# Spectral control of high order harmonics through non-linear propagation
effects

M. Hussain<EMAIL_ADDRESS>LIDYL, CEA, CNRS, Université
Paris-Saclay, CEA Saclay 91191, Gif-sur-Yvette, France.  GoLP/Instituto de
Plasmas e Fusão Nuclear-Laboratório Associado, Instituto Superior Técnico,
Universidade de Lisboa, 1049-001 Lisboa, Portugal.  S. Kaassamani  LIDYL, CEA,
CNRS, Université Paris-Saclay, CEA Saclay 91191, Gif-sur-Yvette, France.  T.
Auguste  LIDYL, CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191, Gif-sur-
Yvette, France.  W. Boutu  LIDYL, CEA, CNRS, Université Paris-Saclay, CEA
Saclay 91191, Gif-sur-Yvette, France.  D. Gauthier  LIDYL, CEA, CNRS,
Université Paris-Saclay, CEA Saclay 91191, Gif-sur-Yvette, France.  M.
Kholodtsova  LIDYL, CEA, CNRS, Université Paris-Saclay, CEA Saclay 91191, Gif-
sur-Yvette, France.  J-T. Gomes  Novae, 15 Rue Sismondi, 87000 Limoges,
France.  L. Lavoute  Novae, 15 Rue Sismondi, 87000 Limoges, France.  D.
Gaponov  Novae, 15 Rue Sismondi, 87000 Limoges, France.  N. Ducros  Novae, 15
Rue Sismondi, 87000 Limoges, France.  S. Fevrier  Novae, 15 Rue Sismondi,
87000 Limoges, France.  Univ. Limoges, CNRS, XLIM, UMR 7252, 87000 Limoges,
France.  R. Nicolas  LIDYL, CEA, CNRS, Université Paris-Saclay, CEA Saclay
91191, Gif-sur-Yvette, France.  Natural Sciences Department, Lebanese American
University, Beirut, Lebanon.  T. Imran  Research Laboratory of Lasers
(RLL)-Group of Laser Development (GoLD), Department of Physics, COMSATS
University Islamabad, Park Road 45550, Islamabad, Pakistan.  G. O. Williams
GoLP/Instituto de Plasmas e Fusão Nuclear-Laboratório Associado, Instituto
Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal.  M.
Fajardo  GoLP/Instituto de Plasmas e Fusão Nuclear-Laboratório Associado,
Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal.
H. Merdji<EMAIL_ADDRESS>LIDYL, CEA, CNRS, Université Paris-Saclay, CEA
Saclay 91191, Gif-sur-Yvette, France.

###### Abstract

High harmonic generation (HHG) in crystals has revealed a wealth of
perspectives such as all-optical mapping of the electronic band structure,
ultrafast quantum information and the creation of novel all-solid-state
attosecond sources. Significant efforts have been made to understand the
microscopic aspects of HHG in crystals, whereas the macroscopic effects, such
as non-linear propagation effects of the driving pulse inside the dense solid
media and its impact on the HHG process is often overlooked. In this work, we
study macroscopic effects by comparing two materials with distinct optical
properties, silicon (Si) and zinc oxide (ZnO). By scanning the focal position
of 85 fs, 2.123 $\mu$m wavelength pulses inside the crystals (Z-scan) we
reveal spectral shifts in the generated harmonics. We interpret the overall
blueshift of the emitted harmonic spectrum as an imprint of the driving field
spectral modulation occurring during the propagation inside the crystal. This
is supported with numerical simulations. This study demonstrates that through
manipulation of the fundamental driving field through non-linear propagation
effects, precise control of the emitted HHG spectrum in solids can be
realised. This method could offer a robust way to tailor HHG spectra for a
range of spectroscopic applications.

††preprint: AIP/APL

##  I Introduction

In the last decade, solid-state HHG has been studied in a wide variety of
crystals and amorphous solids Ghimire  _et al._ (2011); You, Reis, and Ghimire
(2017); You  _et al._ (2017). The physical picture of solid-state HHG is in
its infancy and is the subject of ongoing discussion Golde, Meier, and Koch
(2008); Higuchi, Stockman, and Hommelhoff (2014); Hawkins, Ivanov, and
Yakovlev (2015); Wu  _et al._ (2016); Tancogne-Dejean  _et al._ (2017); Floss
_et al._ (2018). A better understanding of the physical process of solid-state
HHG will not only lead to improved secondary sources, but to an ability to map
the nonlinear properties and possibly allow all-optical mapping of the
electronic structure of solids Marangos (2011); Schubert  _et al._ (2014);
Hohenleutner  _et al._ (2015); Luu  _et al._ (2015); Vampa  _et al._ (2015a);
Ghimire and Reis (2018). HHG in solids must be realised in materials
transparent to the driving laser pulse wavelength, which is generally in the
near to mid infra-red region. Therefore, insulating and semi-conducting solids
are the materials of choice. The HHG process in solids shares some
similarities with HHG in gases, at least conceptually Vampa  _et al._ (2015a);
Vampa and Brabec (2017). The first step is an inter-band transition in the
material that promotes a charge carrier to a conduction band. Once there, this
electron is driven by the fundamental field, and can emit harmonics due to
oscillations within a band (intra-band), or excitation and relaxation between
bands (inter-band) Vampa  _et al._ (2015b); Ghimire and Reis (2018).

The initial inter-band excitation requires tunnelling or a multi-photon
absorption process, and the rate of free carrier generation is therefore
inversely proportional to the band gap. Apart from the microscopic physics of
carrier generation and motion, the properties of the HHG emission can be
influence by the propagation of the electric fields inside the crystal. This
macroscopic propagation is nonlinear for the driving field and linear for the
harmonics that have a much lower intensity. The emitted HHG spectrum can
reflect the available electronic states of the material. This makes the HHG
spectrum an intricate fingerprint of the complex interplay of microscopic and
macroscopic processes. For example, HHG in ZnO Ghimire  _et al._ (2011);
Gholam-Mirzaei, Beetar, and Chini (2017); Gholam-Mirzaei  _et al._ (2018) has
been shown to reveal both the inter and intra-band mechanisms coupled with the
non-linear response of the crystal. Similarly, harmonics have been generated
in the most widely used material in electronics, silicon (Si), at 2.1
$\mathrm{\mu m}$ driving wavelength Vampa  _et al._ (2016, 2017) and used to
disentangle the surface and bulk contributions Vampa  _et al._ (2019). HHG in
Si and ZnO can be controlled and monitored by manipulating the crystal
surfaces Sivis  _et al._ (2017); Franz  _et al._ (2019). For example, spatial
shaping of the crystal surface has been used to create HHG beams that carry an
orbital angular momentum (OAM) Gauthier  _et al._ (2019) without the need of
expensive device for spatial phase control. These generated OAM beams have
many applications varying from the manipulation of nanoparticles to quantum
cryptography Torres and Torner (2011).

The propagation of the intense driving field has been previously linked with
redshifts in the HHG spectrum in plasmas and gases Brandi, Giammanco, and
Ubachs (2006); Bian and Bandrauk (2013); Du  _et al._ (2015). This effect is
attributed to the delayed emission of HHG from excited states or resonant
states Bian and Bandrauk (2011) or dislocation of molecules Bian and Bandrauk
(2014). In solids, numerically computed HHG spectra have shown redshifts in
the "higher plateaus" due to higher band transitions Jia, Huang, and Bian
(2017). In sapphire, intensity-dependent redshifts and blueshifts were
observed in the spectrum of harmonics and attributed to the long and short
electron/hole trajectories, respectively Kim  _et al._ (2019). These studies
showed the electronic effects on the HHG process, while the macroscopic or
propagation effects of the driving field on the spectral profile of harmonics
hasn’t been fully explored.

In the present study, we investigate the impact of the non-linear macroscopic
beam propagation in the HHG process in Si and ZnO. We have generated 3rd (H3),
5th (H5) and 7th (H7) order-harmonics in Si and ZnO crystals at 2.123
$\mathrm{\mu m}$ driving central wavelength in the sub-TW/cm2 intensity
regime. The spectral and spatial profiles of these high order harmonics are
measured. This letter is structured as follows; the experimental setup to
generate HHG and to observe the propagation effects of the driving field is
described in section II. HHG in Si and ZnO, and their spectral shift are
reported in sections III.1 and III.2, respectively. Section IV presents a
discussion of the experimental results linked with numerical propagation of a
focused driving laser field in Si. Finally, the conclusion is reported in
section V.

##  II Experimental setup

Figure 1: Schematic representation of the experimental setup to generate high
order harmonics (H3, H5 and H7) in the 300 $\mathrm{\mu m}$ thick silicon (Si)
and 200 $\mathrm{\mu m}$ thick-(100) oriented ZnO crystal. The convention for
the Z-scan notation is explained in the inset. Note that, P: polarizer, L:
lens, T: telescope, S: sample, HF: harmonic filters, FM: flip mirror, CCD:
charged coupled device.

A schematic representation of the crystal HHG experimental setup is shown in
Fig. 1. We have employed a fibre laser (NOVAE Company) operating at 2.123
$\mathrm{\mu m}$ with a pulse energy of $\sim$ 8 nJ and a repetition rate of
19 MHz. To enable a tighter focus and thus achieve higher intensities, the
size of the laser beam is magnified by a factor 3.75 times using a telescope
(T). The 85 fs duration pulses are focused in the crystals by a convex lens of
3 cm focal length up to a maximum peak intensity of $\sim$ 0.38 TW/cm2 well
below the damage threshold of crystals. The generated diverging harmonics are
further focused by a convex lens of 15 cm focal length either on a UV-enhanced
CCD camera or a UV-VIS spectrometer, for spatial and spectral beam
characterization, respectively. Band-pass filters centred at each harmonic
wavelength are used to separate and characterise the HHG properties. Using the
optimized coupled spectrometer, Z-scans are performed to localize the
longitudinal medium response as illustrated in the inset of figure 1.

##  III Results and discussions

###  III.1 HHG in Si and spectral shifting of harmonics

We have generated harmonics in a 300 $\mathrm{\mu m}$ thick, (100) oriented,
Si crystal. The spectral and spatial profiles of H3, H5 and H7 harmonics are
shown in Fig. 2(a). The z-scans for H3 and H5 are recorded with steps of 10
$\mathrm{\mu m}$ as shown in Fig. 2(b). The signal of H7 is very weak and is
therefore excluded from the Z-scan study. The yield of H3 and H5 reaches their
maximum when the focus position of the driving field on the back surface of
the Si crystal. This is attributed to the absorption of the harmonics
generated at the front surface and inside the Si crystal.Compared to H3, the
generation region of H5 is translated $\sim$ 5 $\mathrm{\mu m}$ towards the
back surface due to its stronger absorption in the crystal (see Fig. 2(b)). In
parallel, we have measured the intensity-dependent spectral shifts of the
harmonics generated in Si.

(a)

(b)

Figure 2:  Spectral and spatial measurements of high order harmonics in a 300
$\mathrm{\mu m}$ thick Si crystal. The spectral and spatial profile of each
harmonic measured separately with the corresponding harmonic filters for
different acquisition time. (a) H5 multiplied by 100 and H7 by 1000 to make
them visible in the combined graph. (b) Z-scan measurements of H3 (red dots)
and H5 (blue dots) in the Si crystal. Figure 3: The intensity-dependent
spectral blueshifts of the harmonic signal generated in the 300 $\mathrm{\mu
m}$ thick Si crystal. (a) The driving intensity evolution ($I_{z}$) at the
rear face of the crystal with the propagation distance "z". (b) spectral shift
in H3 (c) spectral shift in H5.

The driving intensity ($I_{z}$) at the surface of the crystal is estimated as
$I_{z}$=$2P/\pi w_{z}^{2}$, where $w_{z}$ is the beam waist at z calculated
using hyperbolic function Bandres and Gutiérrez-Vega (2004) by considering the
non-linear refractive indices of crystals with a confocal parameter b=46
$\mathrm{\mu m}$ and $P$ is the peak power which is 0.05 MW in our case shown
in Fig. 3a. An intensity-dependent spectral shift of about 3.38 nm for H3 and
3.9 nm for H5 is observed in Si (see. Fig. 3b and 3c) show the intensity-
dependent spectral shift of H3 and H5 generated from Si. As the laser focus is
scanned from outside the crystal (at the position -60 $\mathrm{\mu m}$) to
inside the bulk (at the position of +60 $\mathrm{\mu m}$) a spectral blue
shift in H3 and H5 is observed. Indeed, when the laser focus is outside the
crystal, the spectrum of H3 is centered at 704 nm, while that of H5 is
centered at 425 nm. In this configuration, we expect that nonlinear
propagation effects are minimal. However when the laser focus passes inside
the crystal, the spectrum of H3 shifts to 701 nm, while that of H5 is shifts
to 421 nm. In this case, the local field intensity is very high and triggers
nonlinear effects.

###  III.2 HHG in ZnO and spectral shifting of harmonics

(a)

(b)

Figure 4: (a) Spectrum of the harmonic emission in ZnO at z=0 for different
acquisition time, with the harmonic beam profiles in insets, (b) Z-scan
measurements of H3 (red dots), H5 (blue diamonds) and H7 (violet hexagons)
order harmonics.

The combined harmonic spectrum (H3, H5 and H7) generated in ZnO and the
corresponding spatial profiles are shown in Fig. 4(a). The measured harmonic
signal (H3, H5, and H7) as a function of the focus position of the driving
beam in the ZnO crystal is shown in Fig. 4(b). The Z-scan of H3 exhibits a
double peak structure. The first peak corresponds to the generation from the
front side of the crystal, while the second peak at z=0 corresponds to
generation from the rear side of the crystal. The signal of H3 is reduced when
the laser peak intensity is in the centre of the crystal. H3 generated before
and after the focus point in the bulk contribute destructively when exiting
the surface. As a result, the signal of H3 drops when the laser peak intensity
is at the centre of the crystal while the signal of H3 is maximum when the
laser is focused near the surfaces as shown in Fig. 4(b). We observe a single
peak for H5 and H7 which shows the detected harmonics originate from the rear
surface of the crystal. Indeed, the harmonics H5 and H7 generated from the
front surface and bulk of the crystal are absorbed. As the harmonic order
increases, the region of efficient harmonic generation moves closer to the
back surface of the crystal.

We now turn to the corresponding spectral response of the harmonics.
Blueshifts in the spectral profile of the harmonics are observed with the
translation of the laser focus. Fig. 5a shows the spectral response of H3 for
different relative focus positions. A spectral modulation (shifting) and
splitting of H3 peaks (double peak spectrum is observed) which are separated
up to 12 nm when the focus of the driving field is in the middle of the ZnO
crystal. The first peak of H3 begins to decrease while the second peak of H3
builds up with the translation of the driving focus towards the front surface.
The intensity of H3 increases with the spectral blueshift while we move from
outside the crystal towards the back surface of the crystal (from z = - 60
$\mathrm{\mu m}$ = - 1.30b to z = 0, triangle data points) and reached a
maximum at z = 0 which corresponds to the rear surface of the ZnO crystal.

Figure 5: (a) The intensity-dependent spectral shifting and splitting of H3 in
ZnO (two curve statistics before and after the middle). (b) The evolution of
the driving intensity ($I_{z}$) at the rear surface of crystal for different
propagation distance (z) (green line). (c) H3 generated from the front surface
(red data points) and at a back surface (triangle data points) of ZnO crystal.
H3 exhibits a blueshift as the focus of the driving field translates towards
the back and front surfaces. The z = 0 corresponds to the front surface for
the red data points and the back surface of crystal for the triangle data
points. The Z-scan depth for HHG in terms of confocal parameters "b" is in the
range of -1.3b to +0.9b, where b=46 $\mathrm{\mu m}$. (d) spectral shift in
H5, (e) spectral shift in H7.

When further translating the laser focus inside the crystal (z = + 40
$\mathrm{\mu m}$ = +0.9b), the intensity of H3 decreases. However, the width
of the spectral shift increases even more. H3 generated from the front surface
(red rectangle) and back surface (black triangle) of the crystal undergoes
significant blueshifts (8.2 nm and 12.2 nm respectively) as shown in Fig. 5c.
This different spectral shift of H3 attributed to the different propagation
distance of the intense driving beam at the front and back surface of crystal.
We have observed spectral blueshifts in the spectrum of H5 and H7 as shown in
Figs. 5d and 5e, respectively. There were no spectral modulations and
splitting as observed for H3. The signals of H5 and H7 (Fig. 4(b)) has
increased when translating the laser peak intensity towards the rear surface.
By further translating the focus of the driving field inside the crystal, the
signal of H5 and H7 decreases. The total spectral shift across our measurement
range is of about 3.91 nm for H5 and 3.89 nm for H7 and exhibits spectral
blueshifts as shown in Figs. 5d and 5e, respectively.

##  IV Modelling and Discussions

To understand the physical processes that give rise to the harmonic spectral
shift, we model the propagation of the driving field inside the Si crystal. We
restrict the calculation to z>>zR, where z is the focus position inside the
crystal and zR is the Rayleigh length. We have used the model described in
Ref. Couairon  _et al._ (2005), with the exception that the instantaneous Kerr
effect is added to our model. The group velocity dispersion and higher-order
dispersion are calculated using a Taylor expansion in
$\omega-\omega\textsubscript{0}$ up to the fifth order of the frequency-
dependent $k(\omega$)=$n(\omega)\omega\textsubscript{0}/c$. We then solve the
paraxial wave equation in cylindrical geometry, coupled to the evolution
equation of electron density for a 85 fs pulse duration, $\sim$ 7 mm Gaussian
beam diameter focused onto the 300 $\mathrm{\mu m}$ Si crystal to 2.86
$\mathrm{\mu m}$ by a 3 cm focal length lens. Initially, we have assumed that
the conduction band is empty. The spatially integrated laser input spectrum
(purple line) and the spectrum obtained after propagation through the 300
$\mathrm{\mu m}$ Si crystal for focus position z=300 $\mathrm{\mu m}$ (yellow
line) is shown in Fig. 6(a).

The macroscopic non-linear propagation of the laser pulses in the crystal
induces a spectral blueshift of the driving field. This shift will results in
a shift of the central wavelength of any high order harmonics generated at the
rear side of the crystal. The experimentally measured central wavelengths of
the generated harmonics after the propagation of the driving field through a
300 $\mathrm{\mu m}$ thick Si crystal are 702.4 nm and 423.5 nm for H3 and H5,
respectively. Fig. 6(b) compare the measured spectral shift at z=0 for the
different harmonic orders with the expected values from the numerical
propagation of the fundamental pulses. Note that the spectral shift observed
for H3 is larger in Si compared to ZnO, while the behaviour is the opposite
for higher orders. However, the measured shift for H3 in Si correspond to the
expected value, there is a discrepancy between the numerical simulation and
the measurement for H5 and H7 as illustrated in Fig. 6(b).

(a)

(b)

Figure 6: (a) Spatially integrated normalized laser spectrum after propagating
through Si crystal, input spectrum (purple line) central wavelength at 2.123
$\mathrm{\mu m}$, and for focus position z= 300 $\mathrm{\mu m}$ (yellow
line), respectively. (b) Central wavelength dependence of HHG, HHG according
to non-shifted driving wavelength (blue cube), HHG observed in Si (red
rhombus), HHG central wavelength observed in ZnO (green square) and as per
modelling central wavelength HHG in Si (sky blue circle).

We attribute the HHG spectral shifts to the non-linear propagation of the
intense driving field inside the crystal. Indeed, the strong photoionization
contributes to the generation of free-electrons during the transition from the
valence band to the conduction band. This induces an intensity-dependent phase
delay that results in a blue shift of the generated frequencies. We have found
that the distance of propagation inside the crystal is enough to cause a
significant blue shift of the HHG spectrum. This effect is more pronounced
when beam is focused at the centre of the crystal. Due to the tight focusing
geometry, the intensity dependent interaction region is quite small, therefore
the contribution of self-phase modulation would be negligible. Note that it
has been reported that interference of H3 generated from the interface and
throughout the bulk medium can cause spectral modulations in the third
harmonics generated in ZnO Franz (2018). We have calculated a fringe spacing
of 18 nm in 200 $\mathrm{\mu m}$ ZnO while measured peaks are separated by 12
nm. Furthermore, the width of the bandgap is shown to affect the re-absorption
of the harmonic spectrum. Wider bandgap materials (such as ZnO used in this
study) exhibit less re-absorption and more control of the harmonic spectrum as
compared to Si. Through non-linear propagation modelling, we find that the
generation of free-carrier density is the principle physical mechanism causing
these shifts which induces the change in refractive index due to the
contributions of free electron, $n_{e}(t)\sim$
$1-e^{2}N_{e}(t)/(8\pi^{2}m_{e}\varepsilon_{0}v^{2})$, where $m_{e}$,$e$ are
the electron mass and charge, respectively,$\varepsilon_{0}$ is the vacuum
dielectric constant and $N_{e}(t)$ is the free electron density. During the
photoionization process of valence band, $N_{e}(t)$ increases which result in
decreases of $n_{e}(t)$ and as a result blueshift induces in the harmonics.

##  V Conclusion

We have observed front and back surface emission due to phase matching of the
H3 in ZnO while back surface emission observed due to re-absorption of over
bandgap harmonics in Si and ZnO. In ZnO, the generation of the H3 is efficient
from the front and rear surfaces of the crystal and a modulated spectrum
observed when focusing at the centre of the sample. We attribute this to the
interference of harmonics generated at the front surface and in the bulk
crystal. We have been demonstrated the changes in the emitted harmonic
spectrum, as a function of driving laser focal position in the crystals. The
general trend we observe is that the harmonic peaks are blueshifted as the
intensity of driving pulses is translated into crystals. Due to the dependence
on the focal position, and hence the propagation length in the crystal, we
attribute the observed blueshifts to non-linear propagation effects of the
driving laser. This is supported by calculations of the non-linear propagation
of the driving field in Si which exhibits a blueshift of the central
frequency, attributed to the photoionization of valence band electrons
promoted to the conduction band by the driving field. Even in samples that
strongly absorb the HHG spectrum, we show that since the emitted harmonics
come from a small region at the end of the crystal, modulation of the driving
pulse prior to the point of generation can be used to control the HHG
spectrum. For materials with bandgaps higher than the harmonic photon energy,
further control of the spectrum can be obtained from front and back surface
emission, as we have demonstrated with the third harmonic in ZnO. This study
reveals the fine control of the HHG emission spectrum through macroscopic non-
linear effects in the bulk crystal alone. This technique could be used for
tailoring frequencies for ultrafast spectroscopic techniques or spectral
shaping for attosecond pulse generation.

## Funding

This work was partially supported by the Fundação para a Ciência e Tecnologia
(FCT) under the grant number PD/BD/135224/2017 in the framework of the
Advanced Program in Plasma Science and Engineering (APPLAuSE).
We acknowledge support from the PETACom FET Open H2020 grant number 829153,
OPTOLogic FET Open H2020 grant number 899794, from the European Union’s
Horizon 2020 research and innovation program under grant agreement no 871124
Laserlab-Europe, support from the French ministry of research through the DGA
RAPID grant “SWIM”, and the LABEX “PALM” (ANR-10-LABX-0039-PALM) through the
grants ”Plasmon-X”, “STAMPS” and “HILAC”. We acknowledge the financial support
from the French ASTRE program through the “NanoLight” grant.

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#  Whittaker category for the Lie algebra of polynomial vector fields

Yufang Zhao and Genqiang Liu

###### Abstract.

For any positive integer $n$, let $A_{n}=\mathbb{C}[t_{1},\dots,t_{n}]$,
$W_{n}=\text{Der}(A_{n})$ and
$\Delta_{n}=\text{Span}\\{\frac{\partial}{\partial{t_{1}}},\dots,\frac{\partial}{\partial{t_{n}}}\\}$.
Then $(W_{n},\Delta_{n})$ is a Whittaker pair. A $W_{n}$-module $M$ on which
$\Delta_{n}$ operates locally finite is called a Whittaker module. We show
that each block $\Omega_{\mathbf{a}}^{\widetilde{W}}$ of the category of
$(A_{n},W_{n})$-Whittaker modules with finite dimensional Whittaker vector
spaces is equivalent to the category of finite dimensional modules over
$L_{n}$, where $L_{n}$ is the Lie subalgebra of $W_{n}$ consisting of vector
fields vanishing at the origin. As a corollary, we classify all simple non-
singular Whittaker $W_{n}$-modules with finite dimensional Whittaker vector
spaces using $\mathfrak{gl}_{n}$-modules. We also obtain an analogue of
Skryabin’s equivalence for the non-singular block $\Omega_{\mathbf{a}}^{W}$.

Keywords: Whittaker module, Smash product, weighting functor, Weyl algebra,
Skryabin’s equivalence.

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

##  1\. Introduction

Whittaker modules are important objects in the representation theory since its
emergence in the study of complex semisimple Lie algebras. Let
$\mathfrak{g}=\mathfrak{n}_{-}\oplus\mathfrak{h}\oplus\mathfrak{n}_{+}$ be a
complex semisimple Lie algebra and let
$\eta:\mathfrak{n}_{+}\rightarrow\mathbb{C}$ be a Lie algebra homomorphism. A
$\mathfrak{g}$-module is called a Whittaker module if $x-\eta(x)$ acts locally
nilpotent on it for any $x\in\mathfrak{n}_{+}$. Whittaker modules were first
defined for $\mathfrak{sl}_{2}$ by Arnal and Pinzcon [2]. Kostant [18] defined
Whittaker modules for all complex semisimple Lie algebras. When $\eta$ is non-
singular, Kostant showed that simple Whittaker modules are in bijective
correspondence with maximal ideals of the center $Z(\mathfrak{g})$ of the
universal enveloping algebra of $\mathfrak{g}$. For quantum groups, Whittaker
modules have been studied in [26, 34, 35]. Whittaker modules have also been
studied for the Virasoro algebra in [27, 28, 15, 22, 23], for Heisenberg
algebras in [12], for Weyl algebras and generalized Weyl algebras in [7], for
affine Lie algebras in [1, 10] and for classical Lie-super algebras in [8]. In
[6], the authors gave a more general picture of Whittaker modules for many Lie
algebras which includes, in particular, Lie algebras with triangular
decomposition and simple Lie algebras of Cartan type. They described some
basic properties of Whittaker modules, including a block decomposition of the
category of Whittaker modules and some properties of simple Whittaker modules
under some assumptions.

Among the theory of infinite dimensional Lie algebras, the Lie algebras of
polynomial vector fields on irreducible affine algebraic varieties is an
important class of Lie algebras. These algebras contain the Cartan type Lie
algebra $W_{n}$ which is the derivation Lie algebra of
$A_{n}=\mathbb{C}[t_{1},\dots,t_{n}]$. Weight modules with finite dimensional
weight spaces (also called Harish-Chandra modules) for $W_{n}$ has been well
developed. In 1974-1975, A. Rudakov addressed the classification of a class of
irreducible $W_{n}$-representations that satisfy some natural topological
conditions, see [31, 32]. Tensor modules $T(P,V)$ over $W_{n}$ were introduced
by Shen and Larsson, see [33, 19], for a $\mathcal{D}_{n}$-module $P$ and
$\mathfrak{gl}_{n}$-module V, where $\mathcal{D}_{n}$ is the Weyl algebra. In
[20], the structure of any $T(P,V)$ was completely determined. When $V$ is
finite dimensional, the simplicity of $T(P,V)$ over the Witt algebra on an
$n$-dimensional torus was given by Rao, see [13]. The classification of simple
Harish-Chandra modules over $W_{1}$ was given in [24]. Very recently, based on
the classification of simple uniformly bounded $W_{n}$ modules (see [9] for
$n=2$ and [36] for any $n$) and the description of the weight set of simple
weight $W_{n}$ modules (see [29]), D. Grantcharov and V. Serganova completed
the classification of simple Harish-Chandra $W_{n}$-modules, see [17]. They
proved that every such nontrivial module $M$ is a simple sub-quotient of some
tensor module $T(P,V)$. In [11], tilting modules and their character formulas
for category $\mathcal{O}$ of $W_{n}$ were described. However non-weight
modules and weight modules that are not Harish-Chandra modules for $W_{n}$ are
not well developed.

In this paper, we study the category of Whittaker modules for $W_{n}$. Let
$\Delta_{n}=\text{Span}\\{\frac{\partial}{\partial{t_{1}}},\dots,\frac{\partial}{\partial{t_{n}}}\\}$
which is a commutative subalgebra of $W_{n}$. Then $(W_{n},\Delta_{n})$ is a
Whittaker pair in the sense of [6]. A $W_{n}$-module $M$ on which $\Delta_{n}$
operates locally finite is called a Whittaker module. We denote the semidirect
Lie algebra $W_{n}\ltimes A_{n}$ by $\widetilde{W}_{n}$. A
$\widetilde{W}_{n}$-module $M$ is called an $(A_{n},W_{n})$-module, if the
action of $A_{n}$ on $M$ is associative. This kind of modules play an
important role in the classification of simple modules for Cartan type Lie
algebras, see [3, 4, 16, 14, 36]. For an $\mathbf{a}\in\mathbb{C}^{n}$ and a
Whittaker $W_{n}$-module $M$, a nonzero $v\in M$ is called a Whittaker vector
if $\frac{\partial}{\partial{t_{i}}}v=a_{i}v$ for any $i\in\\{1,\cdots,n\\}$.
Let $\Omega_{\mathbf{a}}^{W}$ (resp. $\Omega_{\mathbf{a}}^{\widetilde{W}}$) be
the category consisting of $W_{n}$-modules (resp. $(A_{n},W_{n})$-modules) $M$
on which $\frac{\partial}{\partial{t_{i}}}-a_{i}$ acts locally nilpotent for
any $i\in\\{1,\cdots,n\\}$ and having finite dimensional Whittaker vector
spaces. We show that each block $\Omega_{\mathbf{a}}^{\widetilde{W}}$ is
equivalent to the category of finite dimensional modules over $L_{n}$, where
$L_{n}=\text{Span}\\{f(t)\frac{\partial}{\partial{t_{i}}}\mid f(t)\in
A_{n},f(0)=0,i=1,\dots,n\\}$, see Theorem 3.4. When
$\mathbf{a}\in(\mathbb{C}^{*})^{n}$, $\Omega_{\mathbf{a}}^{W}$ is said to be
non-singular. We show that any module in the non-singular block
$\Omega_{\mathbf{a}}^{W}$ is a free $U(\mathfrak{h}_{n})$-module of finite
rank when restricted to $U(\mathfrak{h}_{n})$, where $\mathfrak{h}_{n}$ is a
Cartan subalgebra of $W_{n}$. Using the covering method introduced in [4], we
show that any simple module in the non-singular block
$\Omega_{\mathbf{a}}^{W}$ is isomorphic to some simple sub-quotient of
$T(A_{n}^{\mathbf{a}},V)$ for some finite dimensional simple
$\mathfrak{gl}_{n}$-module $V$, see Theorem 4.6. We also define the universal
Whittaker module
$Q_{\mathbf{a}}=U(W_{n})\otimes_{U(\Delta_{n})}\mathbb{C}_{\mathbf{a}}$ and
its endomorphism algebra
$H_{\mathbf{a}}=\text{End}_{W_{n}}(Q_{\mathbf{a}})^{\text{op}}$. When
$\mathbf{a}\in(\mathbb{C}^{*})^{n}$, we show that $\Omega_{\mathbf{a}}^{W}$
and the category of finite dimensional $H_{\mathbf{a}}$-modules are
equivalent, see Theorem 4.8. A similar equivalence for finite $W$-algebras
$W(\mathfrak{g},e)$ was shown by Serge Skryabin, see the appendix in [30].

We denote by $\mathbb{Z}$, $\mathbb{N}$, $\mathbb{Z}_{\geq 0}$, $\mathbb{C}$
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

###  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 $\\{\mathbf{e}_{1},\mathbf{e}_{2},\dots,\mathbf{e}_{n}\\}$.

Denote by $A_{n}=\mathbb{C}[t_{1},t_{2},\cdots,t_{n}]$ 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
$t^{\mathbf{m}}=t_{1}^{m_{1}}\cdots t_{n}^{m_{n}}$,
$h^{\mathbf{m}}=h_{1}^{m_{1}}\cdots h_{n}^{m_{n}}$ and
$|\mathbf{m}|=m_{1}+\cdots+m_{n}$ for any
$\mathbf{m}=(m_{1},\dots,m_{n})\in\mathbb{Z}_{\geq 0}^{n}$. Recall that the
classical Witt algebra
$W_{n}=\text{Der}(A_{n})=\oplus_{i=1}^{n}{A}_{n}\partial_{i}$ is a free
$A_{n}$-module of rank $n$. 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 $\mathbf{m},\mathbf{r}\in\mathbb{Z}_{\geq 0}^{n}$.

It is well-known that $W_{n}$ is a simple Lie algebra. The subspace
$\mathfrak{h}_{n}=\mathbb{C}h_{1}\oplus\cdots\oplus\mathbb{C}h_{n}$ is a
Cartan subalgebra (a maximal abelian subalgebra that is diagonalizable on
$W_{n}$ with respect to the adjoint action) of $W_{n}$.

Note that $A_{n}$ is a $W_{n}$-module under the derivation action of $W_{n}$.
We denote the semidirect Lie algebra $W_{n}\ltimes A_{n}$ by
$\widetilde{W}_{n}$ which is called the extended Witt algebra. Let $W_{n}$-Mod
be the category of finitely generated left $U(W_{n})$-modules.

######  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.,
$M=\bigoplus_{\lambda\in\mathfrak{h}_{n}^{*}}M_{\lambda},$ where

$M_{\lambda}=\\{v\in M\mid hv=\lambda(h)v,\forall\ h\in\mathfrak{h}_{n}\\}.$

A weight $W_{n}$-module $M$ is uniformly bounded if the dimension of each
weight space $M_{\lambda}$ is smaller than a fixed integer.

######  Definition 2.2.

A module $M$ over $\widetilde{W}_{n}$ is called an $(A_{n},W_{n})$-module if
the action of $A_{n}$ on $M$ is associative, i.e., $g(fv)=(gf)v$, for any
$f,g\in A_{n},v\in M$.

######  Example 2.1.

  1. $($a$)$

The natural module $A_{n}$ is an $(A_{n},W_{n})$-module.

  2. $($b$)$

The adjoint $W_{n}$-module $W_{n}$ is an $(A_{n},W_{n})$-module under the
following action:

$f(t)\cdot g(t)\partial_{i}=(f(t)g(t))\partial_{i},\ \ f(t),g(t)\in A_{n},$

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

###  2.2. Whittaker modules

Denote $\Delta_{n}=\text{span}\\{\partial_{1},\dots,\partial_{n}\\}$ which is
an abelian subalgebra of $W_{n}$. Since the adjoint action of $\Delta_{n}$ on
$W_{n}/\Delta_{n}$ is locally nilpotent. Thus $(W_{n},\Delta_{n})$ is a
Whittaker pair in the sense of [6]. A $W_{n}$-module $M$ is called a Whittaker
module if the action of $\Delta_{n}$ on $M$ is locally finite. For an
$\mathbf{a}=(a_{1},\cdots,a_{n})\in\mathbb{C}^{n}$, we can define a Lie
algebra homomorphism $\phi_{\mathbf{a}}:\Delta_{n}\rightarrow\mathbb{C}$ such
that $\phi_{\mathbf{a}}(\partial_{i})=a_{i}$ for any $i\in\\{1,\cdots,n\\}$. A
Whittaker module $M$ is of type $\phi_{\mathbf{a}}$ if for any $v\in M$ there
is a $k\in\mathbb{N}$ such that $(x-\phi_{\mathbf{a}}(x))^{k}v=0$ for all
$x\in\Delta_{n}$. We also define the subspace of $M$ as follows:

$\text{Wh}_{\mathbf{a}}(M)=\\{v\in M\mid xv=\phi_{\mathbf{a}}(x)v,\ \forall\
x\in\Delta_{n}\\}$

of $M$. An element in $\text{Wh}_{\mathbf{a}}(M)$ is called a Whittaker
vector.

We consider Whittaker modules under some natural finite condition. Let
$\Omega_{\mathbf{a}}^{W}$ be the category consisting of Whittaker
$W_{n}$-modules of type $\phi_{\mathbf{a}}$ such that
$\dim\text{Wh}_{\mathbf{a}}(M)<+\infty$. Similarly let
$\Omega_{\mathbf{a}}^{\widetilde{W}}$ be the category consisting of Whittaker
$(A_{n},W_{n})$-modules of type $\phi_{\mathbf{a}}$ such that
$\dim\text{Wh}_{\mathbf{a}}(M)<+\infty$.

For a vector $\mathbf{a}\in\mathbb{C}^{n}$, we call it non-singular if
$a_{i}\neq 0$ for any $i\in\\{1,\cdots,n\\}$.

###  2.3. Weighting functor

In this subsection, we recall the weighting functor introduced in [25], which
establishes a relationship between weight modules and non-weight modules. For
an $\mathbf{r}=(r_{1},\dots,r_{n})\in\mathbb{C}^{n}$, let $I_{\mathbf{r}}$ be
the maximal ideal of $U(\mathfrak{h}_{n})$ generated by

$h_{1}-r_{1},\dots,h_{n}-r_{n}.$

For any $W_{n}$-module $M$ and $\mathbf{r}\in\mathbb{C}^{n}$, set
$M^{\mathbf{r}}:=M/I_{\mathbf{r}}M$. Denote

$\mathfrak{W}(M):=\bigoplus_{\mathbf{r}\in\mathbb{Z}^{n}}M^{\mathbf{r}}.$

By Proposition 8 in [25], we have the following construction.

######  Proposition 2.3.

The vector space $\mathfrak{W}(M)$ becomes a weight $W_{n}$-module under the
following action:

(2.1)
$t^{\mathbf{m}+\mathbf{e}_{i}}\partial_{i}\cdot(v+I_{\mathbf{r}}M):=t^{\mathbf{m}+\mathbf{e}_{i}}\partial_{i}v+I_{\mathbf{r}+\mathbf{m}}M,v\in
M,\mathbf{r}\in\mathbb{Z}^{n},$

where $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ with $m_{i}\geq-1$.

In Section 4, we will use the weighting functor to show that the
$(A_{n},W_{n})$-cover $\widehat{M}$ has a finite composition series as an
$(A_{n},W_{n})$-module for any module $M$ in $\Omega_{\mathbf{a}}^{W}$ when
$\mathbf{a}$ is non-singular.

##  3\. Whittaker $(A_{n},W_{n})$-modules

In this section, we study Whittaker $(A_{n},W_{n})$-modules. Explicitly, we
establish an equivalence between $\Omega_{\mathbf{a}}^{\widetilde{W}}$ and the
category of finite dimensional $L_{n}$-modules.

###  3.1. Whittaker modules over $\mathcal{D}_{n}$

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

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

A $\mathcal{D}_{n}$-module $M$ is called a Whittaker module of type
$\mathbf{a}$ if for any $v\in M$ there is a $k\in\mathbb{N}$ such that
$(\partial_{i}-a_{i})^{k}v=0$ for all $i\in\\{1,\cdots,n\\}$. Let
$\mathcal{V}_{\mathbf{a}}^{D}$ be the category consisting of finite length
Whittaker $\mathcal{D}_{n}$-modules of type $\mathbf{a}$.

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 the
action of $\mathcal{D}_{n}$ on $A_{n}^{\mathbf{a}}$ is defined by

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

######  Lemma 3.1.

  1. $($a$)$

Any simple module in $\mathcal{V}_{\mathbf{a}}^{D}$ is isomorphic to
$A_{n}^{\mathbf{a}}$.

  2. $($b$)$

The category $\mathcal{V}_{\mathbf{a}}^{D}$ is semi-simple.

###### Proof.

(a) For any $f(t)\in A_{n}^{\mathbf{a}}$, the formula
$(\partial_{i}-a_{i})\cdot t^{\mathbf{m}}=m_{i}t^{\mathbf{m}-\mathbf{e}_{i}}$
tells us that the degree of $t_{i}$ in $(\partial_{i}-a_{i})f(t)$ is smaller
than that of $t_{i}$ in $f(t)$. By induction on the degree of $f(t)$, any
submodule of $A_{n}^{\mathbf{a}}$ must contain $1$ which generates
$A_{n}^{\mathbf{a}}$. So $A_{n}^{\mathbf{a}}$ is a simple
$\mathcal{D}_{n}$-module. If $V$ is a simple module in
$\mathcal{V}_{\mathbf{a}}^{D}$, then there is a nonzero $v\in V$ such that
$\partial_{i}v=a_{i}v$ for any $i$. Then the map $V\rightarrow
A_{n}^{\mathbf{a}},t^{\mathbf{m}}v\mapsto t^{\mathbf{m}}$ defines a
$\mathcal{D}_{n}$-module isomorphism between $V$ and $A_{n}^{\mathbf{a}}$.

(b) Since $A^{\mathbf{a}}_{n}$ is equivalent to $A_{n}$ up to the algebra
automorphism $\sigma_{\mathbf{a}}$ of $\mathcal{D}_{n}$ defined by
$t_{i}\mapsto t_{i},\partial_{i}\mapsto\partial_{i}+a_{i}$,
$\text{Ext}^{1}_{\mathcal{D}_{n}}(A_{n}^{\mathbf{a}},A_{n}^{\mathbf{a}})\simeq\text{Ext}^{1}_{\mathcal{D}_{n}}(A_{n},A_{n})$.
By Lemma 3 in [21], $\text{Ext}^{1}_{\mathcal{D}_{n}}(A_{n},A_{n})=0$. So
$\mathcal{V}_{\mathbf{a}}^{D}$ is semi-simple. ∎

###  3.2.  The Whittaker category $\Omega_{\mathbf{a}}^{\widetilde{W}}$

Since $A_{n}$ is a left module algebra over the Hopf algebra $U(W_{n})$, we
have the smash product algebra $A_{n}\\#U(W_{n})$. An $(A_{n},W_{n})$-module
is actually a module over $A_{n}\\#U(W_{n})$.

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 $|\mathbf{m}|\geq 1$ which is called the jet Lie algebra of $W_{n}$.

The following isomorphism was given in [36] or [5].

######  Theorem 3.2.

The linear map $\phi:A_{n}\\#U(W_{n})\rightarrow\mathcal{D}_{n}\otimes
U(L_{n})$ defined by

$\displaystyle\phi(t^{\mathbf{m}}\partial_{k})$
$\displaystyle=t^{\mathbf{m}}\partial_{k}\otimes 1+\sum_{\mathbf{r}\neq
0,\atop r_{i}\leq
m_{i}}\binom{\mathbf{m}}{\mathbf{r}}t^{\mathbf{m}-\mathbf{r}}\otimes
t^{\mathbf{r}}\partial_{k},$ $\displaystyle\phi(t^{\mathbf{m}})$
$\displaystyle=t^{\mathbf{m}}\otimes 1,\ \ \ \mathbf{m}\in\mathbb{Z}_{\geq
0}^{n},k\in\\{1,\cdots,n\\},$

is an associative algebra isomorphism, where
$\binom{\mathbf{m}}{\mathbf{r}}=\binom{m_{1}}{r_{1}}\cdots\binom{m_{n}}{r_{n}}$.

For any $\mathcal{D}_{n}$-module $P$, and $L_{n}$-module $V$, the tensor
product $P\otimes V$ becomes an $(A_{n},W_{n})$-module $T(P,V)$ under the map
$\phi$.

The isomorphism in Theorem 3.2 tells us that there is a close relationship
between vector field Lie algebras and Weyl algebras.

Let $L_{n}\text{-mod}$ be the category of finite dimensional $L_{n}$-modules.
When $V$ is finite dimensional, we can check that
$T(A_{n}^{\mathbf{a}},V)\in\Omega_{\mathbf{a}}^{\widetilde{W}}$ and the
Whittaker vectors space $\text{Wh}_{\mathbf{a}}(T(A_{n}^{\mathbf{a}},V))=V$.

######  Lemma 3.3.

For any $M\in\Omega_{\mathbf{a}}^{\widetilde{W}}$, the Whittaker vector
subspace $\text{Wh}_{\mathbf{a}}(M)$ is an $L_{n}$-module $V$ and
$T(A_{n}^{\mathbf{a}},V)\cong M$.

###### Proof.

Through the isomorphism $\phi$ in Theorem 3.2, we can view $M$ as a module
over $\mathcal{D}_{n}\otimes U(L_{n})$. In $\mathcal{D}_{n}\otimes U(L_{n})$,
$[\mathcal{D}_{n},L_{n}]=0$. So $\text{Wh}_{\mathbf{a}}(M)$ is an
$L_{n}$-module. We claim that the $\mathcal{D}_{n}\otimes U(L_{n})$-module
homomorphism

$\displaystyle\xi:A_{n}^{\mathbf{a}}\otimes\text{Wh}_{\mathbf{a}}(M)$
$\displaystyle\rightarrow M,$ $\displaystyle t^{\mathbf{m}}\otimes v$
$\displaystyle\mapsto t^{\mathbf{m}}v,\ \mathbf{m}\in\mathbb{Z}_{\geq 0}^{n},$

is an isomorphism. By Lemma 3.1, $\xi$ is surjective. Suppose that
$w=\sum_{\mathbf{s}\in\Lambda}t^{\mathbf{s}}\otimes v_{\mathbf{s}}\in\ker\xi$,
where $\Lambda$ is a finite subset of $\mathbb{Z}_{\geq 0}^{n}$. For any
$\mathbf{m}\in\Lambda$, by the simplicity of $A_{n}^{\mathbf{a}}$ and the
density theorem, there is some $X\in\mathcal{D}_{n}$ such that
$Xt^{\mathbf{m}}=1$ and $Xt^{\mathbf{s}}=0$ for any $\mathbf{s}\in\Lambda$
with $\mathbf{s}\neq\mathbf{m}$. Then $X\xi(w)=\xi(Xw)=\xi(1\otimes
v_{\mathbf{m}})=v_{\mathbf{m}}=0$. Consequently $w=0$. So $\xi$ is also
injective. Therefore, $\xi$ is an isomorphism. ∎

In the following theorem, we show that each block
$\Omega_{\mathbf{a}}^{\widetilde{W}}$ is equivalent to $L_{n}\text{-mod}$.

######  Theorem 3.4.

The functor

$\displaystyle\mathcal{F}:L_{n}\text{-mod}$
$\displaystyle\rightarrow\Omega_{\mathbf{a}}^{\widetilde{W}},$ $\displaystyle
V$ $\displaystyle\mapsto T(A_{n}^{\mathbf{a}},V),$

is an equivalence of the two categories.

###### Proof.

By Schur’s Lemma,
$\text{End}_{\mathcal{D}_{n}}(A_{n}^{\mathbf{a}})=\mathbb{C}$. From Lemma 3.3,
for any $M\in\Omega_{\mathbf{a}}^{\widetilde{W}}$, there is an $L_{n}$-module
$\text{Wh}_{\mathbf{a}}(M)$ such that $\mathcal{F}(\text{Wh}(M))\cong M$. From
$\text{Hom}_{\mathcal{D}_{n}\otimes U(L_{n})}(A_{n}^{\mathbf{a}}\otimes
V,A_{n}^{\mathbf{a}}\otimes W)\cong\text{Hom}_{U(L_{n})}(V,W)$, we see that
the homomorphism

$\mathcal{F}_{V,W}:\text{Hom}_{L_{n}}(V,W)\rightarrow\text{Hom}_{\widetilde{W}_{n}}(\mathcal{F}(V),\mathcal{F}(W))$

is an isomorphism, for any modules $V,W\in L_{n}\text{-mod}$. Therefore
$\mathcal{F}$ is an equivalence of the two categories. ∎

Next we give all simple Whittaker $(A_{n},W_{n})$-modules using the
characterization of finite dimensional simple $L_{n}$-modules. Let
$\mathfrak{m}$ be the maximal ideal of $A_{n}$ generated by
$t_{1},\cdots,t_{n}$. Then $\mathfrak{m}\Delta_{n}$ is a Lie subalgebra of
$W_{n}$ which is equal to $L_{n}$.

The following lemma was given in [36].

######  Lemma 3.5.

  1. $($a$)$

The linear map
$\pi:\mathfrak{m}\Delta_{n}/\mathfrak{m}^{2}\Delta_{n}\rightarrow\mathfrak{gl}_{n}$
such that

$\displaystyle t_{i}\partial_{j}$ $\displaystyle\mapsto e_{ij},\ \
i,j\in\\{1,\cdots,n\\},$

is a Lie algebra isomorphism.

  2. $($b$)$

If $V$ is a finite dimensional simple $L_{n}$-module, then
$\mathfrak{m}^{2}\Delta_{n}V=0$. So $V$ is a simple $\mathfrak{gl}_{n}$-module
via the isomorphism in (a).

By Lemma 3.5 and Theorem 3.2, for any simple finite dimensional
$\mathfrak{gl}_{n}$-module $V$, we can define an $(A_{n},W_{n})$-module
structure on $T(A_{n}^{\mathbf{a}},V)$ as follows:

$\displaystyle t^{\mathbf{m}}\partial_{k}(p\otimes v)$
$\displaystyle=t^{\mathbf{m}}\partial_{k}p\otimes
v+\sum_{i=1}^{n}m_{i}t^{\mathbf{m}-\mathbf{e}_{i}}p\otimes e_{ik}v,$
$\displaystyle t^{\mathbf{m}}(p\otimes v)$
$\displaystyle=(t^{\mathbf{m}}p)\otimes v,\ \ \ \mathbf{m}\in\mathbb{Z}_{\geq
0}^{n},k\in\\{1,\cdots,n\\},$

where $p\in A_{n}^{\mathbf{a}},v\in V$.

Combining Theorem 3.4 with Lemma 3.5, we obtain the following description of
simple Whittaker $(A_{n},W_{n})$-modules.

######  Corollary 3.6.

If $M$ is a simple Whittaker $(A_{n},W_{n})$-module of type
$\phi_{\mathbf{a}}$ such that $\dim\text{Wh}_{\mathbf{a}}(M)<\infty$, then
$M\cong T(A_{n}^{\mathbf{a}},V)$ for some finite dimensional simple
$\mathfrak{gl}_{n}$-module $V$.

##  4\. Whittaker $W_{n}$-modules

Throughout this section, we always assume that $\mathbf{a}$ is non-singular,
i.e., $\mathbf{a}\in(\mathbb{C}^{*})^{n}$. In this case, a module in
$\Omega_{\mathbf{a}}^{W}$ is said to be non-singular. In this section, we
classify simple $W_{n}$-modules in $\Omega_{\mathbf{a}}^{W}$. We also obtain
an analogue of Skryabin’s equivalence for the category
$\Omega_{\mathbf{a}}^{W}$.

###  4.1. Preliminary lemmas

The following result gives an interesting property of non-singular Whittaker
modules.

######  Lemma 4.1.

If $M$ is a Whittaker $W_{n}$-module of type $\phi_{\mathbf{a}}$, then

$M=U(\mathfrak{h}_{n})\text{Wh}_{\mathbf{a}}(M):=\text{Span}\\{h^{\mathbf{m}}v\mid
v\in\text{Wh}_{\mathbf{a}}(M),\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}\\}.$

Moreover, if $\\{v_{i}\mid i\in\Lambda\\}$ is a basis of
$\text{Wh}_{\mathbf{a}}(M)$, then

$\\{h^{\mathbf{m}}v_{i}\mid\mathbf{m}\in\mathbb{Z}_{\geq
0}^{n},i\in\Lambda\\}$

is a basis of $M$.

###### Proof.

For an $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$, define

$\partial_{\mathbf{m}}=(\partial_{1}-a_{1})^{m_{1}}\cdots(\partial_{n}-a_{n})^{m_{n}}.$

Note that the elements $\partial_{\mathbf{m}}$ with
$\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$ form a basis for $U(\Delta_{n})$.
Consider the total order on $\mathbb{Z}_{\geq 0}^{n}$ satisfying the
condition: $\mathbf{r}<\mathbf{m}$ if $|\mathbf{r}|<|\mathbf{m}|$ or
$|\mathbf{r}|=|\mathbf{m}|$ and there is an $l\in\\{1,\cdots,n\\}$ such that
$r_{i}=m_{i}$ when $1\leq i<l$ and $r_{l}<m_{l}$. For each $\mathbf{m}$, the
set $\\{\mathbf{r}\in\mathbb{Z}_{\geq 0}^{n}\mid\mathbf{r}<\mathbf{m}\\}$ is a
finite set. So $\mathbb{Z}_{\geq 0}^{n}$ is isomorphic to $\mathbb{Z}_{\geq
0}$ as an ordered set. For a nonzero $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$,
denote by $\mathbf{m}^{\prime}$ the predecessor of $\mathbf{m}$.

Using the condition that $a_{i}\neq 0$ for any $i$ and induction on
$\mathbf{m}$, we can show that for any
$\mathbf{m},\mathbf{s}\in\mathbb{Z}_{\geq 0}^{n}$ and nonzero
$v\in\text{Wh}_{\mathbf{a}}(M)$, we have that
$\partial_{\mathbf{s}}h^{\mathbf{m}}v=0$ whenever $\mathbf{s}>\mathbf{m}$,
$\partial_{\mathbf{m}}h^{\mathbf{m}}v=k_{\mathbf{m}}v$ for some nonzero scalar
$k_{\mathbf{m}}$.

For each $\mathbf{m}\in\mathbb{Z}_{\geq 0}^{n}$, denote by
$\mathbb{I}_{\mathbf{m}}$ the ideal of $U(\Delta_{n})$ spanned by
$\partial_{\mathbf{s}}$ with $\mathbf{s}>\mathbf{m}$, and
$M_{\mathbf{m}}=\\{w\in M\mid\mathbb{I}_{\mathbf{m}}w=0\\}$. Clearly
$\text{Wh}_{\mathbf{a}}(M)=M_{\mathbf{0}}$. For any nonzero $w\in M$, by the
definition of Whittaker modules, there is an $\mathbf{m}\in\mathbb{Z}_{\geq
0}^{n}$ such that $w\in M_{\mathbf{m}}\setminus M_{\mathbf{m}^{\prime}}$,
i.e., $\partial_{\mathbf{m}}w\neq 0$ and $\partial_{\mathbf{s}}w=0$ for any
$\mathbf{s}>\mathbf{m}$. So
$\partial_{\mathbf{m}}w\in\text{Wh}_{\mathbf{a}}(M)$. By the above discussion,
$\partial_{\mathbf{m}}h^{\mathbf{m}}\partial_{\mathbf{m}}w=k_{\mathbf{m}}\partial_{\mathbf{m}}w$.
We call $\mathbf{m}$ the degree of $w$. We use induction on the degree
$\mathbf{m}$ of $w$ to show that $w\in
U(\mathfrak{h}_{n})\text{Wh}_{\mathbf{a}}(M)$. Let
$w^{\prime}=w-\frac{1}{k_{\mathbf{m}}}h^{\mathbf{m}}\partial_{\mathbf{m}}w$.
Then

$\partial_{\mathbf{m}}w^{\prime}=\partial_{\mathbf{m}}w-\frac{1}{k_{\mathbf{m}}}\partial_{\mathbf{m}}h^{\mathbf{m}}\partial_{\mathbf{m}}w=0.$

This implies that the degree of $w^{\prime}$ is smaller than $\mathbf{m}$. By
the induction hypothesis, $w^{\prime}\in
U(\mathfrak{h}_{n})\text{Wh}_{\mathbf{a}}(M)$. Hence $w\in
U(\mathfrak{h}_{n})\text{Wh}_{\mathbf{a}}(M)$.

Suppose that
$w:=\sum_{\mathbf{r}\leq\mathbf{m}}\sum_{i=1}^{k}c_{\mathbf{r},i}h^{\mathbf{r}}v_{i}=0$
in $M$, where $c_{\mathbf{r},i}\in\mathbb{C}$. From
$\partial_{\mathbf{m}}w=0$, we see that $c_{\mathbf{m},i}=0$. Then by
induction on $\mathbf{m}$, we have that $c_{\mathbf{r},i}=0$ for any
$\mathbf{r}<\mathbf{m}$ and $i$. Thus
$\\{h^{\mathbf{m}}v_{i}\mid\mathbf{m}\in\mathbb{Z}_{\geq
0}^{n},i\in\Lambda\\}$ is linearly independent. In conclusion, we complete the
proof.

∎

From Lemma 4.1, if $M$ is a Whittaker $W_{n}$-module of type
$\phi_{\mathbf{a}}$, then $M$ is a free $U(\mathfrak{h}_{n})$-module and
$\text{Wh}_{\mathbf{a}}(M)$ is a basis of $M$ as a free
$U(\mathfrak{h}_{n})$-module.

######  Corollary 4.2.

Any nontrivial $W_{n}$-module $M\in\Omega_{\mathbf{a}}^{W}$ is a free
$U(\mathfrak{h}_{n})$-module of finite rank.

######  Corollary 4.3.

The category $\Omega_{\mathbf{a}}^{W}$ is an abelian category.

###### Proof.

Suppose that $M\in\Omega_{\mathbf{a}}^{W}$ and $N$ is an arbitrary
$W_{n}$-submodule of $M$. Since
$\text{Wh}_{\mathbf{a}}(N)\subset\text{Wh}_{\mathbf{a}}(M)$,
$N\in\Omega_{\mathbf{a}}^{W}$. By Lemma 4.1, $M/N$ is a free
$U(\mathfrak{h}_{n})$ module whose rank is not bigger than that of $M$. Thus
$\dim\text{Wh}_{\mathbf{a}}(M/N)<\infty$ and hence
$M/N\in\Omega_{\mathbf{a}}^{W}$. ∎

Next let us recall the $(A_{n},W_{n})$-cover introduced in [4] in a slightly
different form. Let $\Delta$ be the co-multiplication of
$U(\widetilde{W}_{n})$, that is

$\Delta:U(\widetilde{W}_{n})\rightarrow U(\widetilde{W}_{n})\otimes
U(\widetilde{W}_{n}),X\mapsto X\otimes 1+1\otimes X,\ \
X\in\widetilde{W}_{n}.$

Let $\pi:U(\widetilde{W}_{n})\rightarrow U(W_{n})$ be the homomorphism such
that $\pi|_{W_{n}}=\text{id}_{W_{n}},\pi|_{A_{n}}=0$. Denote the composition

$(\text{id}_{\widetilde{W}_{n}}\otimes\pi)\cdot\Delta:U(\widetilde{W}_{n})\rightarrow
U(\widetilde{W}_{n})\otimes U(W_{n})$

by $\Delta^{\prime}$. Then for any $(A_{n},W_{n})$-module $P$ and
$W_{n}$-module $M$, $P\otimes M$ becomes an $(A_{n},W_{n})$-module under the
pulling back of $\Delta^{\prime}$. In particular, $W_{n}\otimes M$ can be
defined to be an $(A_{n},W_{n})$-module by

$t^{\mathbf{r}}\cdot(t^{\mathbf{m}}\partial_{i}\otimes
w)=t^{\mathbf{m}+\mathbf{r}}\partial_{i}\otimes w,\ \ w\in
M,\mathbf{r},\mathbf{m}\in\mathbb{Z}_{+}^{n},i\in\\{1,\cdots,n\\}.$

Define a linear map $\theta:W_{n}\otimes M\rightarrow M$ such that

$t^{\mathbf{m}}\partial_{i}\otimes w\mapsto t^{\mathbf{m}}\partial_{i}w,\ w\in
M,\mathbf{m}\in\mathbb{Z}_{+}^{n},i\in\\{1,\cdots,n\\}.$

One can check directly that $\theta$ is a $W_{n}$-module homomorphism. Define

$K(M)=\\{v\in\ker\theta\mid A_{n}\cdot v\subset\ker\theta\\}$

which is an $(A_{n},W_{n})$-submodule of $W_{n}\otimes M$. Let
$\widehat{M}=(W_{n}\otimes M)/K(M)$ which is called the $(A_{n},W_{n})$-cover
of $M$. It is natural that $\theta$ induces a $W_{n}$-module homomorphism
$\hat{\theta}:\widehat{M}\rightarrow M$. For any $X\in W_{n}$, $v\in M$,
denote the image of $X\otimes v$ in $\widehat{M}$ by $X\boxtimes v$.

###  4.2. Simple Whittaker modules

For any $m\in\mathbb{Z}_{\geq 0}$, $j\in\\{1,\dots,n\\}$ and
$\mathbf{\alpha},\mathbf{\beta}\in\mathbb{Z}^{n}_{\geq 0}$, in $U(W_{n})$, we
denote the following operator:

$\omega_{\mathbf{\alpha},\mathbf{\beta}}^{m,j,l,p}:=\sum_{i=0}^{m}(-1)^{i}\binom{m}{i}t^{\mathbf{\alpha}+(m-i)\mathbf{e}_{j}}\partial_{l}\cdot
t^{\mathbf{\beta}+i\mathbf{e}_{j}}\partial_{p}.$

The following lemma was given by Xue and Lu, see [36].

######  Lemma 4.4.

If $M$ is a uniformly bounded weight module over $W_{n}$, then there is an
$m\in\mathbb{N}$ such that
$\omega_{\mathbf{\alpha},\mathbf{\beta}}^{m,j,l,p}M=0$ for all
$j,l,p\in\\{1,\dots,n\\}$ and
$\mathbf{\alpha},\mathbf{\beta}\in\mathbb{Z}^{n}_{\geq 0}$.

Using Lemma 4.4, we show that for any simple module $M$ in
$\Omega_{\mathbf{a}}^{W}$, $\text{Wh}_{\mathbf{a}}(\widehat{M})$ is finite
dimensional.

######  Lemma 4.5.

Suppose that $M$ is a simple module in $\Omega_{\mathbf{a}}^{W}$. Then
$\text{Wh}_{\mathbf{a}}(\widehat{M})$ is finite dimensional.

###### Proof.

By Lemma 4.1 and Corollary 4.2, $M$ is a free $U(\mathfrak{h}_{n})$-module of
finite rank. Thus $\mathfrak{W}(M)$ is a uniformly bounded weight
$W_{n}$-module. In fact each weight space of $\mathfrak{W}(M)$ has dimension
no bigger than the rank of $M$ as a free $U(\mathfrak{h}_{n})$-module. From
Lemma 4.4, there exists $m\in\mathbb{N}$ such that

$\omega_{\mathbf{r},\mathbf{s}}^{m,j,l,p}\mathfrak{W}(M)=0,$

for all all $j,l,p\in\\{1,\dots,n\\}$ and
$\mathbf{r},\mathbf{s}\in\mathbb{Z}^{n}_{\geq 0}$.

By the definition of $\mathfrak{W}(M)$, we deduce that

$\omega_{\mathbf{r},\mathbf{s}}^{m,j,l,p}M\subseteq\bigcap_{\mathbf{\alpha}\in\mathbb{Z}^{n}}I_{\mathbf{\alpha}}M=0,\
\forall\ j,l,p\in\\{1,\dots,n\\},\mathbf{r},\mathbf{s}\in\mathbb{Z}^{n}_{\geq
0},$

where $I_{\mathbf{\alpha}}$ is the maximal ideal of $U(\mathfrak{h}_{n})$
generated by $h_{1}-\alpha_{1},\dots,h_{n}-\alpha_{n}$. The result
$\bigcap_{\mathbf{\alpha}\in\mathbb{Z}^{n}}I_{\mathbf{\alpha}}M=0$ follows
from that $M$ is a finitely generated free $U(\mathfrak{h}_{n})$-module.

The simplicity of $M$ implies that $M=W_{n}M$. Next we prove by induction on
$|\mathbf{r}|$ that

$t^{\mathbf{r}}\partial_{l}\boxtimes
t^{\mathbf{s}}\partial_{p}v\in\sum_{i=1}^{n}\sum_{|\mathbf{\alpha}|\leq
mn}t^{\mathbf{\alpha}}\partial_{i}\boxtimes M,$

for all $v\in
M,\mathbf{r},\mathbf{s}\in\mathbb{Z}^{n},p,l\in\\{1,\cdots,n\\}.$ This is
clear for $\mathbf{r}\in\mathbb{Z}^{n}_{\geq 0}$ with $|\mathbf{r}|\leq mn$.
Now we assume that $|\mathbf{r}|>mn$. Without lose of generality, we may
assume that $r_{1}>m$.

For any $\mathbf{r}\in\mathbb{Z}^{n}_{\geq 0}$, we have

$\displaystyle\theta(\sum_{i=0}^{m}(-1)^{i}{m\choose
i}t^{\mathbf{r}-i\mathbf{e}_{1}}\partial_{l}\otimes(t^{\mathbf{s}+i\mathbf{e}_{1}}\partial_{p}v))$
$\displaystyle=$ $\displaystyle\sum_{i=0}^{m}(-1)^{i}{m\choose
i}t^{\mathbf{r}-i\mathbf{e}_{1}}\partial_{l}t^{\mathbf{s}+i\mathbf{e}_{1}}\partial_{p}v$
$\displaystyle=$ $\displaystyle\omega_{\mathbf{r},\mathbf{s}}^{m,1,l,p}(v)=0,$

which implies

$\sum_{i=0}^{m}(-1)^{i}{m\choose
i}t^{\mathbf{r}-i\mathbf{e}_{1}}\partial_{l}\otimes
t^{\mathbf{s}+i\mathbf{e}_{1}}\partial_{p}v\in K(M).$

This means that

$\displaystyle t^{\mathbf{r}}\partial_{l}\boxtimes
t^{\mathbf{s}}\partial_{p}v=\sum_{i=1}^{m}(-1)^{i}{m\choose
i}t^{\mathbf{r}-i\mathbf{e}_{1}}\partial_{l}\boxtimes
t^{\mathbf{s}+i\mathbf{e}_{1}}\partial_{p}v,$

in $\widehat{M}$ which belongs to $\sum_{i=1}^{n}\sum_{|\mathbf{r}|\leq
mn}t^{\mathbf{r}}\partial_{i}\boxtimes M$ by the induction hypothesis.

Therefore $\widehat{M}$ is a finitely generated $U(\mathfrak{h}_{n})$-module.
By Lemma 4.1, $\text{Wh}_{\mathbf{a}}(\widehat{M})$ is finite dimensional.
Otherwise, $\widehat{M}$ is a free $U(\mathfrak{h}_{n})$-module of infinite
rank. ∎

######  Theorem 4.6.

If $M$ is a simple module in $\Omega_{\mathbf{a}}^{W}$, then $M$ is isomorphic
to some simple sub-quotient of $T(A_{n}^{\mathbf{a}},V)$ for some finite
dimensional simple $\mathfrak{gl}_{n}$-module $V$.

###### Proof.

From Lemma 4.5, there is an $(A_{n},W_{n})$-module $M_{1}$ such that
$\text{Wh}_{\mathbf{a}}(M_{1})$ is finite dimensional and a $W_{n}$-module
epimorphism $\psi:M_{1}\rightarrow M$. We may choose $M_{1}$ such that
$\dim\text{Wh}_{\mathbf{a}}(M_{1})$ is minimal. The minimality of $M_{1}$
forces that $M_{1}$ is a simple $(A_{n},W_{n})$-module. Otherwise, $M_{1}$ has
a nonzero maximal $(A_{n},W_{n})$-submodule $M_{2}$. Then from corollary 4.2,
$M_{2}$ and $M_{1}/M_{2}$ are free $U(\mathfrak{h}_{n})$ module of rank both
less than $\dim\text{Wh}_{\mathbf{a}}(M_{1})$. Because $M$ is a simple
$W_{n}$-module, we must have either $\psi(M_{2})=M$ or $\psi(M_{2})=0$. Then
either $M_{2}$ or $M_{1}/M_{2}$ has a simple $W_{n}$-quotient module
isomorphic to $M$, contradicting the minimality of $M_{1}$. From Corollary
3.6, we know that $M_{1}\cong T(A_{n}^{\mathbf{a}},V)$ for a finite
dimensional simple $\mathfrak{gl}_{n}$-module $V$. Hence $M$ is a simple
$W_{n}$-quotient of $T(A_{n}^{\mathbf{a}},V)$.∎

The structure of $T(A_{n}^{\mathbf{a}},V)$ was determined in [20]. Let
$\mathbb{C}^{n}$ be the natural $n$-dimensional representation of
$\mathfrak{gl}_{n}$ and let $V(\delta_{k},k)$ be its $k$-th exterior power,
$k=1,\cdots,n$. We use $V(\delta_{0},0)$ to denote 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 $X\in\mathfrak{g},v\in V$. For a
finite dimensional simple $\mathfrak{gl}_{n}$-module $V$,
$T(A_{n}^{\mathbf{a}},V)$ is a simple module over $W_{n}$ if $V\not\cong
V(\delta_{k},k)$ for any $k\in\\{1,\cdots,n\\}$. Moreover we have the
following complex of $W_{n}$-modules:

(4.1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T(A_{n}^{\mathbf{a}},V(\delta_{0},0))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\textstyle{T(A_{n}^{\mathbf{a}},V(\delta_{1},1))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
28.45274pt\pi_{1}}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n-2}\hskip
56.9055pt}$$\textstyle{{T(A_{n}^{\mathbf{a}},V(\delta_{n-1},n-1))}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76228pt\pi_{n-1}}$$\textstyle{T(A_{n}^{\mathbf{a}},V(\delta_{n},n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\pi_{k-1}$ is the $W_{n}$-module homomorphism defined by

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

for all $p\in A_{n}^{\mathbf{a}}$ and $v\in V(\delta_{k-1},k-1)$, where
$T(A_{n}^{\mathbf{a}},V(\delta_{-1},-1))=0$, $k\in\\{0,1,\cdots,n\\}$.

######  Lemma 4.7.

  1. $($a$)$

The complex (4.1) is exact.

  2. $($b$)$

The $W_{n}$-modules $\text{Im}\pi_{0},\cdots,\text{Im}\pi_{n-1}$ are simple.

###### Proof.

Then from Theorem 3.5 in [20], the $W_{n}$-modules $\text{Im}\pi_{k-1}$ are
simple, for $k=1,2,\ldots,n$. And $\text{Ker}\pi_{k}/\text{Im}\pi_{k-1}$,
$T(A_{n}^{\mathbf{a}},V(\delta_{n},n))/\text{Im}\pi_{n-1}$ are trivial
$W_{n}$-modules, for $k=1,2,\ldots,n-1$. Since $a_{i}\neq 0$ for any $i$, a
trivial module in $\Omega_{\mathbf{a}}^{W}$ must be a zero module. So
$\text{Ker}\pi_{k}=\text{Im}\pi_{k-1}$. Hence the complex (4.1) is exact. ∎

###  4.3. An analogue of Skryabin’s equivalence

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

Let

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

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

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

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

######  Theorem 4.8.

  1. $($a$)$

As a right $H_{\mathbf{a}}$-module, $Q_{\mathbf{a}}$ is free. More precisely,
the subset

$\\{h^{\mathbf{m}}\otimes 1_{\mathbf{a}}\mid\mathbf{m}\in\mathbb{Z}_{\geq
0}^{n}\\}$

is a basis of $Q_{\mathbf{a}}$ as an $H_{\mathbf{a}}$-module.

  2. $($b$)$

The functors $M\mapsto\text{Wh}_{\mathbf{a}}(M)$ and $V\mapsto
Q_{\mathbf{a}}\otimes_{H_{\mathbf{a}}}V$ are inverse equivalence between
$\Omega_{\mathbf{a}}^{W}$ and $H_{\mathbf{a}}\text{-mod}$ .

###### Proof.

(a). By adjunction and the definition of Whittaker Vectors, we have

$\text{End}_{W_{n}}(Q_{\mathbf{a}})\cong\text{Hom}_{\Delta_{n}}(\mathbb{C}_{\mathbf{a}},Q_{\mathbf{a}})\cong\text{Wh}_{\mathbf{a}}(Q_{\mathbf{a}}).$

Then $\text{Wh}_{\mathbf{a}}(Q_{\mathbf{a}})$ is a free
$H_{\mathbf{a}}$-module of rank $1$.

By Lemma 4.1,
$Q_{\mathbf{a}}=U(\mathfrak{h}_{n})\text{Wh}_{\mathbf{a}}(Q_{\mathbf{a}})$. By
PBW Theorem, $Q_{\mathbf{a}}$ is $U(\mathfrak{h}_{n})$-free. Then (a) follows.

(b). By (a) and Lemma 4.1, for any $M\in\Omega_{\mathbf{a}}^{W}$,

$Q_{\mathbf{a}}\otimes_{H_{\mathbf{a}}}\text{Wh}_{\mathbf{a}}(M)\cong M.$

On the other side, for any $V\in H_{\mathbf{a}}\text{-mod}$,

$\text{Wh}_{\mathbf{a}}(Q_{\mathbf{a}}\otimes_{H_{\mathbf{a}}}V)=\text{Wh}_{\mathbf{a}}(U(\mathfrak{h}_{n})\otimes
V)\cong V.$

Then we can complete the proof. ∎

For finite $W$-algebras $W(\mathfrak{g},e)$, a similar category equivalence
was shown by Serge Skryabin, see the appendix in [30].

######  Remark 4.9.

By Theorem 4.6 and Theorem 4.8, we can obtain all finite dimensional simple
$H_{\mathbf{a}}$-modules. It will be interesting to characterize the category
$H_{\mathbf{a}}\text{-mod}$ of finite dimensional $H_{\mathbf{a}}$-modules and
find the PBW basis of $H_{\mathbf{a}}$. For finite $W$-algebras
$W(\mathfrak{g},e)$, the PBW basis of $W(\mathfrak{g},e)$ was given in [30].
By Proposition 35 in [6], $H_{\mathbf{a}}$ is isomorphic to a subalgebra of
$U(L_{n})$.

Acknowledgments

G.L. is partially supported by NSF of China(Grants 11771122).

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