# The carbon cost of a PhD in solar cell materials

Rachel Woods-Robinson*† * University of California and Lawrence Berkeley
National Laboratory, Berkeley, CA, 94720 USA† National Renewable Energy
Laboratory, Golden, CO 80401, USA

###### Abstract

Scientific research can bring about important progress in developing renewable
energy technology such as solar cells. However, in addition to positive
impact, research also comes at a cost, and one such cost is the CO2 footprint
associated with doing research and producing technologies. Here, as an example
of such a cost, I estimate the associated greenhouse gas (GHG) emissions
associated with my PhD, in which I researched computational and experimental
design of thin film materials for photovoltaics. I consider supercomputer
power, synchrotron beam time, laboratory synthesis and characterization,
travel associated with research, and personal emissions over the course of my
PhD. This analysis demonstrates that doing science increases my carbon
footprint to much higher than an average American. Because our consumption is
so high, scientists have a moral imperative to reduce our own emissions in our
laboratories and in the technologies we create.

##  I Introduction

The intention of much of our research within the science and engineering
community is to develop and optimize materials necessary for our transition to
a renewable energy society, and ultimately to contribute to a reduction of CO2
emissions. However, this research comes at a cost: time, resources, energy,
and ultimately CO2 and other greenhouse gases (GHGs). As a scientist, I
believe it is important to assess how our actions and our research activities
impact society, and to acknowledge that research can have both positive and
negative impacts. With an awareness of our impact on society, we can then take
action and make changes to ensure that our science is done a way that
minimizes GHG footprint as well as minimizes other forms of harm. In order to
reach our renewable energy goals, scientists need to think critically about
how we are using energy and resources within the work we do.

In this analysis, I focus on a personal case study to quantify GHG emissions
due to my own research activities and contrast them with emissions of other
members of society. Specifically, I seek to estimate the GHG emissions
incurred by my PhD research studying computational and experimental materials
discovery of inorganic semiconductors for solar energy applications. I find
that the primary energy-intensive aspects of my research have been (1)
supercomputer time, (2) synchrotron beam time, (3) laboratory synthesis and
characterization, and (4) travel associated with research and conferences, and
I include an estimate of my personal GHG emissions for comparison. I also
estimate a few alternate scenarios as a way of contextualizing these
emissions, and compare these scenarios to my actual emissions by source.

I acknowledge that I have limited access to granular data associated with my
research and do not expect these results to be perfectly accurate; rather, my
intention is to quantify the order of magnitude of the GHG emissions and to
open a dialogue within the scientific community about how much energy and CO2
is used doing renewable energy research. Before getting into the analysis,
however, it must be acknowledged that Big Oil coined the term “carbon
footprint” and for decades has championed personal carbon accounting to shift
blame to individuals and evade changing hostile business practices.[1] I still
see such analysis as a useful tool for self-assessment and as a means to scale
up emissions across an individual industry, but this cannot blind-sight us
from the urgent need for massive systemic change.

##  II Methodology

In this analysis, various components of CO2 emissions related to my five-year
PhD are estimated as described in the following sections. Throughout this
discussion, there are two key metrics. The first is energy use. Most of this
is electricity, but there is also energy associated with heating and
transportation. When possible I will focus on energy consumption, rather than
production or capacity. I will report energy in units of kWh or MWh. Some of
this analysis uses explicitly quantified energy consumption metrics, and in
other areas this has been estimated. The second is associated GHG emissions.
Most of the emissions discussed here are from CO2, but other GHG include
methane (CH4) and nitrous oxide (N2), among others. The impact of other GHGs
is quantified in terms of units of CO2-eq., which is derived from their global
warming potential (GWP). The metrics of energy consumption and CO2-eq. are
connected by the CO2-eq. emissions factor for a given energy source or mix,
which is measured in units of mT CO2-eq. per MWh or g CO2-eq. per kWh.
Accurate accounting of carbon emissions associated with a given unit energy is
nontrivial; attribution depends on whether one is considering the physical
origin of the energy (“locational method”) or “market-based” power purchasing
agreements, among other factors.[2, 3] For the purposes of this simple
assessment, we will assume the market-based emission factors as reported in
Table I and Table II from various references,[4, 5, 6, 7, 8, 9, 10, 4, 11]
though we acknowledge that considering consumption-based economic metrics
alone can lead to misallocation of emissions.

##  III Results and Discussion

Table I: GHG emission factors of various electricity sources used in this
analysis.

Source | Emissions factor (g CO2-eq / kWh) | Ref.
---|---|---
Coal | 990–1,100 | [4]
Combined cycle gas power plant | 330–449 | [5]
Wind | 4–14 | [6]
PV | 5–50 | [6]
Ultra low-carbon PV | 5 | [7]
Nuclear power plant | 3.5 | [6]

Table II: GHG emission factors for the grid at a particular location.

Location | Emissions factor (g CO2-eq / kWh) | Assumptions | Ref.
---|---|---|---
NREL (Xcel CO) | 544 | 2019 levels | Xcel[8]
California grid | 220 | 2018 levels | CAISO[9, 10]
SLAC before Nov. 2019 | 1100 | 2011 Navajo coal plant | EPA[4]
Switzerland | 100 | 2009 estimate | Chevalley[11]

Figure 1: Estimate of GHG emissions over the course of my five-year PhD, as
approximated in the text.

###  III-A Research-related emissions

####  III-A1 Supercomputer emissions

The increase of computational capabilities have enabled a wide array of new
research frontiers. In addition to materials discovery, supercomputers can be
used for complex climate change simulations, and to solve other difficult
problems related to our sustainable development. For example, in my research I
use supercomputers and ab inition methods to simulate the structure and
properties of new contact materials for solar energy applications, and
increased computer power enables high-throughput capabilities so I can screen
through large quantities of materials.[12] However, this creates a new
dilemma. The more complex a simulation is, the more energy it uses, and
currently supercomputers and data serves are incredible energy intensive. If
our computers and data centers are not powered by renewable energy, using them
for research may come at a steep CO2 cost. Although many of us are working to
combat the climate crisis, rarely do we discuss our own carbon footprint from
computing.

The computational research in my PhD was performed using the supercomputer
Cori, which is one of the National Energy Research Scientific Computing Center
(NERSC) machines. My research at NERSC has used approximately 8 million CPU
hours over five years.[13] This is very likely at the low end for my
computational materials scientists, since I only spent about half of my
research time doing calculations. For 1,126 of the 1,527 jobs I ran in the
year 2020, 1.05 million CPU hours used a total of 2,431 kWh of electricity.
These jobs are mostly density functional theory calculations using VASP,
though I also run MCSQS and some bash scripts directly. This corresponds to
approximately 0.0024 kWh associated with each CPU hour for my research.
Assuming a similar energy use rate applies to all computational jobs I have
run during my PhD, this approximation allows me to estimate the energy use
associated with other years of computation time, and approximate that
8,169,603 CPU hours corresponds to $\sim$18.83 MWh over five years and
$\sim$3.766 MWh per year (less energy per CPU hour than the average NERSC
user). Assuming the emissions factor of the California grid from the
California Independent System Operator (CAISO),[9, 10] reported in Table II as
$\sim$0.362 mT CO2-eq./MWh,111This is an estimate from CAISO based on granular
data from the electricity grid.[9] this amounts to 6–8 mT CO2-eq. across my
five year PhD due to supercomputer time.

####  III-A2 Synchrotron beam time emissions

Experiments at synchrotron particle accelerators can enable scientists to
probe fundamental properties of materials, and in my research I used the
Stanford Synchrotron Radiation Lightsource (SSRL) to perform X-ray diffraction
and X-ray absorption on thin film materials for solar cells. To my knowledge,
SSRL does not share data about facility and beam line energy and GHG usage, so
here I will estimate emissions during my synchrotron experiments based on
macroscopic energy data and power-purchasing agreements. The synchrotron at
SSRL uses a $\sim$2.5 MW X-ray radiation source called SPEAR,222Energy
information about SPEAR is from personal correspondence with Dr. David Chassin
and Cezary Jach. and operates around the clock for 10.5 months/year (7,560
hr/year). This corresponds to a typical energy use of 2.5 MW $\times$ 7,560
hr/year = 18.9 MWh/year. The SSRL synchrotron has 30 stations and 14
beamlines.[14] I will assume as a user that I am responsible for energy
associated with an entire beam line, although it is sometimes shared during
operation so this estimate is an upper limit. During my PhD, I was granted and
was responsible for 790 hours of beam time, although I was only present at
about 50% of these beam times. Thus, assuming an error bar of $\pm$25% to
account for uncertainties in this estimate, I used approximately 50 MWh total
of synchrotron beam energy during my PhD.

Until November 18, 2019, SSRL’s power-purchasing agreement with the Western
Area Power Administration (WAPA) marketed power from the Navajo Generating
Station in Arizona. This was a coal plant (that has since been
decommissioned),[4] and since all of my beam time was performed before this
date, “economically” all of the energy I used came from coal. From Stanford’s
Western Interconnection Data Analytics Project (WIDAP) database,[15, 16]
emissions from this power plant were approximately 1.1 mT CO2/MWh in 2016–2018
(as well as 2 mT NO_$x$ and 1.6 mT SO_$x$). Thus, my 50 MWh energy consumption
corresponds to approximately 55 $\pm$ 14 mT CO2-eq. from synchrotron beam
time.333A more accurate estimate of synchrotron emissions would use real-time
location-based data, however such data was not available at the time of this
assessment.

Although synchrontron emissions reported here are associated with coal, the
power purchasing agreement has changed and currently power for the synchrotron
is 100% from wind power. According to the 2018 Intergovernmental Panel on
Climate Change (IPCC) report, the CO2-eq. emissions associated with wind power
range from 8–20 g / kWh (0.014 $\pm$ 0.006 mT CO2-eq. / MWh).[17] Therefore,
assuming this range, if I was to start my PhD today and perform the same
amount of work at SSRL my associated emissions would be about two orders of
magnitude lower: approximately 0.7 $\pm$ 0.3 mT CO2-eq.

Figure 2: Alternative scenarios of emissions during my PhD.

####  III-A3 Emissions associated with laboratory work

Estimating emissions associated with laboratory work is nontrivial, since I
have used a variety of tools at a variety of institutions, and associated
energy contributions spans not just the time I used the tool but the energy
required for keeping the facility running, the energy source of the particular
facility, the specific experiment run, and the number of users, among other
factors. LBNL collects and shares macroscopic data about its facility energy
consumption, from 2014 onward,[18] however I was unable to find similar
aggregated data for the other labs I worked at. Therefore I will make a rough
assumption here that LBNL energy use data is representative of my total lab
use. Total energy use at LBNL in 2019 was 142,230,864 kWh, and dry labs used
$\sim$16% of this, i.e. a total of 18,490 MWh/year across the entire lab.
Using a variety of assumptions based on number of experimentalists at LBNL, I
estimate I am responsible for $\sim$15 MWh/year from dry lab usage, or
$\sim$75 MWh over five years. To roughly account for the variability in the
energy source at the different labs I worked at based on their electricity
grids (see Table II), together, this amounts to a weighted average of 0.3875
mT CO2-eq. / MWh, and $\sim$5 mT CO2-eq. / year from dry labs. Since this
quantity is based on a series of assumptions, I will assume an error
propagation of $\pm$50%. In total this amounts to $\sim$25$\pm$12 mT CO2-eq.
over my PhD from dry lab use.

####  III-A4 Office building emissions

Using a similar framework as above, I estimate the associated emissions with
working in an office using LBNL’s aggregated energy use data.[18] In 2019,
office work contributed towards 9% of energy use at LBNL, i.e. 12,800
MWh/year. Assuming emission factors from Table II and Table I, as well as
similar attribution assumptions as discussed for laboratory emissions, this
corresponds to approximately 1.74 mT at NREL, 2.99 mT at LBNL, and 0.053 mT at
EPFL. In total, I can attribute $\sim$4.8 mT CO2-eq. to my use of office
buildings during my PhD.

###  III-B Work-related travel and transport emissions

Work-related travel during my PhD included travel for collaborative research
projects, summer schools and workshops, and international conferences. In
total, I traveled 134,159 miles (215,908 km) by plane, and 102,816 of these
miles (165,464 km) was for work-related travel. Emission factors for
commercial aviation vary depending on the specific aircraft, its weight, its
engine, weather, and number of passengers; reported literature emission values
range from approximately 90–100 g CO2-eq. per passenger per km.[19,
graver2020]. Assuming a more conservative estimate of 100 g CO2-eq. per
passenger per km, I estimate I am accountable for approximately 21.6 mT
CO2-eq. emissions total, and thus 16.5 mT CO2-eq. from my work-related travel.
In addition, I traveled 2859 miles (4601 km) by train. Assuming train travel
emissions of approximately 41 g per km per passenger,[20] my associated train
travel emissions are approximately 0.19 mT CO2-eq.. I do not own a car but
have traveled approximately 2400 miles (3860 km) by car for work. Assuming
automobile emissions of 411 g ofCO2-eq. per mile from the EPA’s GHG calculator
from 2019,[4] my associated automobile emissions from travel are quite low at
0.99 mT CO2-eq. I commuted to work by bicycle, with the occasional $\sim$1
mile shuttle ride to LBNL or bus ride to NREL. Approximating one shuttle ride
every week for five years amounts to 260 miles of bus transport, and, using
the automobile emissions factor as an upper limit, this results in 0.11 mT
CO2-eq from bus transport.

###  III-C Personal emissions

Here, I use the EPA carbon footprint calculator to estimate my personal carbon
footprint.[21] As inputs, I used the data from my electricity and gas bills in
Berkeley, CA, estimating $\sim$100 kWh/month for electricity and $\sim$14
therms/month for gas. According to this estimate, I am responsible personally
for 3,102 lbs CO2-eq. per year, i.e. 1.4 mT CO2-eq. / year and 7 mT CO2-eq.
over the course of my five year PhD. Accounting for the associated emissions
from personal flight, train, bus, and automobile transportation, this adds
approximately 5.04, 0.13, 0.04, and 0.88 mT, respectively, with a total of 6.1
mT. Thus, my estimate of personal emissions amounts to 13.1 mT CO2-eq. over
the course my PhD. For comparison, the emissions of an average American are
$\sim$16.2 mT CO2-eq. per year. [20] I expect in reality my personal emissions
are less than this because of my vegetarian diet, because I do not own a car,
and because I live in a location that does not require much heating or
cooling.

##  IV Discussion

In Figure 1, I plot all of the emissions that have been estimated previously.
The biggest sources of emissions during my PhD has been from laboratory
research and computation rather than from flights. Synchrotron beam time
accounts for the largest portion of my personal emissions. This is due
primarily to the power purchasing agreement at SSRL that was primarily coal
power, though this has since been decommissioned. These cumulative emissions
are also plotted in the first bar of Figure 2, showing total emissions of
$\sim$128 mT CO2-eq. over my five year PhD. This is approximately 63% higher
than the average per capita US emissions over five years of $\sim$81 mT
CO2-eq. ($\sim$16.2 mT CO2-eq. annually).[20] For comparison, the global
average is 4.8, or $\sim$24 mT CO2-eq., and this is shown in the plot as well.

Here, I estimate a few alternate scenarios as a way of contextualizing these
emissions, and compare these scenarios to my actual emissions by source in
Figure 2. First, I estimate what these emissions would be have been if my PhD
had been solely experimental or solely computational research. For an
experimental PhD, I multiply the lab-based emissions by a factor of 2.5, and
remove computational emissions. For a purely computational PhD, I use an
estimate based on the total allocated time of my group. This reduces emissions
by nearly half; however, it is important to note that more complex
computational simulations, e.g. machine learning and artificial intelligence
models, can use significantly more energy and thus would be attributed to far
more emissions from supercomputer time. Additionally, this number is very
dependent on the emissions factor of the supercomputer. For example, it has
been shown that the astronomy community in Australia on average contributes to
22 mT CO2-eq. per person per year, or 110 mT CO2-eq. over five years.[22]
“Computational PhD (Aus. approx.)” is plotted assuming these emissions (with
all other sources unchanged), showing a significantly higher contribution from
compute time.

Next, I estimate alternate scenarios given different emissions factors. I
estimate emissions if, rather than flying and driving, all of my
transportation had been on trains. This somewhat reduces transportation-
associated emissions, although other research associated emissions still keep
the total significantly higher than the US average. I then estimate the impact
of switching to a 100% solar-powered electrical grid. This would impact nearly
every emissions source, bringing my impact down to nearly half of the US
average, though still higher than the global average with most of my emissions
due to flying. And lastly, I estimate the impact of a solar grid and only
traveling by train. This would cut my emissions by nearly 5 times, bringing
them to approximately the global average. The takeaway here is that,
regardless of research task, switching to a renewable grid and offsetting
travel to use trains rather than planes (or traveling less frequently) is
extremely impactful on the carbon footprint of a researcher.

##  V Conclusion

In this paper I have estimated my carbon footprint during my five-year PhD,
showing that my total emissions are $\sim$128 mT CO2-eq. or $\sim$25.6 mT
CO2-eq. annually. This is approximately 63% higher than the average per capita
US emissions over five years of $\sim$16.2 mT CO2-eq. annually[20] and over 25
times higher than average global per capita emissions. Switching to a less
carbon-intensive electricity grid and cleaner methods of transportation should
lower research-related emissions. However, to get to our goals of net-zero
carbon emission by 2050 as set by the IPCC, global per capita emissions must
decrease to less than 2.5 mT CO2-eq. annually. The decreases from an entirely
solar-powered grid and only train transportation are still not enough to reach
this 2050 net-zero target! The key takeaway is that research and activities
associated with a materials science career significantly increase my carbon
footprint beyond that of a typical citizen. Our research has a significant
carbon cost.

I expect that this finding applies generally to scientists across the US, and
around the world, and I expect it to be an underestimate. This is because this
assessment does not include the carbon impact that the technologies we produce
will have, nor the damage from mineral extraction and waste products such
technologies may induce, though such life cycle assessments are also critical.
It is optimistic that switching to renewable power sources within our
laboratories and decreasing travel drastically reduces emissions, and our
energy grids are becoming more renewable overtime, but this is not happening
fast enough. Using energy and CO2 to address our energy CO2 challenges is
indeed necessary, and our research is essential to making progress in the
energy transition. However, because our consumption is so high, scientists
across the planet have a moral imperative to take action and to demand that
the energy used for our research comes from renewable sources.

##  VI Acknowledgments

I would like to acknowledge Dr. Sridutt Bhalachandra and Dr. Norman Bourassa
who helped query NERSC supercomputer data, as well as Dr. David Chassin and
Cezary Jach who assisted with the SSRL synchrotron energy and CO2
approximation. Dr. Soren Scott provided inspiration for this analysis.[23]
Special thanks to Kyle Graycar, Dr. Sebastian Husein, Dr. Matthew Horton, Dr.
Kristin A. Persson, and Dr. Andriy Zakutayev for reviewing this text in
various capacities. I would also like to acknowledge and honor people across
the planet working together to combat climate change and environmental
destruction.

## References

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# A category equivalence on the Lie algebra of polynomial vector fields

Genqiang Liu, Yufang Zhao

###### Abstract.

For any positive integer $n$, let
$W_{n}=\text{Der}(\mathbb{C}[t_{1},\dots,t_{n}])$. The subspaces
$\mathfrak{h}_{n}=\text{Span}\\{t_{1}\frac{\partial}{\partial{t_{1}}},\dots,t_{n}\frac{\partial}{\partial{t_{n}}}\\}$
and
$\Delta_{n}=\text{Span}\\{\frac{\partial}{\partial{t_{1}}},\dots,\frac{\partial}{\partial{t_{n}}}\\}$
are two abelian subalgebras of $W_{n}$. We show that a full subcategory
$\Omega_{\mathbf{1}}$ of the category of $W_{n}$-modules $M$ which are locally
finite over $\Delta_{n}$ is equivalent to some full subcategory of weight
$W_{n}$-modules $M$ which are cuspidal modules when restricted to the
subalgebra $\mathfrak{sl}_{n+1}$ of $W_{n}$.

Keywords: Whittaker module, Weyl algebra, equivalence, cuspidal module.

2020 Math. Subj. Class.: 17B10, 17B20, 17B65, 17B66, 17B68

##  1\. Introduction

The original motivation for this paper is to find relationship between non-
weight modules and weight modules over Lie algebras. In [1], Batra and
Mazorchuk introduced Whittaker pairs. A pair $(\mathfrak{g},\mathfrak{n})$ is
called a Whittaker pair if $\mathfrak{n}$ is a quasi-nilpotent subalgebra of
the Lie algebra $\mathfrak{g}$ and the adjoint action of $\mathfrak{n}$ on the
$\mathfrak{n}$-module $\mathfrak{g}/\mathfrak{n}$ is locally nilpotent. For a
pair $(\mathfrak{g},\mathfrak{n})$, we can study $\mathfrak{g}$-modules that
are locally finite over $\mathfrak{n}$. Apart from Kostant’s Whittaker modules
(see [7]), this kind of modules also include classical weight modules when
$\mathfrak{n}$ is chosen to be a Cartan subalgebra of $\mathfrak{g}$, and
classical Harish-Chandra and generalized Harish-Chandra modules, see e.g.
[16]. Let $\mathfrak{W}^{\mathfrak{g}}_{\mathfrak{n}}$ be the category which
consists of all $\mathfrak{g}$-modules, on which the action of $\mathfrak{n}$
is locally finite. A block decomposition of
$\mathfrak{W}^{\mathfrak{g}}_{\mathfrak{n}}$ and certain properties of simple
Whittaker modules under some rather mild assumptions were described in [1]. We
want to find possible equivalence between
$\mathfrak{W}^{\mathfrak{g}}_{\mathfrak{n}}$ and
$\mathfrak{W}^{\mathfrak{g}}_{\mathfrak{n}^{\prime}}$ for two different pairs
$(\mathfrak{g},\mathfrak{n})$, $(\mathfrak{g},\mathfrak{n}^{\prime})$. For
example, if $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ are two Cartan
subalgebras of a complex semisimple Lie algebra $\mathfrak{g}$, then by the
conjugate relationship between $\mathfrak{h}$ and $\mathfrak{h}^{\prime}$,
$\mathfrak{W}^{\mathfrak{g}}_{\mathfrak{h}}$ and
$\mathfrak{W}^{\mathfrak{g}}_{\mathfrak{h}^{\prime}}$ are equivalent.

Among the theory of infinite dimensional Lie algebras, the derivation Lie
algebra $W_{n}$ of $A_{n}=\mathbb{C}[t_{1},\dots,t_{n}]$ is an important
simple Lie algebra, see [8, 12]. The subspaces
$\mathfrak{h}_{n}:=\text{Span}\\{t_{1}\frac{\partial}{\partial{t_{1}}},\dots,t_{n}\frac{\partial}{\partial{t_{n}}}\\}$
and
$\Delta_{n}:=\text{Span}\\{\frac{\partial}{\partial{t_{1}}},\dots,\frac{\partial}{\partial{t_{n}}}\\}$
are two abelian subalgebras of $W_{n}$. A $W_{n}$-module $M$ with
diagonalizable action of $\mathfrak{h}_{n}$ is called a weight module. Tensor
modules $T(P,V)$ over $W_{n}$ were introduced by Shen and Larsson, see [17,
9], for a $\mathcal{D}_{n}$-module $P$ and $\mathfrak{gl}_{n}$-module V, where
$\mathcal{D}_{n}$ is the Weyl algebra. In [10], the simplicity of any $T(P,V)$
was completely determined. When $V$ is finite dimensional and
$P=\mathbb{C}[t_{1}^{\pm 1},\dots,t_{n}^{\pm 1}]$, the simplicity of $T(P,V)$
over the Witt algebra on an $n$-dimensional torus was given by Rao, see [3].
Weight modules with finite dimensional weight spaces are also called Harish-
Chandra modules, see [12]. The classification of simple Harish-Chandra modules
over $W_{1}$ was given in [12]. Recently, built on the classification of
simple uniformly bounded $W_{n}$ modules (see [2] for $n=2$ and [18] for any
$n$) and the characterization of the weight set of simple weight $W_{n}$
modules (see [14]), Grantcharov and Serganova completed the classification of
all simple Harish-Chandra $W_{n}$-modules, see [5]. They proved that every
such nontrivial module $M$ is a simple sub-quotient of some tensor module
$T(P,V)$, where $P$ is a simple weight $\mathcal{D}_{n}$-module. In [19],
under a finite condition, we show that every simple $W_{n}$-module $M$ with
locally finite action of $\Delta_{n}$ is a simple quotient of some tensor
module $T(A_{n}^{\mathbf{a}},V)$, where $A_{n}^{\mathbf{a}}$ is a simple
$\mathcal{D}_{n}$-module twisted from $A_{n}$ by an isomorphism, see (2.3).
From these known results, it seems that there is a relation between a
subcategory of $W_{n}$-modules $M$ which are locally finite over $\Delta_{n}$
and some subcategory of uniformly bounded weight $W_{n}$-modules. In this
paper, we will address this problem.

The paper is organized as follows. In Section 2, we collect all necessary
preliminaries, including the Whittaker category $\Omega_{\mathbf{1}}$ and the
cuspidal category $\mathcal{C}_{\alpha}$. The category $\Omega_{\mathbf{1}}$
is the category consisting of $W_{n}$-modules $M$ such that $\partial_{i}-1$
acts locally nilpotently on $M$ for any $i\in\\{1,\dots,n\\}$, and the
subspace $\text{wh}_{\mathbf{1}}(M)=\\{v\in M\mid\partial_{i}v=v,\
i=1,\dots,n\\}$ is finite dimensional. A weight module $M$ with finite
dimensional weight spaces over $W_{n}$ is called a cuspidal module if it is
cuspidal when restricted to the subalgebra $\mathfrak{sl}_{n+1}$, see
Definition 2.4. For an $\alpha\in\mathbb{C}^{n}$, $\mathcal{C}_{\alpha}$ is
the category of cuspidal $W_{n}$-modules $M$ such that
$\text{Supp}M\subset\mathbf{\alpha}+\mathbb{Z}^{n}$. In Section 3, we study
the localization $U_{\partial}$ of $U(W_{n})$ with respect to the Ore subset
generated by $\partial_{1},\dots,\partial_{n}$. Let $H_{n}$ be the centralizer
of $\mathfrak{h}_{n}\oplus\Delta_{n}$ in $U_{\partial}$. We show that $H_{n}$
is a tensor product factor of $U_{\partial}$, and $H_{n}$ is isomorphic to the
opposite algebra of the endomorphism algebra of a universal Whittaker module
$Q_{\mathbf{1}}$, see Theorems 3.3 and 3.7. In Section 4, using
$H_{n}$-modules, we study the categories $\Omega_{\mathbf{1}}$ and
$\mathcal{C}_{\alpha}$. Every dominant integral weight $\gamma$ of
$\mathfrak{gl}_{n}$ can define a full subcategory
$\Omega_{\mathbf{1}}^{[\gamma]}$ of $\Omega_{\mathbf{1}}$ and a subcategory
$\mathcal{C}^{[\gamma]}_{\alpha}$ of $\mathcal{C}_{\alpha}$. We show that
there exists a suitable $\alpha\in\mathbb{C}^{n}$ such that
$\Omega_{\mathbf{1}}^{[\gamma]}$ is equivalent to
$\mathcal{C}^{[\gamma]}_{\alpha}$, see Proposition 4.2 and Theorems 4.9 and
4.10.

We denote by $\mathbb{Z}$, $\mathbb{Z}_{>0}$, $\mathbb{Z}_{\geq 0}$,
$\mathbb{C}$ and $\mathbb{C}^{*}$ the sets of integers, positive integers,
nonnegative integers, complex numbers, and nonzero complex numbers,
respectively. For a Lie algebra $L$ over $\mathbb{C}$, we use $U(L)$ to denote
the universal enveloping algebra of $L$.

##  2\. Preliminaries and known results

###  2.1. Witt algebras $W_{n}$

Throughout this paper, $n$ denotes a positive integer. We fix the vector space
$\mathbb{C}^{n}$ of $n$-dimensional complex vectors. Denote its standard basis
by $\\{\mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n}\\}$. Let
$|\mathbf{m}|=m_{1}+\cdots+m_{n}$ for any
$\mathbf{m}=(m_{1},\dots,m_{n})\in\mathbb{C}^{n}$.

Let $A_{n}=\mathbb{C}[t_{1},t_{2},\dots,t_{n}]$ be the polynomial algebra in
the commuting variables $t_{1},t_{2},\cdots,t_{n}$. Let
$\partial_{i}=\frac{\partial}{\partial{t_{i}}},h_{i}=t_{i}\partial_{i}$ for
any $i=1,2,\cdots,n$. For the convenience, set
$t^{\mathbf{m}}=t_{1}^{m_{1}}\cdots t_{n}^{m_{n}}$,
$h^{\mathbf{m}}=h_{1}^{m_{1}}\cdots h_{n}^{m_{n}}$ for any
$\mathbf{m}=(m_{1},\dots,m_{n})\in\mathbb{Z}_{\geq 0}^{n}$. Recall that the
Witt algebra $W_{n}=\text{Der}(A_{n})=\oplus_{i=1}^{n}{A}_{n}\partial_{i}$ is
a free $A_{n}$-module of rank $n$, which is also called the Lie algebra of
polynomial vector fields on $\mathbb{C}^{n}$. It has the following Lie
bracket:

$[t^{\mathbf{m}}\partial_{i},t^{\mathbf{r}}\partial_{j}]=r_{i}t^{\mathbf{m}+\mathbf{r}-\mathbf{e}_{i}}\partial_{j}-m_{j}t^{\mathbf{m}+\mathbf{r}-\mathbf{e}_{j}}\partial_{i},$

where all $\mathbf{m},\mathbf{r}\in\mathbb{Z}_{\geq 0}^{n},i,j=1,\dots,n$.

It is well-known that $W_{n}$ is a simple Lie algebra. Let
$\mathfrak{h}_{n}=\text{Span}\\{h_{1},\dots,h_{n}\\}$ which is a Cartan
subalgebra (a maximal abelian subalgebra that is diagonalizable on $W_{n}$
with respect to the adjoint action) of $W_{n}$. Denote
$\Delta_{n}=\text{Span}\\{\partial_{1},\dots,\partial_{n}\\}$ which is also an
$n$-dimensional abelian subalgebra of $W_{n}$.

######  Definition 2.1.

A $W_{n}$-module $M$ is called a weight module if the action of
$\mathfrak{h}_{n}$ on $M$ is diagonalizable, i.e.,
$M=\bigoplus_{\lambda\in\mathbb{C}^{n}}M_{\lambda},$ where

$M_{\lambda}=\\{v\in M\mid h_{i}v=\lambda_{i}v,\forall\ i=1,\dots,n\\}.$

For a weight $W_{n}$-module $M$, denote its weight set by
$\text{Supp}M=\\{\lambda\in\mathbb{C}^{n}\mid M_{\lambda}\neq 0\\}$. For
example, the adjoint module $W_{n}$ has the weight set
$\text{Supp}W_{n}=\mathbb{Z}^{n}$. If $M$ is indecomposable, then there is a
$\lambda\in\mathbb{C}^{n}$ such that
$\text{Supp}M\subset\lambda+\mathbb{Z}^{n}$.

###  2.2.  Finite-dimensional $\mathfrak{gl}_{n}$-modules

We denote by $E_{ij}$ the $n\times n$ square matrix with $1$ as its
$(i,j)$-entry and 0 as other entries. Then $\\{E_{ij}\mid 1\leq i,j\leq n\\}$
is a basis of the general linear Lie algebra $\mathfrak{gl}_{n}$.

We can view a $\lambda\in\mathbb{C}^{n}$ as a linear function on
$\oplus_{i=1}^{n}\mathbb{C}E_{ii}$ such that $\lambda(E_{ii})=\lambda_{i}$ for
all $i\in\\{1,\dots,n\\}$. Let
$\Lambda_{+}=\\{\lambda\in\mathbb{C}^{n}\,|\,\lambda_{i}-\lambda_{i+1}\in\mathbb{Z}_{\geq
0}\text{ for }i=1,2,\dots,n-1\\}$ be the set of dominant integral weights of
$\mathfrak{gl}_{n}$. It is well known that the simple highest weight
$\mathfrak{gl}_{n}$-module $V(\lambda)$ is finite dimensional if and only if
$\lambda\in\Lambda_{+}$. Define the fundamental weights
$\delta_{i}\in\Lambda_{+}$ by
$\delta_{i}=\mathbf{e}_{1}+\cdots+\mathbf{e}_{i}$ for all $i=1,2,\dots,n$. For
convenience, we define $\delta_{0}=0\in\Lambda_{+}$. So $V(\delta_{0})$ is the
$1$-dimensional trivial $\mathfrak{gl}_{n}$-module. A module $V$ over a Lie
algebra $\mathfrak{g}$ is called a trivial module if $Xv=0$ for any
$X\in\mathfrak{g},v\in V$. It is well-known that the module $V(\delta_{1})$ is
isomorphic to the natural module $\mathbb{C}^{n}$ of $\mathfrak{gl}_{n}$. The
exterior product
$\bigwedge^{k}(\mathbb{C}^{n})=\mathbb{C}^{n}\wedge\cdots\wedge\mathbb{C}^{n}\
\ (k\ \mbox{times})$ is a $\mathfrak{gl}_{n}$-module with the action given by

$X(v_{1}\wedge\cdots\wedge v_{k})=\sum\limits_{i=1}^{k}v_{1}\wedge\cdots
v_{i-1}\wedge Xv_{i}\cdots\wedge v_{k},\,\,\forall\,\,X\in\mathfrak{gl}_{n}.$

Moreover, ${\bigwedge}^{k}(\mathbb{C}^{n})\cong V(\delta_{k}),\forall\ 1\leq
k\leq n$, see Exercise 21.11 in [6]. We can set
$V(\delta_{0})=\bigwedge^{0}(\mathbb{C}^{n})=\mathbb{C}$ and $v\wedge a=av$
for any $v\in\mathbb{C}^{n},a\in\mathbb{C}$.

###  2.3. Whittaker modules and cuspidal modules

We recall the definition of Whittaker pair introduced in [1]. A pair
$(\mathfrak{g},\mathfrak{n})$ is called a Whittaker pair if $\mathfrak{n}$ is
a quasi-nilpotent subalgebra (a generalization of the nilpotent algebra) of
the Lie algebra $\mathfrak{g}$ and the adjoint action of $\mathfrak{n}$ on
$\mathfrak{g}/\mathfrak{n}$ is locally nilpotent. For the Lie algebra $W_{n}$,
$\Delta_{n}=\text{Span}\\{\partial_{1},\dots,\partial_{n}\\}$ is an abelian
subalgebra, whence is a nilpotent subalgebra. Since the adjoint action of
$\Delta_{n}$ on $W_{n}/\Delta_{n}$ is locally nilpotent, $(W_{n},\Delta_{n})$
is a Whittaker pair. A $W_{n}$-module $M$ is called a Whittaker module if the
action of $\Delta_{n}$ on $M$ is locally finite.

######  Definition 2.2.

For any $\mathbf{a}=(a_{1},\cdots,a_{n})\in\mathbb{C}^{n}$, let
$\Omega_{\mathbf{a}}$ be the category consisting of $W_{n}$-modules $M$ such
that:

  1. $($a$)$

For any $v\in M$, $1\leq i\leq n$, there is a $k\in\mathbb{Z}_{>0}$ such that
$(\partial_{i}-a_{i})^{k}v=0$.

  2. $($b$)$

The subspace $\emph{wh}_{\mathbf{a}}(M)=\\{v\in M\mid\partial_{i}v=a_{i}v,\
\forall\ i\\}$ is finite dimensional.

An element in $\emph{wh}_{\mathbf{a}}(M)$ is called a Whittaker vector. For a
vector $\mathbf{a}\in\mathbb{C}^{n}$, we call it non-singular if $a_{i}\neq 0$
for any $i\in\\{1,\cdots,n\\}$.

Denote $\mathbf{1}=(1,\dots,1)\in\mathbb{C}^{n}$. For a $W_{n}$-module $M$,
and an automorphism $\sigma$ of $W_{n}$, $M$ can be twisted by $\sigma$ to be
a new $W_{n}$-module $M^{\sigma}$. As a vector space $M^{\sigma}=M$, the
action of $W_{n}$ on $M^{\sigma}$ is defined by $u\cdot v=\sigma(u)v,\forall\
u\in W_{n},v\in M$.

######  Lemma 2.3.

If $\mathbf{a}\neq 0$, then $\Omega_{\mathbf{a}}$ is equivalent to
$\Omega_{\mathbf{1}}$.

###### Proof.

Any $X=(x_{ij})_{n\times n}\in GL(n,\mathbb{C})$ can define an automorphism
$\sigma_{X}$ of $W_{n}$ such that
$\sigma_{X}(\partial_{i})=\sum_{j=1}^{n}x_{ij}\partial_{j}$. Choosing a proper
$X$, we can obtain that $\partial_{i}-1$ acts locally nilpotently on
$M^{\sigma_{X}}$, i.e., $M^{\sigma_{X}}\in\Omega_{\mathbf{1}}$ for any
$M\in\Omega_{\mathbf{a}}$. ∎

Let $E_{n}=\sum_{i=1}^{n}t_{i}\partial_{i}$. It is known that the subalgebra
of $W_{n}$ spanned by $\partial_{i},t_{i}\partial_{j},t_{i}E_{n},1\leq i,j\leq
n$ is isomorphic to $\mathfrak{sl}_{n+1}$, see page 578 in [13]. We denote
this subalgebra still by $\mathfrak{sl}_{n+1}$. A weight module $M$ with
finite dimensional weight spaces over $\mathfrak{sl}_{n+1}$ is called a
cuspidal module if every root vector of $\mathfrak{sl}_{n+1}$ acts injectively
on $M$, see [13, 4].

######  Definition 2.4.

A weight module $M$ with finite dimensional weight spaces over $W_{n}$ is
called a cuspidal module if it is cuspidal when restricted to the subalgebra
$\mathfrak{sl}_{n+1}$, i.e., $\partial_{i},t_{i}\partial_{j},t_{i}E_{n}$ act
injectively on $M$, for any $1\leq i\neq j\leq n$. For an
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{C}^{n}$, let
$\mathcal{C}_{\alpha}$ be the category of cuspidal $W_{n}$-modules $M$ such
that $\emph{Supp}M\subset\mathbf{\alpha}+\mathbb{Z}^{n}$.

In the remainder of this section, we give all simple modules in
$\Omega_{\mathbf{1}}$ and $\mathcal{C}_{\alpha}$ using the results in [18,
19].

###  2.4. Shen-Larsson’s modules

The Weyl algebra $\mathcal{D}_{n}$ over $A_{n}$ is the unital associative
algebra over $\mathbb{C}$ generated by $t_{1},\dots,t_{n}$,
$\partial_{1},\dots,\partial_{n}$ subject to the relations

$[\partial_{i},\partial_{j}]=[t_{i},t_{j}]=0,\qquad[\partial_{i},t_{j}]=\delta_{i,j},\
1\leq i,j\leq n.$

The following homomorphism was well known, see [17] or [10].

######  Theorem 2.5.

The linear map $\phi:U(W_{n})\rightarrow\mathcal{D}_{n}\otimes
U(\mathfrak{gl}_{n})$ such that

(2.1) $\displaystyle\phi(t^{\mathbf{m}}\partial_{k})$
$\displaystyle=t^{\mathbf{m}}\partial_{k}\otimes
1+\sum_{i=1}^{n}m_{i}t^{\mathbf{m}-\mathbf{e}_{i}}\otimes E_{ik},$

is an associative algebra homomorphism.

The homomorphism in Theorem 2.5 implies that there is a close relationship
between $W_{n}$-modules and $\mathcal{D}_{n}$-modules. For any simple
$\mathcal{D}_{n}$-module $P$, and simple $\mathfrak{gl}_{n}$-module $V$, the
tensor product $P\otimes V$ can be lifted to a $W_{n}$-module denoted by
$T(P,V)$ under the map $\phi$.

The simplicity of $T(P,V)$ was determined in [10], for any simple
$\mathcal{D}_{n}$-module $P$, and simple $\mathfrak{gl}_{n}$-module $V$. For a
finite dimensional simple $\mathfrak{gl}_{n}$-module $V$, $T(P,V)$ is a simple
module over $W_{n}$ if $V\not\cong V(\delta_{k})$ for any
$k\in\\{0,1,\cdots,n\\}$. Moreover we have the following complex of
$W_{n}$-modules:

(2.2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(P,V(\delta_{0}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{P}_{0}}$$\textstyle{T(P,V(\delta_{1}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\pi^{P}_{1}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{P}_{n-2}\hskip
56.9055pt}$$\textstyle{{T(P,V(\delta_{n-1}))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76228pt\pi^{P}_{n-1}}$$\textstyle{T(P,V(\delta_{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\pi^{P}_{k-1}$ is the $W_{n}$-module homomorphism defined by

$\begin{array}[]{lrcl}\pi^{P}_{k-1}:&T(P,V(\delta_{k-1}))&\rightarrow&T(P,V(\delta_{k})),\\\
&p\otimes v&\mapsto&\sum_{j=1}^{n}\partial_{j}\cdot
p\otimes(\mathbf{e}_{j}\wedge v),\end{array}$

for all $p\in P$ and $v\in V(\delta_{k-1})$, where $T(P,V(\delta_{-1}))=0$,
$k\in\\{0,1,\cdots,n\\}$. By Theorem 3.5 in [10],
$\text{im}\pi^{P}_{0},\text{im}\pi^{P}_{1},\dots,\text{im}\pi^{P}_{n-1}$ are
simple $W_{n}$-modules. $T(P,V(\delta_{0}))$ is not simple if and only if
$P\cong A_{n}$. Moreover $T(P,V(\delta_{n}))$ is irreducible if and only if
$\sum_{k=1}^{n}\partial_{k}P=P$. If $T(P,V(\delta_{n}))$ is reducible, then
$T(P,V(\delta_{n}))/\text{im}\pi^{P}_{n-1}$ is a trivial module.

For a simple $\mathcal{D}_{n}$-module $P$, and simple
$\mathfrak{gl}_{n}$-module $V(\lambda)$, we denote

$L(P,V(\lambda)):=\left\\{\begin{array}[]{l}T(P,V(\lambda)),\ \text{if}\
\lambda\neq\delta_{i},i=0,1,\dots,n,\\\ \text{im}\pi^{P}_{i-1},\ \text{if}\
\lambda=\delta_{i},i=1,\dots,n.\end{array}\right.$

And $L(P,V(\delta_{0})):=L(P,V(\delta_{1}))$. Hence $L(P,V(\lambda))$ are
simple $W_{n}$-modules, for any $\lambda\in\Lambda_{+}$.

###  2.5. Simple modules in $\Omega_{\mathbf{1}}$

For an $\mathbf{a}\in\mathbb{C}^{n}$, let $\sigma_{\mathbf{a}}$ be the algebra
automorphism of $\mathcal{D}_{n}$ defined by

$t_{i}\mapsto t_{i},\partial_{i}\mapsto\partial_{i}+a_{i}.$

The $\mathcal{D}_{n}$-module $A_{n}$ can be twisted by $\sigma_{\mathbf{a}}$
to be a new $\mathcal{D}_{n}$-module $A_{n}^{\mathbf{a}}$. Explicitly the
action of $\mathcal{D}_{n}$ on $A_{n}^{\mathbf{a}}$ is defined by

(2.3) $X\cdot f(t)=\sigma_{\mathbf{a}}(X)f(t),\ \ f(t)\in
A_{n},X\in\mathcal{D}_{n}.$

By Theorem 4.6 and Lemma 4.7 in [19], we have the following theorem.

######  Theorem 2.6.

Suppose that $\mathbf{a}\in\mathbb{C}^{n}$ is nonsingular, i.e., $a_{i}\neq 0$
for any $i=1,\dots,n$. If $M$ is a simple module in $\Omega_{\mathbf{a}}$,
then $M$ is isomorphic to $L(A_{n}^{\mathbf{a}},V(\lambda))$ for some
$\lambda\in\Lambda_{+}$.

For a $\gamma\in\Lambda_{+}$, let
$[\gamma]=\\{\lambda\in\Lambda_{+}\mid\lambda-\gamma\in\mathbb{Z}^{n}\\}$,
$\Omega_{\mathbf{1}}^{[\gamma]}$ be the Serre subcategory of
$\Omega_{\mathbf{1}}$ generated by the simple modules
$L(A_{n}^{\mathbf{1}},V(\lambda))$, for $\lambda\in[\gamma]$. In Section $4$,
we will show the subcategory decomposition
$\Omega_{\mathbf{1}}=\bigoplus_{[\gamma]}\Omega_{\mathbf{1}}^{[\gamma]}$.

###  2.6. Simple modules in $\mathcal{C}_{\alpha}$

For any $\mu\in\mathbb{C}^{n}$, let $P(\mu)=t_{1}^{\mu_{1}}\cdots
t_{n}^{\mu_{n}}\mathbb{C}[t_{1}^{\pm 1},\dots,t_{n}^{\pm 1}]$. As a
$\mathcal{D}_{n}$-module, $P(\mu)$ is simple if and only if
$\mu_{i}\not\in\mathbb{Z}$ for any $i=1,\dots,n$. Since
$\phi(h_{k})=t_{k}\partial_{k}\otimes 1+1\otimes E_{kk}$,
$\text{Supp}T(P(\mu),V(\lambda))\subset\mu+\lambda+\mathbb{Z}^{n}$.

######  Lemma 2.7.

For $\lambda\in\Lambda_{+}$ and $\mu\in\mathbb{C}^{n}$, $T(P(\mu),V(\lambda))$
is cuspidal if and only if
$\mu_{i}\not\in\mathbb{Z},|\mu+\lambda|+\lambda_{i}\not\in\mathbb{Z}$ for any
$i=1,\dots,n$.

###### Proof.

By (2.1), for any $v\in V(\lambda),\mathbf{m}\in\mathbb{Z}^{n}$, we have

$\displaystyle\partial_{i}(t^{\mu+\mathbf{m}}\otimes v)=$ $\displaystyle\
(\mu_{i}+m_{i})t^{\mu+\mathbf{m}-\mathbf{e}_{i}}\otimes v,$ $\displaystyle
t_{i}\partial_{j}(t^{\mu+\mathbf{m}}\otimes v)=$ $\displaystyle\
(\mu_{j}+m_{j})t^{\mu+\mathbf{m}+\mathbf{e}_{i}-\mathbf{e}_{j}}\otimes
v+t^{\mu+\mathbf{m}}\otimes E_{ij}v,\ i\neq j,$ $\displaystyle
t_{i}E_{n}(t^{\mu+\mathbf{m}}\otimes v)=$ $\displaystyle\
t^{\mu+\mathbf{m}+\mathbf{e}_{i}}\otimes(|\mu+\lambda+\mathbf{m}|+E_{ii})v+\sum_{j=1,j\neq
i}^{n}t^{\mu+\mathbf{m}+\mathbf{e}_{j}}\otimes E_{ij}v.$

Then it can be checked that $\partial_{i},t_{i}\partial_{j}$ act injectively
on $T(P(\mu),V(\lambda))$ if and only if $\mu_{i},\mu_{j}\not\in\mathbb{Z}$,
and $t_{i}E_{n}$ acts injectively on $T(P(\mu),V(\lambda))$ if and only if
$|\mu+\lambda|+\lambda_{i}\not\in\mathbb{Z}$. ∎

The following theorem was shown in [18].

######  Theorem 2.8.

If $M$ is a simple cuspidal $W_{n}$-module, then $M$ is isomorphic to
$L(P(\mu),V(\lambda))$ for some $\lambda\in\Lambda_{+}$ and
$\mu\in\mathbb{C}^{n}$ with
$\mu_{i}\not\in\mathbb{Z},|\mu+\lambda|+\lambda_{i}\not\in\mathbb{Z}$ for any
$i=1,\dots,n$.

For a $\gamma\in\Lambda_{+}$, let $\mathcal{C}_{\alpha}^{[\gamma]}$ be the
Serre subcategory of $\mathcal{C}_{\alpha}$ generated by the simple modules
$L(P(\mu),V(\lambda))$, for $\lambda\in[\gamma],\mu\in\mathbb{C}^{n}$ with
$\lambda+\mu\in\alpha+\mathbb{Z}^{n}$. We will show that
$\Omega_{\mathbf{1}}^{[\gamma]}$ is equivalent to some
$\mathcal{C}_{\alpha}^{[\gamma]}$ in Section 4.

######  Remark 2.9.

By Lemma 2.7, for a $\gamma\in\Lambda_{+}$, $\mathcal{C}_{\alpha}^{[\gamma]}$
is nonempty if and only if
$\alpha_{i}-\gamma_{i}\not\in\mathbb{Z},|\alpha|+\gamma_{i}\not\in\mathbb{Z}$
for any $i=1,\dots,n$.

##  3\. Localization of $U(W_{n})$

Throughout the paper, we denote $U=U(W_{n})$. In this section, we study the
structure of $U$, mainly the centralizer of $\Delta_{n}\oplus\mathfrak{h}_{n}$
in a bigger algebra $U_{\partial}$.

###  3.1. Tensor product decomposition of $U_{\partial}$

Since each $\text{ad}\partial_{i}$ is locally nilpotent on $U$, the following
set

$\partial:=\\{\partial_{1}^{i_{1}}\dots\partial_{n}^{i_{n}}\mid
i_{1},\dots,i_{n}\in\mathbb{Z}_{\geq 0}\\}$

is an Ore subset of $U$, see Lemma 4.2 in [13]. Let $U_{\partial}$ be the
localization of $U$ with respect to $\partial$. Define

$H_{n}=\\{u\in U_{\partial}\mid[u,\partial_{i}]=[u,h_{i}]=0,\ \forall\
i=1,\dots,n\\},$

which is a subalgebra of $U_{\partial}$. Define the following elements in
$U_{\partial}$:

(3.1) $\displaystyle z_{i,j}=$ $\displaystyle\
(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}-t_{i}\partial_{i},$
$\displaystyle z_{i,l,j}=$ $\displaystyle\
(t_{i}t_{l}\partial_{j})\partial_{i}\partial_{l}\partial_{j}^{-1}-(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}(t_{l}\partial_{l})$
$\displaystyle-(t_{l}\partial_{j})\partial_{l}\partial_{j}^{-1}(t_{i}\partial_{i})+(t_{l}\partial_{l})(t_{i}\partial_{i})+\delta_{il}t_{i}\partial_{l},$
$\displaystyle z_{i}=$ $\displaystyle\
(t_{i}^{3}\partial_{i})\partial_{i}^{2}-3(t_{i}^{2}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}+2(t_{i}\partial_{i})(t_{i}\partial_{i}-1)(t_{i}\partial_{i}-2),$

where $i,j,l\in\\{1,\dots,n\\}$.

The following Lemma is used to construct generators of the associative algebra
$H_{n}$.

######  Lemma 3.1.

For any $i,j,l\in\\{1,\dots,n\\}$, $z_{i,j},z_{i,l,j},z_{i}\in H_{n}$.

###### Proof.

It is easy to see that $[h_{k},z_{i,j}]=[h_{l},z_{i,j,k}]=[h_{k},z_{i}]=0$.

We can compute that

$\displaystyle\
[\partial_{k},(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}-t_{i}\partial_{i}]=\delta_{ki}\partial_{j}\partial_{i}\partial_{j}^{-1}-\delta_{ki}\partial_{i}=0,$

and

$\displaystyle[\partial_{k},z_{i,l,j}]$ $\displaystyle=$ $\displaystyle\
\delta_{ki}(t_{l}\partial_{j})\partial_{i}\partial_{l}\partial_{j}^{-1}+\delta_{kl}(t_{i}\partial_{j})\partial_{i}\partial_{l}\partial_{j}^{-1}-\delta_{ki}\partial_{j}\partial_{i}\partial_{j}^{-1}(t_{l}\partial_{l})-\delta_{kl}(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}\partial_{l}$
$\displaystyle\
-\delta_{kl}\partial_{j}\partial_{l}\partial_{j}^{-1}(t_{i}\partial_{i})-\delta_{ki}(t_{l}\partial_{j})\partial_{l}\partial_{j}^{-1}\partial_{i}+\delta_{kl}\partial_{l}(t_{i}\partial_{i})+\delta_{ki}(t_{l}\partial_{l})\partial_{i}+\delta_{ki}\delta_{li}\partial_{l}$
$\displaystyle=$
$\displaystyle-\delta_{ki}\partial_{i}(t_{l}\partial_{l})-\delta_{kl}\partial_{l}(t_{i}\partial_{i})+\delta_{kl}\partial_{l}(t_{i}\partial_{i})+\delta_{ki}(t_{l}\partial_{l})\partial_{i}+\delta_{ki}\delta_{li}\partial_{l}$
$\displaystyle=$ $\displaystyle\
-\delta_{ki}\partial_{i}(t_{l}\partial_{l})+\delta_{ki}(t_{l}\partial_{l})\partial_{i}+\delta_{ki}\delta_{li}\partial_{l}$
$\displaystyle=$ $\displaystyle\ 0.$

Finally we can also obtain that

$\displaystyle[\partial_{i},z_{i}]$ $\displaystyle=$ $\displaystyle\
[\partial_{i},(t_{i}^{3}\partial_{i})\partial_{i}^{2}-3(t_{i}^{2}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}+2(t_{i}\partial_{i})(t_{i}\partial_{i}-1)(t_{i}\partial_{i}-2)]$
$\displaystyle=$ $\displaystyle\
3(t_{i}^{2}\partial_{i})\partial_{i}^{2}-6(t_{i}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}-3(t_{i}^{2}\partial_{i})\partial_{i}^{2}+2\partial_{i}(t_{i}\partial_{i}-1)(t_{1}\partial_{i}-2)$
$\displaystyle\
+2(t_{i}\partial_{i})\partial_{i}(t_{i}\partial_{i}-2)+2(t_{i}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}$
$\displaystyle=$ $\displaystyle\
-6(t_{i}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}+6(t_{i}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}$
$\displaystyle=$ $\displaystyle\ 0.$

∎

The following lemma is well known. For the convenience of proving Lemma 3.5,
we give its proof.

######  Lemma 3.2.

When $n>1$, $W_{n}$ is generated $t^{\mathbf{m}}\partial_{j}$, $j=1,\dots,n$,
$\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with $|\mathbf{m}|=0,1,2$.

###### Proof.

For any $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with $|\mathbf{m}|\geq 3$. If
$m_{j}=0$, then there must exists $i\neq j$ such that $m_{i}\neq 0$. From

(3.2)
$t^{\mathbf{m}}\partial_{j}=\left\\{\begin{array}[]{l}[t^{\mathbf{m}-\mathbf{e}_{i}}\partial_{j},t_{j}t_{i}\partial_{j}],\text{if}\
m_{j}=0,\\\
\frac{1}{(3-m_{j})}[t^{\mathbf{m}-\mathbf{e}_{j}}\partial_{j},t_{j}^{2}\partial_{j}],\text{if}\
m_{j}\neq 0,3,\\\ \
[t_{j}\partial_{i},t_{j}^{2}t_{i}\partial_{j}]+t_{j}^{2}t_{i}\partial_{i},\text{if}\
m_{j}=3,|\mathbf{m}|=3,\\\ \
\frac{1}{2}[t^{\mathbf{m}-2\mathbf{e}_{j}}\partial_{j},t_{j}^{3}\partial_{j}],\text{if}\
m_{j}=3,|\mathbf{m}|>3,\end{array}\right.$

by induction on $|\mathbf{m}|$, we can complete the proof. ∎

Let $B_{n}$ be the subalgebra of $U_{\partial}$ generated by
$\partial_{i},h_{i},\partial_{i}^{-1},i=1,\dots,n$. The following result gives
a tensor product decomposition of $U_{\partial}$.

######  Theorem 3.3.

$U_{\partial}=B_{n}H_{n}\cong B_{n}\otimes H_{n}$.

###### Proof.

Clearly $B_{n}H_{n}\subseteq U_{\partial}$. To show that
$U_{\partial}\subseteq B_{n}H_{n}$, we need to explain that the generators of
$W_{n}$ belong to $B_{n}H_{n}$. When $n=1$, $W_{1}$ is generated by
$\partial_{1},t_{1}\partial_{1},t_{1}^{2}\partial_{1},t_{1}^{3}\partial_{1}$,
and

$\displaystyle
t_{1}^{2}\partial_{1}=z_{1,1,1}\partial_{1}^{-1}+(t_{1}\partial_{1})^{2}\partial_{1}^{-1}-(t_{1}\partial_{1})\partial_{1}^{-1},\partial_{1},t_{1}\partial_{1}\in
B_{1}H_{1},$
$\displaystyle(t_{1}^{3}\partial_{1})=z_{1}\partial_{1}^{-2}+3(t_{1}^{2}\partial_{1})(t_{1}\partial_{1}-1)\partial_{1}^{-1}-2(t_{1}\partial_{1})(t_{1}\partial_{1}-1)(t_{1}\partial_{1}-2)\partial_{1}^{-2}\in
B_{1}H_{1}.$

By Lemma 3.2, when $n>1$, $W_{n}$ is generated $t^{\mathbf{m}}\partial_{i}$,
$i=1,\dots,n$, $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with
$|\mathbf{m}|=0,1,2$. These generators belong to $B_{n}H_{n}$, since

$\displaystyle t_{i}\partial_{j}=$ $\displaystyle
z_{i,j}\partial_{j}\partial_{i}^{-1}+(t_{i}\partial_{i})\partial_{j}\partial_{i}^{-1}\in
B_{n}H_{n},$ $\displaystyle t_{i}t_{l}\partial_{j}=$ $\displaystyle
z_{i,l,j}\partial_{i}^{-1}\partial_{l}^{-1}\partial_{j}+(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}(t_{l}\partial_{l})\partial_{i}^{-1}\partial_{l}^{-1}\partial_{j}+(t_{l}\partial_{j})\partial_{l}\partial_{j}^{-1}(t_{i}\partial_{i})\partial_{i}^{-1}\partial_{l}^{-1}\partial_{j}$
$\displaystyle-(t_{l}\partial_{l})(t_{i}\partial_{i})\partial_{i}^{-1}\partial_{l}^{-1}\partial_{j}-\delta_{il}(t_{i}\partial_{l})\partial_{i}^{-1}\partial_{l}^{-1}\partial_{j}\in
B_{n}H_{n}.$

Since $[B_{n},H_{n}]=0$, we have that $B_{n}H_{n}\cong B_{n}\otimes H_{n}$. ∎

###  3.2. Isomorphism between $H_{n}$ and the endomorphism algebra of a
universal Whittaker module

Let $\mathbb{C}_{\mathbf{1}}=\mathbb{C}v_{\mathbf{1}}$ be the one dimensional
$U(\Delta_{n})$-module such that $\partial_{i}v_{\mathbf{1}}=v_{\mathbf{1}}$
for any $i\in\\{1,\dots,n\\}$.

Let

$Q_{\mathbf{1}}=U(W_{n})\otimes_{U(\Delta_{n})}\mathbb{C}_{\mathbf{1}},$

an induced $W_{n}$-module, and

$H_{n}^{\mathbf{1}}=\text{End}_{W_{n}}(Q_{\mathbf{1}})^{\text{op}},$

an associative algebra over $\mathbb{C}$. Then $Q_{\mathbf{1}}$ is both a left
$U(W_{n})$-module and a right $H_{n}^{\mathbf{1}}$-module. Let
$H_{n}^{\mathbf{1}}\text{-mod}$ be the category of finite dimensional
$H_{n}^{\mathbf{1}}$-modules.

In [19], we showed the following theorem.

######  Theorem 3.4.

The functors $M\mapsto\mathrm{wh}_{\mathbf{1}}(M)$ and $V\mapsto
Q_{\mathbf{1}}\otimes_{H_{\mathbf{1}}}V$ are inverse equivalence between
$\Omega_{\mathbf{1}}$ and $H_{n}^{\mathbf{1}}\text{-mod}$ .

Inspired by the constructions of $z_{i,j},z_{i,l,j}$ in (3.1), we can give
more useful elements in $H_{n}$.

######  Lemma 3.5.

For any $j\in\\{1,\dots,n\\}$ and $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with
$|\mathbf{m}|\geq 1,\mathbf{m}\neq\mathbf{e}_{j}$, there is an
$X_{\mathbf{m},j}\in H_{n}$ such that

(3.3)
$X_{\mathbf{m},j}=(t^{\mathbf{m}}\partial_{j})\partial^{\mathbf{m}-\mathbf{e}_{j}}+\sum_{0<|\mathbf{r}|<|\mathbf{m}|}(t^{\mathbf{r}}\partial_{j})g_{\mathbf{r}}(h)\partial^{\mathbf{r}-\mathbf{e}_{j}},$

where $g_{\mathbf{r}}(h)$ is a polynomial in $h_{1},\cdots,h_{n}$ of the
degree $|\mathbf{m}|-|\mathbf{r}|$.

###### Proof.

When $n=1$, denote $d_{i}:=t_{1}^{i+1}\partial_{1}$, for any
$i\in\mathbb{Z}_{\geq-1}$. Define

$X_{m+1,1}:=d_{m}d_{-1}^{m}+\sum_{k=1}^{m-1}(-1)^{m-k}\binom{m+1}{k+1}d_{k}(\prod_{i=1}^{m-k}(d_{0}-i))d_{-1}^{k}+(-1)^{m}m\prod_{i=0}^{m}(d_{0}-i),$

for any $m\in\mathbb{Z}_{\geq 1}$. It can be directly verified that
$X_{m+1,1}\in H_{1}$.

When $n>1$, we do induction on $|\mathbf{m}|$. If $|\mathbf{m}|=1,2$, then we
set $X_{\mathbf{e}_{i},j}=z_{i,j}$ and
$X_{\mathbf{e}_{i}+\mathbf{e}_{l},j}=z_{i,l,j}$, where $z_{i,j},z_{i,l,j}$ are
defined in (3.1). For any $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with
$|\mathbf{m}|>2$, there exists $m_{i}>0$. We set that

(3.4)
$X_{\mathbf{m},j}=\left\\{\begin{array}[]{l}[X_{\mathbf{m}-\mathbf{e}_{i},j},X_{\mathbf{e}_{j}+\mathbf{e}_{i},j}],\
\ \text{if}\ m_{j}=0,\\\
\frac{1}{(3-m_{j})}[X_{\mathbf{m}-\mathbf{e}_{j},j},X_{2\mathbf{e}_{j},j}],\ \
\text{if}\ m_{j}\neq 0,3,\\\ \
[X_{\mathbf{e}_{j},i},X_{2\mathbf{e}_{j}+\mathbf{e}_{i},j}]+X_{2\mathbf{e}_{j}+\mathbf{e}_{i},i},\
\ \text{if}\ m_{j}=3,|\mathbf{m}|=3,\\\
\frac{1}{2}[X_{\mathbf{m}-2\mathbf{e}_{j},j},X_{3\mathbf{e}_{j},j}],\ \
\text{if}\ m_{j}=3,|\mathbf{m}|>3,\end{array}\right.$

By induction on $|\mathbf{m}|$ and the bracket (3.2), we can see that
$X_{\mathbf{m},j}\in H_{n}$ and it is of the form (3.3). ∎

By Theorem 3.3 and Lemma 3.5, we obtain a PBW type basis of $H_{n}$.

######  Corollary 3.6.

The ordered monomials in $X_{\mathbf{m},j}\in H_{n}$, for
$j\in\\{1,\dots,n\\}$ and $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with
$|\mathbf{m}|\geq 1,\mathbf{m}\neq\mathbf{e}_{j}$, form a basis of $H_{n}$.

Following the original Premet’s proof of the PBW theorem for finite
$W$-algebras (see [15]), we will show the following algebra isomorphism.

######  Theorem 3.7.

We have that $H_{n}\cong H_{n}^{\mathbf{1}}$.

###### Proof.

Let $L_{n}$ be the Lie subalgebra of $W_{n}$ spanned by
$t^{\mathbf{m}}\partial_{i}$, $i=1,\dots,n,\mathbf{m}\in\mathbb{Z}^{n}_{\geq
0}$ with $|\mathbf{m}|\geq 1$. Clearly as a linear space, $Q_{\mathbf{1}}\cong
U(L_{n})$. Any $\theta\in\text{End}_{W_{n}}(Q_{\mathbf{1}})$ is determined by
$\theta(v_{\mathbf{1}})$ and
$\theta(v_{\mathbf{1}})\in\text{wh}_{\mathbf{1}}(Q_{\mathbf{1}})$. Since
$[H_{n},\Delta_{n}]=0$, we can define an algebra homomorphism

$\Theta:H_{n}\rightarrow
H_{n}^{\mathbf{1}}=\text{End}_{W_{n}}(Q_{\mathbf{1}})^{\text{op}},X\mapsto\Theta(X),$

where $\Theta(X)\in\text{End}_{W_{n}}(Q_{\mathbf{1}})^{\text{op}}$ such that
$\Theta(X)(v_{\mathbf{1}})=Xv_{\mathbf{1}}$. By Lemma 3.5 and Corollary 3.6,
$\Theta$ is injective. To show that $\Theta$ is surjective, we need to prove
that $\text{End}_{W_{n}}(Q_{\mathbf{1}})^{\text{op}}$ is generated by
$\Theta(X_{\mathbf{m},j})$, for $j\in\\{1,\dots,n\\}$ and
$\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with $|\mathbf{m}|\geq
1,\mathbf{m}\neq\mathbf{e}_{j}$. For convenience, denote

$\\{y_{k}\mid k\in\mathbb{Z}_{\geq
1}\\}:=\\{t_{i}\partial_{j}-t_{i}\partial_{i},t^{\mathbf{m}}\partial_{j}\mid
1\leq i\neq j\leq n,\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n},|\mathbf{m}|\geq
2\\},$

and

$\\{Y_{k}\mid k\in\mathbb{Z}_{\geq 1}\\}:=\\{\Theta(X_{\mathbf{m},j})\mid
j=1,\dots,n,\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n},|\mathbf{m}|\geq
1,\mathbf{m}\neq\mathbf{e}_{j}\\}.$

Then $\\{y_{k}\mid k\in\mathbb{Z}_{\geq 1}\\}\cup\\{h_{1},\dots,h_{n}\\}$ is a
basis of $L_{n}$. Let $n_{k}$ be the integer such that
$[E_{n},y_{k}]=n_{k}y_{k}$. For an
$\tilde{l}=(l_{1},l_{2},\dots)\in\mathbb{Z}_{\geq 0}^{\infty}$ with only
finite nonzero entries and
$\mathbf{r}=(r_{1},\dots,r_{n})\in\mathbb{Z}_{\geq}^{n}$, denote

$\displaystyle\text{wt}(\mathbf{r}+\tilde{l}):=n_{1}l_{1}+\cdots+n_{q}l_{q}+\cdots,$
$\displaystyle|\mathbf{r}+\tilde{l}|:=|\mathbf{r}|+(n_{1}+1)l_{1}+\cdots+(n_{q}+1)l_{q}+\cdots,$
$\displaystyle y^{\tilde{l}}h^{\mathbf{r}}:=y_{1}^{l_{1}}y_{2}^{l_{2}}\cdots
y_{q}^{l_{q}}\cdots h_{1}^{r_{1}}\cdots h_{n}^{r_{n}}.$

We say that the weight of $y^{\tilde{l}}h^{\mathbf{r}}$ is
$\text{wt}(\mathbf{r}+\tilde{l})$, the degree of $y^{\tilde{l}}h^{\mathbf{r}}$
is $|\mathbf{r}+\tilde{l}|$. For any
$0\neq\theta\in\text{End}_{W_{n}}(Q_{\mathbf{1}})$, we can write

$\theta(v_{\mathbf{1}})=\sum_{|\tilde{l}+\mathbf{r}|\leq
d}a_{\tilde{l},\mathbf{r}}y^{\tilde{l}}h^{\mathbf{r}}v_{\mathbf{1}},$

where $d$ is called the degree of $\theta$, and $a_{\tilde{l},\mathbf{r}}\neq
0$ for at least one $(\tilde{l},\mathbf{r})$ with $|\tilde{l}+\mathbf{r}|=d$.
Let $\Lambda^{d}=\\{(\tilde{l},\mathbf{r})\mid a_{\tilde{l},\mathbf{r}}\neq
0,|\tilde{l}+\mathbf{r}|=d\\}$, and $\Lambda^{d}_{\text{max}}$ be the subset
$\Lambda^{d}$ of all $(\tilde{p},\mathbf{s})$ such that
$\text{wt}(\mathbf{s}+\tilde{p})$ assumes its maximum value. This maximal
value will be denoted by $N(\theta)$. We define $(d,N(\theta))$ to be the
Whittaker degree of $\theta$.

Claim: If $(\tilde{p},\mathbf{s})\in\Lambda^{d}_{\text{max}}$, then
$\mathbf{s}=0$.

Suppose the contrary. Then there is a
$(\tilde{p},\mathbf{s})\in\Lambda^{d}_{\text{max}}$ such that $\mathbf{s}\neq
0$. Hence $s_{k}>0$ for some $k\in\\{1,\dots,n\\}$. From
$[\partial_{k},h^{\mathbf{r}}]=\sum_{i=1}^{r_{k}}\binom{r_{k}}{i}h^{\mathbf{r}-i\mathbf{e}_{k}}$
and $(\partial_{k}-1)v_{\mathbf{1}}=0$, we have:

$\displaystyle(\partial_{k}-1)\theta(v_{\mathbf{1}})=\sum_{|\tilde{l}+\mathbf{r}|\leq
d}a_{\tilde{l},\mathbf{r}}[\partial_{k},y^{\tilde{l}}h^{\mathbf{r}}]v_{\mathbf{1}}$
$\displaystyle=$ $\displaystyle\sum_{|\tilde{l}+\mathbf{r}|\leq
d}a_{\tilde{l},\mathbf{r}}[\partial_{k},y^{\tilde{l}}]h^{\mathbf{r}}v_{\mathbf{1}}+\sum_{|\tilde{l}+\mathbf{r}|\leq
d}a_{\tilde{l},\mathbf{r}}y^{\tilde{l}}[\partial_{k},h^{\mathbf{r}}]v_{\mathbf{1}}$
$\displaystyle=$ $\displaystyle\sum_{|\tilde{l}+\mathbf{r}|\leq
d}a_{\tilde{l},\mathbf{r}}[\partial_{k},y^{\tilde{l}}]h^{\mathbf{r}}v_{\mathbf{1}}+\sum_{|\tilde{l}+\mathbf{r}|\leq
d}\sum_{i=1}^{r_{k}}\binom{r_{k}}{i}a_{\tilde{l},\mathbf{r}}y^{\tilde{l}}h^{\mathbf{r}-i\mathbf{e}_{k}}v_{\mathbf{1}}$
$\displaystyle=$ $\displaystyle\sum_{|\tilde{l}+\mathbf{r}|\leq
d-1}b_{\tilde{l},\mathbf{r}}y^{\tilde{l}}h^{\mathbf{r}}v_{\mathbf{1}}.$

We can see that
$s_{k}a_{\tilde{p},\mathbf{s}}y^{\tilde{p}}h^{\mathbf{s}-\mathbf{e}_{k}}v_{\mathbf{1}}$
is a nonzero term of degree $d-1$ in the above sum whose weight assumes
$\text{wt}(\mathbf{s}+\tilde{p})$. Hence
$(\partial_{k}-1)\theta(v_{\mathbf{1}})\neq 0$, which is a contradiction.

Let $\theta^{\prime}=\theta-\sum_{|\tilde{l}|=d}a_{\tilde{l}}Y^{\tilde{l}}$,
where $Y^{\tilde{l}}:=Y_{1}^{l_{1}}Y_{2}^{l_{2}}\cdots Y_{q}^{l_{q}}\cdots$.
Order the elements in $\mathbb{Z}_{\geq 0}^{2}$ lexicographically. By the
above claim, $\theta^{\prime}$ has the Whittaker degree less than $\theta$. By
induction on the Whittaker degree of $\theta$, we can conclude that
$\text{End}_{W_{n}}(Q_{\mathbf{1}})^{\text{op}}$ is generated by $\\{Y_{k}\mid
k\in\mathbb{Z}_{\geq 1}\\}$. ∎

##  4\. Equivalences between different blocks

In this section, using $H_{n}$-modules, we will show that
$\Omega_{\mathbf{1}}^{[\gamma]}$ is equivalent to
$\mathcal{C}^{[\gamma]}_{\alpha}$ for some $\alpha\in\mathbb{C}^{n}$.

###  4.1. Equivalence between $\Omega_{\mathbf{1}}$ and $H_{n}\text{-mod}$

Let $H_{n}\text{-mod}$ be the category of finite dimensional $H_{n}$-modules.
By Theorems 3.4 and 3.7, the category $\Omega_{\mathbf{1}}$ is equivalent to
$H_{n}\text{-mod}$. In this subsection, we give a more essential proof of this
equivalence using the tensor product decomposition of $U_{\partial}$.

We define the functor $F:\Omega_{\mathbf{1}}\rightarrow H_{n}\text{-mod}$ such
that $F(M)=\text{wh}_{\mathbf{1}}(M)$. Since $[H_{n},\partial_{i}]=0$ and that
$\text{wh}_{\mathbf{1}}(M)$ is finite dimensional, $F(M)\in H_{n}\text{-mod}$.
Conversely, for a finite dimensional $H_{n}$-module $V$, define the action of
$\Delta_{n}$ on $V$ such that $\partial_{i}v=v$ for any $v\in V$,
$i=1,\dots,n$. Let

$G(V):=\text{Ind}_{H_{n}U(\Delta_{n})_{\partial}}^{U_{\partial}}V=U_{\partial}\otimes_{H_{n}U(\Delta_{n})_{\partial}}V.$

It is easy to see that $G(V)=U(\mathfrak{h}_{n})\otimes V$, since
$U_{\partial}=B_{n}H_{n}$ and $B_{n}$ is generated by
$\partial_{i},h_{i},\partial_{i}^{-1},i=1,\dots,n$.

The following lemma in [19] is a key observation on modules in
$\Omega_{\mathbf{\mathbf{1}}}$.

######  Lemma 4.1.

Let $M$ be a module in $\Omega_{\mathbf{\mathbf{1}}}$. Then as a vector space,
$M\cong U(\mathfrak{h}_{n})\otimes\mathrm{wh}_{\mathbf{1}}(M)$.

The following proposition reduces modules in $\Omega_{\mathbf{1}}$ to finite
dimensional $H_{n}$-modules.

######  Proposition 4.2.

The category $\Omega_{\mathbf{1}}$ is equivalent to $H_{n}\text{-mod}$.

###### Proof.

Since $FG(V)=\text{wh}_{\mathbf{\mathbf{1}}}(U(\mathfrak{h}_{n})\otimes V)=V$
for any $V\in H_{n}\text{-mod}$, we have $FG=\text{id}_{H_{n}\text{-mod}}$.
For any $M\in\Omega_{\mathbf{1}}$, by Lemma 4.1, as a vector space, $M\cong
U(\mathfrak{h}_{n})\otimes\mathrm{wh}_{\mathbf{1}}(M)$. By the isomorphism
$U_{\partial}\cong B_{n}\otimes H_{n}$, we have that $M\cong
G(\mathrm{wh}_{\mathbf{1}}(M))=GF(M)$. So
$GF=\text{id}_{\Omega_{\mathbf{1}}}$. ∎

The following corollary gives a classification of all finite dimensional
irreducible $H_{n}$-modules in terms of finite dimensional simple
$\mathfrak{gl}_{n}$-modules. In $T(A_{n}^{\mathbf{1}},V)$, we identify
$1\otimes V$ with $V$.

######  Corollary 4.3.

If $M$ is a finite dimensional irreducible $H_{n}$-module, then
$M\cong\mathrm{wh}_{\mathbf{1}}(T(A_{n}^{\mathbf{1}},V))=V$, or
$M\cong\mathrm{wh}_{\mathbf{1}}(\text{im}\pi_{i})$, where $i=0,\dots,n-1$, $V$
is a finite dimensional simple $\mathfrak{gl}_{n}$-module which is not
isomorphic to $k$-th exterior power $V(\delta_{k},k)$ of $\mathbb{C}^{n}$, for
$k=0,\cdots,n$. Moreover, the action of $H_{n}$ on $M$ satisfies that:

(4.1) $\displaystyle z_{i,j}v=$ $\displaystyle\ (E_{ij}-E_{ii})v,$
$\displaystyle z_{i,l,k}v=$ $\displaystyle\
\Big{(}E_{ll}E_{ii}-E_{ik}E_{ll}-E_{lk}E_{ii}+\delta_{il}E_{il}$
$\displaystyle+(\delta_{lk}-\delta_{il})E_{ik}+(\delta_{ik}-\delta_{il})E_{lk}\Big{)}v,$
$\displaystyle z_{i,j,j}v=$ $\displaystyle\ (E_{ij}-E_{ij}E_{jj})v,$
$\displaystyle z_{i}v=$ $\displaystyle\ 2(E_{ii}^{3}-3E_{ii}^{2}+2E_{ii})v,$

where the operators $z_{i,j},z_{i,l,k},z_{i}$ are defined in (3.1),
$i,j=1,\cdots,n,v\in M$.

###### Proof.

By the action of $W_{n}$ on $T(A_{n}^{\mathbf{1}},V)$, see (2.1), we have that

$\displaystyle z_{i,j}(1\otimes v)$
$\displaystyle=((t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}-t_{i}\partial_{i})(1\otimes
v)$ $\displaystyle=t_{i}\otimes v+1\otimes E_{ij}v-t_{i}\otimes v-1\otimes
E_{ii}v$ $\displaystyle=1\otimes(E_{ij}-E_{ii})v,$

and

$\displaystyle z_{i,l,k}(1\otimes v)=$ $\displaystyle\
((t_{i}t_{l}\partial_{k})\partial_{i}\partial_{l}\partial_{k}^{-1}-(t_{i}\partial_{k})\partial_{i}\partial_{k}^{-1}(t_{l}\partial_{l})$
$\displaystyle-(t_{l}\partial_{k})\partial_{l}\partial_{k}^{-1}(t_{i}\partial_{i})+(t_{l}\partial_{l})(t_{i}\partial_{i})+\delta_{il}t_{i}\partial_{l})(1\otimes
v)$ $\displaystyle=$ $\displaystyle\ t_{i}t_{l}\otimes v+t_{i}\otimes
E_{lk}v+t_{l}\otimes
E_{ik}v-(t_{i}\partial_{k})\partial_{i}\partial_{k}^{-1}(t_{l}\otimes
v+1\otimes E_{ll}v)$
$\displaystyle-(t_{l}\partial_{k})\partial_{l}\partial_{k}^{-1}(t_{i}\otimes
v+1\otimes E_{ii}v)+(t_{l}\partial_{l})(t_{i}\otimes v+1\otimes E_{ii}v)$
$\displaystyle+\delta_{il}(t_{i}\otimes v+1\otimes E_{il}v)$ $\displaystyle=$
$\displaystyle\ t_{i}t_{l}\otimes v+t_{i}\otimes E_{lk}v+t_{l}\otimes
E_{ik}v-t_{i}(t_{l}+\delta_{il})\otimes v-t_{i}\otimes E_{ll}v$
$\displaystyle-(t_{l}+\delta_{il}-\delta_{lk})\otimes E_{ik}v-1\otimes
E_{ik}E_{ll}v-t_{l}(t_{i}+\delta_{il})\otimes v-t_{l}\otimes E_{ii}v$
$\displaystyle-(t_{i}-\delta_{ik}+\delta_{il})\otimes E_{lk}v-1\otimes
E_{lk}E_{ii}v+t_{l}(t_{i}+\delta_{il})\otimes v+t_{l}\otimes E_{ii}v$
$\displaystyle+t_{i}\otimes E_{ll}v+1\otimes
E_{ll}E_{ii}v+\delta_{il}(t_{i}\partial_{l}\otimes v+1\otimes E_{il}v)$
$\displaystyle=$ $\displaystyle\
1\otimes\big{(}E_{ll}E_{ii}-E_{ik}E_{ll}-E_{lk}E_{ii}+\delta_{il}E_{il}$
$\displaystyle+(\delta_{lk}-\delta_{il})E_{ik}+(\delta_{ik}-\delta_{il})E_{lk}\big{)}v.$

We can also obtain that

$\displaystyle z_{i}(1\otimes v)$
$\displaystyle=\Big{(}(t_{i}^{3}\partial_{i})\partial_{i}^{2}-3(t_{i}^{2}\partial_{i})(t_{i}\partial_{i}-1)\partial_{i}+2(t_{i}\partial_{i})(t_{i}\partial_{i}-1)(t_{i}\partial_{i}-2)\Big{)}(1\otimes
v)$ $\displaystyle=1\otimes 2(E_{ii}^{3}-3E_{ii}^{2}+2E_{ii})v.$

This completes the proof. ∎

######  Remark 4.4.

For a finite dimensional simple $\mathfrak{gl}_{n}$-module $V(\lambda)$, we
denote

$W(\lambda)=\left\\{\begin{array}[]{l}V(\lambda),\ \text{if}\
\lambda\neq\delta_{i},i=1,\dots,n,\\\
\mathrm{wh}_{\mathbf{1}}(\text{im}\pi^{A_{n}^{\mathbf{1}}}_{i-1}),\text{if}\
\lambda=\delta_{i},i=1,\dots,n.\end{array}\right.$

By Corollary 4.3, under the action (4.1), $W(\lambda)$ exhaust all finite
dimensional simple $H_{n}$-modules. Note that $W(\delta_{0})\cong
W(\delta_{1})$ since $\pi_{0}^{A_{n}^{\mathbf{1}}}$ is injective.

By Corollary 4.3, the actions of $z_{i,i,i}$ and $z_{i}$ on $W(\lambda)$ are
diagonalizable. We will use the weight sets of $z_{i,i,i}$ and $z_{i}$ to give
a subcategory decomposition of $H_{n}\text{-mod}$.

######  Lemma 4.5.

For $\gamma,\lambda\in\Lambda_{+}$, if $\lambda-\gamma\not\in\mathbb{Z}^{n}$,
then $\mathrm{Ext}^{1}_{H_{n}}(W(\gamma),W(\lambda))=0$.

###### Proof.

Since $z_{i,i,i}v=(E_{ii}-E_{ii}^{2})v$ and
$z_{i}v=2(E_{ii}^{3}-3E_{ii}^{2}+2E_{ii})v$ for any $v\in W(\gamma)\cup
W(\lambda)$, both $z_{i,i,i}$ and $z_{i}$ act diagonally on
$W(\gamma),W(\lambda)$ for any $i\in\\{1,\dots,n\\}$. We consider eigenvalues
of $z_{i,i,i}$ and $z_{i}$ on $W(\gamma)$ and $W(\lambda)$. Let
$S_{\lambda+\mathbb{Z}^{n}}=\\{\big{(}\beta(z_{1,1,1}),\dots,\beta(z_{n,n,n}),\beta(z_{1}),\dots,\beta(z_{n})\big{)}\mid\beta\in\lambda+\mathbb{Z}^{n}\\}$.
Note that the eigenvalues of $z_{i,i,i}$ and $z_{i}$ on $W(\lambda)$
constitute a subset of $S_{\lambda+\mathbb{Z}^{n}}$. To show that
$\text{Ext}^{1}_{H_{n}}(W(\gamma),W(\lambda))=0$, we will prove that
$S_{\gamma+\mathbb{Z}^{n}}\cap S_{\lambda+\mathbb{Z}^{n}}=\emptyset$ on the
contrary. Suppose that there exist $\mathbf{r},\mathbf{s}\in\mathbb{Z}^{n}$
such that $(\lambda+\mathbf{r})(z_{i,i,i})=(\gamma+\mathbf{s})(z_{i,i,i})$ and
$(\lambda+\mathbf{r})(z_{i})=(\gamma+\mathbf{s})(z_{i})$, for any $i$. Then
$\lambda_{i}+r_{i}-(\lambda_{i}+r_{i})^{2}=\gamma_{i}+s_{i}-(\gamma_{i}+r_{i})^{2}$
and
$(\lambda_{i}+r_{i})^{3}-(\lambda_{i}+r_{i})^{2}=(\gamma_{i}+s_{i})^{3}-(\gamma_{i}+s_{i})^{2}$,
consequently $\lambda_{i}+r_{i}=\gamma_{i}+s_{i}$ or
$\\{\lambda_{i}+r_{i},\gamma_{i}+s_{i}\\}=\\{0,1\\}$ for any $i$. This is a
contradiction, since $\lambda-\gamma\not\in\mathbb{Z}^{n}$. ∎

For a $\gamma\in\Lambda_{+}$, let $H_{n}^{[\gamma]}\text{-mod}$ be the Serre
subcategory of $H_{n}\text{-mod}$ generated by simple $H_{n}$-modules
$W(\lambda)$, for $\lambda\in[\gamma]$. By Lemma 4.5, we obtain the following
decomposition of $H_{n}\text{-mod}$.

######  Corollary 4.6.

We have the subcategory decomposition
$H_{n}\text{-mod}=\bigoplus_{[\gamma]}H_{n}^{[\gamma]}\text{-mod}$.

By Proposition 4.2, we have the following subcategory equivalence.

######  Corollary 4.7.

For each $\gamma\in\Lambda_{+}$, the subcategory
$\Omega_{\mathbf{1}}^{[\gamma]}$ is equivalent to $H_{n}^{\gamma}\text{-mod}$.
Moreover we have the subcategory decomposition
$\Omega_{\mathbf{1}}=\bigoplus_{[\gamma]}\Omega_{\mathbf{1}}^{[\gamma]}$.

###  4.2. Equivalence between $H_{n}\text{-mod}$ and a full subcategory of
weight $W_{n}$-modules

For a finite dimensional $H_{n}$-module $V$, let each $h_{i}$ act on it as the
scalar $\alpha_{i}$. Define the induced $U_{\partial}$-module

$G_{1}(V)=\text{Ind}_{U(\mathfrak{h}_{n})H_{n}}^{U_{\partial}}V=U_{\partial}\otimes_{U(\mathfrak{h}_{n})H_{n}}V.$

It is clear that $G_{1}(V)=\mathbb{C}[\partial_{1}^{\pm
1},\dots,\partial_{n}^{\pm 1}]\otimes V$. Restricted to $W_{n}$, $G_{1}(V)$
becomes a $W_{n}$-module which is still denoted by $G_{1}(V)$.

######  Lemma 4.8.

Let $V$ be a finite dimensional $H_{n}$-module, the action of $W_{n}$ on
$G_{1}(V)$ satisfies:

(4.2) $\displaystyle h_{k}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=(\alpha_{k}-r_{k})(\partial^{\mathbf{r}}\otimes v),$
$\displaystyle\partial_{k}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=\partial^{\mathbf{r}+\mathbf{e}_{k}}\otimes v,$ $\displaystyle
t_{i}\partial_{j}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=\partial^{\mathbf{r}+\mathbf{e}_{j}-\mathbf{e}_{i}}\otimes
z_{i,j}v+(\alpha_{i}-r_{i}+1-\delta_{ij})\partial^{\mathbf{r}+\mathbf{e}_{j}-\mathbf{e}_{i}}\otimes
v,$ $\displaystyle t_{i}t_{j}\partial_{j}\cdot(\partial^{\mathbf{r}}\otimes
v)$
$\displaystyle=\partial^{\mathbf{r}-\mathbf{e}_{i}}\otimes\Big{(}z_{i,j,j}+(\alpha_{j}-r_{j})(z_{i,j}+\alpha_{i}-r_{i}+1)\Big{)}v,$

where
$\partial^{\mathbf{r}}=\partial_{1}^{r_{1}}\dots\partial_{n}^{r_{n}},\mathbf{r}\in\mathbb{Z}^{n},1\leq
l,j,k\leq n$.

###### Proof.

It is easy to see that

$\displaystyle h_{k}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=(\alpha_{k}-r_{k})(\partial^{\mathbf{r}}\otimes v),$
$\displaystyle\partial_{k}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=\partial^{\mathbf{r}+\mathbf{e}_{k}}\otimes v.$

From

$\displaystyle t_{i}\partial_{j}$
$\displaystyle=z_{i,j}\partial_{j}\partial_{i}^{-1}+h_{i}\partial_{j}\partial_{i}^{-1},$
$\displaystyle t_{i}t_{j}\partial_{j}$
$\displaystyle=z_{i,j,j}\partial_{i}^{-1}-\delta_{ij}(t_{i}\partial_{j})\partial_{i}^{-1}+(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}(h_{j})\partial_{i}^{-1},$

and $[z_{i,j},\partial_{k}]=[z_{i,j,j},\partial_{k}]=0$, we can obtain that

$\displaystyle t_{i}\partial_{j}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=\
(z_{i,j}\partial_{j}\partial_{i}^{-1}+h_{i}\partial_{j}\partial_{i}^{-1})\cdot(\partial^{\mathbf{r}}\otimes
v)$ $\displaystyle=\
\partial^{\mathbf{r}+\mathbf{e}_{j}-\mathbf{e}_{i}}\otimes
z_{i,j}v+(\alpha_{i}-r_{i}+1-\delta_{ij})\partial^{\mathbf{r}+\mathbf{e}_{j}-\mathbf{e}_{i}}\otimes
v,$

and

$\displaystyle t_{i}t_{j}\partial_{j}\cdot(\partial^{\mathbf{r}}\otimes v)=$
$\displaystyle\
(z_{i,j,j}\partial_{i}^{-1}-\delta_{ij}(t_{i}\partial_{j})\partial_{i}^{-1}+(t_{i}\partial_{j})\partial_{i}\partial_{j}^{-1}(h_{j})\partial_{i}^{-1})\cdot(\partial^{\mathbf{r}}\otimes
v)$ $\displaystyle=$ $\displaystyle\
\partial^{\mathbf{r}-\mathbf{e}_{i}}\otimes\big{(}z_{i,j,j}+(\alpha_{j}-r_{j})(z_{i,j}+\alpha_{i}-r_{i}+1)\big{)}v.$

This completes the proof. ∎

Thus we have a functor $G_{1}$ from $H_{n}\text{-mod}$ to the category of
weight modules over $W_{n}$. In the following theorem, we show that each
$H_{n}^{[\gamma]}\text{-mod}$ is equivalent to some block of the cuspidal
category.

######  Theorem 4.9.

For each $\gamma\in\Lambda_{+}$, there exists an $\alpha\in\mathbb{C}^{n}$
such that $H_{n}^{[\gamma]}\text{-mod}$ is equivalent to
$\mathcal{C}^{[\gamma]}_{\alpha}$.

###### Proof.

We choose an $\alpha\in\mathbb{C}^{n}$ such that
$\alpha_{i}-\gamma_{i}\not\in\mathbb{Z},|\alpha|+\gamma_{i}\not\in\mathbb{Z}$
for any $i\in\\{1,\cdots,n\\}$.

Claim. $G_{1}(V)\in\mathcal{C}^{[\gamma]}_{\alpha}$ for any $V\in
H_{n}^{[\gamma]}\text{-mod}$.

We prove this claim by induction on the length of the $H_{n}$-module $V$. By
Corollary 4.3 and Remark 4.4, any simple $H_{n}$-module in
$H_{n}^{[\gamma]}\text{-mod}$ is isomorphic to $W(\lambda)$,
$\lambda\in[\gamma]$. The action of $H_{n}$ on $W(\lambda)$ satisfies (4.1).
By (4.1) and (4.2), the action of $W_{n}$ on $G_{1}(W(\lambda))$ is such that:

$\displaystyle t_{i}\partial_{j}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=\partial^{\mathbf{r}+\mathbf{e}_{j}-\mathbf{e}_{i}}\otimes(E_{ij}-E_{ii}+\alpha_{i}-r_{i}+1)v,\
i\neq j,$ $\displaystyle t_{i}E_{n}\cdot(\partial^{\mathbf{r}}\otimes v)$
$\displaystyle=\partial^{\mathbf{r}-\mathbf{e}_{i}}\otimes\sum_{j=1}^{n}\big{(}(E_{ij}-E_{ij}E_{jj})+(\alpha_{j}-r_{j})(E_{ij}-E_{ii}+\alpha_{i}-r_{i}+1)\big{)}v,$

where $1\leq i,j\leq n,v\in W(\lambda)$.

To show $G_{1}(W(\lambda))\in\mathcal{C}_{\alpha}^{[\gamma]}$, we need to
prove that $\partial_{i},t_{i}\partial_{j},t_{i}E_{n}$ act injectively on
$G_{1}(W(\lambda))$, for any $1\leq i\neq j\leq n$. By the second formula in
(4.2), $\partial_{i}$ acts injectively on $G_{1}(W(\lambda))$. Note that
$W(\lambda)\subset V(\lambda)$ and
$V(\lambda)=\oplus_{\mu\in\mathbb{C}^{n}}V(\lambda)_{\mu}$, where
$V(\lambda)_{\mu}:=\\{w\in V(\lambda)\mid E_{kk}w=\mu_{k}w,k=1\dots,n\\}$. For
any nonzero $v=\sum_{\mu\in I}v_{\mu}\in W(\lambda)$, where
$I\subset\mathbb{C}^{n}$ is a finite subset, $0\neq v_{\mu}\in
V(\lambda)_{\mu}$, we have that

$\displaystyle(E_{ij}-E_{ii}+\alpha_{i}-r_{i}+1)v$ $\displaystyle=\sum_{\mu\in
I}E_{ij}v_{\mu}+\sum_{\mu\in I}(-\mu_{i}+\alpha_{i}-r_{i}+1)v_{\mu}$
$\displaystyle=\sum_{\mu\in
I}v^{\prime}_{\mu+\mathbf{e}_{i}-\mathbf{e}_{j}}+\sum_{\mu\in
I}(-\mu_{i}+\alpha_{i}-r_{i}+1)v_{\mu},$

where $v^{\prime}_{\mu+\mathbf{e}_{i}-\mathbf{e}_{j}}:=E_{ij}v_{\mu}\in
V(\lambda)_{\mu+\mathbf{e}_{i}-\mathbf{e}_{j}}$. Since $I$ is a finite set, we
can choose $\mu^{\prime}\in I$ such that $\mu^{\prime}_{i}$ is minimal. By the
choice of $\alpha$, we have that $-\mu^{\prime}_{i}+\alpha_{i}-r_{i}+1\neq 0$.
So $(E_{ij}-E_{ii}+\alpha_{i}-r_{i}+1)v\neq 0$, and hence $t_{i}\partial_{j}$
acts injectively on $G_{1}(W(\lambda))$, for any $1\leq i\neq j\leq n$.
Furthermore,

$\displaystyle\sum_{j=1}^{n}\big{(}(E_{ij}-E_{ij}E_{jj})+(\alpha_{j}-r_{j})(E_{i,j}-E_{i,i}+\alpha_{i}-r_{i}+1)\big{)}v$
$\displaystyle=\sum_{j=1,j\neq i}^{n}\sum_{\mu\in
I}(1-\mu_{j}+\alpha_{j}-r_{j})v^{\prime}_{\mu+e_{i}-e_{j}}$ $\displaystyle\ \
\ +\sum_{\mu\in I}(\mu_{i}-\mu_{i}^{2})v_{\mu}+\sum_{\mu\in
I}(\alpha_{i}-r_{i})\mu_{i}v_{\mu}$ $\displaystyle\ \ \ +\sum_{\mu\in
I}(|\alpha|-|\mathbf{r}|)(-\mu_{i}+\alpha_{i}-r_{i}+1)v_{\mu}$
$\displaystyle=\sum_{j=1,j\neq i}^{n}\sum_{\mu\in
I}(1-\mu_{j}+\alpha_{j}-r_{j})v^{\prime}_{\mu+e_{i}-e_{j}}$ $\displaystyle\ \
\ +\sum_{\mu\in
I}(|\alpha|-|\mathbf{r}|+\mu_{i})(-\mu_{i}+\alpha_{i}-r_{i}+1)v_{\mu}.$

The choice of $\alpha$ forces that
$(|\alpha|-|\mathbf{r}|+\mu_{i})(-\mu_{i}+\alpha_{i}-r_{i}+1)\neq 0$. Hence
$t_{i}E_{n}$ acts injectively on $G_{1}(W(\lambda))$, for any $1\leq i\leq n$.
Therefore $G_{1}(W(\lambda))\in\mathcal{C}^{[\gamma]}_{\alpha}$. By induction
on the length of the $H_{n}$-module $V$, we can show that
$G_{1}(V)\in\mathcal{C}_{\alpha}^{[\gamma]}$ for any $V\in
H_{n}^{[\gamma]}\text{-mod}$.

We define the functor $F_{1}:\mathcal{C}^{[\gamma]}_{\alpha}\rightarrow
H_{n}^{[\gamma]}\text{-mod}$ such that $F_{1}(M)=M_{\alpha}$. From
$[H_{n},\mathfrak{h}_{n}]=0$, we have $H_{n}M_{\alpha}\subset M_{\alpha}$.
Hence $M_{\alpha}\in H_{n}^{[\gamma]}\text{-mod}$. We can check that
$G_{1}F_{1}=\text{id}_{\mathcal{C}^{[\gamma]}_{\alpha}}$ and
$F_{1}G_{1}=\text{id}_{H_{n}^{[\gamma]}\text{-mod}}$. Therefore
$H_{n}^{[\gamma]}\text{-mod}$ is equivalent to
$\mathcal{C}^{[\gamma]}_{\alpha}$. ∎

By Corollary 4.7 and Theorem 4.9, we obtain the main result of the present
paper.

######  Theorem 4.10.

For each $\gamma\in\Lambda_{+}$, there exists an $\alpha\in\mathbb{C}^{n}$
such that $\Omega_{\mathbf{1}}^{[\gamma]}$ is equivalent to
$\mathcal{C}^{[\gamma]}_{\alpha}$.

Acknowledgments

G.L. is partially supported by NSF of China(Grants 12371026).

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G.L.: School of Mathematics and Statistics, and Institute of Contemporary
Mathematics, Henan University, Kaifeng 475004, China. Email:
<EMAIL_ADDRESS>
Y.Z.: School of Mathematics and Statistics, Henan University, Kaifeng 475004,
China. Email<EMAIL_ADDRESS>