# Temporal coupled-mode theory for thermal emission from multiple arbitrarily
coupled resonators

Xin Huang  Christopher Yeung  Aaswath P. Raman<EMAIL_ADDRESS>Department of
Materials Science and Engineering, University of California, Los Angeles, Los
Angeles, CA 90095

###### Abstract

Controlling the spectral response of thermal emitters has become increasingly
important for a range of energy and sensing applications. Conventional
approaches to achieving arbitrary spectrum selectivity in photonic systems
have entailed combining multiple resonantly emissive elements together to
achieve a range of spectral profiles through numerical optimization, with a
universal theoretical framework lacking. Here, we develop a temporal coupled
mode theory for thermal emission from multiple, arbtirarily-coupled
resonators. We validate our theory against numerical simulations of complex
two- and three-dimensional nanophotonic thermal emitters, highlighting the
anomalous thermal emission spectra that can emerge when multiple resonators
with arbitrary properties couple to each other with varying strengths.

##  I Introduction

Thermal emission is a fundamental physical process whose control is essential
for a broad range of imaging, sensing and energy technologies. Conventional
thermal emitters are typically incoherent, broadband, unpolarized and lack
directionality. Over the last two decades, a range of photonic strategies have
by contrast shown that it is indeed possible to create narrowband, polarized
and/or directional thermal emitters Shchegrov  _et al._ (2000); Pralle  _et
al._ (2002); Greffet  _et al._ (2002); Celanovic  _et al._ (2005); Laroche
_et al._ (2005); Lee  _et al._ (2005); Puscasu and Schaich (2008); Shen  _et
al._ (2009). The spectral characteristics of thermal emission in particular
have been sculpted through a range of nanophotonic approaches including
metallic nanoantennas Liu  _et al._ (2011, 2017), photonic crystals Lin  _et
al._ (2000); Luo  _et al._ (2004); Lin  _et al._ (2003); Inoue  _et al._
(2014); Cornelius and Dowling (1999); Zoysa  _et al._ (2012) and semiconductor
nanorods Mason  _et al._ (2011). Tailoring the frequency and bandwidth
response of emitted thermal radiation using nanophotonic strategies has in
turn proved critical for improved performance in many emerging applications,
including thermophotovoltaics Guo and Fan (2016); Lin  _et al._ (2003);
Narayanaswamy and Chen (2003); Nagpal  _et al._ (2008); Rephaeli and Fan
(2009); Messina and Ben-Abdallah (2013); Lenert  _et al._ (2014), radiative
cooling Raman  _et al._ (2014); Chen  _et al._ (2016); Kou  _et al._ (2017);
Hossain  _et al._ (2015); Shi  _et al._ (2015); Gentle and Smith (2015);
Goldstein  _et al._ (2017) and thermal management Hsu  _et al._ (2016); Tong
_et al._ (2015); Catrysse  _et al._ (2016); Jafar-Zanjani  _et al._ (2017).

Conventional strategies to achieve an arbitrary degree of control over
spectral selectivity for a thermal emitter rely on either numerical
optimization or exploiting known electromagnetic mode behavior in conventional
photonic systems. In this context, a resonance based approach to understanding
thermal emission offers the potential of both understanding and designing
desired thermal emission spectra in a more systematic way. Temporal coupled-
mode theory Haus (1984); Suh  _et al._ (2004) in particular is a widely-used
semi-analytical method that has been shown to provide excellent approximations
and physical insight into the behavior of a range of resonance- and mode-
driven nanophotonic structures and devices Khan  _et al._ (1999); Liu and
Yariv (2011); Fan  _et al._ (2003); Yanik and Fan (2004); Ramirez  _et al._
(2011); Ghebrebrhan  _et al._ (2011).

Recently, temporal coupled mode theory for single-mode thermal emitters was
developed to analytically model the thermal emission from such an emitter with
high accuracy Zhu  _et al._ (2013). Further work has extended this coupled-
mode theory to include coupling between multiple identical resonant thermal
emitters, and shown that thermal emission decreases with increasing number of
identical thermal emitters in analogy to quantum superradiance effects M.Zhou
_et al._ (2015). As a semi-analytical model, coupled-mode theory is able to
offer deeper insight on the effect of material and structural parameters and
the resulting spectral nature of their thermal emission, an insight that
simulations alone cannot offer. However, prior work has been limited to
describing a constrained set of thermal emitters due to the resonators being
identical in character. For many practical scenarios, highly complex and
selective thermal emission spectra with multiple frequency peaks and varying
bandwidths are desirable, but difficult to achieve with single-mode
resonators, and more generally are challenging to elucidate. One particularly
promising mechanism to achieving complex thermal emission spectra is to
consider systems supporting multiple arbitrary resonators Liu  _et al._ (2011,
2017) which may additionally couple or hybridize with each other, thereby
enhancing or suppressing thermal emission with several degrees of potential
freedom to control the resulting spectral response. Such a scenario, while
intriguing, is however challenging to model analytically and can require many
complex and slow simulations. A universal theoretical framework for
understanding the full range of complex coupling, and resulting thermal
emission that may be possible in multi-resonant photonic structures is
currently lacking, but could enable deeper insights into the behavior of such
structures and enable more rapid design of their arrangements.

In this Letter we introduce an extended temporal coupled-mode theory framework
to derive an analytical formalism to model thermal emission from an arbitrary
number of resonators which can arbitrarily couple to each other. We validate
the theory against simulation-based calculations of thermal emission from a
range of physically realizable systems. We first demonstrate the coupled-mode
theory’s accuracy for multiple two-dimensional slit resonators, where the
dielectric permittivity of each slit can vary, along with the distance between
each slit, thereby altering both the real and imaginary part of the coupling
coefficient. We also demonstrate the coupled-mode theory’s capabilities with a
three slit system, highlighting how both subradiant enhancement and
superradiant suppression of thermal emission can be selectively engineered at
different frequencies. Finally, we demonstrate the accuracy of the coupled-
mode theory in predicting the thermal emission spectra of supercell three-
dimensional metal-dielectric-metal resonators for a range of inter-resonator
distances. Collectively, our results offer a general theoretical framework
capable of taking the response of individual resonators and using them to
determine their complex spectral response when integrated and hybridized with
each other.

##  II Extended Coupled-mode Theory

We first develop an extended temporal coupled-mode theory capable of
describing the collective thermal emission from $N$ different emitters that
are coupled to each other with varying degrees of strength. Each resonant
emitter is assumed to have an amplitude
$\textbf{a}=(a_{1},a_{2},...,a_{N})^{T}$. The energy stored inside an emitter
may decay through three pathways. The first pathway is through intrinsic
absorption which is represented by the intrinsic decay rate
$\bm{\Gamma_{0}}$=diag($\gamma_{01}$, $\gamma_{02}$ …$\gamma_{0N}$). The
second is to decay to the external free space channels, which is described by
the external decay rate $\bm{\Gamma_{e}}$ =diag($\gamma_{e1}$, $\gamma_{e2}$
…$\gamma_{eN}$). The third is through coupling with other resonators, which is
expressed by a complex coupling coefficient matrix $K$ defined as

$\mathbf{K}=\left(\begin{array}[]{ccccc}0&\kappa_{12}+i\beta_{12}&\cdots&\kappa_{1(N-1)}+i\beta_{1(N-1)}&\kappa_{1N}+i\beta_{1N}\\\
\kappa_{21}+i\beta_{21}&0&\cdots&\kappa_{2(N-1)}+i\beta_{2(N-1)}&\kappa_{2N}+i\beta_{2N}\\\
\vdots&&\ddots&&\vdots\\\
\kappa_{(N-1)1}+i\beta_{(N-1)1}&\kappa_{(N-1)2}+i\beta_{(N-1)2}&\cdots&0&\kappa_{(N-1)N}+i\beta_{(N-1)N}\\\
\kappa_{N1}+i\beta_{N1}&\kappa_{N2}+i\beta_{N2}&\cdots&\kappa_{N(N-1)}+i\beta_{N(N-1)}&0\end{array}\right)$
(1)

. To capture arbitrary coupling between resonating elements, we define
$\kappa$ as the real part of the coupling coefficient, which in our context
can capture scenarios such as variable distance between resonators, and
$\beta$ which is the imaginary part of the coupling strength, and captures the
phase difference between resonators. In the framework of the fluctuation-
dissipation theorem, the absorption process is balanced by a random thermal
excitation source $\bm{n}$. With all these considerations, the dynamic
equations for resonance amplitudes can be written in the following form:

$\displaystyle\frac{{d\bm{a}}}{dt}=(j\bm{\Omega_{0}}-{\bm{\Gamma_{0}}-\bm{\Gamma_{e}}})\
\bm{a}+\sqrt{2\ \bm{\Gamma_{0}}}\ \bm{n}-\bm{Ka}$ (2)

where $\bm{\Omega_{0}}$ = diag($\omega_{1}$, $\omega_{2}$ … $\omega_{N}$)
describes the resonant frequency of each resonator. We can explicitly
calculate $\bm{a(\omega)}$ in the frequency domain,

$\displaystyle\bm{a(\omega)}=(j(\omega-\bm{\Omega_{0}})+{\bm{\Gamma_{0}}+\bm{\Gamma_{e}}}+\bm{K})^{-1}\sqrt{2\
\bm{\Gamma_{0}}}\ \bm{n}$ (3)

We normalize the amplitude so that the mode energy is given by $|\bm{a}|^{2}$.
Here we have introduced a noise source vector $\bm{n}$ in Eq. (1), to
compensate for the intrinsic resonator loss and maintain thermal equilibrium
Zhu  _et al._ (2013). Following the fluctuation-dissipation theorem this noise
source is defined by a correlation function (see Supplementary Information):

$\displaystyle\langle\bm{n}^{*}(\omega)\bm{n}(\omega^{\prime})\rangle=\frac{1}{2N\pi}\bm{\Theta}(\omega,T)\delta(\omega-\omega^{\prime})$
(4)

where
$\bm{\Theta}(\omega,T)=\frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$. The
total power emitted as thermal radiation, $\langle\boldsymbol{P}\rangle$, can
then be calculated as

$\displaystyle\langle\bm{P}(t)\rangle$ $\displaystyle=$ $\displaystyle
2\bm{\Gamma_{e}}\langle\bm{a}^{*}(t)\bm{a}(t)\rangle$ (5) $\displaystyle=$
$\displaystyle
2\bm{\Gamma_{e}}\int_{0}^{\infty}d\omega\int_{0}^{\infty}d\omega^{\prime}e^{-j(\omega-\omega^{\prime})t}\langle\boldsymbol{a}^{*}(\omega)\boldsymbol{a}(\omega^{\prime})\rangle$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}d\omega\frac{\bm{\Theta}(\omega,\bm{T})}{2N\pi}4\bm{\Gamma_{e}}((j(\omega-\bm{\Omega_{0}})+\bm{\Gamma_{0}}+\bm{\Gamma_{e}}+\bm{K})^{-1})^{2}\bm{\Gamma_{0}}$

Solving for the power spectral density of thermal emission we find that

$\displaystyle\bm{P}(\omega)=\frac{\bm{\Theta}(\omega,\bm{T})}{2N\pi}4\bm{\Gamma_{e}}((j(\omega-\bm{\Omega_{0}})+\bm{\Gamma_{0}}+\bm{\Gamma_{e}}+\bm{K})^{-1})^{2}\
\bm{\Gamma_{0}}$ (6)

Here, as in conventional expositions of temporal coupled mode theory for
multi-port systems, we assume that each resonances can decay into every other
port in the system, described by a total value $d_{ij}$ and encompassed by a
coupling matrix $\mathbf{D}$:

$\mathbf{D}=\left(\begin{array}[]{ccccc}d_{11}&d_{12}&\cdots&d_{1(N-1)}&d_{1N}\\\
d_{21}&d_{22}&\cdots&d_{2(N-1)}&d_{2N}\\\ \vdots&&\ddots&&\vdots\\\
d_{(N-1)1}&d_{(N-1)2}&\cdots&d_{(N-1)(N-1)}&d_{(N-1)N}\\\
d_{N1}&d_{N2}&\cdots&d_{N(N-1)}&d_{NN}\end{array}\right).$ (7)

.

By energy conservation the total coupling matrix $\mathbf{D}$ bounds the
values of the coupling coefficients to each relevant channel (see
Supplementary Information):

$\displaystyle 2(\bm{\Gamma_{0}}+\bm{\Gamma_{e}}+\bm{K})=\bm{D^{+}D}\ $ (8)

Eq. (6) is a key result of this paper and provides a general expression for
thermal emission from $N$ resonators that are coupled to each other
arbitrarily. To elucidate the power of this result, we first write analytical
forms for small $N$ scenarios of typical interest, beginning with the $N$ = 1
scenario where coupling is not relevant:

$\displaystyle
P(\omega)=\frac{\bm{\Theta}(\omega,\bm{T})}{2\pi}\frac{4\gamma_{01}\gamma_{e1}}{(\omega-\omega_{1})^{2}+(\gamma_{01}+\gamma_{e1})^{2}}$
(9)

This expression is simply a standard Lorentzian form of power emitted due to a
single resonance previously derived in Ref. Zhu  _et al._ (2013). For two
resonators ($N$ = 2), however, we must include complex coupling terms
$\kappa_{ij}+i\beta_{ij}$ for generality. This results in the following
expression for the radiated power:

$\displaystyle P(\omega)$ $\displaystyle=$
$\displaystyle\frac{\bm{\Theta}(\omega,\bm{T})}{4\pi}(\frac{4\gamma_{01}\gamma_{e1}\alpha_{2}^{2}+4\gamma_{e1}\gamma_{02}\kappa_{12}^{2}+4\gamma_{e1}\gamma_{02}\beta_{12}^{2}-8\alpha_{2}\kappa_{12}\gamma_{e1}\sqrt{\gamma_{01}\gamma_{02}}}{(\alpha_{1}\alpha_{2}-(\kappa_{12}+i\beta_{12})(\kappa_{21}+i\beta_{21}))^{2}}$
(10)
$\displaystyle+\frac{4\gamma_{e2}\gamma_{02}\alpha_{1}^{2}+4\gamma_{e2}\gamma_{01}\kappa_{21}^{2}+4\gamma_{e2}\gamma_{01}\beta_{21}^{2}-8\alpha_{1}\kappa_{21}\gamma_{e2}\sqrt{\gamma_{01}\gamma_{02}}}{(\alpha_{1}\alpha_{2}-(\kappa_{12}+i\beta_{12})(\kappa_{21}+i\beta_{21}))^{2}})$

Here for convenience we have introduced variables $\alpha_{j}$ defined as:

$\alpha_{j}=i(\omega-\omega_{j})+\gamma_{0j}+\gamma_{ej}$

$\kappa_{12}$ and $\kappa_{21}$ are the real parts of the coupling term
determined by the spatial distance between the resonances, while the complex
terms $\beta_{12}$ and $\beta_{21}$ describe the phase mismatch between
resonances. Due to energy conservation, a two resonator system is constrained
by Eq. (8), resulting in the following expression:

$\cos(\theta_{12}-\theta_{11})+i\sin(\theta_{12}-\theta_{11})+\cos(\theta_{22}-\theta_{21})+i\sin(\theta_{22}-\theta_{21})=\frac{2\kappa_{12}+2j\beta_{12}}{\sqrt{(\gamma_{01}+\gamma_{e1})(\gamma_{02}+\gamma_{e2})}}$
(11)

Here, $\theta_{ij}$ is the phase angle of $d_{ij}$. As in conventional
coupled-mode theory, this energy conservation relation extends to arbitrary
numbers of resonators and fundamentally links the various coupling terms
external to each resonator: coupling to neighboring resonators or to free
space. In the two resonator case, we can first consider the scenario when the
two resonators are spatially closed each other. In this case, as the two
resonators move closer and closer, $\theta_{12}$ will be getting closer to
$\theta_{11}$ and $\theta_{21}$ will be getting closer to $\theta_{22}$.
Therefore, cos($\theta_{12}-\theta_{11}$) is near 1 and
sin($\theta_{12}-\theta_{11}$) is near 0, and the total thermal emission peak
power is strongly driven by the real part of the coupling coefficients. When
the two resonators are physically far away from each other, as $\theta_{12}$
and $\theta_{21}$ is small, the coupling strength is determined by the phase
angle of $\theta_{11}$ and $\theta_{22}$. If $\theta_{11}$ and $\theta_{22}$
closes to $\pi/2$, the total thermal emission peak is mainly determined by the
imaginary portion of the coupling strength which encodes phase mismatch. If
$\theta_{11}$ and $\theta_{22}$ closes to 0, the total thermal emission peak
power is strongly driven by the real part of the coupling coefficients. More
generally, the behavior of each resonant peak in an arbitrary multiple-
resonance system is determined by both real and imaginary parts of the
coupling coefficients.

##  III Numerical Results

To validate the extended couple-mode theory developed above, we consider an
exemplary system consisting of multiple narrow dielectric slits of arbitrary
permittivity which are introduced into a perfect electric conductor (PEC)
layer, as shown in Fig. 1(a). In the system, the optical fields are confined
in the slit and can couple with each other. This system is an extension of the
slit resonator system considered in Ref. M.Zhou  _et al._ (2015) which
consider multiple identical resonators that were sufficiently close to each
other to enable near-field coupling.

Figure 1: The effect of varying distance between dissimilar resonators. a) The
structure of resonant emitters consisting of two slits in two PEC slabs. The
length and width of the slit are 1.4 $\mu$m and 5 nm, respectively. They are
filled with two different emissive materials with a dielectric constant of
$\varepsilon_{1}$ = 12.5 + 0.001$i$ and $\varepsilon_{2}$ = 12.53 + 0.001$i$.
b) FDFD-simulated spectra of emission cross section for two resonant emitters
with a distance of 400 nm, 800 nm and 1690 nm, separately. c) Coupled mode
theory thermal emission power prediction for two different emitters with a
coupling strength of $\kappa$ = 0.000145, $\kappa$ = 0.00015, $\kappa$ =
0.000185

We first consider the two resonator (two dielectric slit) scenario and
demonstrate how the coupled-mode theory accurately predicts the effect of
varying distance between dissimilar resonators, as well as differing
permittivities at a fixed separation distance. In this scenario, the slits are
1.4 $\mu$m wide, and 5 nm long, and contain dielectric media with
permittivities $\varepsilon_{1}=12.5+0.001i$ and
$\varepsilon_{2}=12.53+0.001i$ respectively. Since the permittivity of each
slit is different the system supports resonances at slightly different
frequencies, $\omega_{1}$ and $\omega_{2}$ as can be seen in Fig. 1(b). We
then calculate the emission cross section of this system using the finite-
difference frequency domain method for different values of the slit separation
distance $d$ as shown in Fig. 1(b). Simulations reveal a notable behavior as
we decrease the distance between two resonators: the resonator with lower
dielectric permittivity sees a large enhancement in its total associated
emission while a large decrease for the resonator with higher permittivity is
observed.

We next model the same system using the extended coupled-mode theory and
compare its predictions to the FDFD simulations. By defining a purely real
inter-resonator coupling constant, $\kappa$ and examining a range of its
values we are able to systematically replicate the trend observed in Fig. 1(b)
in Fig. 1(c) remarkably well. In particular, as the two resonators are moved
closer, the real part of the coupling strength further increases. This in turn
influences the internal coupling coefficients $\gamma_{01}$ and $\gamma_{02}$
and external coupling coefficients $\gamma_{e1}$ and $\gamma_{e2}$ due to
energy conservation. As detailed in the Supplementary Information, further
increasing the real part of coupling constant will need to increase
$\sqrt{(\gamma_{01}+\gamma_{e1})(\gamma_{02}+\gamma_{e2})}$ thereby resulting
in a decrease of the lower frequency peak, and an increase of the higher
frequency peak. The exact same behavior is observed in the FDFD simulations,
with a small variation in resonance frequency the only notable difference.
This result suggests that once coupling coefficients have been established for
a particular multi-resonator system of interest, the temporal coupled-mode
theory can be used as a rapid simulator of thermal emission from the class of
nanophotonic structures being examined.

Figure 2: The effect of changing permittivity in one of the two dielectric
slits at a fixed, large distance. a) The structure of resonant emitters
consisting of two slits in two PEC slabs. The length and width of the slit are
1.4 $\mu$m and 5 nm, respectively. They are filled with two different
dielectric materials with non-zero emissivity that are separated by a constant
distance $d$ = 800 nm. b) FDFD-simulated spectra of emission cross section for
two resonant emitters. The dielectric constant difference of
$\Delta\epsilon_{1}$ = 0.0008, $\Delta\epsilon_{2}$ = 0.03 and
$\Delta\epsilon_{3}$ = 0.05 separately. c) Coupled mode theory thermal
emission power prediction for two different emitters with a different
imaginary coupling strength of $\beta$=0.00001, $\beta$ = 0.0002, $\beta$ =
0.00015.

Next, we examine the effect of changing the permittivity in one of the two
dielectric slits at a fixed, large distance $d$ = 800 nm (Fig. 2(a)). As the
permittivity of one of the slits is changed, the imaginary part of inter-
resonator coupling coefficient, $\beta$ changes due to phase difference
between each resonator. Since the resonators are far apart, we expect minimal
contribution from the real part of the coupling coefficient $\kappa$. In our
system, we maintain the permittivity of $\varepsilon_{1}$ as 12.5 + 0.001$i$
and change the permittivity of $\varepsilon_{2}$ to 12.5008 + 0.001$i$, 12.53
+ 0.001$i$, and 12.55 + 0.001$i$ respectively. FDFD simulations shown in Fig.
2(b) highlight that as $\varepsilon_{2}$ changes the emission peak associated
with resonator 2 increases first, reaching a peak at $\Delta\varepsilon=0.03$
but then decreasing as $\varepsilon_{2}$ is further increased. We explore the
same system using coupled-mode theory in Fig. 2(c) and find a range of $\beta$
values which result in exactly the same behavior observed in the FDFD
simulations, further demonstrating the utility of this coupled-mode theory
model. Indeed, we emphasize that a purely real coupling coefficient would be
insufficient to capture the scenarios shown here. Referring back to Eq. (11)
we observe that as the sin($\theta$) term purely influences $\beta$ it thereby
encodes the phase difference between the resonators, which peaks at
$\theta=\pi/2$ which corresponds to a particular $\beta$ value depending on
the strengths of the internal coupling coefficients $\gamma_{0}$ and external
coupling coefficients of the resonances $\gamma_{e}$.

To highlight the generality and flexibility of the developed coupled-mode
theory we now examine a more complex systems involving three dielectric slits
separated by some arbitrary distance in a finite thickness PEC slab, and
filled with dielectric materials of arbitrary permittivity, as is shown in
Fig. 3(a). We consider the scenario where the first and third resonators have
the same permittivity $\varepsilon_{1}=\varepsilon_{3}=$ 12.5 + 0.001$i$ while
the middle resonator has a slightly different permittivity of
$\varepsilon_{2}=$ 12.53 + 0.001$i$. In Fig. 3(b), we compare the emission
cross section as simulated by FDFD for this three slit resonator systems
against a two resonator only system with $\varepsilon_{1}=$ 12.5 + 0.001$i$
and $\varepsilon_{2}=$ 12.53 + 0.001$i$ and without $\varepsilon_{3}$ .
Remarkably, we observe that the addition of the third resonator suppresses the
lower frequency peak (associated with $\varepsilon_{1}$ and $\varepsilon_{3}$)
while greatly enhancing the emission peak associated with the middle resonator
(with permittivity $\varepsilon_{2}$). Coupling between the resonators 1 and 3
results in a super-radiant suppression of emission analogous to that observed
in Ref. M.Zhou  _et al._ (2015) for the lower frequency peak. However,
coupling between resonator 2 and its neighbors results in a dramatic sub-
radiant enhancement in its thermal emission peak. We can model both the
baseline two resonator, and the more complex three resonator system using the
coupled-mode theory and find that it replicates the result observed in the
full-wave electromagnetic simulation, as is shown in Fig. 3(c). Our results,
in addition to highlighting the capabilities of the coupled-mode theory, show
a new mechanism to develop high power, narrow bandwidth thermal emitters
through inter-resonator coupling.

Figure 3:  Comparison between two resonant emitters system and three resonant
emitters system. a) The structure of resonant emitters consisting of three
slits in three PEC slabs. The length and width of the slit are 1.4 $\mu$m and
5 nm, respectively. They are filled with three emissive materials. b) FDFD-
simulated spectra of emission cross section for two resonators with a
permittivity $\varepsilon_{1}$ = 12.5 + 0.001i and $\varepsilon_{2}$ =
12.53+0.001i, and for a three resonant emitter scenario where the dielectric
permittivities are $\varepsilon_{1}$ = 12.5 + 0.001$i$, $\varepsilon_{2}$ =
12.53 + 0.001$i$ and $\varepsilon_{3}$ = 12.5 + 0.001$i$. c) Coupled mode
theory thermal emission power prediction for both the two resonator and three
resonator scenarios, showing strong alignment with the numerical simulations
of b).

Finally, we demonstrate the ability of the coupled-mode theory to accurately
model thermal emission from complex, three-dimensional nanophotonic
structures. In particular we consider complex supercells of metal-insulator-
metal (MIM) metasurfaces Liu  _et al._ (2011); Yeung  _et al._ (2021)
featuring cross-shaped resonators made of two different noble metals,
schematically shown in Fig. 4(a). The structures have $\pi$/2 rotational
symmetry and consist of three layers. The top layer consists of a pair of gold
metasurface elements, and a pair of silver metasurface elements with arms
lengths of $l_{1}=2.5\ \mu$m, $l_{2}=2.1\ \mu$m, and arm widths $w_{1}=0.6\
\mu$m , $w_{2}=0.4\ \mu$m respectively, with a thickness of 0.1 $\mu$m. The
permittivities of both silver and gold are modeled with a Drude fit that is
accurate to the long-wave infrared target wavelength range Ordal  _et al._
(1983). The metasurfaces lie atop a 0.2 $\mu$m Al2O3 layer whose permittivity
is defined in our wavelength region of interest with a Drude-Lorentz model
$\varepsilon=\varepsilon_{\infty}-\sigma/(\omega^{2}-i\omega\gamma-\omega_{0}^{2})$,
with $\varepsilon_{\infty}$ = 2.228, $\sigma$ = 0.008385$(2\pi c/a)$, $\gamma$
= 0.04$(2\pi c/a)$, and $\omega_{0}$ = $0.08(2\pi c/a)$, where $a=1\ \mu m$.
The bottom layer is assumed to be gold as well at a thickness of 0.2 $\mu m$.

Figure 4: Couple-mode theory predictions vs. full-field simulations of a
complex supercell metasurface. a) Geometry of the supercell metasurface
architecture evaluated. The metasurface elements have dimensions of
$l_{1}=2.5\ \mu$m, $l_{2}=2.1\ \mu$m, and $w_{1}=0.6\ \mu$m , $w_{2}=0.4\
\mu$m respectively, with a layer thickness of 0.1 $\mu$m. The central layer is
Al2O3 with thickness 0.2 $\mu m$ while the bottom layer is gold with thickness
of 0.2 $\mu$m. b) Simulation thermal emission power when d=1.2 $\mu$m versus
analytical coupled-mode theory prediction when $\kappa$=0. c) Simulation
thermal emission power when d=0.4 $\mu$m versus analytical coupled-mode theory
prediction when $\kappa$=5.31$\times$ $10^{-4}$(2$\pi$c/a) d) Simulation
thermal emission power when d=0.2 $\mu$m versus analytical coupled-mode theory
prediction when $\kappa$=0.0016 (2$\pi$c/a)

We performed full-field electromagnetic simulations of a range of MIM
supercell metasurfaces where the distance between the resonators in each
supercell is modified, in analogy to the scenario explored in Fig. 1 in a two-
dimensional systems. We simulated each geometry across both polarizations and
all angles of incidence to calculate the total spectral hemispherical
emittance. Given the four-fold symmetry of the supercells, to find the total
emitted power we integrate the incident angle $\theta$ from 0 to $\pi$/2 at
$\pi$/12 step and the azimuth angle $\phi$ from 0 to $\pi$/4 with $\pi$/12
steps:

$\displaystyle\varepsilon(\lambda)=\frac{\int_{0}^{2\pi}\int_{0}^{\pi/2}\varepsilon(\lambda,\theta,\phi)\cos\theta\sin\theta~{}d\theta
d\phi}{\int_{0}^{2\pi}\int_{0}^{\pi/2}\cos\theta\sin\theta~{}d\theta d\phi}$
(12)

As shown in Fig. 4(b-d), the simulations show that the supercell structure
supports two non-degenerate modes at $\omega_{1}$=0.131 (2$\pi$c/a) and
$\omega_{2}$=0.143 (2$\pi$c/a). We then use the coupled-mode theory to fit
each resonator’s intrinsic decay rate and external decay rate by simulating
its hemispherical emittance with a single resonator model, finding
$\gamma_{01}$ = 0.0019 (2$\pi$c/a), $\gamma_{02}$ = 0.0017 (2$\pi$c/a),
$\gamma_{e1}$ = 0.0038 (2$\pi$c/a), and $\gamma_{e2}$= 0.0030 (2$\pi$c/a). As
the distance $d$ is varied, the numerically simulated total hemispherical
emittance is then compared against the coupled-mode theory predictions and
shows remarkably strong agreement, as can be seen in Fig. 4(b-d) for a range
of resonator distances. Our results highlight a powerful capability enabled by
the coupled-mode theory framework in the context of thermal emission from such
complex supercell systems: simulating the response of each metasurface
resonator in a periodic configuration provides sufficient information to
rapidly model the behavior of complex arrangements of metasurface resonators
through the use of the complex coupling coefficients $\bm{K}$.

##  IV Concluding Remarks

In conclusion, we have developed a coupled mode theory that accurately models
thermal emission from complex photonic structures composed of multiple
resonators that can be arbitrarily coupled to each other. We demonstrated that
the coupled-mode theory accurately models anomalous, complex behavior such as
combined sub- and super-radiant thermal emission in a single nanophotonic
structure, as well as the response of supercell three-dimensional MIM
metasurface thermal emitters. Furthermore, the coupled mode theory provides
accurate predictions while taking far less time than full-field simulations
which takes hours to simulateYeung  _et al._ (2021), and in the process
provides physical insight into the original of the complex spectral behavior
that can result. Although we only fit our theoretical framework using regular
nanophotonic systems, this theory also implies that, once intrinsic decay
rate, external decay rate and coupling coefficients for a particular multi-
resonator system of interest have been determined, the temporal coupled-mode
theory can be utilized as a quick simulator of thermal emission from the class
of complex nanophotonic structures under consideration. As a semi-analytical
model, it can provide deeper insight into the effect of material and
structural characteristics on the spectral nature of their thermal emission,
which simulations alone cannot provide. One can thus imagine building
libraries of individual resonator responses which can then rapidly be assessed
for their integration into coupled arrangements based on the developed
coupled-mode theory. As demands on the complexity of the spectral response of
thermal photonic emitters grow, such a framework may prove critical to further
enhance and rapidly model their capabilities.

##  V Acknowledgments

This material is based upon work supported by the National Science Foundation
(NSF CAREER) under Grant No. 2146577, and the Sloan Research Fellowship
(Alfred P. Sloan Foundation).

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# The Strong Lefschetz Property of Gorenstein Algebras Generated by Relative
Invariants

Takahiro Nagaoka  and  Akihito Wachi  (Akihito Wachi) Department of
Mathematics, Hokkaido University of Education, Kushiro, 085-8580, Japan.

###### Abstract.

We prove the strong Lefschetz property for Artinian Gorenstein algebras
generated by the relative invariants of prehomogeneous vector spaces of
commutative parabolic type.

###### Key words and phrases:

Lefschetz property, Gorenstein algebra, Prehomogeneous vector space, Relative
invariant

###### 2020 Mathematics Subject Classification:

13E10, 11S90, 17B10

##  1\. Introduction

Let $Q=k[x_{1},x_{2},\ldots,x_{n}]$ be a polynomial ring over a field $k$ of
characteristic zero, and $R=k[\partial/\partial x_{1},\partial/\partial
x_{2},\ldots,\partial/\partial x_{n}]$ the ring of partial differential
operators with constant coefficients. Set $\operatorname{Ann}_{R}(F)=\\{P\in
R\mid P(F)=0\\}$ for $F\in Q$. Then $R/\operatorname{Ann}_{R}(F)$ is a graded
Artinian Gorenstein algebra. Conversely, any graded Artinian Gorenstein
algebra is constructed this way (Macaulay’s dual annihilator theorem. See [6,
Theorem 2.71]). $R/\operatorname{Ann}_{R}(F)$ is called a Gorenstein algebra
generated by $F$.

The graded Artinian Gorenstein algebra
$R/\operatorname{Ann}_{R}(F)=\bigoplus_{i=0}^{c}A_{i}$ is a Poincaré duality
algebra, which is an algebra such that the map $A_{i}\times A_{c-i}\to
A_{c}\;(\simeq k)$ ($(a,b)\mapsto ab$) forms a perfect paring for any
$i=0,1,\ldots,\lfloor c/2\rfloor$. Therefore, graded Artinian Gorenstein
algebras come as cohomology rings in certain categories. The following
definition is an algebraic abstraction of the hard Lefschetz theorem for the
cohomology rings of compact Kähler manifolds.

######  Definition 1 (Strong Lefschetz property).

A graded Artinian algebra $A=\bigoplus_{i=0}^{c}A_{i}$ ($A_{0}\simeq k$,
$A_{c}\neq 0$) over $k$ is said to have the strong Lefschetz property if there
exists $L\in A_{1}$ such that $\times L^{c-2i}:A_{i}\to A_{c-i}$ is bijective
for every $i=0,1,\ldots,\lfloor c/2\rfloor$. In this case $L$ is called a
Lefschetz element.

When a graded Artinian algebra $A$ is a complete intersection, which is a
special case of Gorenstein algebras, there is a long-standing conjecture that
$A$ has the strong Lefschetz property. In general, to check whether $A$ has
the strong Lefschetz property or not is difficult, and therefore to check for
which $F$ the algebra $R/\operatorname{Ann}_{R}(F)$ has the strong Lefschetz
property is also difficult.

There are related studies on this question in terms of the Hessian of $F$. By
Maeno-Watanabe [9] the multiplication map $\times
L^{c-2}:(R/\operatorname{Ann}_{R}(F))_{1}\to(R/\operatorname{Ann}_{R}(F))_{c-1}$
is bijective for some $L\in R_{1}$ if and only if the Hessian of $F$ is not
identically zero. Moreover, $R/\operatorname{Ann}_{R}(F)$ has the strong
Lefschetz property if and only if every ‘higher Hessian’ of $F$ is not
identically zero [9, Theorem 3.1].

Using this criterion Maeno-Numata [7] proved the strong Lefschetz property
when $F$ is the basis generating function of a matroid if the lattice of flats
of the matroid is modular geometric. They also conjectured the strong
Lefschetz property for any matroid.

Nagaoka-Yazawa [12, 13] proved the bijectivity of $\times
L^{c-2}:(R/\operatorname{Ann}_{R}(F))_{1}\to(R/\operatorname{Ann}_{R}(F))_{c-1}$
for some $L\in R_{1}$ when $F$ is the Kirchhoff polynomial of any simple
graph. In particular, if $F$ is the Kirchhoff polynomial of a complete graph,
then $F$ is the determinant of a symmetric matrix, which is the case of $({\rm
C}_{n},n)$ in Table 1. For this reason this paper is motivated by their work.

Murai-Nagaoka-Yazawa [11] generalized this result to the case when $F$ is the
basis generating function of any matroid (the basis generating function of a
graphic matroid is same as the Kirchhoff polynomial of a graph).

From a viewpoint of prehomogeneous vector spaces the Hessian of a relative
invariant is not identically zero if and only if the prehomogeneous vector
spaces is regular (Sato-Kimura [16, Definition 7]).

As another approach to this question, Gondim-Russo [3], Gondim [2] and Gondim-
Zappalà [4, 5] study polynomials whose Hessians are identically zero.

In this paper we give a new family of polynomials $F$ such that
$R/\operatorname{Ann}_{R}(F)$ has the strong Lefschetz property. The family
consists of the relative invariants of regular prehomogeneous vector spaces of
commutative parabolic type (see Definition 3). This family contains
determinants, determinants of symmetric matrices, Pfaffians of alternating
matrices of even size, $x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}$ and a polynomial
of degree three in 27 variables. The family also contains powers of the above
polynomials.

This paper is organized as follows. In Section 2 we review the definition of
prehomogeneous vector spaces of commutative parabolic type. In Section 3 we
state our main theorem and prove it in Section 4.

Acknowledgment. The authors thank Masatoshi Kitagawa for a critical comment on
the proof made in a seminar given by the first author, which is at Waseda
University in December, 2019. The authors also thank Satoshi Murai for advice
on how we write the paper. This work was supported by JSPS KAKENHI Grant
Numbers 22K03347, 20K03508.

##  2\. Prehomogeneous vector spaces of commutative parabolic type

In this section we review the definition of prehomogeneous vector spaces of
commutative parabolic type. Our main theorem shows that their relative
invariants $F$ give Artinian Gorenstein algebras $R/\operatorname{Ann}_{R}(F)$
that have the strong Lefschetz property.

######  Definition 2 (Prehomogeneous vector space).

For simplicity we define prehomogeneous vector spaces over the complex number
field.

Let $G$ be a complex Lie group, and $(G,\pi,V)$ a representation of $G$ on a
$\mathbb{C}$-vector space $V$. $(G,\pi,V)$ is called a prehomogeneous vector
space if there exists a Zariski open $G$-orbit on $V$.

A polynomial function $F$ on $V$ is called a relative invariant of $(G,\pi,V)$
if there exists a group character $\chi:G\to\mathbb{C}^{\times}$ such that
$F(gv)=\chi(g)F(v)$ ($g\in G$, $v\in V$).

######  Definition 3 (Prehomogeneous vector space of commutative parabolic
type).

Let $\mathfrak{g}$ be a complex simple Lie algebra, $\mathfrak{p}$ its
parabolic subalgebra, and $\mathfrak{n}^{+}$ the nilpotent radical of
$\mathfrak{p}$. Let $\mathfrak{k}$ be a Levi subalgebra of $\mathfrak{p}$,
which is, by definition, a subalgebra that is a complement to
$\mathfrak{n}^{+}$ in $\mathfrak{p}$, and $K$ the complex Lie subgroup of $G$
corresponding to $\mathfrak{k}$. Let $\mathfrak{h}$ be a Cartan subalgebra of
$\mathfrak{k}$, which is, by definition, a maximal commutative subalgebra of
$\mathfrak{k}$. $\mathfrak{h}$ is also a Cartan subalgebra of $\mathfrak{g}$.

Then it is known that
$(K,\operatorname{Ad},\mathfrak{n}^{+}/[\mathfrak{n}^{+},\mathfrak{n}^{+}])$
is a prehomogeneous vector space [MR0506488], where
$\operatorname{Ad}:K\to\operatorname{Aut}(\mathfrak{n}^{+})$ is the adjoint
action. This is called a prehomogeneous vector space of parabolic type. In
particular, when $\mathfrak{p}$ is a maximal parabolic subalgebra, and
therefore, $\mathfrak{n}^{+}$ is a commutative subalgebra,
$(K,\operatorname{Ad},\mathfrak{n}^{+})$ is called a prehomogeneous vector
space of commutative parabolic type (see also [14, Corollaire 4.1.11]).

Prehomogeneous vector spaces of commutative parabolic type are classified as
in Table 1. Type in Table 1 means the pair of the type of the Lie algebra
$\mathfrak{g}$ and the index of the simple root that characterizes the maximal
parabolic subalgebra $\mathfrak{p}$.

> Dynkin diagrams are used for the classification of complex simple Lie
> algebras, and there are types from A to G. For example, the complex simple
> Lie algebra of type Am is $\mathfrak{sl}_{m+1}$, and it is the semisimple
> part of $\mathfrak{g}$. White nodes in the above Dynkin diagrams specify
> maximal parabolic subalgebra $\mathfrak{p}$. The semisimple part of the Levi
> subalgebra $\mathfrak{k}$ of $\mathfrak{p}$ corresponds to the Dynkin
> diagram obtained by removing the white node and its adjacent edges from the
> Dynkin diagram of $\mathfrak{g}$.

Table 1. Prehomogeneous vector spaces of commutative parabolic type

Among them relative invariants exist only for types $({\rm A}_{2n-1},n)$,
$({\rm B}_{m},1)$, $({\rm C}_{n},n)$, $({\rm D}_{m},1)$, $({\rm D}_{2m},2m)$
and $({\rm E}_{7},7)$, which are called regular prehomogeneous vector spaces
of commutative parabolic type . In each type there exists an irreducible
relative invariant, which is called a basic relative invariant. The other
relative invariants are powers of the basic relative invariant. We list below
the basic relative invariant for each type. In the list, Lie algebras, Lie
groups and spaces of matrices are defined over $\mathbb{C}$.

#### $({\rm A}_{2n-1},n)$

$\mathfrak{g}=\mathfrak{gl}_{2n}$, $K\simeq GL_{n}\times GL_{n}$ and
$\mathfrak{n}^{+}\simeq\operatorname{Mat}_{n}$. The action of $K$ on
$\mathfrak{n}^{+}$ is given by

$\displaystyle(g_{1},g_{2}).X=g_{1}Xg_{2}^{-1}\qquad((g_{1},g_{2})\in
GL_{n}\times GL_{n},\;X\in\operatorname{Mat}_{n}),$

and the basic relative invariant is $f(X)=\det X$.

####  $({\rm B}_{m},1)$ and $({\rm D}_{m},1)$

For type $({\rm B}_{m},1)$, $\mathfrak{g}$ is the complex orthogonal Lie
algebra $\mathfrak{o}_{2m+1}$, $K\simeq GL_{1}\times O_{2m}$ and
$\mathfrak{n}^{+}\simeq\mathbb{C}^{2m}$. For type $({\rm D}_{m},1)$,
$\mathfrak{g}=\mathfrak{o}_{2m}$, $K\simeq GL_{1}\times O_{2m-1}$ and
$\mathfrak{n}^{+}\simeq\mathbb{C}^{2m-1}$. Set $n=2m$ or $2m-1$ as the
dimension of $\mathfrak{n}^{+}$, then the action of $K$ on $\mathfrak{n}^{+}$
is given by

$\displaystyle(a,g).v=agv\qquad((a,g)\in GL_{1}\times
O_{n},\;v\in\mathbb{C}^{n}),$

and the basic relative invariant is $f=x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}$,
where $x_{1},\ldots,x_{n}$ are linear coordinate functions on
$\mathbb{C}^{n}$.

#### $({\rm C}_{n},n)$

$\mathfrak{g}$ is the complex symplectic Lie algebra $\mathfrak{sp}_{2n}$ of
size $2n$, $K\simeq GL_{n}$ and
$\mathfrak{n}^{+}\simeq\operatorname{Sym}_{n}$, which is the space of
symmetric matrices of size $n$. The action of $K$ on $\mathfrak{n}^{+}$ is
given by

$\displaystyle g.X=gX\,{}^{t}\\!g\qquad(g\in
GL_{n},\;X\in\operatorname{Sym}_{n}),$

and the basic relative invariant is $f(X)=\det X$.

#### $({\rm D}_{2m},2m)$

$\mathfrak{g}=\mathfrak{o}_{4m}$, $K\simeq GL_{2m}$ and
$\mathfrak{n}^{+}\simeq\operatorname{Alt}_{2m}$, which is the space of
alternating matrices of size $2m$. The action of $K$ on $\mathfrak{n}^{+}$ is
given by

$\displaystyle g.X=gX\,{}^{t}\\!g\qquad(g\in
GL_{2m},\;X\in\operatorname{Alt}_{2m}),$

and the basic relative invariant is the Pfaffian $f(X)=\operatorname{Pf}X$.

#### $({\rm E}_{7},7)$

In this type $K$ is isomorphic to the product of the group of type E6 and the
group $GL_{1}$. The basic relative invariant is a homogeneous polynomial of
degree three on 27-dimensional vector space $\mathfrak{n}^{+}$. We omit the
details (see [16, Example 39]).

If $(K,\operatorname{Ad},\mathfrak{n}^{+})$ is a regular prehomogeneous vector
space of commutative parabolic type, then its contragredient representation
$(K,\operatorname{Ad},\mathfrak{n}^{-})$ with respect to the Killing form is
also a regular prehomogeneous vector space of commutative parabolic type, and
there is a basic relative invariant in $\mathbb{C}[\mathfrak{n}^{-}]$, where
$\mathfrak{n}^{-}\;(\subset\mathfrak{g})$ is the opposite Lie algebra of
$\mathfrak{n}^{+}$. Take type $({\rm C}_{n},n)$ for example, the action of
$(K,\operatorname{Ad},\mathfrak{n}^{-})$ is $g.X=\,{}^{t}\\!g^{-1}Xg^{-1}$
($g\in GL_{n}$, $X\in\operatorname{Sym}_{n}$), and the basic relative
invariant $\overline{f}\in\mathbb{C}[\mathfrak{n}^{-}]$ is
$\overline{f}(X)=\det X$.

In our main theorem (Theorem 4), we deal with regular prehomogeneous vector
spaces of commutative parabolic type described above, and take the basic
relative invariant $\overline{f}\in Q=\mathbb{C}[\mathfrak{n}^{-}]$. The ring
$R$ of differential operators with constant coefficients, which acts on $Q$ by
differentiation, is identified with $\mathbb{C}[\mathfrak{n}^{+}]$ via the
Killing form. We prove the strong Lefschetz property of
$R/\operatorname{Ann}_{R}(\overline{f}^{s})$ for any positive integer $s$. See
the next section for details.

##  3\. Main theorem

Set $Q=\mathbb{C}[\mathfrak{n}^{-}]$. For $f\in\mathbb{C}[\mathfrak{n}^{+}]$
we define a constant-coefficient differential operator $f(\partial)$ on
$\mathfrak{n}^{-}$ by

(1) $\displaystyle
f(\partial)\exp(x,y)=f(x)\exp(x,y)\qquad(x\in\mathfrak{n}^{+},\;y\in\mathfrak{n}^{-}),$

where $(x,y)$ denotes the Killing form on $\mathfrak{g}$. By this
identification we regard $\mathbb{C}[\mathfrak{n}^{+}]$ as the ring of partial
differential operators with constant coefficients on $\mathfrak{n}^{-}$, and
set $R=\mathbb{C}[\mathfrak{n}^{+}]$. The homogeneous component $R_{1}$ of
degree one can be identified with $\mathfrak{n}^{-}$.

We have our main theorem, which is proved in Section 4.

######  Theorem 4.

Let $\overline{f}\in\mathbb{C}[\mathfrak{n}^{-}]$ be the basic relative
invariant (see Section 2) of $(K,\operatorname{Ad},\mathfrak{n}^{-})$ which is
a regular prehomogeneous vector space of commutative parabolic type. Set
$R=\mathbb{C}[\mathfrak{n}^{+}]$, and $F=\overline{f}^{s}$ for a positive
integer $s$. Then the Artinian Gorenstein algebra
$R/\operatorname{Ann}_{R}(F)$ generated by $F$ has the strong Lefschetz
property.

Moreover, $L\in R_{1}\simeq\mathfrak{n}^{-}$ is a Lefschetz element if and
only if $L$ is in the open $K$-orbit of the prehomogeneous vector space
$(K,\operatorname{Ad},\mathfrak{n}^{-})$ independent of $s$.

######  Example 5 (Type (C${}_{n},n$)).

We see an example of type $({\rm C_{n}},n)$ in Table 1. Set

$\displaystyle\mathfrak{g}$
$\displaystyle=\mathfrak{sp}_{2n}=\left\\{\begin{pmatrix}A&B\\\
C&-\,{}^{t}\\!A\end{pmatrix}\;\Big{|}\;A\in\mathfrak{gl}_{n},\;B,C\in\operatorname{Sym}_{n}\right\\},$
$\displaystyle\mathfrak{p}$ $\displaystyle=\left\\{\begin{pmatrix}A&B\\\
0&-\,{}^{t}\\!A\end{pmatrix}\;\Big{|}\;A\in\mathfrak{gl}_{n},\;B\in\operatorname{Sym}_{n}\right\\},$
$\displaystyle\mathfrak{k}$ $\displaystyle=\left\\{\begin{pmatrix}A&0\\\
0&-\,{}^{t}\\!A\end{pmatrix}\;\Big{|}\;A\in\mathfrak{gl}_{n},\right\\}\simeq\mathfrak{gl}_{n},$
$\displaystyle\mathfrak{n}^{+}$ $\displaystyle=\left\\{\begin{pmatrix}0&B\\\
0&0\end{pmatrix}\;\Big{|}\;B\in\operatorname{Sym}_{n}\right\\}\simeq\operatorname{Sym}_{n},$
$\displaystyle\mathfrak{n}^{-}$ $\displaystyle=\left\\{\begin{pmatrix}0&0\\\
C&0\end{pmatrix}\;\Big{|}\;C\in\operatorname{Sym}_{n}\right\\}\simeq\operatorname{Sym}_{n},$

then $\mathfrak{p}$ is a maximal parabolic subalgebra of $\mathfrak{g}$,
$\mathfrak{k}$ is a Levi subalgebra of $\mathfrak{p}$, and $\mathfrak{n}^{+}$
is the nilpotent radical of $\mathfrak{p}$. The complex Lie group
corresponding to $\mathfrak{k}$ is

$\displaystyle K$ $\displaystyle=\left\\{\begin{pmatrix}g&0\\\
0&\,{}^{t}\\!g^{-1}\end{pmatrix}\;\Big{|}\;g\in GL_{n},\right\\}\simeq
GL_{n}.$

The adjoint actions of $K$ on $\mathfrak{n}^{+}$ and $\mathfrak{n}^{-}$ are
given by

$\displaystyle\begin{pmatrix}g&0\\\
0&\,{}^{t}\\!g^{-1}\end{pmatrix}\begin{pmatrix}0&B\\\
0&0\end{pmatrix}\begin{pmatrix}g&0\\\ 0&\,{}^{t}\\!g^{-1}\end{pmatrix}^{-1}$
$\displaystyle=\begin{pmatrix}0&gB\,{}^{t}\\!g\\\ 0&0\end{pmatrix},$
$\displaystyle\begin{pmatrix}g&0\\\
0&\,{}^{t}\\!g^{-1}\end{pmatrix}\begin{pmatrix}0&0\\\
C&0\end{pmatrix}\begin{pmatrix}g&0\\\ 0&\,{}^{t}\\!g^{-1}\end{pmatrix}^{-1}$
$\displaystyle=\begin{pmatrix}0&0\\\ \,{}^{t}\\!g^{-1}Cg^{-1}&0\end{pmatrix}.$

Thus the action of the prehomogeneous vector space
$(K,\operatorname{Ad},\mathfrak{n}^{+})$ (resp.
$(K,\operatorname{Ad},\mathfrak{n}^{-})$) is written simply as
$g.B=gB\,{}^{t}\\!g$ (resp. $g.C=\,{}^{t}\\!g^{-1}Cg^{-1}$) ($g\in GL_{n}$,
$B,C\in\operatorname{Sym}_{n}$) as already seen in Section 2.

Since $\overline{f}=\det C\in\mathbb{C}[\mathfrak{n}^{-}]$ is just multiplied
by $\det g^{-2}$ under the action $g.C=\,{}^{t}\\!g^{-1}Cg^{-1}$,
$\overline{f}$ is a relative invariant. Moreover, $\overline{f}$ is the basic
invariant, since it is an irreducible polynomial.

Set $R=\mathbb{C}[\mathfrak{n}^{+}]$. Then Theorem 4 says that
$R/\operatorname{Ann}_{R}(\overline{f}^{s})$ has the strong Lefschetz property
for any positive integer $s$. A Lefschetz element $L\in\mathfrak{n}^{-}$ is
any symmetric matrix of rank $n$, since the open orbit of
$(K,\operatorname{Ad},\mathfrak{n}^{-})$ is equal to the set of the matrices
of full rank.

######  Remark 6 (The set of Lefschetz elements).

Although Theorem 4 gives the set of Lefschetz elements completely, to
determine it is very difficult in general, and there are only a few such
examples.

The simplest example of such graded Artinian algebras that has the strong
Lefschetz property is a monomial complete intersection
$\mathbb{C}[x_{1},x_{2},\ldots,x_{n}]/\langle
x_{1}^{a_{1}},x_{2}^{a_{2}},\ldots,x_{n}^{a_{n}}\rangle$. Another known
example is the case of coinvariant rings of Weyl groups and real reflection
groups (Maeno-Numata-Wachi [8]).

In these two known cases the complement of the set of Lefschetz elements is a
union of hyperplanes, but in our main theorem the complement is a union of
hypersurfaces.

##  4\. Proof of the main theorem

In the rest of this paper we prove Theorem 4. We do not use Hessians (see
Introduction for Hessians), but use the theory of generalized Verma modules of
Lie algebras. In this section we use the notation of Definition 3, and suppose
that $(K,\operatorname{Ad},\mathfrak{n}^{+})$ is regular.

###  4.1. $\operatorname{ad}(\mathfrak{k})$-module structure of
$\mathbb{C}[\mathfrak{n}^{+}]$

######  Definition 7 (strongly orthogonal roots).

Let $\Delta$ be the root system of $(\mathfrak{g},\mathfrak{h})$. Two roots
$\alpha,\beta\in\Delta$ are said to be strongly orthogonal if $\alpha$ is not
proportional to $\beta$, and neither $\alpha+\beta$ nor $\alpha-\beta$ belongs
to $\Delta$.

If $\alpha$ and $\beta$ are strongly orthogonal, then $\alpha$ is orthogonal
to $\beta$, since $(\alpha,\beta)<0$ implies $\alpha-\beta\in\Delta$.

Let $\alpha_{i_{0}}$ be the simple root that characterizes the maximal
parabolic subalgebra $\mathfrak{p}$. Namely, $i_{0}$ is the index of the white
circle in Table 1. Let $\Delta_{N}^{+}$ be the set of roots corresponding to
$\mathfrak{n}^{+}$. We take a sequence
$\gamma_{1},\gamma_{2},\ldots,\gamma_{r}$ of mutually strongly orthogonal
roots in $\Delta_{N}^{+}$ as follows. Set $\gamma_{1}=\alpha_{i_{0}}$. When we
have defined $\gamma_{1},\ldots,\gamma_{i}$, let
$\gamma_{i+1}\in\Delta_{N}^{+}$ be the lowest root that is strongly orthogonal
to all $\gamma_{1},\ldots,\gamma_{i}$ if there exists such a root.

Set $\lambda_{i}=-(\gamma_{1}+\gamma_{2}+\cdots+\gamma_{i})$
($i=1,2,\ldots,r$). $\lambda_{i}$ is an integral weight of $\mathfrak{g}$, and
it can be also considered as that of $\mathfrak{k}$, since we can take a
common Cartan subalgebra $\mathfrak{h}$ for $\mathfrak{g}$ and $\mathfrak{k}$.
Then we have the structure theorem of $\mathbb{C}[\mathfrak{n}^{+}]$ as
follows.

######  Lemma 8 (Decomposition of $\mathbb{C}[\mathfrak{n}^{+}]$ as an
$\operatorname{ad}(\mathfrak{k})$-module, Schmid [17]).

$(\mathfrak{k},\operatorname{ad},\mathbb{C}[\mathfrak{n}^{+}])$ decomposes
multiplicity-freely into simple modules as follows.

$\displaystyle\mathbb{C}[\mathfrak{n}^{+}]=\bigoplus_{k_{1},\ldots,k_{r}\geq
0}V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}},$

where $V_{\lambda}$ denotes the finite-dimensional simple
$\operatorname{ad}(\mathfrak{k})$-module of highest weight $\lambda$. ∎

It is known that there exist homogeneous polynomials $f_{i}$ of degree $i$
($1\leq i\leq r$), and $V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ contains
$f_{1}^{k_{1}}f_{2}^{k_{2}}\cdots f_{r}^{k_{r}}$ ($k_{i}\geq 0$) as a highest
weight vector.

Moreover, $f_{r}$ in a one-dimensional vector space $V_{\lambda_{r}}$ is the
basic relative invariant of the regular prehomogeneous vector space
$(K,\operatorname{Ad},\mathfrak{n}^{+})$ [18, Lemma 6.4]. Other relative
invariants $f_{r}^{k_{r}}$ ($k_{r}\geq 1$) are highest weight vectors of one-
dimensional vector spaces $V_{k_{r}\lambda_{r}}$.

Thus $V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ consists of homogeneous
polynomial of degree $k_{1}+2k_{2}+\cdots+rk_{r}$.

######  Example 9 (Type (C${}_{n},n$)).

See Example 5 for the notation. For Type (C${}_{n},n$), $r$ is equal to $n$.
Strongly orthogonal roots in $\Delta_{N}^{+}$ are $\gamma_{1}=(0,\ldots,0,2)$,
$\gamma_{2}=(0,\ldots,0,2,0),\ldots$, $\gamma_{n}=(2,0,\ldots,0)$, and
integral weights $\lambda_{i}$’s are $\lambda_{1}=(0,\ldots,0,-2)$,
$\lambda_{2}=(0,\ldots,0,-2,-2),\ldots$, $\lambda_{n}=(-2,\ldots,-2,-2)$.

The decomposition into simple $\operatorname{ad}(\mathfrak{gl}_{n})$-modules
is as follows:

$\mathbb{C}[\mathfrak{n}^{+}]=\mathbb{C}[x_{ij}\mid 1\leq i\leq j\leq
n]=\bigoplus_{k_{1},\ldots,k_{n}\geq
0}V_{k_{1}\lambda_{1}+\cdots+k_{n}\lambda_{n}}\\\ =\bigoplus_{0\leq
l_{1}\leq\cdots\leq
l_{n}}V_{(-2l_{1},-2l_{2},\ldots,-2l_{n})}=\bigoplus_{k_{1},\ldots,k_{n}\geq
0}\langle f_{1}^{k_{1}}f_{2}^{k_{2}}\cdots
f_{n}^{k_{n}}\rangle_{\operatorname{ad}(\mathfrak{gl}_{n})},$

where $x_{ij}$ denotes the linear coordinate function on $\mathfrak{n}^{+}$,
and $f_{t}=\det(x_{ij})_{n-t+1\leq i,j\leq n}$. We use the convention
$x_{ij}=x_{ji}$ for $i>j$, and $\langle
f\rangle_{\operatorname{ad}(\mathfrak{gl}_{n})}$ is the
$\operatorname{ad}(\mathfrak{gl}_{n})$-submodule of
$\mathbb{C}[\mathfrak{n}^{+}]$ generated by
$f\in\mathbb{C}[\mathfrak{n}^{+}]$.

In the above decomposition $f_{1}^{k_{1}}f_{2}^{k_{2}}\cdots f_{n}^{k_{n}}$ is
the highest weight vector in $V_{k_{1}\lambda_{1}+\cdots+k_{n}\lambda_{n}}$.
$f_{n}$ is the basic relative invariant of the prehomogeneous vector space
$(K,\operatorname{Ad},\mathfrak{n}^{+})=(GL_{n},\operatorname{Ad},\operatorname{Sym}_{n})$.

The $\mathfrak{k}$-module
$(\mathfrak{k},\operatorname{ad},\mathbb{C}[\mathfrak{n}^{-}])$ is dual to
$(\mathfrak{k},\operatorname{ad},\mathbb{C}[\mathfrak{n}^{+}])$, and has a
decomposition into simple $\operatorname{ad}(\mathfrak{k})$-modules similar to
Lemma 8:

$\displaystyle\mathbb{C}[\mathfrak{n}^{-}]=\bigoplus_{k_{1},\ldots,k_{r}\geq
0}V_{k_{1}\overline{\lambda}_{1}+\cdots+k_{r}\overline{\lambda}_{r}},$

where $\overline{\lambda}_{i}=\gamma_{r-i+1}+\gamma_{r-i+1}+\cdots+\gamma_{r}$
is the highest weight of the contragredient representation of
$V_{\lambda_{i}}$. Let $\overline{f}_{i}\in\mathbb{C}[\mathfrak{n}^{-}]$ be a
highest weight vector of $V_{\overline{\lambda}_{i}}$. Then similarly to the
case of $(\mathfrak{k},\operatorname{ad},\mathbb{C}[\mathfrak{n}^{+}])$,
$\overline{f}_{1}^{k_{1}}\overline{f}_{2}^{k_{2}}\cdots\overline{f}_{r}^{k_{r}}\in\mathbb{C}[\mathfrak{n}^{-}]$
is a highest weight vector of $V_{k_{1}\overline{\lambda}_{1}+\cdots
k_{r}\overline{\lambda}_{r}}$.

###  4.2. $\operatorname{ad}(\mathfrak{k})$-module structure of
$R/\operatorname{Ann}_{R}(F)$

######  Proposition 10 (Decomposition of $R/\operatorname{Ann}_{R}(F)$ as an
$\operatorname{ad}(\mathfrak{k})$-module).

Set $R=\mathbb{C}[\mathfrak{n}^{+}]$. The annihilator of a relative invariant
$F=\overline{f}_{r}^{\,s}\in\mathbb{C}[\mathfrak{n}^{-}]$ of
$(K,\operatorname{Ad},\mathfrak{n}^{-})$ for a positive integer $s$ has the
following decomposition.

$\displaystyle\operatorname{Ann}_{R}(F)=\bigoplus_{\begin{subarray}{c}k_{1},\ldots,k_{r}\geq
0\\\
k_{1}+\cdots+k_{r}>s\end{subarray}}V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}.$

Therefore the decomposition of the Gorenstein algebra generated by $F$ is
given by

$\displaystyle
R/\operatorname{Ann}_{R}(F)\simeq\bigoplus_{\begin{subarray}{c}k_{1},\ldots,k_{r}\geq
0\\\ k_{1}+\cdots+k_{r}\leq
s\end{subarray}}V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}.$

###### Proof.

The second decomposition follows from the first one. We prove the first
decomposition.

Since $F$ is a relative invariant under the action of $\operatorname{Ad}(K)$,
$\operatorname{Ann}_{R}(F)\subset\mathbb{C}[\mathfrak{n}^{+}]$ is an
$\operatorname{ad}(\mathfrak{k})$-submodule, and is decomposed into a sum of
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$. Since
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ is irreducible,
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}\subset\operatorname{Ann}_{R}(F)$
if and only if a nonzero polynomial in
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ annihilates $F$ by using the
identification of $\mathbb{C}[\mathfrak{n}^{+}]$ with differential operators
(Equation (1)).

Consider the differentiation $(f_{1}^{k_{1}}f_{2}^{k_{2}}\cdots
f_{r}^{k_{r}})(\partial)\overline{f}_{r}^{\,s}$. We repeatedly use the formula
(see [15, Lemme 5.6], [18, Equation (10.3)])

(2) $\displaystyle
f_{r-i}(\partial)\overline{f}_{1}^{m_{1}}\overline{f}_{2}^{m_{2}}\cdots\overline{f}_{i}^{m_{i}}\overline{f}_{r}^{m+1}=b_{r-i}(m)\overline{f}_{1}^{m_{1}}\overline{f}_{2}^{m_{2}}\cdots\overline{f}_{i-1}^{m_{i-1}}\overline{f}_{i}^{m_{i}+1}\overline{f}_{r}^{m}$

(up to nonzero scaling) for non-negative integers $m_{1},\ldots,m_{i}$ and an
integer $m$, where the polynomial $b_{r-i}(m)$ in $m$ is the $b$-function of
$\overline{f}_{r-i}$, and we have

$\displaystyle(f_{1}^{k_{1}}f_{2}^{k_{2}}\cdots
f_{r}^{k_{r}})(\partial)\overline{f}_{r}^{s}=B(s)\overline{f}_{1}^{k_{r-1}}\overline{f}_{2}^{k_{r-2}}\cdots\overline{f}_{r-1}^{k_{1}}\overline{f}_{r}^{s-(k_{1}+\cdots+k_{r})},$

where a polynomial $B(s)$ in $s$ is a product of $b$-functions of
$\overline{f}_{1},\overline{f}_{2},\ldots,\overline{f}_{r}$ ($b_{i}$ appears
$k_{i}$ times) evaluated at $s-k$ ($1\leq k\leq k_{1}+\cdots+k_{r}$). Since it
is known that zeros of $b$-functions are negative, and $m=-1$ is always a
(maximum integral) zero of any $b$-function, $B(s)$ is equal to zero if and
only if $s-k$ can be $-1$. Namely, $B(s)=0$ if and only if
$s-(k_{1}+\cdots+k_{r})<0$

Thus we have proved that
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}\subset\operatorname{Ann}_{R}(F)$
if and only if $k_{1}+k_{2}+\cdots+k_{r}>s$. ∎

######  Remark 11 (Generating set of $\operatorname{Ann}_{R}(F)$).

When $F=\overline{f}_{r}^{s}$ for a positive integer $s$,
$\operatorname{Ann}_{R}(F)$ is generated by $V_{(s+1)\lambda_{1}}$ as an ideal
of $R=\mathbb{C}[\mathfrak{n}^{+}]$.

Indeed, $\operatorname{Ann}_{R}(F)$ is generated by
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ with
$k_{1}+k_{2}+\cdots+k_{r}>s$ by Proposition 10, and this condition is weakened
to $k_{1}+k_{2}+\cdots+k_{r}=s+1$, since the highest weight vector of
$V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ is
$f_{1}^{k_{1}}f_{2}^{k_{2}}\cdots f_{r}^{k_{r}}$. By using Equation (2) we can
prove that

$\displaystyle\mathbb{C}[\mathfrak{n}^{+}]_{1}V_{k_{1}\lambda_{1}+\cdots+k_{i-2}\lambda_{i-2}+(k_{i-1}+1)\lambda_{i-1}+k_{i}\lambda_{i}}\supset
V_{k_{1}\lambda_{1}+\cdots+k_{i-1}\lambda_{i-1}+(k_{i}+1)\lambda_{i}}.$

For the proof of the above formula we need to consider ‘lower-rank version’ of
Equation (2), but we omit the details (see [18, Section 8], [10, Section 2]
for the ‘lower-rank’ setting). Therefore, by repeated use of the above
formula, it follows that
$\mathbb{C}[\mathfrak{n}^{+}]V_{(s+1)\lambda_{1}}\supset
V_{k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}}$ whenever
$k_{1}+k_{2}+\cdots+k_{r}=s+1$, and we have proved $\operatorname{Ann}_{R}(F)$
is generated by $V_{(s+1)\lambda_{1}}$ as an ideal of $R$.

Moreover, we can conclude that $\operatorname{Ann}_{R}(F)$ is generated by
$f_{1}^{s+1}$ as an $\operatorname{ad}(\mathfrak{k})$-stable ideal of $R$. For
example, in Type (C${}_{n},n$) (see Example 9 for the notation)
$\operatorname{Ann}_{R}(\overline{f}_{n}^{s})$ is generated by $x_{nn}^{s+1}$
as an $\operatorname{ad}(\mathfrak{gl}_{n})$-stable ideal of
$R=\mathbb{C}[\mathfrak{n}^{+}]=\mathbb{C}[\operatorname{Sym}_{n}]$.

######  Example 12 (Narayana numbers).

In Type (C${}_{n},n$) (see Example 9 for the notation), when we consider the
basic relative invariant $F=\overline{f}_{n}\in\mathbb{C}[\mathfrak{n}^{-}]$,
it follows from Proposition 10 that

$\displaystyle R/\operatorname{Ann}_{R}(\overline{f}_{n})$
$\displaystyle\simeq
V_{0}+V_{\lambda_{1}}+V_{\lambda_{2}}+\cdots+V_{\lambda_{n}}$
$\displaystyle=V_{(0,\ldots,0)}+V_{(0,\ldots,0,-2)}+V_{(0,\ldots,0,-2,-2)}+\cdots+V_{(-2,\ldots,-2)}.$

Since this decomposition coincides with homogeneous decomposition as a graded
algebra, we can compute the Hilbert function of
$R/\operatorname{Ann}_{R}(\overline{f}_{n})$, which is written as a sequence
of dimensions of homogeneous components in this paper, using the Weyl
dimension formula for irreducible representations of $\mathfrak{gl}_{n}$:

$\displaystyle\operatorname{Hilb}(R/\operatorname{Ann}_{R}(\overline{f}_{n}))$
$\displaystyle\quad=\left(\tfrac{1}{n+1}\tbinom{n+1}{1}\tbinom{n+1}{0},\;\tfrac{1}{n+1}\tbinom{n+1}{2}\tbinom{n+1}{1},\;\ldots,\tfrac{1}{n+1}\tbinom{n+1}{n+1}\tbinom{n+1}{n}\right).$

This sequence consists of Narayana numbers, which are originated in
combinatorics, and defined as $N(n,k)=\frac{1}{n}\binom{n}{k}\binom{n}{k-1}$.

###  4.3. Generalized Verma modules

Define $\mathfrak{g}$, $\mathfrak{p}$, $\mathfrak{k}$, $K$ and
$\mathfrak{n}^{+}$ as in Definition 3. In addition, let $\mathfrak{n}^{-}$ be
the opposite of $\mathfrak{n}^{+}$. Suppose
$(K,\operatorname{Ad},\mathfrak{n}^{+})$ is regular.

Let $(\mathfrak{p},\mu,\mathbb{C}_{\mu})$ be a one-dimensional representation
of $\mathfrak{p}$ ($\mathbb{C}_{\mu}=\mathbb{C}$), and set
$M(\mu)=U(\mathfrak{g})\otimes_{U(\mathfrak{p})}\mathbb{C}_{\mu}$, which is
called a generalized Verma module of $\mathfrak{g}$ induced from $\mu$. Since
$(\mathfrak{p},\mu,\mathbb{C}_{\mu})$ is one-dimensional, $\mu$ is a multiple
of $\varpi_{i_{0}}$, which is the fundamental weight of $\mathfrak{g}$
corresponding to the simple root $\alpha_{i_{0}}$ that characterizes the
maximal parabolic subalgebra $\mathfrak{p}$. In addition, $\lambda_{r}$ is
also a multiple of $\varpi_{i_{0}}$, since
$(\mathfrak{k},\operatorname{ad},V_{\lambda_{r}})$ is one-dimensional. Indeed,
$\lambda_{r}=-2\varpi_{i_{0}}$ [18, Lemma 6.4].

Then it follows from $\mathfrak{g}=\mathfrak{n}^{-}\oplus\mathfrak{p}$ that

$\displaystyle
M(\mu)=U(\mathfrak{g})\otimes_{U(\mathfrak{p})}\mathbb{C}_{\mu}=U(\mathfrak{n}^{-})U(\mathfrak{p})\otimes_{U(\mathfrak{p})}\mathbb{C}_{\mu}\simeq
U(\mathfrak{n}^{-})$

as $\mathbb{C}$-vector spaces. Since $\mathfrak{n}^{-}$ is a commutative Lie
algebra, we have $M(\mu)\simeq
S(\mathfrak{n}^{-})\simeq\mathbb{C}[\mathfrak{n}^{+}]$ as vector spaces, where
$S(\mathfrak{n}^{-})$ is the symmetric algebra of $\mathfrak{n}^{-}$, and the
second isomorphism is by $\mathfrak{n}^{-}\simeq(\mathfrak{n}^{+})^{*}$ via
the Killing form on $\mathfrak{g}$. Thus we have the action of $\mathfrak{g}$
on $\mathbb{C}[\mathfrak{n}^{+}]$, and we denote this representation by
$(\mathfrak{g},\psi_{\mu},\mathbb{C}[\mathfrak{n}^{+}])$.

The explicit form of the action of
$(\mathfrak{g},\psi_{\mu},\mathbb{C}[\mathfrak{n}^{+}])$ is given in the
following lemma. The action of $\mathfrak{n}^{+}$ is, in fact, a differential
operator of second order with polynomial coefficients, though we do not need
it, and omit the explicit form.

######  Lemma 13 (Actions of generalized Verma modules, [18, Lemma 3.2]).

The action of $\mathfrak{n}^{-}$ and $\mathfrak{k}$ under the representation
$(\mathfrak{g},\psi_{\mu},\mathbb{C}[\mathfrak{n}^{+}])$ is given as follows:

$\displaystyle\psi_{\mu}(X)$ $\displaystyle=\times X$
$\displaystyle(X\in\mathfrak{n}^{-}),$ $\displaystyle\psi_{\mu}(X)$
$\displaystyle=\operatorname{ad}(X)+\mu(X),$
$\displaystyle(X\in\mathfrak{k}),$

where $\times X$ denotes the multiplication map on
$\mathbb{C}[\mathfrak{n}^{+}]$ by a linear polynomial
$X\in\mathfrak{n}^{-}\simeq(\mathfrak{n}^{+})^{*}\simeq\mathbb{C}[\mathfrak{n}^{+}]_{1}$,
and $\mu(X)$ denotes the multiplication map by a scalar $\mu(X)$.

###### Proof.

Let $f\in\mathbb{C}[\mathfrak{n}^{+}]$. We regard $f$ as an element in
$S(\mathfrak{n}^{-})$, and $f\otimes 1_{\mu}\in M(\lambda)$, where
$1_{\mu}=1\in\mathbb{C}_{\mu}$ is the basis of $\mathbb{C}_{\mu}$.

If $X\in\mathfrak{n}^{-}$, then $X(f\otimes 1_{\mu})=Xf\otimes 1_{\mu}$, and
$Xf\in S(\mathfrak{n}^{-})$. Therefore $\psi_{\mu}(X)f=Xf$.

If $X\in\mathfrak{k}$, then

$\displaystyle X(f\otimes 1_{\mu})=Xf\otimes 1_{\mu}=(fX+[X,f])\otimes
1_{\mu}$ $\displaystyle=f\otimes\mu(X)1_{\mu}+\operatorname{ad}(X)f\otimes
1_{\mu}$ $\displaystyle=(\mu(X)f+\operatorname{ad}(X)f)\otimes 1_{\mu}.$

Therefore $\psi_{\mu}(X)f=\operatorname{ad}(X)f+\mu(X)f$. ∎

######  Remark 14 (Decomposition of $\mathbb{C}[\mathfrak{n}^{+}]$ as a
$\psi_{\mu}(\mathfrak{k})$-module).

The decomposition of $\mathbb{C}[\mathfrak{n}^{+}]$ into simple
$\psi_{\mu}(\mathfrak{k})$-modules coincides with that as
$\operatorname{ad}(\mathfrak{k})$-modules, since these two actions differ only
by the constant multiplication by $\mu(X)$. But highest weights of irreducible
components change by the constant $\mu(X)$.

###  4.4. Maximal submodules of $M(\mu)$

We continue to use the notation of the previous subsection.

It is known that there exists a unique maximal submodule of $M(\mu)$, and
denote the submodule by $Y_{\mu}$. If we regard $Y_{\mu}$ as an
$\operatorname{ad}(\mathfrak{k})$-submodule of $\mathbb{C}]\mathfrak{n}^{+}]$,
$Y_{\mu}$ must decompose into a sum of simple modules $V_{\lambda}$.

######  Lemma 15 (Maximal submodules of $M(\mu)$, [18, Lemma 10.3, Proposition
10.7]).

Let $\mu=s\varpi_{i_{0}}$ ($s\in\mathbb{C}$) be a one-dimensional
representation of $\mathfrak{p}$, and consider the generalized Verma module
$M(\mu)$. For $\lambda=k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}$,
$V_{\lambda}\subset Y_{\mu}$ if and only if $q_{\mu}(\lambda)=0$, where the
polynomial $q_{\mu}(\lambda)$ in $k_{1},k_{2},\ldots,k_{r}$ is defined as

$\displaystyle
q_{\mu}(\lambda)=\prod_{i=0}^{r-1}\prod_{l=0}^{k_{i+1}+\cdots+k_{r}-1}(\frac{id}{2}+s-l),$

where $d=1,2,4,2m-3,2m-4$ for $({\rm C}_{n},n)$, $({\rm A}_{2n-1},n)$, $({\rm
E}_{7},7)$, $({\rm B}_{m},1)$, $({\rm D}_{m},1)$, respectively. ∎

######  Proposition 16 ($\operatorname{Ann}_{R}(F)$ is a submodule of
$M(\mu)$).

Let $s$ be a positive integer, and $\mu=s\varpi_{i_{0}}$. Then the maximal
submodule $Y_{\mu}$ of the generalized Verma module
$M(\mu)\simeq(\mathfrak{g},\psi_{\mu},\mathbb{C}[\mathfrak{n}^{+}])$ is equal
to
$\operatorname{Ann}_{R}(\overline{f}_{r}^{s})\subset\mathbb{C}[\mathfrak{n}^{+}]$.

Therefore $\mathfrak{g}$ acts on
$R/\operatorname{Ann}_{R}(\overline{f}_{r}^{s})$ via $\psi_{\mu}$.

###### Proof.

By Lemma 15 an irreducible component $V_{\lambda}$
($\lambda=k_{1}\lambda_{1}+\cdots+k_{r}\lambda_{r}$) of
$(\mathfrak{g},\psi_{\mu},\mathbb{C}[\mathfrak{n}^{+}])$ is contained in
$Y_{\mu}$ if and only if $id/2+s-l=0$ for some $i=0,1,\ldots,r-1$ and
$l=0,1,\ldots,k_{i+1}+k_{i+2}+\cdots+k_{r}-1$. This is equivalent to that the
minimum of $id/2-l$ is less than or equal to $-s$. $id/2$ takes the minimum
value when $i=0$, and $-l$ takes the minimum value when $i=0$ and
$l=k_{1}+k_{2}+\cdots+k_{r}-1$. Thus the minimum of $id/2-l$ is equal to
$-(k_{1}+k_{2}+\cdots+k_{r}-1)$. Therefore $V_{\lambda}\subset Y_{\mu}$ if and
only if $-(k_{1}+k_{2}+\cdots+k_{r}-1)\leq-s$, that is,
$k_{1}+k_{2}+\cdots+k_{r}>s$.

On the other hand
$V_{\lambda}\subset\operatorname{Ann}_{R}(\overline{f}_{r}^{s})$ if and only
if $k_{1}+k_{2}+\cdots+k_{r}>s$ by Proposition 10. Thus we have proved
$\operatorname{Ann}_{R}(\overline{f}_{r}^{s})=Y_{\mu}$. ∎

###  4.5. Proof of the main theorem

The following lemma is essentially the same as [6, Theorem 3.32], where the
multiplication map by $L$ corresponds to $X\in\mathfrak{sl}_{2}$, but $\times
L$ corresponds to $Y\in\mathfrak{sl}_{2}$ in our lemma for the proof of
Theorem 4.

######  Lemma 17 (Condition for the strong Lefschetz property).

Let $I$ be a homogeneous ideal of $R=\mathbb{C}[x_{1},x_{2},\ldots,x_{n}]$.
Let $\mathfrak{sl}_{2}=\mathbb{C}X+\mathbb{C}Y+\mathbb{C}H$ be the complex
simple Lie algebra, where $[X,Y]=H$, $[H,X]=2X$, and $[H,Y]=-2Y$.

When $A=R/I$ is a graded Artinian algebra with a symmetric Hilbert function,
the following two conditions are equivalent:

  1. (1)

$A=R/I$ has the strong Lefschetz property, and $L\in A_{1}$ is a Lefschetz
element.

  2. (2)

There exists an action of $\mathfrak{sl}_{2}$ on $A$ such that

    1. (a)

The weight space decomposition of $A$ coincides with the homogeneous
decomposition of $A$, and

    2. (b)

The action of $Y\in\mathfrak{sl}_{2}$ on $A$ coincides with the multiplication
map by $L\in A_{1}$. ∎

Set $R=\mathbb{C}[\mathfrak{n}^{+}]$, and
$F=\overline{f}_{r}^{s}\in\mathbb{C}[\mathfrak{n}^{-}]$, where $s$ is a
positive integer. Set $\mu=s\varpi_{i_{0}}$ so that $\mathfrak{g}$ acts on
$R/\operatorname{Ann}_{R}(F)$ through $\psi_{\mu}$ (Proposition 16). To prove
the strong Lefschetz property of $R/\operatorname{Ann}_{R}(F)$, we will take
an $\mathfrak{sl}_{2}$-triple $X,Y,H\in\mathfrak{g}$ so that the action of
$\mathfrak{sl}_{2}$ on $R/\operatorname{Ann}_{R}(F)$ via $\psi_{\mu}$
satisfies the condition (2) of Lemma 17.

First, $H\in\mathfrak{h}$ is uniquely determined. Indeed, by the condition (2)
(a) of Lemma 17, the action of $\operatorname{ad}(H)$ on
$\mathfrak{n}^{-}\;(\simeq\mathbb{C}[\mathfrak{n}^{+}]_{1})$ should be the
multiplication by $-2$, since the eigenspaces of
$\psi_{\mu}(H)=\operatorname{ad}(H)+\mu(H)$ on $R/\operatorname{Ann}_{R}(F)$
coincide with the homogeneous spaces of $R/\operatorname{Ann}_{R}(F)$, and the
eigenvalues must decrease by two when the degrees of homogeneous spaces
increase by one. Therefore $H$ is the unique element in the one-dimensional
center of $\mathfrak{k}$, which is contained in the Cartan subalgebra
$\mathfrak{h}$, satisfying

$\displaystyle\alpha_{i_{0}}(H)=2,$ $\displaystyle\alpha(H)=0\quad(\text{for
any simple root $\alpha$ other than $\alpha_{i_{0}}$}).$

The existence of $X\in\mathfrak{n}^{+}$ and $Y\in\mathfrak{n}^{-}$ such that
$[X,Y]=H$ is classified using weighted Dynkin diagrams (the Dynkin-Kostant
classification, [1, Theorem 3.5.4]). But in our setting we can find $X$ and
$Y$ without argument about the classification. Namely, we can take $X$ and $Y$
as

$\displaystyle
X=X_{\gamma_{1}}+X_{\gamma_{2}}+\cdots+X_{\gamma_{r}}\in\mathfrak{n}^{+},\qquad
Y=Y_{\gamma_{1}}+Y_{\gamma_{2}}+\cdots+Y_{\gamma_{r}}\in\mathfrak{n}^{-},$

where $X_{\gamma_{i}}\in\mathfrak{n}^{+}$ and
$Y_{\gamma_{i}}\in\mathfrak{n}^{-}$ are root vectors corresponding to
$\pm\gamma_{i}$ such that $X,Y,H$ forms an $\mathfrak{sl}_{2}$-triple.

Then the action of $\psi_{\mu}(Y)$ on $R/\operatorname{Ann}_{R}(F)$ is the
multiplication by $Y\in\mathfrak{n}^{-}\simeq\mathbb{C}[\mathfrak{n}^{+}]_{1}$
(Lemma 13), and the weight space decomposition with respect to $\psi_{\mu}(H)$
on $R/\operatorname{Ann}_{R}(F)$ coincides with the homogeneous decomposition.
Therefore it follows from Lemma 17 that $R/\operatorname{Ann}_{R}(F)$ has the
strong Lefschetz property, and we have proved the first paragraph of Theorem
4.

A linear coordinate change by the action of $\operatorname{Ad}(K)$ on
$\mathbb{C}[\mathfrak{n}^{+}]$ causes an automorphism on
$\mathbb{C}[\mathfrak{n}^{+}]/\operatorname{Ann}_{R}(F)$, and clearly
preserves the strong Lefschetz property. Therefore every element of
$\operatorname{Ad}(K)$-orbit on $\mathfrak{n}^{-}$ through
$Y\in\mathfrak{n}^{-}$ is a Lefschetz element. Conversely, representatives of
orbits of lower dimensions are
$Y_{\gamma_{1}}+Y_{\gamma_{2}}+\cdots+Y_{\gamma_{i}}\in\mathfrak{n}^{-}$
($0\leq i<r$) [10, Théorème 2.8], and this element never gives an
$\mathfrak{sl}_{2}$-triple containing $H$. This means that the set of
Lefschetz element is equal to the open $\operatorname{Ad}(K)$-orbit on
$\mathfrak{n}^{-}$.

######  Example 18 (Type (C${}_{n},n$)).

In the case of type $({\rm C}_{n},n)$ (see Example 9 for the notation),
$\mathfrak{sl}_{2}\subset\mathfrak{g}=\mathfrak{sp}_{2n}$ is given by

$\displaystyle H$ $\displaystyle=\begin{pmatrix}1_{n}&0\\\
0&-1_{n}\end{pmatrix},$ $\displaystyle X$
$\displaystyle=\begin{pmatrix}0&1_{n}\\\ 0&0\end{pmatrix},$ $\displaystyle Y$
$\displaystyle=\begin{pmatrix}0&0\\\ 1_{n}&0\end{pmatrix},$

where $1_{n}$ denotes the identity matrix of size $n$.

The set of Lefschetz elements is the set of full-rank matrices in
$\mathfrak{n}^{-}$, which is the $GL_{n}$-orbit through $Y$. In particular,
$Y\in\mathfrak{n}^{-}$ is a Lefschetz element, and, in a form of polynomial,
it is $x_{11}+x_{22}+\cdots+x_{nn}\in\mathbb{C}[\mathfrak{n}^{+}]$.

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