Hermann  _et al._ [2010] A. Hermann, J. Furthmüller, H. W. Gäggeler, and P. Schwerdtfeger, Spin-orbit effects in structural and electronic properties for the solid state of the group-14 elements from carbon to superheavy element 114, Phys. Rev. B 82, 155116 (2010).
  * Oliveira and Gonze [2013a] M. J. T. Oliveira and X. Gonze, Spin–orbit effects in the bismuth atom and dimer: tight-binding and density functional theory comparison, Journal of Physics B: Atomic, Molecular and Optical Physics 46, 095101 (2013a).
  * Díaz-Sánchez  _et al._ [2007] L. E. Díaz-Sánchez, A. H. Romero, M. Cardona, R. K. Kremer, and X. Gonze, Effect of the spin-orbit interaction on the thermodynamic properties of crystals: Specific heat of bismuth, Phys. Rev. Lett. 99, 165504 (2007).
  * Hohenberg and Kohn [1964] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136, B864 (1964).
  * Kohn and Sham [1965] W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  * Philipsen and Baerends [2000] P. H. T. Philipsen and E. J. Baerends, Relativistic calculations to assess the ability of the generalized gradient approximation to reproduce trends in cohesive properties of solids, Phys. Rev. B 61, 1773 (2000).
  * Lejaeghere  _et al._ [2014] K. Lejaeghere, V. V. Speybroeck, G. V. Oost, and S. Cottenier, Error estimates for solid-state density-functional theory predictions: An overview by means of the ground-state elemental crystals, Critical Reviews in Solid State and Materials Sciences 39, 1–24 (2014).
  * Oliveira and Gonze [2013b] M. J. T. Oliveira and X. Gonze, Spin–orbit effects in the bismuth atom and dimer: tight-binding and density functional theory comparison, Journal of Physics B: Atomic, Molecular and Optical Physics 46, 095101 (2013b).
  * Zeng Tao  _et al._ [2011] Zeng Tao, Fedorov Dmitri G., Schmidt Michael W., and Klobukowski Mariusz, Effects of Spin–Orbit Coupling on Covalent Bonding and the Jahn–Teller Effect Are Revealed with the Natural Language of Spinors, Journal of Chemical Theory and Computation 7, 2864 (2011), doi: 10.1021/ct200457q.
  * Lejaeghere  _et al._ [2016] K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. Dal Corso, S. de Gironcoli, T. Deutsch, J. K. Dewhurst, I. Di Marco, C. Draxl, M. Dułak, O. Eriksson, J. A. Flores-Livas, K. F. Garrity, L. Genovese, P. Giannozzi, M. Giantomassi, S. Goedecker, X. Gonze, O. Grånäs, E. K. U. Gross, A. Gulans, F. Gygi, D. R. Hamann, P. J. Hasnip, N. A. W. Holzwarth, D. Iuşan, D. B. Jochym, F. Jollet, D. Jones, G. Kresse, K. Koepernik, E. Küçükbenli, Y. O. Kvashnin, I. L. M. Locht, S. Lubeck, M. Marsman, N. Marzari, U. Nitzsche, L. Nordström, T. Ozaki, L. Paulatto, C. J. Pickard, W. Poelmans, M. I. J. Probert, K. Refson, M. Richter, G.-M. Rignanese, S. Saha, M. Scheffler, M. Schlipf, K. Schwarz, S. Sharma, F. Tavazza, P. Thunström, A. Tkatchenko, M. Torrent, D. Vanderbilt, M. J. van Setten, V. Van Speybroeck, J. M. Wills, J. R. Yates, G.-X. Zhang, and S. Cottenier, Reproducibility in density functional theory calculations of solids, Science 351, aad3000 (2016).
  * Singh [1994] D. Singh,  _Planewaves, Pseudopotentials and the LAPW Method_ (Kluwer Academic Publishers, Boston/Dordrecht/London, 1994) ISBN 0–7923–9412–7.
  * Gulans  _et al._ [2014] A. Gulans, S. Kontur, C. Meisenbichler, D. Nabok, P. Pavone, S. Rigamonti, S. Sagmeister, U. Werner, and C. Draxl, exciting: a full-potential all-electron package implementing density-functional theory and many-body perturbation theory, J. Phys.: Condens. Matter 26, 363202 (2014).
  * Gulans  _et al._ [2018] A. Gulans, A. Kozhevnikov, and C. Draxl, Microhartree precision in density functional theory calculations, Phys. Rev. B 97, 161105 (2018).
  * van Lenthe  _et al._ [1993] E. van Lenthe, E. J. Baerends, and J. G. Snijders, Relativistic regular two-component hamiltonians, The Journal of Chemical Physics 99, 4597 (1993).
  * van Lenthe  _et al._ [1994] E. van Lenthe, E. J. Baerends, and J. G. Snijders, Relativistic total energy using regular approximations, The Journal of Chemical Physics 101, 9783 (1994).
  * Dyall and van Lenthe [1999] K. G. Dyall and E. van Lenthe, Relativistic regular approximations revisited: An infinite-order relativistic approximation, The Journal of Chemical Physics 111, 1366 (1999).
  * Loucks [1965] T. L. Loucks, Relativistic Electronic Structure in Crystals. I. Theory, Phys. Rev. 139, A1333 (1965).
  * Koelling [1974] D. D. Koelling, On the efficient solution of the relativistic apw secular equation for both eigenvalues and eigenvectors, International Journal of Quantum Chemistry 8, 473 (1974).
  * MacDonald  _et al._ [1980] A. H. MacDonald, W. E. Picket, and D. D. Koelling, A linearised relativistic augmented-plane-wave method utilising approximate pure spin basis functions, Journal of Physics C: Solid State Physics 13, 2675 (1980).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Gulans  _et al._ [a] A. Gulans, S. Lubeck, and C. Draxl, Delta test Science 2016 exciting, http://dx.doi.org/10.17172/NOMAD/2016.09.20-1 (a).
  * Gulans  _et al._ [b] A. Gulans, S. Lubeck, and C. Draxl, Delta test 2019 exciting, http://dx.doi.org/ (b).
  * Slater and Koster [1954] J. C. Slater and G. F. Koster, Simplified LCAO Method for the Periodic Potential Problem, Phys. Rev. 94, 1498 (1954).
  * Lee [2004] Y. S. Lee, Chapter 6 - Two-component Relativistic Effective Core Potential Calculations for Molecules, in  _Relativistic Electronic Structure Theory_, Theoretical and Computational Chemistry, Vol. 14, edited by P. Schwerdtfeger (Elsevier, 2004) p. 352–416.
  * Zhou  _et al._ [2013] L.-S. Zhou, B. Yan, and M.-X. Jin, Chinese Physics B 22, 043102 (2013).
  * Jeffrey and Vlasse [1967] G. A. Jeffrey and M. Vlasse, Crystal structures of the red, yellow, and orange forms of mercuric iodide, Inorganic Chemistry 6, 396 (1967), https://doi.org/10.1021/ic50048a048 .
  * Bader [1990] R. F. W. Bader, Atoms in molecules: A quantum theory, (1990).
  * Yu and Trinkle [2011] M. Yu and D. R. Trinkle, Accurate and efficient algorithm for bader charge integration, The Journal of Chemical Physics 134, 064111 (2011), https://doi.org/10.1063/1.3553716 .
  * Yue  _et al._ [2013] Q. Yue, S. Chang, S. Qin, and J. Li, Functionalization of monolayer MoS2 by substitutional doping: A first-principles study, Physics Letters A 377, 1362 (2013).


# A local converse theorem for $\widetilde{\mathrm{Sp}}_{2n}$ : the generic
case

Jaeho Haan  School of Mathematics, Yonsei University, Seoul, Korea
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove the local converse theorem and stability of local
gamma factors for $\widetilde{\mathrm{Sp}}_{2n}$ via the precise local theta
correspondence between $\widetilde{\mathrm{Sp}}_{2n}$ and
$\mathrm{SO}_{2n+1}$. As an application, we prove the rigidity theorem for
irreducible generic cuspidal automorphic representations of
$\widetilde{\mathrm{Sp}}_{2n}$.

###### Key words and phrases:

gamma factors, metaplectic groups, local converse theorem, theta
correspondence

###### 2020 Mathematics Subject Classification:

11F70, 22E50

##  1\. Introduction

Let $F$ be a $p$-adic local field of characteristic zero and $W$ be a
$2n$-dimensional symplectic space over $F$. Let
$G_{n}=\widetilde{\mathrm{Sp}}(W)$ be the metaplectic group of $W$, that is
the double covering group of $\mathrm{Sp}(W)$, the isometry group of $W$. Let
$\psi$ be a non-trivial additive character of $F$. In this paper, we prove the
following.

######  Local Converse Theorem for $\widetilde{\mathrm{Sp}}_{2n}$ 1.

Let $\widetilde{\pi}$ and $\widetilde{\pi}^{\prime}$ be irreducible
$\psi$-generic representations of $G_{n}(F)$ such that

$\gamma(s,\widetilde{\pi}\times\sigma,\psi)=\gamma(s,\widetilde{\pi}^{\prime}\times\sigma,\psi)$

holds for any irreducible supercuspidal representation $\sigma$ of $GL_{i}(F)$
with $1\leq i\leq n$. Then

$\widetilde{\pi}\simeq\widetilde{\pi}^{\prime}.$

Here, $\gamma$ is the local gamma factor (see Section 2.3 for more explanation
of it) and for the definition of $\psi$-generic, see Section 2.2.

There are various versions of the local converse theorem (LCT) concerning the
relevant groups. We now briefly review some celebrated works regarding it.

It seems that Henniart was the first who considered this type of the problem.
In [He], he proved a (weak) version of the theorem for general linear groups.
Building upon the Henniart result, Jiang and Soudry [JS03] proved a (weak) LCT
for odd special orthogonal groups. They combined the global weak functorial
lift from $\mathrm{SO}_{2n+1}$ to $\mathrm{GL}_{2n}$, the local descent from
$\mathrm{GL}_{2n}$ to $\widetilde{\mathrm{Sp}}_{2n}$ and the Howe lifts from
$\mathrm{SO}_{2n+1}$ to $\widetilde{\mathrm{Sp}}_{2n}$. In their proof, a
global-to-local argument is involved to pull-back Henniart’s result.
Thereafter, D. Jiang [Jng] proposed LCT for all classical groups.

Around 2017, almost at the same time, Chai [Cha19] and Jacquet and Liu [JL18]
independently sharpened the result of Henniart to the best. In turn, this
might also sharpen the result of Jiang and Soudry [JS03] to the best though
the proof is not written explicitly in the literature.

Quite recently, Q. Zhang [Q18, Q19] proved the supercuspidal cases of the
conjectures for $\mathrm{Sp}_{2n}$ and $U_{2n+1}$ via a theory of partial
Bessel functions developed by Cogdell, Shahidi and Tsai in [CST17]. Along in
the same line with Q. Zhang’s work, Y. Jo [Jo] extended the results of Jiang
and Soudry [JS03] and Zhang [Q18] to the generic cases not only for the local
fields of characteristic zero but also the local fields of positive
characteristics. It is noteworthy that their proof is purely local and does
not invoke the weak Langlands functorial lifts to general linear groups to use
the recent results of [Cha19] and [JL18]. For other classical groups, Morimoto
[M] proved the theorem for even unitary groups using the local descent methods
and B. Liu and A. Hazeltine [LH] proved it for split even special orthogonal
groups.

Taken together, to attack the metaplectic cases of the LCT, it seems plausible
to pull back the LCT for general linear groups ([Cha19] and [JL18]) via the
weak Langlands functorial lift or the local descent methods. Or it is also
possible to use the partial Bessel functions theory [CST17] as in [Q18, Q19]
and [Jo]. In either approach, some labor seems to be required.

In this paper, we prove the LCT for metaplectic groups with a relatively cheap
price. Instead of invoking the results concerning general linear groups or
developing some non-trivial works on the partial Bessel functions, we use the
precise local theta correspondence (LTC) between $\mathrm{SO}_{2n+1}$ and
$\widetilde{\mathrm{Sp}}_{2n}$ established by many people’s efforts. The
advantage of the LTC is to transfer the problems of one group to those of
another group. This technique is not new and already used by many people. For
instance, by establishing the precise LTC between almost equal rank of unitary
groups, Gan and Ichino [GI] proved the local Gan–Gross–Prasad (GGP) conjecture
for the Fourier–Jacobi cases of skew-hermitian unitary groups from the Bessel
cases of hermitian unitary groups. Similarly, Atobe [At] proved the GGP
conjecture for metaplectic–symplectic groups from that of the orthogonal
groups using the same method. Inspired by their work, we transfer the recent
LCT result of Jo [Jo] for $\mathrm{SO}_{2n+1}$ to that of metaplectic groups
via the LTC between $\mathrm{SO}_{2n+1}$ and $\widetilde{\mathrm{Sp}}_{2n}$.
To the best of the author’s knowledge, this might be the first time to exploit
the precise LTC to prove the LCT within the context of classical (and
metaplectic) groups themselves. Using the similar method in this paper, we can
extend the recent result in [LH], the LCT for split $\mathrm{SO}_{2n}$, to
quasi-split $\mathrm{SO}_{2n}$. To avoid making notation too complicated, we
shall treat it separately in a sequel to this paper.

Since we shall use the LCT for $\mathrm{SO}_{2n+1}$ throughout the paper, we
now recall Jo’s result for $\mathrm{SO}_{2n+1}$, which sharpens Jiang and
Sourdry’s original result ([JS03].)

######  Theorem 1.1.

([Jo]) Let $\pi$ and $\pi^{\prime}$ be irreducible generic representations of
$\mathrm{SO}_{2n+1}(F)$ such that

$\gamma(s,\pi\times\sigma,\psi)=\gamma(s,\pi^{\prime}\times\sigma,\psi)$

holds for any irreducible supercuspidal representation $\sigma$ of
$\mathrm{GL}_{i}(F)$ with $1\leq i\leq n$. Then

$\pi\simeq\pi^{\prime}.$

On the other hand, using the LTC between $\widetilde{\mathrm{Sp}}_{2n}$ and
$\mathrm{SO}_{2n+1}$, we can also prove the stability of the local gamma
factors for $\widetilde{\mathrm{Sp}}_{2n}$.

######  Theorem 1.2.

(Stability of local $\gamma$-factors for $\widetilde{\mathrm{Sp}}_{2n}$) Let
$\widetilde{\pi}$ and $\widetilde{\pi}^{\prime}$ be irreducible $\psi$-generic
representations of $G_{n}(F)$. Then there exists
$l=l(\widetilde{\pi},\widetilde{\pi}^{\prime})$ such that for any quasi-
character $\chi$ of $F^{\times}$, with conductor $\text{cond}(\chi)>l$, we
have

$\gamma(s,\widetilde{\pi}\times\chi,\psi)=\gamma(s,\widetilde{\pi}^{\prime}\times\chi,\psi).$

As an application of the LCT and the stability of local $\gamma$-factors for
$\widetilde{\mathrm{Sp}}_{2n}$, we can prove the rigidity theorem for
$\widetilde{\mathrm{Sp}}_{2n}$

######  Theorem 1.3.

(Rigidity theorem for $\widetilde{\mathrm{Sp}}_{2n}$) Let
$\widetilde{\pi}=\otimes_{v}\widetilde{\pi_{v}}$ and
$\widetilde{\pi}^{\prime}=\otimes_{v}\widetilde{\pi_{v}}^{\prime}$ be
irreducible cuspidal generic automorphic representations of
$\widetilde{\mathrm{Sp}}_{2n}(\mathbb{A})$. If
$\widetilde{\pi_{v}}\simeq\widetilde{\pi_{v}}^{\prime}$ for almost all $v$,
then $\widetilde{\pi}\simeq\widetilde{\pi}^{\prime}.$

This paper is organized as follows. In Section 2, we prepare the setting to
state our theorem precisely. In Section 3, we explain the local theta
correspondence and recall some of its results we shall use. In Section 4, we
give a proof of the main theorem and Theorem 1.2. Finally in Section 5, we
prove the rigidity theorem for $\widetilde{\mathrm{Sp}}_{2n}$ as an
application of our theorems.

###  1.1. Notations

  * •

$F$ : a non-Archimedean local field of characteristic zero whose residue field
is of order $q$

  * •

$\psi$ : a non-trivial additive character of $F$

  * •

$\varpi$ : a uniformizer of $F$

  * •

$|\cdot|_{F}$ : a normalized absolute value in $F$ such that
$|\varpi|_{F}=q^{-1}$

  * •

$\operatorname{Irr}(G)$ : the set of isomorphism classes of irreducible
representations of algebraic group (or its covering group) $G$

  * •

$\pi^{\vee}$ : the contragrediant representation of
$\pi\in\operatorname{Irr}(G)$

##  2\. Preliminaries

In this section, we prepare the setting to state our main theorem.

###  2.1. Metaplectic and orthogonal groups

####  2.1.1. Symplectic groups

Let $W$ be a $2n$-dimensional vector space over $F$ equipped with a non-
degenerate symplectic form $\langle\cdot,\cdot\rangle_{W}$ and let
$\mathrm{Sp}(W)$ be the associated orthogonal group. Let
$\\{e_{1},\cdots,e_{n},e_{n}^{*},\cdots,e_{1}^{*}\\}$ be a specific basis of
$W$ satisfying

$\langle e_{i},e_{j}\rangle_{W}=\langle
e_{i}^{*},e_{j}^{*}\rangle_{W}=0,\quad\langle
e_{i},e_{j}^{*}\rangle=\delta_{ij}.$

For $1\leq k\leq n$, let

$X_{k}=\text{Span}\\{e_{1},\cdots,e_{k}\\}\text{ and
}X_{k}^{*}=\text{Span}\\{e_{1}^{*},\cdots,e_{k}^{*}\\},$

so that $W=X_{k}\oplus X_{k}^{*}$. We also set

$W_{n-k}=\text{Span}\\{e_{k+1},\cdots,e_{n},e_{n}^{*},\cdots,e_{k+1}^{*}\\},$

so that $W=X_{k}\oplus W_{n-k}\oplus X_{k}^{*}$.

Next, we consider the flag of isotropic subspaces

$X_{k_{1}}\subset X_{k_{1}+k_{2}}\subset\cdots\subset
X_{k_{1}+\cdots+k_{r}}\subset W.$

The stabilizer of such a flag is a parabolic subgroup $P$ of $\mathrm{Sp}(W)$
whose Levi factor $M$ is

$M\simeq\mathrm{GL}_{k_{1}}\times\cdots\times\mathrm{GL}_{k_{r}}\times\mathrm{Sp}(W_{n-k_{1}-\cdots-
k_{r}}),$

where each $\mathrm{GL}_{k_{i}}$ is the group of invertible linear maps on
Span $\\{e_{k_{i}+1},\cdots,e_{k_{i+1}}\\}$.

####  2.1.2. Metaplectic groups

For a symplectic space $W$, metaplectic group $\widetilde{\mathrm{Sp}}(W)$ is
a two-fold cover of $\mathrm{Sp}(W)$ defined by

$\widetilde{\mathrm{Sp}}(W)=\mathrm{Sp}(W)\times\\{\pm{1}\\}$

with group law given by

$(g_{1},\epsilon_{1})\times(g_{2},\epsilon_{2})=(g_{1}g_{2},\epsilon_{1}\epsilon_{2}\cdot
c(g_{1},g_{2})),$

where $c$ is some 2-cocycle on $\mathrm{Sp}(W)$ whose values lie in $\\{\pm
1\\}$. The covering group $\widetilde{\mathrm{GL}(X_{k})}$ of
$\mathrm{GL}(X_{k})$ is defined by $\mathrm{GL}(X_{k})\times\\{\pm 1\\}$ with
group law

$(g_{1},\epsilon_{1})\cdot(g_{2},\epsilon_{2})=(g_{1}g_{2},\epsilon_{1}\epsilon_{2}\cdot(\det
g_{1},\det g_{2})),\quad\quad g_{1},g_{2}\in\mathrm{GL}(X_{k}),$

where $(\det g_{1},\det g_{2})$ denotes the Hilbert symbol.

For a Levi subgroup $M$ of $\mathrm{Sp}(W)$, denote its double covering group
$\widetilde{M}$ by

$\widetilde{M}\simeq\widetilde{\mathrm{GL}_{k_{1}}}\times_{\\{\pm{1}\\}}\cdots\times_{\\{\pm{1}\\}}\widetilde{\mathrm{GL}_{k_{r}}}\times_{\\{\pm{1}\\}}\widetilde{\mathrm{Sp}}(W_{0}).$

Any unipotent subgroup $N$ of $\mathrm{Sp}(W)$ splits uniquely in
$\widetilde{\mathrm{Sp}}(W)$. Therefore, we denote its splitting image in
$\widetilde{\mathrm{Sp}}(W)$ using the same symbol $N$. If $P=MN$ is a
parabolic subgroup of $\mathrm{Sp}(W)$, we call $\widetilde{P}=\widetilde{M}N$
the double covering group of $P$. Every parabolic subgroup of
$\widetilde{\mathrm{Sp}}(W)$ has this form.

####  2.1.3. Orthogonal groups

Let $V$ be a $2m+1$-dimensional vector space over $F$ equipped with a non-
degenerate symmetric bilinear form $(\cdot,\cdot)_{V}$ of discriminant 1.
There are two isomorphism classes of $V$ depending on the dimension of maximal
isotropic subspaces of $V$. If it is $m$, it is denoted by $V^{+}$ and call it
a split quadratic space. Otherwise, the dimension of maximal isotropic
subspaces of $V$ is $m-1$ and it is denoted by $V^{-}$. We call this one non-
split. Define

$\epsilon(V)=\begin{cases}+,\quad\text{if $V$ is split}\\\ -,\quad\text{if $V$
is non-split}.\end{cases}$

Let $\mathrm{O}(V)$ be the associated orthogonal group to $V$. Observe that

$O(V)=\mathrm{SO}(V)\times\\{\pm 1\\},$

where $\mathrm{SO}(V)$ is the special orthogonal group. Then $\mathrm{SO}(V)$
is split if and only if $V$ is split. Given any irreducible representation
$\pi$ of $\mathrm{SO}(V)$, there are exactly two extensions of $\pi$ to
$\mathrm{O}(V)$. It is determined by the action of $-1\in\mathrm{O}(V)$; it
acts on $V$ as a scalar multiple $1$ or $-1$. Denote the two extensions by
$\pi^{+}$ and $\pi^{-}$.

Let $\\{v_{1},\cdots,v_{r}\\}$ and $\\{v_{1}^{*},\cdots,v_{r}^{*}\\}$ be
maximal isotropic subspaces of $V$ satisfying

$(v_{i},v_{j})_{V}=(v_{i}^{*},v_{j}^{*})_{V}=0,\quad(e_{i},e_{j}^{*})_{v}=\delta_{ij}.$

For $1\leq k\leq r$, let

$Y_{k}=\text{Span}\\{v_{1},\cdots,v_{k}\\}\text{ and
}Y_{k}^{*}=\text{Span}\\{v_{1}^{*},\cdots,v_{k}^{*}\\},$

and $V_{m-k}$ be a subspace of $V$ so that $V=Y_{k}\oplus V_{m-k}\oplus
Y_{k}^{*}$.

Next, we consider the flag of isotropic subspaces

$Y_{k_{1}}\subset Y_{k_{1}+k_{2}}\subset\cdots\subset
Y_{k_{1}+\cdots+k_{r}}\subset V.$

The stabilizer of such a flag is a parabolic subgroup $P$ of $\mathrm{O}(V)$
whose Levi factor $M$ is

$M\simeq\mathrm{GL}_{k_{1}}\times\cdots\times\mathrm{GL}_{k_{r}}\times\mathrm{O}(V_{m-k_{1}-\cdots-
k_{r}}),$

where each $\mathrm{GL}_{k_{i}}$ is the group of invertible linear maps on
Span $\\{v_{k_{i}+1},\cdots,v_{k_{i+1}}\\}$.

####  2.1.4. Genuine representations

Let $\gamma_{\psi}$ be the Weil factor associated to $\psi$ (see the appendix
of [Rao93]). For a representation $\tau$ of $\mathrm{GL}_{k}$, the genuine
representation $\tau_{\psi}$ of $\widetilde{GL_{k}}$ is defined by

$\tau_{\psi}(a,\epsilon)=\epsilon\gamma_{\psi}(\det(a))\tau(a),\quad(a,\epsilon)\in
GL_{k}\times\\{\pm 1\\}.$

Every irreducible genuine representation of $\widetilde{\mathrm{GL}_{k}}$
arises from that of $\mathrm{GL}_{k}$ in this way. On the other hand, for an
irreducible representation $\widetilde{\pi}$ of $\widetilde{\mathrm{Sp}}(W)$,
if $-1\in\widetilde{\mathrm{Sp}}(W)$ does not act trivially, we call
$\widetilde{\pi}$ a genuine representation. In this paper, when we deal with
irreducible representations of $\widetilde{\mathrm{Sp}}_{2n}$ or
$\widetilde{\mathrm{GL}_{k}}$, we tacitly understand that they are genuine
representations.

###  2.2. Generic characters and generic representations

Let $U$ be the unipotent radical of a Borel subgroup $B=TU$ of
$\mathrm{SO}(V^{+})$, where $T$ is the $F$-rational torus stabilizing the
lines $Fv_{i}$ for each $i=1,\cdots,m$. Let $\lambda$ be a generic character
of $U$. By definition, a representation $\pi$ of $\mathrm{SO}(V^{+})$ is
generic if $\operatorname{Hom}_{U}(\pi,\lambda)\neq 0$. Similarly, let
$U^{\prime}$ be the unipotent radical of the Borel subgroup
$\widetilde{B}^{\prime}=\widetilde{T}^{\prime}U^{\prime}$ of
$\widetilde{\mathrm{Sp}}(W)$, where $T^{\prime}$ is the $F$-split torus
stabilizing the lines $Fw_{i}$ for each $i=1,\cdots,n$. Let $\lambda^{\prime}$
be a generic character of $U$. A representation $\widetilde{\pi}$ of
$\widetilde{\mathrm{Sp}}(W)$ is called generic if
$\operatorname{Hom}_{U^{\prime}}(\widetilde{\pi},\lambda^{\prime})\neq 0$.
Obviously, these two definitions depend on the $T$-orbits (resp.
$T^{\prime}$-orbits) of generic characters of $U$ (resp. $U^{\prime}$).
Fortunately, there is a unique $T$-orbit of generic characters of $U$ and so
the definition of generic representation of $\mathrm{SO}(V^{+})$ does not
depend on the choices of generic character $\lambda$. However, in the
metaplectic cases, $T^{\prime}$-orbits of generic characters of $U^{\prime}$
are indexed by non-trivial characters of $F$ modulo the action of $F^{\times
2}$ (see [GGP, Sect. 12].) Thus, we denote the generic character of
$U^{\prime}$ corresponding to $\psi$ by $\lambda_{\psi}^{\prime}$. If
$\operatorname{Hom}_{U^{\prime}}(\widetilde{\pi},\lambda_{\psi}^{\prime})\neq
0$, we say that $\widetilde{\pi}$ is $\psi$-generic.

###  2.3. $\gamma$-factors

Let $\pi$ (resp. $\widetilde{\pi}$) be an irreducible generic representation
of $\mathrm{SO}_{2n+1}(F)$ (resp. $\widetilde{\mathrm{Sp}}_{2n}(F)$) and
$\sigma$ an irreducible generic representation of $\mathrm{GL}_{k}(F)$. Then
the local $\gamma$-factors $\gamma(s,\pi\times\sigma,\psi)$ (resp.
$\gamma(s,\widetilde{\pi}\times\sigma,\psi)$) are defined as the
proportionality between the functional equations of local Rankin–Selberg type
integrals (resp. Shimura type integral) for
$\mathrm{SO}_{2n+1}\times\mathrm{GL}_{k}$ (resp.
$\widetilde{\mathrm{Sp}}_{2n}\times\mathrm{GL}_{k}$.) For the precise
definition, see [So93] for orthogonal groups and [GRS] for metaplectic groups.
We recall some properties of $\gamma$-factors which we shall use later.

######  Proposition 2.1.

[Kap15, Theorem 1] Let $\tau_{1}\otimes\cdots\otimes\tau_{r}\otimes\pi_{0}$
(resp.
$(\tau_{1})_{\psi}\otimes\cdots\otimes(\tau_{r})_{\psi}\otimes\widetilde{\pi_{0}}$)
be an irreducible generic representation of
$\mathrm{GL}_{r_{1}}\times\cdots\times\mathrm{GL}_{r_{r}}\times\mathrm{SO}_{2k+1}$
(resp. $\widetilde{\mathrm{GL}_{r_{1}}}\times_{\\{\pm
1\\}}\cdots\times_{\\{\pm 1\\}}\widetilde{\mathrm{GL}_{r_{r}}}\times_{\\{\pm
1\\}}\widetilde{\mathrm{Sp}}_{2k}$). Assume that $\pi$ (resp.
$\widetilde{\pi}$) is the irreducible generic constituent of a parabolically
induced representation from
$\tau_{1}\otimes\cdots\otimes\tau_{r}\otimes\pi_{0}$ (resp.
$(\tau_{1})_{\psi}\otimes\cdots\otimes(\tau_{r})_{\psi}\otimes\widetilde{\pi_{0}}$).
Then for any irreducible generic representation $\sigma$ of $\mathrm{GL}_{k}$,
the followings hold;

  1. $(\mathrm{i})$

(Unramified twist)
$\gamma(s,\pi\times\sigma|\det|^{s_{0}},\psi)=\gamma(s+s_{0},\pi\times\sigma,\psi)$
$\quad\text{ for }s_{0}\in\mathbb{R}.$

  2. $(\mathrm{ii})$

(Multiplicity)

$\displaystyle\gamma(s,\pi\times\sigma,\psi)=\gamma(s,\pi_{0}\times\sigma,\psi)\prod_{i=1}^{r}\gamma(s,\tau_{i}\times\sigma,\psi)\gamma(s,\tau_{i}^{\vee}\times\sigma,\psi),$
$\displaystyle\gamma(s,\widetilde{\pi}\times\sigma,\psi)=\gamma(s,\widetilde{\pi_{0}}\times\sigma,\psi)\prod_{i=1}^{r}\gamma(s,\tau_{i}\times\sigma,\psi)\gamma(s,\tau_{i}^{\vee}\times\sigma,\psi).$

Here, $\gamma(s,\tau_{i}\times\sigma,\psi)$ is the
$\mathrm{GL}_{r_{i}}\times\mathrm{GL}_{k}$ $\gamma$-factor of Jacquet,
Piatetski-Shapiro and Shalika [JSS].

######  Remark 2.2.

In [Kap15], Kaplan showed that these definitions of $\gamma$-factors are equal
to those of ones arising from the Langlands–Shahidi method defined in [Sha90].
Therefore, having defined the local $L$-functions and $\epsilon$-factors as in
[Sha90, Section 7], we have

$\gamma(s,\pi\times\sigma,\psi)=\epsilon(s,\pi\times\sigma,\psi)\cdot\frac{L(1-s,\pi^{\vee}\times\sigma^{\vee},\psi)}{L(s,\pi\times\sigma,\psi)}.$

##  3\. Local theta correspondence

In this section, we introduce the local theta correspondence induced by a Weil
representation $\omega_{\psi,V,W}$ of
$\mathrm{O}(V)\times\widetilde{\mathrm{Sp}}(W)$ when $\dim(W)=2m$ and
$\dim(V)=2n+1$ and recall some basic results related to it.

We fix a non-trivial additive character $\psi$ of $F$. Then the group
$\mathrm{O}(V)\times\widetilde{\mathrm{Sp}}(W)$ has a natural representation
$\omega=\omega_{\psi,V,W}$, so called a Weil representation of
$\mathrm{O}(V)\times\widetilde{\mathrm{Sp}}(W)$. Write the isomorphism classes
of irreducible genuine representations of $\widetilde{\mathrm{Sp}}(W)$ as
$\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))$ and that of $\mathrm{SO}(V)$
as $\operatorname{Irr}(\mathrm{SO}(V))$.

For $\pi\in\operatorname{Irr}(\mathrm{SO}(V))$, the maximal $\pi$-isotypic
quotient of $\omega$ is of the form

$\pi\boxtimes\Theta(\pi),$

for some $\Theta(\pi)=\Theta_{\psi,V,W}(\pi)$ smooth finite length
representation of $\widetilde{\mathrm{Sp}}(W)$ (called the big theta lift of
$\pi$). The maximal semisimple quotient of $\Theta(\pi)$ is called the small
theta lift of $\pi$. Similarly, for
$\widetilde{\pi}^{\prime}\in\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))$,
we obtain a smooth finite length representation
$\Theta(\widetilde{\pi}^{\prime})=\Theta_{\psi,V,W}(\widetilde{\pi}^{\prime})$
of $\mathrm{O}(V)$. The maximal semi-simple quotient of
$\Theta(\widetilde{\pi}^{\prime})$ (resp. $\Theta(\pi)$) is denoted by
$\theta(\widetilde{\pi}^{\prime})=\theta_{\psi,V,W}(\widetilde{\pi}^{\prime})$
(resp. $\theta(\pi)=\theta_{\psi,V,W}(\pi)$). The Howe duality conjecture,
which was proven by Waldspurger [Wa90] when the residue characteristic is not
$2$ and by Gan–Takeda [GT1, GT2] in general, says that if they are nonzero,
then $\theta(\widetilde{\pi}^{\prime})$ and $\theta(\pi)$ are irreducible.
Given
$\widetilde{\pi}^{\prime}\in\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))$,
there is a unique $\epsilon\in\\{\pm 1\\}$ such that
$\theta_{\psi,V^{\epsilon},W}(\widetilde{\pi}^{\prime})$ is nonzero (see [GS,
Theorem 1.4].)

Using the theta correspondence between $\widetilde{\mathrm{Sp}}(W)$ and
$\mathrm{O}(V)$, for each non-trivial additive character
$\psi:F\rightarrow\mathbb{C}^{\times}$, define a map
$\Theta_{\psi}:\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))\longrightarrow\operatorname{Irr}(\mathrm{SO}(V^{+}))\sqcup\operatorname{Irr}(\mathrm{SO}(V^{-}))$
as
$\Theta_{\psi}(\widetilde{\pi}^{\prime}):=\theta_{\psi,V^{\epsilon},W}(\widetilde{\pi}^{\prime})|_{\mathrm{SO}(V^{\epsilon})}$
for
$\widetilde{\pi}^{\prime}\in\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))$.
(Here, $\epsilon\in\\{\pm 1\\}$ is uniquely chosen so that
$\theta_{\psi,V^{\epsilon},W}(\widetilde{\pi}^{\prime})$ is nonzero and
$\theta_{\psi,V^{\epsilon},W}(\widetilde{\pi}^{\prime})|_{\mathrm{SO}(V^{\epsilon})}$
is the restriction of $\theta_{\psi,V^{\epsilon},W}(\widetilde{\pi}^{\prime})$
to $\mathrm{SO}(V^{\epsilon})$.)

One of the main results in [GS] is the following.

######  Theorem 3.1.

[GS, Theorem 1.1] Fix a non-trivial additive character
$\psi:F\rightarrow\mathbb{C}^{\times}$, Then the map $\Theta_{\psi}$

$\Theta_{\psi}:\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))\longrightarrow\operatorname{Irr}(\mathrm{SO}(V^{+}))\sqcup\operatorname{Irr}(\mathrm{SO}(V^{-})).$

is bijective.

The bijection $\Theta_{\psi}$ preserves many natural invariants and
properties. The following is a summary of some results in [GS], which we shall
use in this paper.

######  Theorem 3.2.

Suppose that
$\widetilde{\pi}^{\prime}\in\operatorname{Irr}(\widetilde{\mathrm{Sp}}(W))$
and $\pi\in\operatorname{Irr}(\mathrm{SO}(V^{\epsilon}))$ correspond under
$\Theta_{\psi}$. Then the following holds.

  1. $(\mathrm{i})$

If $\widetilde{\pi}^{\prime}$ is $\psi$-generic and tempered, then $\pi$ is
generic and tempered.

  2. $(\mathrm{ii})$

Assume that $\pi$ is generic and $\widetilde{\pi}^{\prime}$ is $\psi$-generic.
Then for any irreducible representation $\sigma$ of $\mathrm{GL}_{r}$,

$\gamma(s,\pi\times\sigma,\psi)=\gamma(s,\widetilde{\pi}^{\prime}\times\sigma,\psi).$

######  Remark 3.3.

When $n=1$, even though $\widetilde{\pi}$ is $\psi$-generic and supercuspidal
representation, the corresponding $\pi$ might not be supercuspidal . For
example, if $\widetilde{\pi}$ is the odd Weil representation, then $\pi$ is a
Steinberg representation. For such reason, Jiang and Soudry separated the
proof of the LCT for $\mathrm{SO}_{2n+1}$ according to whether $n=1$ or $n\geq
2$. In this paper, to make our arguments also work for $n=1$ cases, we reduce
the proof of our main theorem to the tempered cases, not the supercupidal
cases. This contrasts with the treatments in [JS03], where they reduced it to
the supercuspidal cases. This is one of the delicate parts in the paper.

##  4\. The proof

In this section, we prove our main theorem. The proof consists of two steps;
the first step is to prove the following tempered cases.

######  Theorem 4.1.

Let $\widetilde{\pi}$ and $\widetilde{\pi}^{\prime}$ be irreducible
$\psi$-generic tempered representations of $G_{n}(F)$ such that

$\gamma(s,\widetilde{\pi}\times\sigma,\psi)=\gamma(s,\widetilde{\pi}^{\prime}\times\sigma,\psi)$

holds for any irreducible supercuspidal representation $\sigma$ of
$\mathrm{GL}_{i}(F)$ with $1\leq i\leq n$.

$\widetilde{\pi}\simeq\widetilde{\pi}^{\prime}.$

###### Proof.

Let $V^{+}$ be a split quadratic space of dimension $2n+1$ and discriminant 1.
Using the bijection $\Theta_{\psi}$, consider $\Theta_{\psi}(\widetilde{\pi})$
and $\Theta_{\psi}(\widetilde{\pi}^{\prime})$. Since $\widetilde{\pi}$ and
$\widetilde{\pi}^{\prime}$ are $\psi$-generic and tempered, by Theorem 3.2,
$\Theta_{\psi}(\widetilde{\pi})$ and $\Theta_{\psi}(\widetilde{\pi}^{\prime})$
are nonzero irreducible generic representations of $\mathrm{SO}(V^{+})$. Again
by Theorem 3.2, we have

$\gamma(s,\Theta_{\psi}(\widetilde{\pi})\times\sigma,\psi)=\gamma(s,\widetilde{\pi}\times\sigma,\psi)=\gamma(s,\widetilde{\pi}^{\prime}\times\sigma,\psi)=\gamma(s,\Theta_{\psi}(\widetilde{\pi}^{\prime})\times\sigma,\psi).$

By Theorem 1.1, we have
$\Theta_{\psi}(\widetilde{\pi})\simeq\Theta_{\psi}(\widetilde{\pi}^{\prime})$.
Since $\Theta_{\psi}$ is bijective,
$\widetilde{\pi}\simeq\widetilde{\pi}^{\prime}$.

∎

Consider a unitary induced representation

$\operatorname{Ind}_{\widetilde{Q}}^{\widetilde{\mathrm{Sp}}(W)}((\tau_{1}|\det|_{F}^{s_{1}})_{\psi}\otimes\cdots\otimes(\tau_{r}|\det|_{F}^{s_{r}})_{\psi}\otimes\widetilde{\pi}_{0}),$

where

  * •

$W=W_{2n}$ and $W_{0}=W_{2n_{0}}$ are symplectic spaces of dimension $2n$ and
$2n_{0}$, respectively;

  * •

$\widetilde{Q}$ is a parabolic subgroup of $\widetilde{\mathrm{Sp}}(W)$ with
the Levi subgroup isomorphic to
$\widetilde{\mathrm{GL}}_{n_{1}}(F)\times_{\\{\pm 1\\}}\cdots\times_{\\{\pm
1\\}}\widetilde{\mathrm{GL}}_{n_{r}}(F)\times_{\\{\pm
1\\}}\widetilde{\mathrm{Sp}}(W_{0})$;

  * •

$\tau_{i}$ is an irreducible unitary supercuspidal representation of
$\mathrm{GL}_{n_{i}}(F)$;

  * •

$s_{i}$ is a real number such that $s_{1}\geq\dots\geq s_{r}>0$;

  * •

$\widetilde{\pi}_{0}$ is an irreducible generic tempered representation of
$\widetilde{\mathrm{Sp}}(W_{0})$.

An irreducible generic representation of $G_{n}(F)$ is a subquotient of such
an induced representation (cf. [Jo, Lemma 4.11].) In that case, we say that it
has support $(\tau_{1},\cdots,\tau_{r};\widetilde{\pi_{0}})$ and exponents
$(s_{1},\cdots,s_{r})$. Note that it is different with usual supercuspidal
support because $\widetilde{\pi}_{0}$ is not supercuspidal but tempered.

The following lemma is an extended version of [JS03, Proposition 3.2] and [Jo,
Proposition 4.12] from supercuspidal to tempered cases.

######  Lemma 4.2.

For an irreducible generic tempered representation $\pi$ of
$\mathrm{SO}_{2n+1}(F)$ and supercuspidal representation $\sigma$ of
$\mathrm{GL}_{k}(F)$, the followings hold.

  1. $(\mathrm{i})$

$\gamma(s,\pi\times\sigma,\psi)$ has no zero for $\operatorname{Re}(s)>0$.

  2. $(\mathrm{ii})$

If $\gamma(s,\pi\times\sigma,\psi)$ has a real pole at $s=s_{0}$, then the
pole must be a simple pole at $s_{0}=1$ and $\sigma\simeq\sigma^{\vee}$.

###### Proof.

Since the definition of $\gamma$-factor is equal to that of Shahidi’s
$\gamma$-factor in [Sha90], we have

$\gamma(s,\pi\times\sigma,\psi)=\epsilon(s,\pi\times\sigma,\psi)\cdot\frac{L(1-s,\pi^{\vee}\times\sigma^{\vee},\psi)}{L(s,\pi\times\sigma,\psi)}.$

By [CS, page 573], $L(s,\pi\times\sigma,\psi)$ is holomorphic for
$\operatorname{Re}(s)>0$ and $(\mathrm{i})$ follows from this.

In the proof of [Jo, Proposition 4.12 $(\mathrm{iii})$], the restriction that
$\pi$ is supercuspidal is given to apply ’Casselman–Shahidi Lemma’ ([Jo, Lemma
4.10]). However, ’Casselman–Shahidi Lemma’ also holds for tempered
representations by [Kim, Proposition 12.3]. Except that, other arguments of
[Jo, Proposition 4.12 $(\mathrm{iii})$] equally work for tempered
representations. It completes the proof of $(\mathrm{ii})$. ∎

######  Proposition 4.3.

Let $\sigma$ be an irreducible unitary supercuspidal representation of
${\mathrm{GL}}_{k}(F)$ and $\tau_{i}$, $s_{i}$ and $\widetilde{\pi_{0}}$ are
as in the above. Then we have the followings;

  1. $(\mathrm{i})$

If the product $\prod_{i=1}^{t}\gamma(s+z_{i},\tau_{i}\times\sigma,\psi)$ has
a real pole $($respectively, a real zero$)$ at $s=s_{0}$, then
$\sigma\cong\tau_{i}^{\vee}$ and $s_{0}=1-z_{i}$ $($respectively,
$s_{0}=-z_{i})$ for some $1\leq i\leq t$.

  2. $(\mathrm{ii})$

If the product
$\prod_{i=1}^{t}\gamma(s-z_{i},\tau_{i}^{\vee}\times\sigma,\psi)$ has a real
pole $($respectively, a real zero$)$ at $s=s_{0}$, then $\sigma\cong\tau_{i}$
and $s_{0}=1+z_{i}$ $($respectively, $s_{0}=z_{i})$ for some $1\leq i\leq t$.

  3. $(\mathrm{iii})$

The factor $\gamma(s,\widetilde{\pi}_{0}\times\sigma,\psi)$ has no zero for
${\operatorname{Re}}(s)>0$. If
$\gamma(s,\widetilde{\pi_{0}}\times\sigma,\psi)$ has a real pole at $s=s_{0}$,
then the pole must be a simple pole at $s_{0}=1$ and
$\sigma\cong\sigma^{\vee}$.

###### Proof.

$(\mathrm{i})$ and $(\mathrm{ii})$ follows from the discussion of [JS03,
Section 3.2]. For $(\mathrm{iii})$, consider
$\Theta_{\psi}(\widetilde{\pi_{0}})$. Then by Theorem 3.2,
$\Theta_{\psi}(\widetilde{\pi_{0}})$ is tempered and

$\gamma(s,\widetilde{\pi_{0}}\times\sigma,\psi)=\gamma(s,\Theta_{\psi}(\widetilde{\pi_{0}})\times\sigma,\psi).$

Then by Lemma 4.2, $(\mathrm{iii})$ readily follows . ∎

The following proposition is a consequence of Proposition 4.3. Since the proof
is identical to that of [JS03, Theorem 5.1] (cf. [Jo, Proposition 4.14]), we
omit it.

######  Proposition 4.4.

Let $\widetilde{\pi}$ (resp. $\widetilde{\pi}^{\prime}$) be an irreducible
generic representations of $G_{n}(F)$, which has support
$(\tau_{1},\cdots,\tau_{r};\widetilde{\pi_{0}})$ (resp.
$(\tau_{1}^{\prime},\cdots,\tau_{r^{\prime}};\widetilde{\pi_{0}}^{\prime})$)
and exponents $(s_{1},\cdots,s_{r})$ (resp.
$(s_{1}^{\prime},\cdots,s_{r^{\prime}}^{\prime})$). Suppose that

(4.1)
$\left[\prod_{i=1}^{r}\gamma(s+s_{i},\tau_{i}\times\sigma,\psi)\gamma(s-s_{i},\tau_{i}^{\vee}\times\sigma,\psi)\right]\gamma(s,\widetilde{\pi_{0}}\times\sigma,\psi)\\\
=\left[\prod_{i=1}^{r^{\prime}}\gamma(s+s_{i}^{\prime},\tau_{i}^{\prime}\times\sigma,\psi)\gamma(s-s_{i}^{\prime},(\tau_{i}^{\prime})^{\vee}\times\sigma,\psi)\right]\gamma(s,\widetilde{\pi_{0}}^{\prime}\times\sigma,\psi)$

for all irreducible unitary supercuspidal representations $\sigma$ of
${\mathrm{GL}}_{t}(F)$ with $1\leq t\leq n$. Then $r=r^{\prime}$ and there
exists a permutation $\rho$ of $\\{1,2,\dotsm,r\\}$ such that

  1. $(\mathrm{i})$

$r_{i}=r^{\prime}_{\rho(i)}$ for all $i=1,2,\dotsm,r$;

  2. $(\mathrm{ii})$

$z_{i}=z^{\prime}_{\rho(i)}$ and $\tau_{i}\cong\tau_{\rho(i)}^{\prime}$ for
all $i=1,2,\dotsm,r$;

  3. $(\mathrm{iii})$

$\gamma(s,\widetilde{\pi_{0}}\times\sigma,\psi)=\gamma(s,\widetilde{\pi_{0}}^{\prime}\times\sigma,\psi)$
for all irreducible unitary supercuspidal representations $\sigma$ of
${\mathrm{GL}}_{t}(F)$ with $1\leq t\leq n$.

Now we are ready to prove the following main theorem.

######  Theorem 4.5.

Let $\widetilde{\pi}$ and $\widetilde{\pi}^{\prime}$ be irreducible
$\psi$-generic representations of $G_{n}(F)$. Suppose that

$\gamma(s,\widetilde{\pi}\times\sigma,\psi)=\gamma(s,\widetilde{\pi}^{\prime}\times\sigma,\psi)$

holds for any irreducible supercuspidal representation $\sigma$ of
$\mathrm{GL}_{t}(F)$ with $1\leq t\leq n$. Then

$\widetilde{\pi}\simeq\widetilde{\pi}^{\prime}.$

###### Proof.

Suppose that $\widetilde{\pi}$ and $\widetilde{\pi}^{\prime}$ have supports as
in Proposition 4.4. Then by Theorem 4.1 and Proposition 4.4, we have
$\widetilde{\pi_{0}}\ \simeq\widetilde{\pi_{0}}^{\prime}$ and so
$\widetilde{\pi}$ and $\widetilde{\pi}^{\prime}$ have the same supports. Since
an induced representation has the unique generic constituent,
$\widetilde{\pi}\simeq\widetilde{\pi}^{\prime}$. ∎

Using the similar arguments we used so far, we prove Theorem 1.2 as follows.

###### Proof.

Assume that $\widetilde{\pi}$ (resp. $\widetilde{\pi}^{\prime}$) has support
$(\tau_{1},\cdots,\tau_{r};\widetilde{\pi_{0}})$ (resp.
$(\tau_{1}^{\prime},\cdots,\tau_{r^{\prime}};\widetilde{\pi_{0}}^{\prime})$)
and exponents $(s_{1},\cdots,s_{r})$ (resp.
$(s_{1}^{\prime},\cdots,s_{r^{\prime}}^{\prime})$). Consider the irreducible
generic representation $\pi$ (resp. $\pi^{\prime}$) of $\mathrm{SO}(V^{+})$,
which has support
$(\tau_{1},\cdots,\tau_{r};\Theta_{\psi}(\widetilde{\pi_{0}}))$ (resp.
$(\tau_{1}^{\prime},\cdots,\tau_{r^{\prime}};\Theta_{\psi}(\widetilde{\pi_{0}}^{\prime}))$)
and exponents $(s_{1},\cdots,s_{r})$ (resp.
$(s_{1}^{\prime},\cdots,s_{r^{\prime}}^{\prime})$).

By [CKPSS04, Theorem 4.1] or [CP, Theorem], there is an integer
$l_{0}=l(\pi,\pi^{\prime})>0$ such that for any quasi-character $\chi$ of
$F^{\times}$, with conductor $\text{cond}(\chi)>l$,

$\gamma(s,\pi\times\chi,\psi)=\gamma(s,\pi^{\prime}\times\chi,\psi).$

By Theorem 3.2,

$\gamma(s,\widetilde{\pi_{0}}\times\chi,\psi)=\gamma(s,\pi_{0}\times\chi,\psi),\quad\gamma(s,\widetilde{\pi_{0}}^{\prime}\times\chi,\psi)=\gamma(s,\pi_{0}^{\prime}\times\chi,\psi)$

and thus

$\gamma(s,\widetilde{\pi}\times\chi,\psi)=\gamma(s,\pi\times\chi,\psi),\quad\gamma(s,\widetilde{\pi}^{\prime}\times\chi,\psi)=\gamma(s,\pi^{\prime}\times\chi,\psi).$

If we take $l=l_{0}$, this completes the proof. ∎

##  5\. An application

We give a direct application of our main theorem, that is the rigidity theorem
for metaplectic groups. The discussion in this section is similar with that of
[JS03, Theorem 5.3].

In this section, let $F$ be a number field and $\mathbb{A}$ its adéle ring.
Let $\widetilde{\pi}=\otimes_{v}\widetilde{\pi}_{v}$ and
$\Pi=\otimes_{v}\Pi_{v}$ be irreducible automorphic representations of
$\widetilde{\mathrm{Sp}}_{2n}(\mathbb{A})$ and $\mathrm{GL}_{2n}(\mathbb{A})$,
respectively. Then we say that $\Pi$ is a weak functorial lift of
$\widetilde{\pi}$ if $\Pi_{v}$ is the local Langlands functorial lift of
$\widetilde{\pi_{v}}$ for all archimedean places and for almost all places $v$
of $F$, where $\widetilde{\pi_{v}}$ and $\Pi_{v}$ are unramified.

######  Lemma 5.1.

Let $\widetilde{\pi}=\otimes_{v}\widetilde{\pi_{v}}$ be irreducible cuspidal
generic automorphic representations of
$\widetilde{\mathrm{Sp}}_{2n}(\mathbb{A})$. Let $\Pi=\otimes_{v}\Pi_{v}$ be
any weak functorial lift of $\widetilde{\pi}$. Fix any finite place $v$ of
$F$. Then for every supercuspidal representation $\sigma_{v}$ of
$GL_{t}(F_{v})$ with $1\leq t\leq 2n$, we have

$\gamma(s,\pi_{v}\times\sigma_{v},\psi_{v})=\gamma(s,\Pi_{v}\times\sigma_{v},\psi_{v}).$

###### Proof.

The proof follows almost the same line of [CKPSS, Corollary 4]. The key
ingredient of the proof of [CKPSS, Corollary 4] is the stability of local
$\gamma$-factors for $\mathrm{SO}_{2n+1}$. However, Theorem 1.2 replaces the
role of it and the other arguments equally work as well. ∎

Now we are ready to prove Theorem 1.3.

###### Proof.

Let $\Pi$ and $\Pi^{\prime}$ be a weak functorial lifts of $\widetilde{\pi}$
and $\widetilde{\pi}^{\prime}$ to $\mathrm{GL}_{2n}(\mathbb{A})$. Then by
[GRS, Theorem 11.2], $\Pi$ and $\Pi^{\prime}$ are irreducible generic self-
dual unitary automorphic representations of $\mathrm{GL}_{2n}(\mathbb{A})$ and
by the assumption, $\Pi_{v}\simeq\Pi_{v}^{\prime}$ for almost all places $v$.
Then by the strong multiplicity one theorem for
$\mathrm{GL}_{2n}(\mathbb{A})$, $\Pi\simeq\Pi^{\prime}$ and thus
$\Pi_{v}\simeq\Pi_{v}^{\prime}$ for all places $v$. Then by the property of
the local functorial lifts,
$\widetilde{\pi_{v}}\simeq\widetilde{\pi_{v}}^{\prime}$ for all archimedean
places. Furthermore, by Lemma 5.1, for arbitrary finite place $v$, we have

$\gamma(s,\widetilde{\pi}_{v}\times\sigma,\psi)=\gamma(s,\Pi_{v}\times\sigma,\psi)=\gamma(s,\Pi_{v}^{\prime}\times\sigma,\psi)=\gamma(s,\widetilde{\pi_{v}}^{\prime}\times\sigma,\psi)$

for any irreducible supercuspidal representation $\sigma$ of
$\mathrm{GL}_{t}(F_{v})$ with $1\leq t\leq n$. Then by Theorem 4.5, we have
$\widetilde{\pi_{v}}\simeq\widetilde{\pi_{v}}^{\prime}$. This completes the
proof of the theorem. ∎

### Acknowledgements

The author would like to thank Yeansu Kim for bringing this problem to his
attention. He is also grateful to Dongho Byeon, Haseo Ki, Sugwoo Shin, Youngju
Choie for their constant support and encouragement over the years. Without
their warm supports, this paper would not see the light of the world.
The author learned the use of local theta correspondence to other problems
from the brilliant works of Gan and Ichino. Indeed, this paper is an example
of their methods and we expect their original works to have broad applications
on future research in the field of automorphic forms. Being greatly influenced
by them and their works, we would like to express deep admiration for Gan and
Ichino.
Finally, we thank Yonsei University for providing a wonderful place to conduct
research and its generous support. The author is supported by the National
Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT)
(No. 2020R1F1A1A01048645).

## References

  * [At] H. Atobe, _The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case_ , Math. Annalen., 371, (2018), 225–295.
  * [CS] W. Casselman and F. Shahidi, _On irreducibility of standard modules for generic representations_ , Ann. Sci. École Norm. Sup. 31 (1998), 561–589.
  * [Cha19] Jingsong Chai, _Bessel functions and local converse conjecture of Jacquet_ , J. Eur. Math. Soc., 21, (2019), no. 6, 1703–1728.
  * [CKPSS] J. W. Cogdell, H. Kim, I.I. Piathtski-Shapiro, F. Shahidi, _On lifting from classical groups to $\mathrm{GL}(n)$_, IHES Publ. Math. 93 (2001), 5–30.
  * [CKPSS04] J. W. Cogdell, H. Kim, I.I. Piathtski-Shapiro, F. Shahidi, _Functoriality for the classical groups_ , IHES Publ. Math. 99 (2004), 163–233.
  * [CP] J. W. Cogdell, I.I. Piathtski-Shapiro, _Stability of gamma factors for $\mathrm{SO}(2n+1)$_, Manuscr. Math., 95 (1998), 437–461.
  * [CST17] J. W. Cogdell, F. Shahidi and T.-L. Tsai, _Local Langlands correspondence for $\mathrm{GL}_{n}$ and the exterior and symmetric square $\varepsilon$-factors_, Duke Math. J. 166, (2017), no. 11, 2053–2132.
  * [GGP] W. T. Gan, B. Gross, D. Prasad, _Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups_, Astérisque., 346, (2012), 1–109.
  * [GI] W. T. Gan and A. Ichino, _The Gross–Prasad conjecture and local theta correspondence_ , Invent. Math., 206, (2016), no. 3, 705–799.
  * [GS] W. T. Gan and G. Savin, _Representations of metaplectic groups $\mathrm{I}$: epsilon dichotomy and local Langlands correspondence_, ompo, 148 (2012), 1655–1694.
  * [GRS] D. Ginzburg, S. Rallis and D. Soudry, _The descent map from automorphic representations of $GL(n)$ to classical groups_, World Scientific, Hackensack (2011).
  * [GT1] W. T. Gan and S. Takeda, On the Howe duality conjecture in classical theta correspondence, Advances in the theory of automorphic forms and their $L$-functions, 105–117, Contemp. Math., 664, Amer. Math. Soc., Providence, RI, 2016.
  * [GT2] W. T. Gan and S. Takeda, A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29 (2016), no. 2, 473–493.
  * [GRS] D. Ginzberg, S. Rallis, D. Soudry, _$L$ -functions for symplectic groups_, Bull. Soc. Math. France, 126, (1998), 181–224.
  * [He] G. Henniart, _Caractérisation de la correspondance de Langlands locale par les facteurse de paires_ , Invent. M., 113, (1993), 339–350.
  * [JL18] Hervé Jacquet and Baiying Liu, _On the local converse theorem for $p$-adic $\mathrm{GL}_{n}$_, Amer. J. Math., 140, (2018), no. 5, 1399–1422.
  * [Jng] D. Jiang, On local $\gamma$-factors. In: Arithmetic geometry and number theory, Series on Number Theory and Applications, 1, World Sci. Publ., Hackensack, NJ, (2006), 1-28.
  * [JSS] , H. Jacquet, I.I. Piatetski-Shapiro, J.A. Shalika, _Rankin–Selbeerg convolutions_ , Amer. J. Math., 105, (1983), 367–464.
  * [JS03] Dihua Jiang and David Soudry, _The local converse theorem for ${\mathrm{S}O}(2n+1)$ and applications_, Ann. of Math. (2), 157, (2003), no.3, 743–806.
  * [JS04] Dihua Jiang and David Soudry, _Generic representations and local Langlands reciprocity law for $p$-adic ${\mathrm{S}O}_{2n+1}$_, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp. 457–519.
  * [Jo] Yeongseong Jo, _The local converse theorem for odd special orthogonal and symplectic groups in positive characteristic_ , preprint
  * [Kap15] Eyal Kaplan, _Complementary results on the Rankin-Selberg gamma factors of classical groups_ , J. Number Theory, 146, (2015), 390–447.
  * [Kim] Henry H. Kim, _Automorphic $L$-functions_, Lectures on automorphic $L$-functions, Fields Inst. Monogr., vol.20, Amer. Math. Soc., Providence, RI, (2004), 97–201.
  * [LH] B. Liu and A. Hazeltine, _On the local converse theorem for split SO(2n)_ , (2022), submitted.
  * [M] K. Morimoto, _On the irreducibility of global descents for even unitary groups and its applications_ , Transactions of A.M.S., 370 (2018), 6245-6295.
  * [Rao93] R. Rao, _On some explicit formulas in the theory of weil representation_ , Pacific. J. Math. 157, (1993), 335-371.
  * [Sha90] F. Shahidi, _A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups_, Ann. of. Maath. (2) 132 (1990), 273–330.
  * [So93] D. Soudry, _Rankin–Selberg convolutions for $\mathrm{SO}(2n+1)\times\mathrm{GL}(n)$: local theory_, Mem. Amer. Math. Soc. 105 (1993), 1–100.
  * [Q18] Qing Zhang, _A local converse theorem for ${\mathrm{S}p}_{2r}$_, Math. Ann., 372, (2018), no. 1-2, 451–488.
  * [Q19] Qing Zhang, _A local converse theorem for ${\mathrm{U}}_{2r+1}$_, Trans. Amer. Math. Soc., 371, (2019), no. 8, 5631–5654.
  * [Wa90] J.-L. Waldspurger, Demonstration d’une conjecture de dualite de Howe dans le cas $p$-adique, $p\not=2$, in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, part $\mathrm{I}$ (Ramat Aviv, 1989), Israel Mathematical Conference Proceedings, vol. 2 (Weizmann, Jerusalem, 1990), 267–324.