# Impact of Speech Mode in
Automatic Pathological Speech Detection

Shakeel A. Sheikh, Ina Kodrasi  Signal Processing for Communication Group,
Idiap Research Institute, Martigny, Switzerland
{shakeel.sheikh<EMAIL_ADDRESS>
###### Abstract

Automatic pathological speech detection approaches yield promising results in
identifying various pathologies. These approaches are typically designed and
evaluated for phonetically-controlled speech scenarios, where speakers are
prompted to articulate identical phonetic content. While gathering controlled
speech recordings can be laborious, spontaneous speech can be conveniently
acquired as potential patients navigate their daily routines. Further,
spontaneous speech can be valuable in detecting subtle and abstract cues of
pathological speech. Nonetheless, the efficacy of automatic pathological
speech detection for spontaneous speech remains unexplored. This paper
analyzes the influence of speech mode on pathological speech detection
approaches, examining two distinct categories of approaches, i.e., classical
machine learning and deep learning. Results indicate that classical approaches
may struggle to capture pathology-discriminant cues in spontaneous speech. In
contrast, deep learning approaches demonstrate superior performance, managing
to extract additional cues that were previously inaccessible in non-
spontaneous speech.

###### Index Terms:

pathological speech, spontaneous speech, non-spontaneous speech, deep
learning.

##  I Introduction

Pathological speech resulting from neurological disorders poses a significant
healthcare challenge, where early diagnosis is crucial for delaying
progression and for managing the condition. Pathological speech conditions
such as dysarthria or apraxia of speech are characterized by alterations in
speech production, including imprecise articulation, abnormal pitch, or
irregular rhythm, making it challenging for patients to effectively convey
their intentions through speech [1, 2, 3, 4]. Automatic speech-based
diagnostic approaches emerge as cost-effective methods for early detection
[1], especially compared to traditional diagnostic techniques such as Positron
emission tomography and Dopamine transporter scans, which are expensive and
inaccessible, creating a barrier to timely diagnosis and treatment [5, 6].
Such approaches provide convenient monitoring of disease progression, reducing
the need for costly in-person therapy sessions. As a result, various automatic
approaches have been proposed for pathological speech detection, broadly
falling into two categories: i) those utilizing classical machine learning
techniques with handcrafted acoustic features [7, 8], and ii) those based on
deep learning architectures operating on time-frequency input representations
[9, 10, 11, 12, 13] or self-supervised embeddings [14, 15, 16, 17, 18]. The
first category of approaches often employs support vector machines (SVMs) with
Mel-frequency cepstral coefficients (MFCCs) [19], openSMILE features [15], or
sparsity-based features [20]. In contrast, the second category of approaches
focuses on exploring time-frequency input representations with different
network architectures and training paradigms such as long short-term memory
networks [21], autoencoders (AEs) [22], adversarial training [9], or
convolutional neural networks (CNNs) [23, 9]. More recently, approaches in the
second category focus on using self-supervised embeddings such as wav2vec2 for
automatic pathological speech detection [14, 15].

While both categories of automatic approaches have demonstrated promising
results, they are typically developed and assessed in phonetically-controlled,
non-spontaneous, speech settings. Phonetically-controlled non-spontaneous
speech refers to elicited speech, where speakers are prompted to repeat the
same words, phrases, or sentences, meticulously designed by clinicians to
elicit discernible cues indicative of pathological speech. In contrast,
spontaneous speech reflects natural conversation and can be valuable in
detecting subtle and abstract cues of pathological speech. Furthermore,
spontaneous speech mirrors real-world communication and places a higher
cognitive load on real-time processing of various speech tasks such as
planning, precise sensorimotor execution, and articulation, making it more
susceptible to deficits related to pathology [1]. Yet, to the best of our
knowledge, the performance of state-of-the-art automatic pathological speech
detection approaches on spontaneous speech remains unexplored. Further
investigation is required to assess how these approaches fare in more natural
settings, where speech is less structured and influenced by diverse factors
such as individual variations in communication styles.

This paper analyzes the effect of speech mode, i.e., non-spontaneous and
spontaneous speech, on the performance of numerous state-of-the-art automatic
pathological speech detection approaches. The evaluation is conducted on two
distinct databases, i.e., the Spanish PC-GITA database containing dysarthria
recordings from patients with Parkinson’s disease [24] and the French MoSpeeDi
database [25] containing dysarthria recordings from patients with Parkinson’s
disease or Amytrophic Lateral Sclerosis. The considered classical machine
learning-based approaches include SVMs operating on hand-crafted acoustic
features like openSMILE [26], MFCCs [19], and sparsity-based features [20].
Additionally, the considered deep learning-based approaches include the CNN-
based approach proposed in [23], the AE-based approach proposed in [9], and
the wav2vec2-based approach in [15].

##  II Methods

This section outlines the state-of-the-art pathological speech detection
approaches considered in this paper for analyzing the impact of the speech
mode on performance.

###  II-A Input Representation

In the following, we describe the different input representations used in the
considered approaches.

#### OpenSMILE

OpenSMILE features are commonly used as input to classical machine learning
approaches [15, 27]. For each utterance, we extract a $6373$–dimensional
feature vector using the openSMILE toolkit [26]. Following feature extraction,
we employ principal component analysis for dimensionality reduction similar to
previous works [27, 15]. We retain only the features that explain 95% of the
variance in the training data.

#### MFCCs

MFCC features are also commonly used as input to classical machine learning
approaches [8]. For each utterance, we compute the mean, variance, kurtosis,
and skewness of the first $12$ MFCC coefficients, resulting in a
$48$–dimensional feature vector as in [15]. The MFCC features are extracted
using the openSMILE toolkit [26].

#### Sparsity-based features

Sparsity-based features have also been used with classical machine learning
approaches [28, 20]. Following the methodology outlined in [28, 20], we
compute sparsity-based features for each utterance using the shape parameter
of a Chi distribution. The process involves computing the short-time Fourier
transform (STFT) using a Hamming window of length $32$ ms and a hop size of
$4$ ms. Next, we obtain the maximum likelihood estimate of the shape parameter
for the Chi distribution that best models the spectral magnitude at each
frequency bin. This procedure results in a 257-dimensional feature vector for
each utterance.

#### Mel spectrograms

Mel spectrograms are typically used with deep learning-based approaches [8].
For each speech segment, we compute the STFT coefficients using a Hamming
window of length $32$ ms and a hop size of $4$ ms. The computed coefficients
are converted into Mel-scale representations with $126$ Mel-bands, following
the approach used in [15]. The logarithm of the Mel spectrogram is then used
as the input representation for deep learning-based approaches.

#### Self-supervised embeddings

The wav2vec2 framework has revolutionised the extraction of meaningful
contextual representations from raw speech [29]. These extracted embeddings
have shown promising performance in various pathological speech tasks [30, 14,
31, 32]. For our analysis, we use embeddings extracted from the XLSR-53 model
[33]. It has been shown that the last layers of wav2vec2 models are more
adapted towards non paralinguistic and other phonetic content-related tasks,
while the first layers are more applicable to paralinguistic and prosody-
related tasks [34, 29, 32]. Hence, in this paper, we consider embeddings
extracted from the first transformer layer of the XLSR-53 model. For each
utterance, the final input representation is computed as the mean and standard
deviation of the embeddings across time. This procedure results in a
$2048$–dimensional input representation. It should be noted that the wav2vec2
model is not fine-tuned, but rather utilized as a self-supervised feature
extractor for automatic pathological speech detection.

###  II-B Classifiers

In the following, we describe the different classifiers used in the considered
approaches.

#### Support vector machines

To analyze the performance of classical approaches based on handcrafted
acoustic features, we use SVMs with radial basis kernel function as in [15,
19, 20]. Different SVMs are trained for different acoustic features, i.e., for
openSMILE, MFCCs, and sparsity-based features.

#### Convolutional neural networks

CNNs have been widely used in various speech applications, including automatic
pathological speech detection [12, 35, 36, 37]. For our analysis, we train a
CNN with Mel spectrogram input representations as in [15]. The CNN consists of
a normalization layer, followed by two convolutional layers with $64$ channels
each and kernel sizes of $2\times 2$ and $3\times 3$ respectively. Each
convolutional layer is followed by batch normalisation, max-pooling with a
$2\times 2$ or $3\times 3$ kernel, and a ReLU activation function. A dropout
rate of 30% is applied after the second convolutional layer. Finally, the
output is fed to a fully-connected linear layer (input size: $25600$, output
size: $2$) for pathological speech detection.

#### Autoencoders

Given the promising performance of the AE-based framework for pathological
speech detection in [9], we also consider this approach in our analysis. Also
in this framework Mel spectrograms are used as the input representation. The
encoder $\theta_{e}$ is composed of $4$ convolutional layers with a kernel
size of $3\times 3$. Each convolutional layer is followed by batch
normalization, max-pooling with a $2\times 2$ kernel, and a ReLU activation
function. The output of the encoder is fed to a bottleneck layer of size
$128$. This bottleneck representation is then decoded by the decoder
$\theta_{d}$ to reconstruct the input representation. The decoder mirrors the
encoder components in reverse order, using interpolation and transposed
convolution instead of max-pooling and convolution. In addition, the
bottleneck representation is simultaneously fed to a classifier aiming to
learn a pathology-discriminant representation in a multi-task learning
fashion. The pathology classifier $\theta_{pa}$ is composed of a fully
connected linear layer with $128$ input units and $2$ output units. The
classification loss $\mathcal{L}_{pa}$ and the AE reconstruction loss
$\mathcal{L}_{ae}$ are jointly minimized by optimizing the parameters
$\theta_{e}$, $\theta_{d}$, $\theta_{pa}$ using

$\mathcal{L}(\hat{\theta}_{e},\hat{\theta}_{d},\hat{\theta}_{pd})=\underset{\theta_{e},\theta_{d},\theta_{pd}}{\mathrm{argmin}}~{}\mathcal{L}(\theta_{e},\theta_{d},\theta_{pd}),$
(1)

where

$\mathcal{L}(\theta_{e},\theta_{d},\theta_{pd})=(1-\alpha)\mathcal{L}_{ae}(\theta_{e},\theta_{d})+\alpha\mathcal{L}_{pd}(\theta_{e},\theta_{pd}),$
(2)

with $\alpha$ $\in[0,1]$ being a trade-off parameter between the AE loss and
the classification loss. In our analysis, we use $\alpha=0.01$ as in [15]. For
inference, the decoder is disregarded and the bottleneck representation is
directly fed to the pathology classifier.

#### Fully connected linear layer

Similar to [15], we employ a fully connected linear layer for pathological
speech classification when using wav2vec2 embeddings as the input
representation. The linear layer consists of $2048$ input units and $2$ output
units.

##  III Experimental Settings

In this section, the settings used for the experimental analysis are provided.

###  III-A Databases

Analysis are conducted using two different databases, i.e., the Spanish PC-
GITA database [24] and the French MoSpeeDi database [25].

#### PC-GITA

The PC-GITA database contains Spanish recordings from a gender-balanced group
of $50$ patients diagnosed with Parkinson’s disease along with a gender-
balanced group of $50$ neurotypical speakers. For the non-spontaneous speech
mode, we use recordings of $10$ sentences and a phonetically balanced text.
Using these recordings, the average length of the total available non-
spontaneous speech material across speakers is $55.4$ s. For the spontaneous
speech mode, we use recordings of the speakers talking about what they do in a
normal day. Using these recordings, the average length of the available
spontaneous speech material across speakers is $47.1$ s.

#### MoSpeeDi

From the French MoSpeeDi database, we use recordings from a gender-balanced
group of $35$ patients diagnosed with Parkinson’s disease or Amytrophic
Lateral Sclerosis along with a gender-balanced group of $35$ neurotypical
speakers. For the non-spontaneous speech mode, we use recordings of $8$
sentences. Using these recordings, the average length of the total available
non-spontaneous speech material across speakers is $97.7$ s. For the
spontaneous speech mode, we use recordings of the speakers talking about their
holidays. Using these recordings, the average length of the available
spontaneous speech material across speakers is $153.1$ s.

###  III-B Evaluation and Training

For all considered approaches, evaluation is done within a stratified $K$-fold
cross validation strategy, with $K=10$ for the PC-GITA database and $K=7$ for
the MoSpeeDi database. For each fold of the PC-GITA database, we use $80\%$,
$10\%$, and $10\%$ of the data for training, validation, and testing,
respectively. For each fold of the MoSpeeDi database, we use $72\%$, $14\%$,
and $14\%$ of the data for training, validation, and testing, respectively.

The SVM-based and the wav2vec2-based approaches accept variable length
segments of speech as input, hence, full utterances are used as input to these
approaches. The CNN-based and the AE-based approach accept only fixed-size
segments of speech as input. For these two approaches, we segment available
utterances into $500$ ms segments with a $50\%$ overlap and use these fixed-
size segments as input.

The performance is evaluated in terms of speaker-level accuracy. For the SVM-
based approaches, speaker-level accuracy is computed through majority voting
of the decisions for all utterances belonging to each speaker. For the
remaining approaches, speaker-level accuracy is computed through soft voting
of the probability of decisions for all segments/utterances belonging to each
speaker.

Implementation is done using PyTorch and Torchaudio. For the SVM-based
approaches, separate SVMs are trained for each of the handcrafted acoustic
features, i.e., for openSMILE, MFCCs, and sparsity-based features. By
optimizing the performance on the validation set, the optimal kernel width
$\gamma$ and soft margin constant $\mathcal{C}$ is found from the sets
$\\{10^{-4},10^{-3},10^{-2},10^{-1}]$ and $\\{10,10^{2},10^{3},10^{4}\\}$
respectively. The remaining approaches are trained using the Adam optimizer.
The initial learning rates are $0.0015$ for the AE-based approach, $0.001$ for
the CNN-based approach, and $0.001$ for the wav2vec2-based approach. The
learning rates are reduced after every $15$ epochs using the _MultiStepLR_
scheduler with $\gamma$ = 0.9. Training is terminated if the validation loss
does not decrease for $10$ consecutive epochs, with the last best model saved
and used for testing.

##  IV Experimental Results

TABLE I: Speaker-level classification accuracy [%] of the considered automatic pathological speech detection approaches on the PC-GITA and MoSpeeDi database using non-spontaneous and spontaneous speech recordings. Approach | PC-GITA | MoSpeeDi
---|---|---
| Non-spotaneous | Spontaneous | Non-spotaneous | Spontaneous
SVM with openSMILE input features | 43.0 $\pm$ 17.0 | 57.0 $\pm$ 17.0 | 42.9 $\pm$ 7.76 | 57.1 $\pm$ 19.8
SVM with MFCC input features | 68.0 $\pm$ 09.2 | 62.0 $\pm$ 13.2 | 71.4 $\pm$ 10.7 | 84.3 $\pm$ 15.1
SVM with sparsity-based input features | 73.0 $\pm$ 09.5 | 56.0 $\pm$ 22.2 | 82.8 $\pm$ 11.1 | 78.6 $\pm$ 09.0
CNN with Mel spectrogram input representations | 75.0 $\pm$ 07.0 | 74.0 $\pm$ 12.6 | 90.1 $\pm$ 11.5 | 92.9 $\pm$ 11.1
AE with Mel spectrogram input representations | 71.0 $\pm$ 13.7 | 80.0 $\pm$ 14.1 | 82.9 $\pm$ 16.0 | 85.7 $\pm$ 05.3
Linear layer with wav2vec2 embedding input representations | 77.0 $\pm$ 14.9 | 75.0 $\pm$ 15.8 | 87.1 $\pm$ 11.12 | 87.1 $\pm$ 11.1

In this section, we analyse and discuss the impact of speech modes (i.e., non-
spontaneous and spontaneous) in SVM-based approaches, CNN- and AE-based
approaches, and the wav2vec2-based approach. Results obtained for all
considered approaches and both databases are presented in Table I.

### SVM-based approaches

As shown in Table I, the different SVM-based approaches do not consistently
outperform in one speech mode over the other. On the one hand, using openSMILE
features yields a better performance on spontaneous speech compared to non-
spontaneous speech, regardless of the database. However, openSMILE features
are designed for various tasks like speech emotion recognition or audio
detection, which is not ideal for pathological speech detection. Consequently,
many openSMILE features capture cues unrelated to pathology. This is reflected
in the overall low performance achieved by openSMILE features, regardless of
the speech mode or database used. On the other hand, MFCC features demonstrate
better performance on spontaneous speech than on non-spontaneous speech in the
MoSpeeDi database, while the opposite trend is observed in the PC-GITA
database. In spontaneous speech, the various acoustic, linguistic, spectral,
and phonetic features exhibit a lot more non-pathology related fluctuations
than in non-spontaneous settings. It is expected that MFCC features
inadvertently capture phonetic variations in spontaneous recordings, resulting
in lower performance for spontaneous speech than for non-spontaneous speech in
the PC-GITA database. The absence of this performance trend in the MoSpeeDi
database warrants further investigation. Finally, using sparsity-based
features yields a better performance on non-spontaneous speech than on
spontaneous speech, regardless of the database. This result is expected, given
that sparsity-based features are derived by fitting a distribution to the
speech spectral coefficients. The use of single utterances for this fitting
process influences the quality of the fit, leading to distributions that are
greatly influenced by the phonetic content of the utterance. The fluctuating
phonetic content among spontaneous speech recordings results in unreliable
distribution fits that not only exhibit pathological cues, but also phonetic
content cues.

In summary, results show that the performance of different SVM-based
approaches is differently influenced by the speech mode, depending on the used
handcrafted acoustic features. These findings emphasize the importance of
carefully crafting and selecting acoustic features for classical approaches,
such that undesirable cues are not captured.

### CNN-based and AE-based approaches

While SVM-based approaches are not be powerful enough to ignore pathology-
unrelated fluctuations in spontaneous speech, the CNN-based and AE-based
approaches have a better potential. Results in Table I show that the CNN-based
approach yields a similar performance for both speech modes in both databases
whereas the AE-based approach performs better on spontaneous speech recordings
than on non-spontaneous ones.

These findings indicate that these approaches can not only disregard
pathology-unrelated fluctuations in spontaneous speech, but can also extract
additional cues that were not identifiable in non-spontaneous speech. It
should be noted that unlike the SVM-based approaches operating on a single
utterance, the CNN- and AE-based approaches operate on multiple segments of
speech, even for the spontaneous speech recordings. The availability of more
samples when using the spontaneous speech recordings for the CNN- and AE-based
approaches might contribute to the observed improved performance.

### wav2vec2-based approach

Similar to the CNN- and AE-based approaches, we anticipate the wav2vec2-based
approach to be able to disregard pathology-unrelated variations in spontaneous
speech. As expected, the results in Table I demonstrate that the
wav2vec2-based approach performs similarly for both speech modes and
databases, with a negligible decrease in performance for spontaneous speech in
the PC-GITA database. The XLRS53 variant of the wav2vec2 model is trained on a
diverse range of non-spontaneous and spontaneous multilingual databases [33],
resulting in embeddings that do not exhibit significant fluctuations caused by
the speech mode.

Overall, the results presented in this section show that on the one hand, SVM-
based approaches struggle to perform well in spontaneous speech recordings
depending on the used acoustic features. On the other hand, deep learning-
based approaches not only have the ability to disregard pathology-unrelated
fluctuations in spontaneous speech, but can also extract additional pathology-
discriminant cues that may not be present in non-spontaneous speech. In the
future, we will investigate how the performance of the SVM-based and
wav2vec2-based approaches changes when the spontaneous speech recording is
segmented into shorter fixed-sized segments (as for the CNN- and AE-based
approaches), rather than treating it as a single utterance. Furthermore, we
will investigate the extraction of phonetic-dependent acoustic features from
spontaneous speech to be used in classical approaches.

##  V Conclusion

Pathological speech detection approaches developed so far have been trained
and evaluated only in phonetically-controlled non-spontaneous speech settings.
However, spontaneous speech can be more conveniently recorded and it can be
valuable in detecting subtle and abstract cues of pathological speech. In this
paper, we have examined the performance of various classical and deep
learning-based pathological speech detection approaches using both non-
spontaneous and spontaneous speech recordings. Our findings suggest that
classical approaches employing handcrafted acoustic features may encounter
challenges in capturing pathology-discriminant cues in spontaneous speech.
Conversely, deep learning approaches exhibit superior performance,
successfully extracting additional cues that were previously inaccessible in
non-spontaneous speech. Future work will target further improving the
performance of different approaches on spontaneous speech recordings.

##  VI Acknowledgements

This work was supported by the Swiss National Science Foundation project
CRSII5_202228 on “Characterisation of motor speech disorders and processes”.

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# Non-weight modules over the mirror Heisenberg-Virasoro algebra

Dongfang Gao  D. Gao: School of Mathematical Sciences, University of Science
and Technology of China, Hefei, Anhui 230026, P. R. China
<EMAIL_ADDRESS>,  Yao Ma  Y. Ma: School of Mathematics and
Statistics, Northeast Normal University, Changchun, Jilin 130024, P. R. China
<EMAIL_ADDRESS>and  Kaiming Zhao  K. Zhao: Department of Mathematics,
Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada, and School of
Mathematical Science, Hebei Normal (Teachers) University, Shijiazhuang, Hebei,
050024 P. R. China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study irreducible non-weight modules over the mirror
Heisenberg-Virasoro algebra $\mathcal{D}$, including Whittaker modules,
$\mathcal{U}(\mathbb{C}d_{0})$-free modules, and their tensor products. More
precisely, we give the necessary and sufficient conditions for the Whittaker
modules to be irreducible. We determine all $\mathcal{D}$-module structures on
$\mathcal{U}(\mathbb{C}d_{0})$, and find the necessary and sufficient
conditions for these modules to be irreducible. At last we determine the
necessary and sufficient conditions for the tensor products of Whittaker
modules and $\mathcal{U}(\mathbb{C}d_{0})$-free modules to be irreducible, and
obtain that any two such tensor products are isomorphic if and only if the
corresponding Whittaker modules and $\mathcal{U}(\mathbb{C}d_{0})$-free
modules are isomorphic. These lead to many new irreducible non-weight modules
over $\mathcal{D}$.

###### Key words and phrases:

mirror Heisenberg-Virasoro algebra, tensor product, Whittaker module,
$U(\mathbb{C}d_{0})$-free module, irreducible module

###### 2020 Mathematics Subject Classification:

17B10, 17B20,17B65,17B66,17B68

##  1\. Introduction

It is well-known that the Virasoro algebra $\mathcal{V}$ is one of the most
important Lie algebras in mathematics and in mathematical physics because of
its widespread applications in quantum physics (see [21]), conformal field
theory (see [15]), vertex operator algebras (see [16, 19]), and so on. Many
other interesting and important algebras are closely related to the Virasoro
algebra, such as the Schrödinger-Virasoro algebra (see [23, 24]), the twisted
Heisenberg-Virasoro algebra (see [3, 25, 30]) and the mirror Heisenberg-
Virasoro algebra $\mathcal{D}$ (see [6, 20, 28]) which is the even part of the
mirror $N=2$ superconformal algebra (see [6]). The mirror Heisenberg-Virasoro
algebra $\mathcal{D}$ has a nice structure (see Definition 2.1) which is
similar to the twisted Heisenberg-Virasoro algebra. This algebra is the main
object we concern in this paper.

Harish-Chandra modules and weight modules with infinite-dimensional weight
spaces have been the most popular modules in the representation theory for
many Lie algebras with triangular decomposition
$\mathcal{G}=\mathcal{G}_{+}\oplus\mathfrak{h}\oplus\mathcal{G}_{-}$. To some
extent, Harish-Chandra modules are well understood for many infinite-
dimensional Lie algebras, for example, the affine Kac-Moody algebras in [11,
22], the Virasoro algebra in [4, 17, 33], the twisted Heisenberg-Virasoro
algebra in [3, 30], the Schrödinger-Virasoro algebra in [27, 38], and the
mirror Heisenberg-Virasoro algebra in [28]. There are also some researches
about weight modules with infinite-dimensional weight spaces (see [8, 13, 20,
32]).

Recently, non-weight modules over $\mathcal{G}$ attract more attentions from
mathematicians. In particular, Whittaker modules, and
$\mathcal{U}(\mathfrak{h})$-free $\mathcal{G}$-modules have been widely
studied for many Lie algebras. Whittaker modules for
$\mathrm{sl_{2}(\mathbb{C})}$ were constructed by Arnal and Pinzcon (see [5]).
Whittaker modules for arbitrary finite-dimensional complex semisimple Lie
algebra $\mathfrak{g}$ were introduced and systematically studied by Kostant
in [26], where he proved that these modules with a fixed regular Whittaker
function (Lie homomorphism) on a nilpotent radical are (up to isomorphism) in
bijective correspondence with central characters of
$\mathcal{U}(\mathcal{G})$. In recent years, Whittaker modules for many other
Lie algebras have been investigated (see [1, 2, 7, 9, 14, 34, 35]). The
notation of $\mathcal{U}(\mathfrak{h})$-free modules was first introduced by
Nilsson [36] for the simple Lie algebra $\mathrm{sl}_{n+1}(\mathbb{C})$. At
the same time these modules were introduced in a very different approach in
the paper [37]. Later, $\mathcal{U}(\mathfrak{h})$-free modules for many
infinite-dimensional Lie algebras are determined, for example, the Kac-Moody
algebras in [10], the Virasoro algebra in [31, 37], the Witt algebra in [37],
the twisted Heisenberg-Virasoro algebra and $W(2,2)$ algebra in [12], and so
on. In the present paper, we will study the Whittaker modules and
$\mathcal{U}(\mathbb{C}d_{0})$-free modules over $\mathcal{D}$. Also, we study
the tensor products of Whittaker modules and
$\mathcal{U}(\mathbb{C}d_{0})$-free modules.

The paper is organized as follows. In Section 2, we recall notations related
to the mirror Heisenberg-Virasoro algebra $\mathcal{D}$ and collect some known
results on Whittaker modules and $\mathcal{U}(\mathbb{C}d_{0})$-free modules
over $\mathcal{V}$, including also two important lemmas on tensor product
modules for later use. In Section 3, we give the necessary and sufficient
conditions for the Whittaker modules $W_{\varphi_{m}}$ over $\mathcal{D}$ to
be irreducible, see Theorem 3.4. In Section 4, we determine all
$\mathcal{D}$-module structures on $\mathcal{U}(\mathbb{C}d_{0})$, and find
the necessary and sufficient conditions for these modules to be irreducible,
see Theorem 4.3. In Section 5, we give the necessary and sufficient conditions
for the tensor product of a Whittaker module $W_{\varphi_{m}}$ and a
$\mathcal{U}(\mathbb{C}d_{0})$-free $\mathcal{D}$-module
$\Omega(\lambda,\alpha,\beta)$ to be irreducible, see Theorem 5.3.
Furthermore, we show that two such tensor product modules are isomorphic if
and only if the corresponding Whittaker modules and
$\mathcal{U}(\mathbb{C}d_{0})$-free $\mathcal{D}$-modules are isomorphic, see
Theorem 5.4. Consequently, we obtain a lot of new irreducible non-weight
modules over $\mathcal{D}$.

Throughout this paper, we denote by
$\mathbb{Z},\mathbb{N},\mathbb{Z}_{+},\mathbb{C}$ and $\mathbb{C}^{*}$ the set
of integers, positive integers, non-negative integers, complex numbers and
nonzero complex numbers respectively. All vector spaces and Lie algebras are
over $\mathbb{C}$. We denote by $\mathcal{U}{(\mathcal{G})}$ the universal
enveloping algebra for a Lie algebra $\mathcal{G}$.

##  2\. Notations and Preliminaries

For convenience, in this section we recall some notations and collect some
known results.

######  Definition 2.1.

The mirror Heisenberg-Virasoro algebra $\mathcal{D}$ is a Lie algebra with
basis
$\\{d_{m},h_{r},{\bf{c}},{\bf{l}}~{}|~{}m\in\mathbb{Z},r\in\frac{1}{2}+\mathbb{Z}\\}$
subject to the following commutation relations

$\begin{split}&[d_{m},d_{n}]=(m-n)d_{m+n}+\frac{m^{3}-m}{12}{\delta}_{m+n,0}{{\bf{c}}},\\\
&[d_{m},h_{r}]=-rh_{m+r},\\\ &[h_{r},h_{s}]=r{\delta}_{r+s,0}{\bf{l}},\\\
&[{\bf{c}},\mathcal{D}]=[{\bf{l}},\mathcal{D}]=0,\end{split}$

for $m,n\in\mathbb{Z},r,s\in\frac{1}{2}+\mathbb{Z}$.

The Lie subalgebra spanned by $\\{d_{m},{\bf{c}}~{}|~{}m\in\mathbb{Z}\\}$ is
the Virasoro algebra $\mathcal{V}$, and the Lie subalgebra spanned by
$\\{h_{r},{\bf{l}}~{}|~{}r\in\frac{1}{2}+\mathbb{Z}\\}$ is the twisted
Heisenberg algebra $\mathcal{H}$. Moreover, $\mathcal{H}$ is an ideal of
$\mathcal{D}$ and $\mathcal{D}$ is the semi-direct product of $\mathcal{V}$
and $\mathcal{H}$. It is clear that $\mathcal{D},\mathcal{V},\mathcal{H}$ are
all $\mathbb{Z}$-graded Lie algebras.

######  Definition 2.2.

Let $\mathcal{G}=\oplus_{i\in\mathbb{Z}}{\mathcal{G}}_{i}$ (resp.
$\mathcal{G}=\oplus_{i\in\frac{1}{2}\mathbb{Z}}{\mathcal{G}}_{i}$) be a
$\mathbb{Z}$ (resp. $\frac{1}{2}\mathbb{Z}$)-graded Lie algebra. A
$\mathcal{G}$-module $V$ is called the $\bf{restricted}$ module if for any
$v\in V$ there exists $n\in\mathbb{N}$ such that $\mathcal{G}_{i}v=0$, for
$i>n$ (resp. $i>\frac{1}{2}n$). The category of restricted modules over
$\mathcal{G}$ will be denoted as $\mathcal{R}_{\mathcal{G}}$.

Note that if $V$ is a $\mathcal{V}$-module, then $V$ can be easily viewed as a
$\mathcal{D}$-module by defining $\mathcal{H}V=0$, the resulting module is
denoted by $V^{\mathcal{D}}$. Thanks to [18], for any
$H\in\mathcal{R}_{\mathcal{H}}$ with the action of $\bf l$ as a nonzero scalar
$l$, we can give $H$ a $\mathcal{D}$-module structure denoted by
$H^{{\mathcal{D}}}$ via the following map

$\displaystyle d_{n}$
$\displaystyle\mapsto\frac{1}{2l}\sum_{k\in\mathbb{Z}+\frac{1}{2}}h_{n-k}h_{k},\quad\forall
n\in\mathbb{Z},n\neq 0,$ (2.1) $\displaystyle d_{0}$
$\displaystyle\mapsto\frac{1}{2l}\sum_{k\in\mathbb{Z}+\frac{1}{2}}h_{-|k|}h_{|k|}+\frac{1}{16},$
(2.2) $\displaystyle h_{r}$ $\displaystyle\mapsto h_{r},\quad\forall
r\in\frac{1}{2}+\mathbb{Z},\quad{\bf{c}}\mapsto 1,\quad{\bf{l}}\mapsto l.$
(2.3)

######  Definition 2.3.

Let $\mathcal{G}$ be a Lie algebra. Suppose $f$ is an automorphism of
$\mathcal{G}$ and $V$ is a $\mathcal{G}$-module. The following actions

$x\cdot v=f(x)v,\quad\forall x\in\mathcal{G},v\in V$

give $V$ a new $\mathcal{G}$-module structure, denoted by $V^{\prime}.$ Then
$V,V^{\prime}$ are called equivalent $\mathcal{G}$-modules.

######  Remark 2.4.

It is easy to see that equivalent modules over the Lie algebra have the same
irreducibility.

For convenience, we define the following subalgebras. For any
$m,n\in\mathbb{Z}_{+}$, set

$\begin{split}&\mathcal{D}^{(m,-n)}=\sum_{i\geq
m}\mathbb{C}d_{i}\oplus\sum_{i\in\mathbb{Z}_{+}}\mathbb{C}h_{-n+i+\frac{1}{2}}\oplus\mathbb{C}{\bf
c}\oplus\mathbb{C}{\bf l},\\\
&\mathcal{D}^{(m,-\infty)}=\sum_{i\in\mathbb{Z}_{+}}\mathbb{C}d_{m+i}\oplus\sum_{i\in\mathbb{Z}}\mathbb{C}h_{i+\frac{1}{2}}+\mathbb{C}{\bf
c}+\mathbb{C}{\bf l},\\\
&\mathcal{V}^{(m)}=\sum_{i\in\mathbb{Z}_{+}}\mathbb{C}d_{m+i}+\mathbb{C}{\bf
c},\\\
&\mathcal{H}^{(m)}=\sum_{i\in\mathbb{Z}_{+}}\mathbb{C}h_{m+i+\frac{1}{2}}+\mathbb{C}{\bf
l}.\end{split}$

######  Lemma 2.5.

(cf, [28, Theorem 6.2]) Let $V$ be an irreducible
${\mathcal{D}}^{(0,-q)}$-module for some $q\in\mathbb{Z}_{+}$ such that the
action of c, l on $V$ are the scalars $c$ and $0$ respectively. Assume that
there exists an integer $t\geq-q$ satisfying the following two conditions:

  1. $(a)$

the action of $h_{t+\frac{1}{2}}$ on $V$ is bijective;

  2. $(b)$

$h_{n+\frac{1}{2}}V=0=d_{n+q}V$ for all $n>t$.

Then the induced ${\mathcal{D}}$-module
$\mathrm{Ind}_{\mathcal{D}^{(0,-q)}}^{\mathcal{D}}(V)$ is irreducible.

Recall that for any $m\in\mathbb{Z}_{+}$, let
$\psi_{m}:\mathcal{V}^{(m)}\rightarrow\mathbb{C}$ be a Whittaker function,
i.e., a Lie algebra homomorphism. Then we have the one-dimensional module
$\mathbb{C}w_{\psi_{m}}$ over $\mathcal{V}^{(m)}$ with $x\cdot
w_{\psi_{m}}=\psi_{m}(x)w_{\psi_{m}}$, for $x\in\mathcal{V}^{(m)}$. The
induced $\mathcal{V}$-module

$W_{\psi_{m}}=\mathrm{Ind}_{\mathcal{V}^{(m)}}^{\mathcal{V}}\mathbb{C}w_{\psi_{m}}$
(2.4)

is called the Whittaker module over $\mathcal{V}$ with respect to $\psi_{m}$.

######  Lemma 2.6.

(cf, [29, Theorem 7]) For any $m\in\mathbb{N}$, and any Whittaker function
$\psi_{m}:\mathcal{V}^{(m)}\rightarrow\mathbb{C}$, then the Whittaker module
$W_{\psi_{m}}$ is irreducible if and only if
$(\psi(d_{2m}),\psi(d_{2m-1}))\neq(0,0)$.

For $\lambda\in\mathbb{C}^{*},\alpha\in\mathbb{C}$, denote by
$\Omega(\lambda,\alpha)=\mathbb{C}[t]$ the polynomial algebra over
$\mathbb{C}$. It is well-known that $\Omega(\lambda,\alpha)$ is a
$\mathcal{V}$-module with the actions

${\bf c}(f(t))=0,\ \
d_{m}(f(t))=\lambda^{m}(t+m\alpha)f(t+m),~{}\forall~{}m\in\mathbb{Z}.$

Thanks to [31], we know that $\Omega(\lambda,\alpha)$ is irreducible if and
only if $\alpha\neq 0$, and $\Omega(\lambda,0)$ has an irreducible submodule
$t\Omega(\lambda,\alpha)$. Moreover, we have the following lemma.

######  Lemma 2.7.

(cf, [37, Theorem 3]) Let $V$ be a $\mathcal{V}$-module. Assume that $V$
viewed as a $\mathcal{U}(\mathbb{C}d_{0})$-module is free of rank $1$. Then
$V\cong\Omega(\lambda,\alpha)$ for some
$\lambda\in\mathbb{C}^{*},\alpha\in\mathbb{C}$.

We conclude this section by recalling two results about tensor product for
later use.

######  Lemma 2.8.

(cf, [32, Lemma 8]) Let $V$ be a module over a Lie algebra $\mathcal{G}$,
$\mathfrak{g}$ be a subalgebra of $\mathcal{G}$, and $W$ be a
$\mathfrak{g}$-module. Then the $\mathcal{G}$-module homomorphism
$\tau:\mathrm{Ind}_{\mathfrak{g}}^{\mathcal{G}}(V\otimes W)\rightarrow
V\otimes\mathrm{Ind}_{\mathfrak{g}}^{\mathcal{G}}(W)$ induced from the
inclusion map $V\otimes W\rightarrow
V\otimes\mathrm{Ind}_{\mathfrak{g}}^{\mathcal{G}}(W)$ is a
$\mathcal{G}$-module isomorphism.

######  Lemma 2.9.

(cf, [20, Lemma 3.1]) Suppose $V,W$ are $\mathcal{V}$-modules, and
$H,K\in\mathcal{R}_{\mathcal{H}}$ are irreducible with nonzero action of
$\bf{l}$. Then

  1. $(1)$

any $\mathcal{D}$-submodule of $V^{\mathcal{D}}\otimes H^{\mathcal{D}}$ is of
the form $(V^{\prime})^{\mathcal{D}}\otimes H^{\mathcal{D}}$ for some
$\mathcal{V}$-submodule $V^{\prime}$ of $V$. In particular,
$V^{\mathcal{D}}\otimes H^{\mathcal{D}}$ is an irreducible
$\mathcal{D}$-module if and only if $V$ is an irreducible
$\mathcal{V}$-module;

  2. $(2)$

$V^{\mathcal{D}}\otimes H^{\mathcal{D}}\cong W^{\mathcal{D}}\otimes
K^{\mathcal{D}}$ if and only if $V\cong W$ and $H\cong K$.

##  3\. Whittaker modules over $\mathcal{D}$

In this section, we will determine the necessary and sufficient conditions for
the Whittaker modules over $\mathcal{D}$ to be irreducible.

Assume that $\phi:\mathcal{H}^{(0)}\rightarrow\mathbb{C}$ is a Whittaker
function, and that $\mathbb{C}w_{\phi}$ is the one-dimensional module over
$\mathcal{H}^{(0)}$ defined by $x\cdot w_{\phi}=\phi(x)w_{\phi}$, for
$x\in\mathcal{H}^{(0)}$. Then for the Whittaker $\mathcal{H}$-module

$W_{\phi}=\mathrm{Ind}_{\mathcal{H}^{(0)}}^{\mathcal{H}}\mathbb{C}w_{\phi},$

we have the following result.

######  Lemma 3.1.

The Whittaker $\mathcal{H}$-module $W_{\phi}$ is irreducible if and only if
$\phi({\bf l})\neq 0$.

###### Proof.

We first assume that $\phi({\bf l})\neq 0$. For any nonzero $v\in W_{\phi}$,
$v$ is a linear combination of vectors in the form

$h_{-r-\frac{1}{2}}^{i_{r}}\cdots
h_{-1-\frac{1}{2}}^{i_{1}}h_{-\frac{1}{2}}^{i_{0}}w_{\phi},$

where $i_{0},i_{1},\cdots,i_{r}\in\mathbb{Z}_{+}$ by PBW Theorem. Let $\langle
v\rangle$ denote the submodule generated by $v$ of $W_{\phi}$. It is not hard
to see $w_{\phi}\in\langle v\rangle$. Thus $\langle v\rangle=W_{\phi}$, which
implies that $W_{\phi}$ is irreducible.

Now we assume that $\phi({\bf l})=0$. Then it is easy to see that
$h_{-\frac{1}{2}}w_{\phi}$ is a Whittaker vector, which implies that
$h_{-\frac{1}{2}}w_{\phi}$ generates a nonzero proper submodule of $W_{\phi}$.
So $W_{\phi}$ is not irreducible. ∎

Now for any $m\in\mathbb{Z}_{+}$, let
$\varphi_{m}:\mathcal{D}^{(m,0)}\rightarrow\mathbb{C}$ be a Whittaker
function, and $\mathbb{C}w_{\varphi_{m}}$ be the one-dimensional module over
$\mathcal{D}^{(m,0)}$ defined by $x\cdot
w_{\varphi_{m}}=\varphi_{m}(x)w_{\varphi_{m}}$, for $x\in\mathcal{D}^{(m,0)}$.
In the rest of this section, we will determine the irreducibility of the
Whittaker $\mathcal{D}$-module

$W_{\varphi_{m}}=\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}}\mathbb{C}w_{\varphi_{m}}.$

Each irreducible quotient of $W_{\varphi_{m}}$ (if it exists) is called the
irreducible Whittaker $\mathcal{D}$-module with respect to $\varphi_{m}$. It
is clear that

$\varphi_{m}(d_{2m+j})=\varphi_{m}(h_{m+j-\frac{1}{2}})=0,\,\,\forall
j\in\mathbb{N},$

since $\varphi_{m}([\mathcal{D}^{(m,0)},\mathcal{D}^{(m,0)}])=0$. In
particular, if $m=0$, then $W_{\varphi_{m}}$ is a Verma module, which has been
studied in [28].

First we consider $W_{\varphi_{m}}$ with $\varphi_{m}({\bf l})=l\neq 0$. We
define a new Whittaker function
$\varphi^{\prime}_{m}:\mathcal{V}^{(m)}\rightarrow\mathbb{C}$ as follows

$\displaystyle\varphi^{\prime}_{m}({\bf c})=\varphi_{m}({\bf c})-1,$
$\displaystyle\varphi^{\prime}_{m}(d_{k})=0,\quad\forall k\geq 2m+1,$
$\displaystyle\varphi^{\prime}_{m}(d_{k})=\varphi_{m}(d_{k})-\frac{1}{2l}\sum_{i=0}^{m-1}\varphi_{m}(h_{i+\frac{1}{2}})\varphi_{m}(h_{k-i-\frac{1}{2}})-\delta_{0,k}\frac{1}{16},\,\,\forall
m\leq k\leq 2m.$

Then we have the Whittaker $\mathcal{V}$-module

${W}_{\varphi^{\prime}_{m}}=\mathrm{Ind}_{\mathcal{V}^{(m)}}^{\mathcal{V}}\mathbb{C}w_{\varphi^{\prime}_{m}},$

where $\mathbb{C}w_{\varphi^{\prime}_{m}}$ is the one-dimensional module over
$\mathcal{V}^{(m)}$ defined by $x\cdot
w_{\varphi^{\prime}_{m}}=\varphi^{\prime}_{m}(x)w_{\varphi^{\prime}_{m}}$, for
$x\in\mathcal{V}^{(m)}$.

######  Proposition 3.2.

Suppose that $m\in\mathbb{Z}_{+}$, and $\varphi_{m},\varphi^{\prime}_{m}$ are
given as above with $\varphi_{m}({\bf l})\neq 0$. Let
$H=U(\mathcal{H})w_{\varphi_{m}}$ be in $W_{\varphi_{m}}$. The following
observations hold.

  1. $(1)$

$W_{\varphi_{m}}\cong H^{\mathcal{D}}\otimes
W_{\varphi^{\prime}_{m}}^{\mathcal{D}}$. In particular, this agrees with [28,
Proposition 5.1] if $m=0$.

  2. $(2)$

Suppose $m\in\mathbb{N}$. Then $W_{\varphi_{m}}$ is an irreducible
$\mathcal{D}$-module if and only if

$\varphi_{m}(d_{2m})\neq 0\,\,{\text{or}}\,\,2\varphi_{m}({\bf
l})\varphi_{m}(d_{2m-1})-\varphi_{m}(h_{m-\frac{1}{2}})^{2}\neq 0.$

  3. $(3)$

Each irreducible Whittaker module over $\mathcal{D}$ with respect to
$\varphi_{m}$ is isomorphic to $H^{\mathcal{D}}\otimes Q^{\mathcal{D}}$, where
$Q$ is an irreducible quotient of $\mathcal{V}$-module
$W_{\varphi^{\prime}_{m}}$.

###### Proof.

(1) For $\mathcal{V}^{(m)}$-module $\mathbb{C}w_{\varphi^{\prime}_{m}}$, we
can easily give $\mathbb{C}w_{\varphi^{\prime}_{m}}$ a
$\mathcal{D}^{(m,-\infty)}$-module structure by defining
$\mathcal{H}w_{\varphi^{\prime}_{m}}=0$. The resulting module is denoted by
$\mathbb{C}_{\varphi^{\prime}_{m}}^{\mathcal{D}^{(m,-\infty)}}$. By PBW
Theorem, we can define the linear map :

$\displaystyle\pi:$
$\displaystyle\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}^{(m,-\infty)}}\mathbb{C}w_{\varphi_{m}}\rightarrow
H^{\mathcal{D}}\otimes\mathbb{C}_{\varphi^{\prime}_{m}}^{\mathcal{D}^{(m,-\infty)}},$
$\displaystyle h_{-r-\frac{1}{2}}^{q_{r}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\mapsto
h_{-r-\frac{1}{2}}^{q_{r}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}.$

It is clear that $\pi$ is a bijection.

######  Claim 1.

$\pi(hw_{\varphi_{m}})=hw_{\varphi_{m}}\otimes w_{\varphi^{\prime}_{m}}$ for
$h\in\mathcal{U}(\mathcal{H})$.

For $h\in\mathcal{U}(\mathcal{H})$, $h$ can be written as a linear combination
of vectors in the form

$h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}{\bf
l}^{t}h_{\frac{1}{2}}^{p_{0}}h_{1+\frac{1}{2}}^{p_{1}}\cdots
h_{s_{1}+\frac{1}{2}}^{p_{s_{1}}}$

by PBW Theorem. Without loss of generality, we can assume

$h=h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}{\bf
l}^{t}h_{\frac{1}{2}}^{p_{0}}h_{1+\frac{1}{2}}^{p_{1}}\cdots
h_{s_{1}+\frac{1}{2}}^{p_{s_{1}}},$

where $q_{i},p_{j},t,r_{1},s_{1}\in\mathbb{Z}_{+},$ for $0\leq i\leq
r_{1},0\leq j\leq s_{1}$. We compute

$\begin{split}\pi(hw_{\varphi_{m}})=&\pi(h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}{\bf
l}^{t}h_{\frac{1}{2}}^{p_{0}}h_{1+\frac{1}{2}}^{p_{1}}\cdots
h_{s_{1}+\frac{1}{2}}^{p_{s_{1}}}w_{\varphi_{m}})\\\ =&\varphi_{m}({\bf
l})^{t}\varphi_{m}(h_{\frac{1}{2}})^{p_{0}}\varphi_{m}(h_{1+\frac{1}{2}})^{p_{1}}\cdots\varphi_{m}(h_{s_{1}+\frac{1}{2}})^{p_{s_{1}}}\pi(h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}})\\\
=&\varphi_{m}({\bf
l})^{t}\varphi_{m}(h_{\frac{1}{2}})^{p_{0}}\varphi_{m}(h_{1+\frac{1}{2}})^{p_{1}}\cdots\varphi_{m}(h_{s_{1}+\frac{1}{2}})^{p_{s_{1}}}h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}\\\ =&h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}\varphi_{m}({\bf
l})^{t}\varphi_{m}(h_{\frac{1}{2}})^{p_{0}}\varphi_{m}(h_{1+\frac{1}{2}})^{p_{1}}\cdots\varphi_{m}(h_{s_{1}+\frac{1}{2}})^{p_{s_{1}}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}\\\ =&h_{-r_{1}-\frac{1}{2}}^{q_{r_{1}}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}{\bf
l}^{t}h_{\frac{1}{2}}^{p_{0}}h_{1+\frac{1}{2}}^{p_{1}}\cdots
h_{s_{1}+\frac{1}{2}}^{p_{s_{1}}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}\\\ =&hw_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}.\end{split}$

This proves Claim 1.

Next, we will show that $\pi$ is a $\mathcal{D}^{(m,-\infty)}$-module
homomorphism. Notice that for any

$v=h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\in\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}^{(m,-\infty)}}\mathbb{C}w_{\varphi_{m}}$

where $q_{i},k\in\mathbb{Z}_{+},0\leq i\leq k$, we have the following:

######  Claim 2.

$\pi(h_{r}v)=h_{r}\pi(v){\rm\ and\ }\pi(d_{n}v)=d_{n}\pi(v),$ for any
$r\in\mathbb{Z}+\frac{1}{2},n\in\mathbb{Z}$ with $n\geq m.$

By Claim 1 and definition of $\pi$, we deduce

$\displaystyle\pi(h_{r}v)$
$\displaystyle=\pi(h_{r}h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}})=h_{r}h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}$
$\displaystyle=h_{r}(h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}})=h_{r}\pi(h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}})$
$\displaystyle=h_{r}\pi(v),$ (3.1)
$\displaystyle\pi(d_{n}h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}})$
$\displaystyle=\pi([d_{n},h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}]w_{\varphi_{m}}+h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}d_{n}w_{\varphi_{m}})$
$\displaystyle=[d_{n},h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}]w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}$ (3.2)
$\displaystyle\quad+\varphi_{m}(d_{n})h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}.$

Using definitions of $\pi,\varphi_{m}^{\prime}$ and equalities (2.1), (2.2) we
see

$\begin{split}d_{n}&\pi(h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}})=d_{n}(h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}})\\\ &\quad=d_{n}h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}+h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
d_{n}w_{\varphi^{\prime}_{m}}\\\
&\quad=[d_{n},h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}]w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}+h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}d_{n}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}\\\ &\qquad+h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes\varphi^{\prime}_{m}(d_{n})w_{\varphi^{\prime}_{m}}\\\
&\quad=[d_{n},h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}]w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}+\varphi_{m}(d_{n})h_{-k-\frac{1}{2}}^{q_{k}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w_{\varphi_{m}}\otimes
w_{\varphi^{\prime}_{m}}.\\\ \end{split}$ (3.3)

Combining equalities (3.1), (3.2) and (3.3), we have proven Claim 2.
Furthermore,

$\pi({\bf c}v)={\bf c}\pi(v),\ \pi({\bf l}v)={\bf l}\pi(v),\ \forall
v\in\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}^{(m,-\infty)}}\mathbb{C}w_{\varphi_{m}}$

is clear. So

$\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}^{(m,-\infty)}}\mathbb{C}w_{\varphi_{m}}\cong
H^{\mathcal{D}}\otimes\mathbb{C}_{\varphi^{\prime}_{m}}^{\mathcal{D}^{(m,-\infty)}}$

as $\mathcal{D}^{(m,-\infty)}$-modules. Then by the property of induced
modules and Lemma 2.8, we have

$\begin{split}W_{\varphi_{m}}=\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}}\mathbb{C}w_{\varphi_{m}}&\cong\mathrm{Ind}_{\mathcal{D}^{(m,-\infty)}}^{\mathcal{D}}(\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}^{(m,-\infty)}}\mathbb{C}w_{\varphi_{m}})\cong\mathrm{Ind}_{\mathcal{D}^{(m,-\infty)}}^{\mathcal{D}}(H^{\mathcal{D}}\otimes\mathbb{C}_{\varphi^{\prime}_{m}}^{\mathcal{D}^{(m,-\infty)}})\\\
&\cong
H^{\mathcal{D}}\otimes\mathrm{Ind}_{\mathcal{D}^{(m,-\infty)}}^{\mathcal{D}}(\mathbb{C}_{\varphi^{\prime}_{m}}^{\mathcal{D}^{(m,-\infty)}})\cong
H^{\mathcal{D}}\otimes W_{\varphi^{\prime}_{m}}^{\mathcal{D}}.\end{split}$

(2) From Lemma 3.1 we know that $H^{\mathcal{D}}$ is an irreducible
$\mathcal{D}$-module. Then, from (1), Lemmas 2.6, 2.9, we obtain the statement
in (2).

(3) follows from (1) and Lemmas 2.9, 3.1. ∎

Now we consider the Whittaker $\mathcal{D}$-module $W_{\varphi_{m}}$ with
$\varphi_{m}({\bf l})=0$. Let
$\alpha=\sum_{i\in\mathbb{Z}}\frac{2a_{i}}{2i-1}h_{i-\frac{1}{2}}$, where
$a_{i}\in\mathbb{C}$ and only finitely many of $a_{i}$ are nonzero. Then we
have the automorphism $\theta_{\alpha}=\exp({\rm ad}\alpha)$ of $\mathcal{D}$
such that

$\displaystyle\theta_{\alpha}(d_{n})=d_{n}+\sum_{i\in\mathbb{Z}}a_{i}h_{n+i-\frac{1}{2}}+\frac{1}{2}\sum_{i\in\mathbb{Z}}a_{i}a_{-n-i+1}{\bf
l},$ $\displaystyle\theta_{\alpha}(h_{r})=h_{r}+a_{-r+\frac{1}{2}}{\bf l},$
$\displaystyle\theta_{\alpha}({\bf c})={\bf c},\quad\theta_{\alpha}({\bf
l})={\bf l},$

for $n\in\mathbb{Z},r\in\mathbb{Z}+\frac{1}{2}$.

######  Proposition 3.3.

Suppose that $m\geq 1$ and
$\varphi_{m}:\mathcal{D}^{(m,0)}\rightarrow\mathbb{C}$ is a Whittaker function
with $\varphi_{m}({\bf l})=0$. Then the Whittaker $\mathcal{D}$-module
$W_{\varphi_{m}}$ is irreducible if and only if
$\varphi_{m}(h_{m-\frac{1}{2}})\neq 0$.

###### Proof.

$(\Leftarrow)$. Since $\varphi_{m}(h_{m-\frac{1}{2}})\neq 0$, we may choose
$a_{0},a_{-1},\cdots,a_{-m}\in\mathbb{C}$ such that

$\begin{pmatrix}\varphi_{m}(d_{m})\\\ \varphi_{m}(d_{m+1})\\\ \vdots\\\
\varphi_{m}(d_{2m})\end{pmatrix}=\begin{pmatrix}\varphi_{m}(h_{m-\frac{1}{2}})&\varphi_{m}(h_{m-\frac{3}{2}})&\cdots&\varphi_{m}(h_{-\frac{1}{2}})\\\
0&\varphi_{m}(h_{m-\frac{1}{2}})&\cdots&\varphi_{m}(h_{\frac{1}{2}})\\\
0&0&\cdots&\cdots\\\
0&0&\cdots&\varphi_{m}(h_{m-\frac{1}{2}})\end{pmatrix}\begin{pmatrix}a_{0}\\\
a_{-1}\\\ \vdots\\\ a_{-m}\end{pmatrix},$

where we denote $\varphi_{m}(h_{-\frac{1}{2}})=0$. Then

$0=\varphi_{m}(d_{n})+\sum_{i=-m}^{0}(-a_{i})\varphi_{m}(h_{n+i-\frac{1}{2}})+\frac{1}{2}\sum_{i\in\mathbb{Z}}a_{i}a_{-n-i+1}\varphi_{m}({\bf
l}),~{}\forall~{}n\geq m,$ (3.4)

since $\varphi_{m}({\bf l})=0$ and
$\varphi_{m}(d_{2m+j})=\varphi_{m}(h_{m-\frac{1}{2}+j})=0$, for all
$j\in\mathbb{N}$. Denote
$\alpha=-\sum_{i=-m}^{0}\frac{2a_{i}}{2i-1}h_{i-\frac{1}{2}}$. Then we have
the Lie algebra automorphism of $\theta_{\alpha}:\mathcal{D}\to\mathcal{D}$.
Let $W_{\varphi_{m}}^{\theta_{\alpha}}$ be the new Whittaker module with
action $x\circ w_{\varphi_{m}}=\theta_{\alpha}(x)w_{\varphi_{m}}$ for any
$x\in\mathcal{D}$, which is equivalent to $W_{\varphi_{m}}$. By definition of
$\theta_{\alpha}$ and equality (3.4), it is easy to see that $d_{k}\circ
w_{\varphi_{m}}=0$ for $k>m-1$, and the actions of $\mathcal{H}$ and ${\bf c}$
are unchanged. Therefore, without loss of generality, we may assume that

$\varphi_{m}(d_{k})=0,~{}\forall~{}k>m-1.$ (3.5)

Denote
$W_{0}=\mathrm{Ind}_{\mathcal{D}^{(m,0)}}^{\mathcal{D}^{(0,0)}}\mathbb{C}w_{\varphi_{m}}$.
Then $\\{d_{m-1}^{i_{m-1}}\cdots
d_{0}^{i_{0}}w_{\varphi_{m}}~{}|~{}(i_{m-1},\ldots,i_{0})\in\mathbb{Z}_{+}^{m}\\}$
is a basis of $W_{0}$ by PBW Theorem. It is not hard to see that
$h_{m-\frac{1}{2}}$ acts injectively on $W_{0}$.

######  Claim 3.

$W_{0}$ is an irreducible $\mathcal{D}^{(0,0)}$-module.

By PBW Theorem, for any $v\in W_{0}$, we may write $v$ in the form of

$v=\sum_{{\bf i}\in\mathbb{Z}_{+}^{m}}a_{\bf i}d^{\mathbf{i}}w_{\varphi_{m}},$

where $d^{\mathbf{i}}=d_{m-1}^{i_{m-1}}\cdots d_{0}^{i_{0}}$, and only
finitely many $a_{\mathbf{i}}\in\mathbb{C}$ are nonzero. Denote by
$\mathrm{supp}(v)$ the set of all ${\bf i}\in\mathbb{Z}_{+}^{m}$ such that
$a_{\bf i}\neq 0$. Let $\succ$ denote the reverse lexicographical total order
on $\mathbb{Z}_{+}^{m}$, that is, for any
${\mathbf{i},\mathbf{j}}\in\mathbb{Z}_{+}^{m}$

${\bf j}\succ{\bf i}\Leftrightarrow{\rm\ there\ exists\ }0\leq k\leq
m-1{\text{\ }such\ that\ }(j_{s}=i_{s},\forall~{}0\leq s<k){\text{\ }and\
}j_{k}>i_{k}.$

Let $\deg(v)$ be the maximal element in $\mathrm{supp}(v)$ with respect to the
reverse lexicographical total order on $\mathbb{Z}_{+}^{m}$. Note that
$\deg(0)$ is not defined and whenever we write $\deg(v)$ we mean that $v\neq
0$. Suppose $v\in W_{0}\setminus\mathbb{C}w_{\varphi_{m}},\deg(v)={\bf i}$ and
$p=\min\\{s:i_{s}\neq 0\\}$. Then it is straightforward to check that

$\deg((h_{m-p-\frac{1}{2}}-\varphi_{m}(h_{m-p-\frac{1}{2}}))v)={\bf
i-\epsilon_{p}},$

where $\epsilon_{p}$ denotes the element
$(\cdots,0,1,0,\cdots)\in\mathbb{Z}_{+}^{m}$ with $1$ being in the $(p+1)$-th
position from right. Thus $W_{0}$ is an irreducible
$\mathcal{D}^{(0,0)}$-module. Claim 3 is proved.

Therefore,
$W_{\varphi_{m}}\cong\mathrm{Ind}_{\mathcal{D}^{(0,0)}}^{\mathcal{D}}W_{0}$ is
an irreducible $\mathcal{D}$-module by Lemma 2.5 (with $q=0,t=m-1$).

$(\Rightarrow)$. Assume that $\varphi_{m}(h_{m-\frac{1}{2}})=0$. Let
$w=h_{-\frac{1}{2}}w_{\varphi_{m}}\in W_{\varphi_{m}}$. Then it is clear that
${\bf l}w=0,\ {\bf c}w=\varphi_{m}({\bf c})w$ and

$d_{m+i}w=\varphi_{m}(d_{m+i})w{\rm\,\ and\,\
}h_{i+\frac{1}{2}}w=\varphi_{m}(h_{i+\frac{1}{2}})w,\forall~{}i\in\mathbb{Z}_{+},$

since $\varphi_{m}({\bf l})=0$ and $h_{i-\frac{1}{2}}w_{\varphi_{m}}=0$ for
$i\geq m$. Let $\langle w\rangle$ denote the submodule generated by $w$ of
$W_{\varphi_{m}}$. By PBW Theorem, we know that $\langle w\rangle$ has a basis

$d_{s}^{p_{s}}\cdots
d_{m-2}^{p_{m-2}}d_{m-1}^{p_{m-1}}h_{-r-\frac{1}{2}}^{q_{r}}\cdots
h_{-1-\frac{1}{2}}^{q_{1}}h_{-\frac{1}{2}}^{q_{0}}w,$

where $p_{i},q_{j},r\in\mathbb{Z}_{+},s\in\mathbb{Z}_{<m}$, for $s\leq i\leq
m-1,0\leq j\leq r$. It is easy to see that $w_{\varphi_{m}}\notin\langle
w\rangle$, which implies that $\langle w\rangle$ is a nonzero proper submodule
of $W_{\varphi_{m}}$. So $W_{\varphi_{m}}$ is not irreducible if
$\varphi_{m}(h_{m-\frac{1}{2}})=0$. ∎

Now we combine Proposition 3.2 and Proposition 3.3 into the following main
theorem.

######  Theorem 3.4.

Let $m\in\mathbb{N}$ and
$\varphi_{m}:\mathcal{D}^{(m,0)}\rightarrow\mathbb{C}$ be a Whittaker
function.

  1. $(1)$

If $\varphi_{m}({\bf l})\neq 0$, then the Whittaker $\mathcal{D}$-module
$W_{\varphi_{m}}$ is irreducible if and only if

$\varphi_{m}(d_{2m})\neq 0\,\,{\text{or}}\,\,2\varphi_{m}({\bf
l})\varphi_{m}(d_{2m-1})-\varphi_{m}(h_{m-\frac{1}{2}})^{2}\neq 0.$

  2. $(2)$

If $\varphi_{m}({\bf l})=0$, then the Whittaker $\mathcal{D}$-module
$W_{\varphi_{m}}$ is irreducible if and only if
$\varphi_{m}(h_{m-\frac{1}{2}})\neq 0$.

##  4\. $\mathcal{U}(\mathbb{C}d_{0})$-free modules over $\mathcal{D}$

In this section, we determine the $\mathcal{D}$-module structure on
$\mathcal{U}(\mathbb{C}d_{0})$. Also, we give the necessary and sufficient
conditions for these modules to be irreducible.

For any $\lambda\in\mathbb{C}^{*},\alpha,\beta\in\mathbb{C}$, denote by
$\Omega(\lambda,\alpha,\beta)=\mathbb{C}[t]$ the polynomial algebra over
$\mathbb{C}$ It is not hard to see that we can give
$\Omega(\lambda,\alpha,\beta)$ a $\mathcal{D}$-module structure via the
following actions

$\displaystyle d_{m}(f(t))=\lambda^{m}(t+m\alpha)f(t+m),$ $\displaystyle
h_{r}(f(t))=\beta\lambda^{r}f(t+r),$ $\displaystyle{\bf c}(f(t))={\bf
l}(f(t))=0,$

for $m\in\mathbb{Z},r\in\mathbb{Z}+\frac{1}{2}$ and
$f(t)\in\Omega(\lambda,\alpha,\beta)$.

######  Remark 4.1.

In the above actions, we always fix a $\lambda_{0}\in\mathbb{C}^{*}$ such that
$\lambda=\lambda_{0}^{2}$. Then we denote $\lambda^{r}=\lambda_{0}^{2r}$ for
$r\in\mathbb{Z}+\frac{1}{2}$.

Moreover, we have the following lemma.

######  Lemma 4.2.

  1. $(1)$

$\Omega(\lambda,0,0)$ has an irreducible submodule $t\Omega(\lambda,0,0)$.

  2. $(2)$

$\Omega(\lambda,\alpha,\beta)$ is an irreducible $\mathcal{D}$-module if and
only if $\alpha\neq 0$ or $\beta\neq 0$.

###### Proof.

It is clear from the irreducibility of $\mathcal{V}$-module
$\Omega(\lambda,\alpha)$ and some simple computations. ∎

Now, we state the main result of this section.

######  Theorem 4.3.

Let $M$ be a $\mathcal{U}(\mathcal{D})$-module such that $M$, when considered
as a $\mathcal{U}(\mathbb{C}d_{0})$-module, is free of rank 1. Then
$M\cong\Omega(\lambda,\alpha,\beta)$ for some
$\lambda\in\mathbb{C}^{*},\alpha,\beta\in\mathbb{C}$. Moreover, $M$ is
irreducible if and only if $M\cong\Omega(\lambda,\alpha,\beta)$ for some
$\lambda\in\mathbb{C}^{*},\alpha,\beta\in\mathbb{C}$ with
$(\alpha,\beta)\neq(0,0)$.

###### Proof.

It is clear that $M\cong\Omega(\lambda,\alpha)=\mathbb{C}[t]$ as
$\mathcal{V}$-modules from Lemma 2.7, where
$\lambda\in\mathbb{C}^{*},\alpha\in\mathbb{C}$ and $\mathbb{C}[t]$ is the
polynomial algebra. So, we can assume that

${\bf c}f(t)=0,\ \ d_{m}f(t)=\lambda^{m}(t+m\alpha)f(t+m),$

for $m\in\mathbb{Z},f(t)\in\mathbb{C}[t]$. Now, we consider the action of
$h_{r}$ for $r\in\mathbb{Z}+\frac{1}{2}$. First, it is not hard to see that

$h_{r}(f(t))=h_{r}f(d_{0})(1)=f(t+r)h_{r}(1),$ (4.1)

for any $r\in\mathbb{Z}+\frac{1}{2}$ and $f(t)\in\mathbb{C}[t]$. Next, we
consider the two cases.

Case 1. $h_{r}(1)=0$ for some $r\in\mathbb{Z}+\frac{1}{2}$.

It is easy to get that $h_{r}M=0$ by equality (4.1). Then we have $h_{s}M=0$
for any $s\in\mathbb{Z}+\frac{1}{2}$ by the defining relations of
$\mathcal{D}$. Thus $\mathcal{H}M=0$. It shows that
$M\cong\Omega(\lambda,\alpha,0)$ as $\mathcal{D}$-modules.

Case 2. $h_{r}(1)\neq 0$ for any $r\in\mathbb{Z}+\frac{1}{2}$.

First, we show the following claim.

######  Claim 4.

$h_{r}(1)\in\mathbb{C}^{*}$ for any $r\in\mathbb{Z}+\frac{1}{2}$.

Assume that there exists $r_{0}\in\mathbb{Z}+\frac{1}{2}$ such that
$h_{r_{0}}(1)=\sum_{i=0}^{k_{r_{0}}}a_{i}t^{i}$ with ${k_{r_{0}}}>0$ and
$a_{k_{r_{0}}}\neq 0$. Choose $s\in\mathbb{Z}+\frac{1}{2}$ such that
$sr_{0}<0$ and $s+r_{0}\neq 0$. Denote

$X(t)=h_{r_{0}}(1)=\sum_{i=0}^{k_{r_{0}}}a_{i}t^{i}{\rm\ \ and\ \
}Y(t)=h_{s}(1)=\sum_{j=0}^{k_{s}}b_{j}t^{j},$

where $b_{k_{s}}\neq 0$. Now, we have $[h_{s},h_{r_{0}}](1)=0$. Thus

$\begin{split}0=&h_{s}h_{r_{0}}(1)-h_{r_{0}}h_{s}(1)=h_{s}X(t)-h_{r_{0}}Y(t)\\\
=&X(t+s)Y(t)-Y(t+r_{0})X(t)\\\
=&(\sum_{i=0}^{k_{r_{0}}}a_{i}(t+s)^{i})(\sum_{j=0}^{k_{s}}b_{j}t^{j})-(\sum_{j=0}^{k_{s}}b_{j}(t+r_{0})^{j})(\sum_{i=0}^{k_{r_{0}}}a_{i}t^{i})\\\
=&a_{k_{r_{0}}}b_{k_{s}}(sk_{r_{0}}-r_{0}k_{s})t^{k_{r_{0}}+k_{s}-1}+({\rm
lower\ degree\ terms\ in\ }t),\end{split}$

which implies a contradiction. Claim 4 is proved.

Now it is clear that ${\bf l}M=0$ from Claim 4. Denote
$h_{r}(1)=a_{r}\in\mathbb{C}^{*}$ for $r\in\mathbb{Z}+\frac{1}{2}$. Then

$\begin{split}-ra_{m+r}=&-rh_{m+r}(1)=[d_{m},h_{r}](1)\\\
=&d_{m}h_{r}(1)-h_{r}d_{m}(1)=a_{r}d_{m}(1)-h_{r}d_{m}(1)\\\
=&a_{r}(\lambda^{m}(t+m\alpha))-(\lambda^{m}(t+r+m\alpha))a_{r}\\\
=&-ra_{r}\lambda^{m},\end{split}$

which implies that $a_{m+r}=\lambda^{m}a_{r}$ for
$m\in\mathbb{Z},r\in\mathbb{Z}+\frac{1}{2}$. So, we have a
$\mathcal{D}$-module isomorphism

$M\rightarrow\Omega(\lambda,\alpha,\lambda^{-\frac{1}{2}}a_{\frac{1}{2}}),\ \
t^{i}\rightarrow t^{i}.$

The remaining part follows from Lemma 4.2. In conclusion, we complete the
proof. ∎

##  5\. Tensor products of Whittaker modules and
$\mathcal{U}(\mathbb{C}d_{0})$-free $\mathcal{D}$-modules

In this section, we show that if $\Omega(\lambda,\alpha,\beta)$ and
$W_{\varphi_{m}}$ are irreducible $\mathcal{D}$-modules, then tensor product
module $\Omega(\lambda,\alpha,\beta)\otimes W_{\varphi_{m}}$ is also
irreducible for
$\lambda\in\mathbb{C}^{*},\alpha,\beta\in\mathbb{C},m\in\mathbb{Z}_{+}$.
Moreover, we prove that $\Omega(\lambda,\alpha,\beta)\otimes
W_{\varphi_{m}}\cong\Omega(\lambda_{1},\alpha_{1},\beta_{1})\otimes
W_{\varphi_{m_{1}}}$ if and only if
$\Omega(\lambda,\alpha,\beta)\cong\Omega(\lambda_{1},\alpha_{1},\beta_{1})$
and $W_{\varphi_{m}}\cong W_{\varphi_{m_{1}}}$. In fact, we get the more
general results.

######  Proposition 5.1.

Let $\lambda\in\mathbb{C}^{*},\alpha,\beta\in\mathbb{C}$ with
$(\alpha,\beta)\neq(0,0)$, and $V$ be an irreducible restricted module over
$\mathcal{D}$. Then $\Omega(\lambda,\alpha,\beta)\otimes V$ is an irreducible
$\mathcal{D}$-module.

###### Proof.

It is clear that, for any $v\in V$, there exists $N(v)\in\mathbb{Z}_{+}$ such
that $d_{m}v=h_{r}v=0$ for $m,r\geq N(v)$ by the definition of restricted
module. Now, suppose $M$ is a nonzero submodule of
$\Omega(\lambda,\alpha,\beta)\otimes V$. We only need to show

$M=\Omega(\lambda,\alpha,\beta)\otimes V.$

######  Claim 5.

There exists a $v\in V\setminus\\{0\\}$ such that $1\otimes v\in M$.

Take a nonzero element $w=\sum_{i=0}^{s}t^{i}\otimes v_{i}\in M$ such that
$v_{i}\in V,v_{s}\neq 0$ and $s$ is minimal.

Let $N=max\\{N(v_{i}):i=0,1,\cdots,s\\}$. Then $d_{m}(v_{i})=h_{r}(v_{i})=0$,
for all $m,r\geq N$ and $i=0,1,\cdots,s$. If $\beta\neq 0$, then we get that

$\beta^{-1}\lambda^{-r}h_{r}w=\sum_{i=0}^{s}(t+r)^{i}\otimes v_{i}\in M,$

for $r\geq N$. We can write the right hand side in the form

$\sum_{i=0}^{s}r^{i}w_{i}\in M,\forall r>N,$

where $w_{i}\in\Omega(\lambda,\alpha,\beta)\otimes V$ are independent of $r$.
In particular, $w_{s}=1\otimes v_{s}$. Taking
$r=\frac{1}{2}+N,\frac{1}{2}+(N+1),\cdots,\frac{1}{2}+(N+s)$, we see that the
coefficient matrix of $w_{i}$ is a Vandermonde matrix. Thus each $w_{i}\in M$.
In particular, $w_{s}=1\otimes v_{s}\in M$.

If $\beta=0$, then $\alpha\neq 0$. Similarly, using the action of $d_{m}w,m>N$
we can show that Claim 5 holds as well.

######  Claim 6.

$M=\Omega(\lambda,\alpha,\beta)\otimes V$.

From Claim 5, we have that $1\otimes v\in M$ for some nonzero $v\in V$. Note
that

$\displaystyle d_{m}(t^{j}\otimes v)=\lambda^{m}(t+m\alpha)(t+m)^{j}\otimes
v=\lambda^{m}t(t+m)^{j}\otimes v+\lambda^{m}m\alpha(t+m)^{j}\otimes v,$ (5.1)

for any $j\in\mathbb{Z}_{+},m\geq N(v)$. Thus by induction on $j$, we deduce
that $t^{j}\otimes v\in M$ for all $j\in\mathbb{Z}_{+}$, i.e.,
$\Omega(\lambda,\alpha,\beta)\otimes v\in M$. Define

$W=\\{w\in V:\Omega(\lambda,\alpha,\beta)\otimes w\subseteq M\\}.$

It is clear that $W\neq 0$ since $v\in W$. We need to prove that $W$ is
$\mathcal{D}$-submodule of $V$. For any $w\in
W,m\in\mathbb{Z},r\in\frac{1}{2}+\mathbb{Z}$, we compute that

$\displaystyle d_{m}(\Omega(\lambda,\alpha,\beta)\otimes
w)=d_{m}\Omega(\lambda,\alpha,\beta)\otimes
w+\Omega(\lambda,\alpha,\beta)\otimes d_{m}w\subseteq M,$ $\displaystyle
h_{r}(\Omega(\lambda,\alpha,\beta)\otimes
w)=h_{r}\Omega(\lambda,\alpha,\beta)\otimes
w+\Omega(\lambda,\alpha,\beta)\otimes h_{r}w\subseteq M.$

Thus $\Omega(\lambda,\alpha,\beta)\otimes
d_{m}w,\Omega(\lambda,\alpha,\beta)\otimes h_{r}w\subseteq M$, i.e.,
$d_{m}w,h_{r}w\in W$. These show that $W$ is a nonzero submodule of $V$. Thus
$W=V$ since $V$ is irreducible, which implies
$M=\Omega(\lambda,\alpha,\beta)\otimes V$. We complete the proof. ∎

######  Proposition 5.2.

Let
$\lambda,\lambda_{1}\in\mathbb{C}^{*},\alpha,\alpha_{1},\beta,\beta_{1}\in\mathbb{C}$.
Assume that $V$ and $V_{1}$ are irreducible restricted modules over
$\mathcal{D}$. Then $\Omega(\lambda,\alpha,\beta)\otimes V$ and
$\Omega(\lambda_{1},\alpha_{1},\beta_{1})\otimes V_{1}$ are isomorphic as
$\mathcal{D}$-modules if and only if
$(\lambda,\alpha,\beta)=(\lambda_{1},\alpha_{1},\beta_{1})$ and $V\cong V_{1}$
as $\mathcal{D}$-modules.

###### Proof.

It is clear that “if part” is trivial. We only need to show “only if part”.
Let $\psi:\Omega(\lambda,\alpha,\beta)\otimes
V\to\Omega(\lambda_{1},\alpha_{1},\beta_{1})\otimes V_{1}$ be a module
isomorphism. For any nonzero element $1\otimes
v\in\Omega(\lambda,\alpha,\beta)\otimes V$, suppose

$\psi(1\otimes v)=\sum_{i=0}^{p}t^{i}\otimes w_{i},$

where $w_{i}\in V_{1}$ and $w_{p}\neq 0$. Let
$N_{v}=max\\{N{(v)},N{(w_{0})},N{(w_{1})},\cdots,N{(w_{p})}\\}$. Then

$d_{m}v=d_{m}(w_{i})=h_{r}v=h_{r}(w_{i})=0,$

for $m,r\geq N_{v},i=0,1,\cdots,p$.

######  Claim 7.

$\lambda=\lambda_{1},\alpha=\alpha_{1},\beta=\beta_{1}$.

Take $m,m_{1}\geq N_{v}$, we have

$(\lambda^{-m}d_{m}-\lambda^{-m_{1}}d_{m_{1}})(1\otimes
v)=\alpha(m-m_{1})(1\otimes v).$

By $(\lambda^{-m}d_{m}-\lambda^{-m_{1}}d_{m_{1}})\psi(1\otimes
v)=\alpha(m-m_{1})\psi(1\otimes v)$, we deduce that

$\displaystyle\alpha(m-m_{1})\sum_{i=0}^{p}t^{i}\otimes
w_{i}=\sum_{i=0}^{p}\\{(\frac{\lambda_{1}}{\lambda})^{m}(t+m\alpha_{1})(t+m)^{i}-(\frac{\lambda_{1}}{\lambda})^{m_{1}}(t+m_{1}\alpha_{1})(t+m_{1})^{i}\\}\otimes
w_{i}.$ (5.2)

Thus
$((\frac{\lambda_{1}}{\lambda})^{m}-(\frac{\lambda_{1}}{\lambda})^{m_{1}})t^{p+1}\otimes
w_{p}=0$ for $m,m_{1}\geq N_{v}$. So, $\lambda=\lambda_{1}$. Therefore, the
equation (5.2) becomes

$\alpha(m-m_{1})\sum_{i=0}^{p}t^{i}\otimes
w_{i}=\sum_{i=0}^{p}t\\{(t+m)^{i}-(t+m_{1})^{i}\\}\otimes
w_{i}+\sum_{i=0}^{p}\alpha_{1}\\{m(t+m)^{i}-m_{1}(t+m_{1})^{i}\\}\otimes
w_{i},$

for $m,m_{1}\geq N_{v}$. Then the coefficient of $t^{p}\otimes w_{p}$ in the
right side is $(\alpha_{1}+p)(m-m_{1})$ which implies $\alpha=p+\alpha_{1}$,
i.e., $\alpha-\alpha_{1}=p\geq 0$. Similar, using $\psi^{-1}$ we can get
$\alpha-\alpha_{1}\geq 0$. So $\alpha=\alpha_{1}$ and $p=0$. Thus

$\psi(1\otimes v)=1\otimes w_{0}.$

Note that for any $r\geq N_{v}$, we have

$\displaystyle\beta\lambda^{r}(1\otimes w_{0})=\beta\lambda^{r}\psi(1\otimes
v)=\psi(h_{r}(1\otimes v))=h_{r}(1\otimes
w_{0})=\beta_{1}\lambda_{1}^{r}(1\otimes w_{0}).$

Thus $\beta=\beta_{1}$. Hence, Claim 7 is proved.

Now we define the linear map

$\xi:V\rightarrow V_{1},$

such that $\psi(1\otimes v)=1\otimes\xi(v)$ for $v\in V$. It is clear that
$\xi$ is injective. Note that for any $r\in\mathbb{Z}+\frac{1}{2}$,

$\begin{split}\beta\lambda^{r}(1\otimes\xi(v))+1\otimes\xi(h_{r}(v))&=\psi(h_{r}(1\otimes
v))=h_{r}(\psi(1\otimes v))\\\
&=h_{r}(1\otimes\xi(v))=\beta\lambda^{r}(1\otimes\xi(v))+1\otimes
h_{r}(\xi(v)),\end{split}$

which implies

$\displaystyle\xi(h_{r}(v))=h_{r}(\xi(v)),\forall v\in
V,r\in\mathbb{Z}+\frac{1}{2}.$ (5.3)

Next, we consider the equation $\psi(d_{m}(1\otimes v))=d_{m}(\psi(1\otimes
v))$ for $m\geq N_{v}$. We deduce

$\displaystyle\lambda^{m}\psi(t\otimes
v)+m\alpha\lambda^{m}(1\otimes\xi(v))=\lambda^{m}t\otimes\xi(v)+m\alpha\lambda^{m}(1\otimes\xi(v)).$

So, $\psi(t\otimes v)=t\otimes\xi(v)$. Hence

$\psi(d_{n}(1)\otimes v)=d_{n}(1)\otimes\xi(v),$

for $n\in\mathbb{Z}$. Using the equation $\psi(d_{n}(1\otimes
v))=d_{n}\psi(1\otimes v)$ for $n\in\mathbb{Z},v\in V$, we deduce

$1\otimes\xi(d_{n}(v))=1\otimes d_{n}(\xi(v)).$

That shows

$\displaystyle\xi(d_{n}(v))=d_{n}(\xi(v)),\forall v\in V,n\in\mathbb{Z}.$
(5.4)

It is clear that

$\psi({\bf c}(1\otimes v))={\bf c}\psi(1\otimes v){\rm\ and\ }\psi({\bf
l}(1\otimes v))={\bf l}\psi(1\otimes v),$

imply that

$\displaystyle\xi({\bf c}(v))={\bf c}\xi(v){\rm\ and\ }\xi({\bf l}(v))={\bf
l}\xi(v).$ (5.5)

Therefore, $\xi$ is a $\mathcal{D}$-module homomorphism by equations (5.3),
(5.4), and (5.5). Since $\xi(V)\neq 0$ and $V_{1}$ is irreducible, so
$\xi(V)=V_{1}$. Thus $\xi$ is a $\mathcal{D}$-module isomorphism from $V$ to
$V_{1}$. In conclusion, we complete the proof. ∎

It is clear that Whittaker $\mathcal{D}$-modules $W_{\varphi_{m}}$ are
restricted modules over $\mathcal{D}$. So we have the following two main
theorems from Theorems 3.4, 4.3 and Propositions 5.1, 5.2.

######  Theorem 5.3.

Let $m\in\mathbb{N},\lambda\in\mathbb{C}^{*},\alpha,\beta\in\mathbb{C}$, and
$\varphi_{m}:\mathcal{D}^{(m,0)}\rightarrow\mathbb{C}$ be a Whittaker
function.

  1. $(1)$

If $\varphi_{m}({\bf l})\neq 0$, then $\Omega(\lambda,\alpha,\beta)\otimes
W_{\varphi_{m}}$ is an irreducible $\mathcal{D}$-module if and only if

$(\varphi_{m}(d_{2m}),2\varphi_{m}({\bf
l})\varphi_{m}(d_{2m-1})-\varphi_{m}(h_{m-\frac{1}{2}})^{2})\neq(0,0){\rm\
and\ }(\alpha,\beta)\neq(0,0).$

  2. $(2)$

If $\varphi_{m}({\bf l})=0$, then $\Omega(\lambda,\alpha,\beta)\otimes
W_{\varphi_{m}}$ is an irreducible $\mathcal{D}$-module if and only if

$\varphi_{m}(h_{m-\frac{1}{2}})\neq 0{\rm\ and\ }(\alpha,\beta)\neq(0,0).$

######  Theorem 5.4.

Let
$m,m_{1}\in\mathbb{Z}_{+},\lambda,\lambda_{1}\in\mathbb{C}^{*},\alpha,\alpha_{1},\beta,\beta_{1}\in\mathbb{C}.$
Suppose that $\varphi_{m}:\mathcal{D}^{(m,0)}\rightarrow\mathbb{C}$ and
$\varphi_{m_{1}}:\mathcal{D}^{(m_{1},0)}\rightarrow\mathbb{C}$ be the
Whittaker functions such that the corresponding Whittaker
$\mathcal{D}$-modules $W_{\varphi_{m_{1}}}$ and $W_{\varphi_{m_{1}}}$ are all
irreducible. The following results hold.

  1. $(1)$

$\Omega(\lambda,\alpha,\beta)\cong\Omega(\lambda_{1},\alpha_{1},\beta_{1})$ if
and only if $(\lambda,\alpha,\beta)=(\lambda_{1},\alpha_{1},\beta_{1})$.

  2. $(2)$

$\Omega(\lambda,\alpha,\beta)\otimes
W_{\varphi_{m}}\cong\Omega(\lambda_{1},\alpha_{1},\beta_{1})\otimes
W_{\varphi_{m_{1}}}$ if and only if
$(\lambda,\alpha,\beta)=(\lambda_{1},\alpha_{1},\beta_{1})$ and
$W_{\varphi_{m}}\cong W_{\varphi_{m_{1}}}$.

######  Remark 5.5.

Using Theorems 3.4, 4.3 and 5.3 we can obtain many new irreducible non-weight
modules over $\mathcal{D}$.

Acknowledgments: The research in this paper was carried out during the visit
of the first author at Wilfrid Laurier University in 2019-2021. The first
author is partially supported by CSC of China (No. 201906340096), and NSF of
China (Grants 11771410 and 11931009). The second author is partially supported
by NSF of China (Grant 11801066). The third author is partially supported by
NSF of China (Grant 11871190) and NSERC (311907-2020). The authors would like
to thank the referees for valuable suggestions to improve the paper.

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