# Investigating the Impact of Cross-lingual Acoustic-Phonetic Similarities on
Multilingual Speech Recognition

###### Abstract

Multilingual automatic speech recognition (ASR) systems mostly benefit low
resource languages but suffer degradation in performance across several
languages relative to their monolingual counterparts. Limited studies have
focused on understanding the languages behaviour in the multilingual speech
recognition setups. In this paper, a novel data-driven approach is proposed to
investigate the cross-lingual acoustic-phonetic similarities. This technique
measures the similarities between posterior distributions from various
monolingual acoustic models against a target speech signal. Deep neural
networks are trained as mapping networks to transform the distributions from
different acoustic models into a directly comparable form. The analysis
observes that the languages `closeness' can not be truly estimated by the
volume of overlapping phonemes set. Entropy analysis of the proposed mapping
networks exhibits that a language with lesser overlap can be more amenable to
cross-lingual transfer, and hence more beneficial in the multilingual setup.
Finally, the proposed posterior transformation approach is leveraged to fuse
monolingual models for a target language. A relative improvement of $\sim$8%
over monolingual counterpart is achieved.

Index Terms: automatic speech recognition, multilingual, acoustic-phonetic
similarities, model fusion

##  1 Introduction

Multilingual automatic speech recognition (ASR) systems have attained
significant attention over the past decade. Motivation for multilingual speech
recognition includes;  (i) having a single unified model capable of
recognising speech of diverse languages [1, 2] and  (ii) using shared language
representations to improve ASR performance in the low resource settings [3, 4,
5, 6, 7, 8, 9, 10].

Previous works show that multilingual ASRs do not yield significant reduction
in Word Error Rate (WER) for various languages including English, German,
French and several others [1, 2, 11]. Many open source data sets are available
for these languages and are generally regarded as resource-rich languages.
Most of the multilingual models, which include resource-rich languages, are
trained on unbalanced data. So, degradation in the performance of these
languages is attributed to increased confusion for them in the multilingual
setup [1]. Conneau et al. [11] mention language interference as the reason of
the increased error rate. However, no proof of these concepts could be found
in the literature.

Earlier studies, in the context of efficient data sharing among languages for
multilingual setup, are on Context Independent (CI) acoustic models [12].
Context Dependent (CD) acoustic-phonetic similarity was first studied by
Imperl et al. [13] and later extended by Le et al. [14]. Both of these studies
measure distance between two polyphones as a weighted sum of monophonic
distances of these polyphones. In [14], a knowledge based approach is applied
which needs considerable manual efforts as it requires a hierarchical graph to
measure the distance between monophones. Furthermore, the phonemes are
clustered on IPA symbols.

Recently, some efforts have been made to interpret the learning of
multilingual speech recognition systems [15, 16]. Phoneme Error Rate (PER) of
each phoneme in monolingual ASR was compared with that of multilingual system
[15]. However, no monotonic trend was observed with the growing number of
languages the phoneme shared. The authors described this as ``unexpected"
because the phonemes shared by more languages provide more training data and
thus the expected error trend would be decreasing. Motivated by the fact that
many languages with significant phonemes overlap pose performance degradation
in the multilingual setups [1, 2, 11, 15, 16], the objective and contribution
of this work is to study acoustic-phonetic similarities to understand if the
cross-lingual phoneme sharing is truly a sharing and how does it impact on
multilingual speech recognition setups?

To that end, a novel technique is proposed to estimate the cross-lingual
acoustic-phonetic similarities for CD hybrid DNN-HMM acoustic models. Hybrid
DNN-HMM system is preferred over end-to-end (e2e) modelling to avoid influence
of the entangled language model in e2e speech recognition systems. An equal
amount of three West Germanic languages (English, German and Dutch) is used to
study the impact on multilingual performance. Behaviour of the monolingual
acoustic models against a speech signal of the target language is studied by
differentiating the posterior distributions. To compare distributions of
source and target AMs, a separate regression neural network is trained for
each $<$source, target$>$ pair to map posteriors from a source language AM to
the posteriors of the target language AM.

##  2 Cross-lingual acoustic-phonetic similarities

The motivation of research into multilingual speech recognition is based on an
assumption that the articulatory representations of phonemes are very close
across the languages and can be considered language independent units [17].
However, several languages with substantial cross-lingual phoneme sharing
exhibit poorer performance in multilingual setups. This calls for a study to
understand the reason of degradation or improvement in multilingual setups
when compared with corresponding monolingual systems.

Hybrid DNN-HMM systems yield better performance than the conventional GMM-HMM
based ASRs and outperform e2e ASRs with the limited amounts of training data
[18]. Furthermore, the output from e2e speech recognition systems is
influenced by the entangled language model which adversely effects the
acoustic analysis.

In hybrid speech recognition systems, a deep neural network is trained to
produce a posterior distribution of tied states of HMM models. The total
number of states (and thus output layer dimension of DNN) is reduced by
clustering many polyphonemes together. Each language yields a different
phonetic decision tree in its monolingual ASR. Thus the number of tied states
differs for each language and the posterior distributions are not directly
comparable across the languages.

In this work a data-driven approach, to transform posteriors from diverse
models to a directly comparable form, is proposed. A similarity measure is
calculated between the posterior distributions from different models against a
given speech signal to estimate the cross-lingual acoustic-phonetic
similarities.

Figure 1: Proposed system architecture

###  2.1 Similarity measure

Let $M_{A}$ and $M_{S_{i}}$ be the monolingual acoustic models of target and
source languages respectively. The target language is the language for which
the similarity is being measured against the source languages. A regression
neural network $N_{S_{i}A}$ is trained to translate posteriors $P_{S_{i}}$ of
dimension $d_{S_{i}}$ from $M_{S_{i}}$ to the posteriors $P_{S_{i}A}$ of
dimension $d_{A}$ where $d_{A}$ is the dimension of posteriors from $M_{A}$.
An underlying assumption is that this mapping network is able to learn some
language related relationships between posterior distributions of source and
the target acoustic models. For example, the network could learn the phonemes
of target language which are more amenable to cross-lingual transfer than the
others. A few hours of speech data can give thousands of examples that provide
sufficient training data for mapping network. Kullback-Leibler (KL)
divergence, the most widely used measure to differentiate two posterior
distributions [19], is calculated as a similarity measure between the
posterior distributions from the target language AM and the mapped posterior
distributions from the source language AM. The proposed system architecture is
shown in Fig. 1.

Let $X=\\{x_{1},x_{2},\dotsc,x_{T}\\}$ be a set of observations of target
language, for which posterior distributions
($P^{Z}=\\{p_{1},p_{2},\dotsc,p_{T}\\}$ where $Z\in(A,S_{i}A)$) are attained
from all monolingual acoustic models. Posteriors from source acoustic models
($P^{S_{i}}$) are mapped to target posteriors ($P^{S_{i}A}$) using mapping
network $N_{S_{i}A}$. The similarity between a source and the target language
for a given set of the observations is calculated as

$D_{X}(M_{T},M_{S_{i}})=\frac{\sum_{t=1}^{T}p^{A}_{t}\cdot(\log p^{A}_{t}-\log
p^{S_{i}A}_{t})}{T}$ (1)

##  3 Experimental Setup

###  3.1 Data set

Experiment results are reported using three languages (English en, German de
and Dutch nl) of West Germanic family. From previous works, the performance of
these languages is either degraded or show a very minor improvement in
multilingual setups despite of a sufficient number of the shared phonemes. The
sharing of cross-lingual phonemes is tabulated in Table 1.

The experiments are carried out using portions of the Multilingual LibriSpeech
(MLS) data set [20]. For training and evaluation of monolingual AMs, 30 hours
and 2 hours are randomly sampled from corresponding sets of the MLS corpus
respectively. Limited amount of data for each language is used in this study
due to several reasons.

  * •

The amount of data is restricted as an ablation study to the argument that the
performance of these languages degrades in the multilingual setups due to
their very strong monolingual counterparts (trained on much more data being
resource-rich languages) [1].

  * •

On phonetic level, 30 hours are sufficient for acoustic-phonetic similarities
analysis as they provide millions of examples for mapping network training.

  * •

Experiments with the limited data for model fusion, provides a realistic
scenario for low resource languages and the technique can be extended for
resource-deficient languages if the outcome is encouraging.

Baseline multilingual ASR is trained by mixing data from all the languages and
hence trained on 90 hours of speech data. To train the mapping network, the 30
hours train set is further divided into 29 hours of training and 1 hour for
validation.

Table 1: % Cross-lingual phoneme shares | Shares of
---|---
shared with |  | en | de | nl
en | 100% | 70.58% | 69.48%
de | 72.96% | 100% | 88.59%
nl | 69.8% | 85.14% | 100%

###  3.2 Similarity measure

All acoustic models used for experiments in this work are trained using
lattice-free MMI criterion (LF-MMI) [18]. Though the study can easily be
extended for polyphones of any context, left biphones are modelled for
experimentation here and the term biphones will be used during the remainder
of the paper.

Given the partially overlapped phoneme sets of two languages (the target and a
source language), there are the following subsets of biphones:

  * •

Shared biphones: Due to the overlapping cross-lingual phoneme set, there would
be some biphones which occur in both (source and the target) languages.

    * –

Shared seen biphones (SS): The set of shared biphones which are seen by both
languages during training of their acoustic models.

    * –

Shared unseen biphones (SU): The set of shared biphones which are never seen
by source language in its train set.

  * •

Unshared biphones (U): The biphones of the target language that are never seen
by source language due to non-overlapping phonemes.

Table 2: Baseline ASR performance Language | WER/PER
---|---
mono | multi | mono-lm
English (en) | 43.84/28.10 | 47.48/31.06 | 46.40/31.06
German (de) | 37.77/26.86 | 40.81/28.35 | 38.11/28.35
Dutch (nl) | 37.94/21.40 | 58.84/36.16 | 52.33/36.16

A similarity measure is calculated for each of the aforementioned case and an
analysis of trends in distance measure is carried out to comprehend the
language similarities and their behaviour in the multilingual setups.

###  3.3 Model fusion

The mapping network is further used to map posterior distributions from
several models to a uniform (target language) dimension. The weighted sum of
these mapped posteriors is then used for ASR decoding.

For a given observation at time $t$, the final posterior vector is given as;

$p_{t}^{F}=w_{T}\cdot p_{t}^{A}+\sum_{i=1}^{N}w_{i}\cdot p_{t}^{S_{i}A}$ (2)

where $w$ are the scalar weights assigned to each posterior vector such that
$\sum w=1$ and $N$ is the number of source languages.

Table 3: Posterior distribution similarity for the en test set AM |  Biphone subsets |  % Correct SAMC |  KL-Div (SAMC) |  Entropy (SAMC)
---|---|---|---|---
de |  SS |  37.08 |  1.32 (0.32) |  2.43 (1.4)
RSS |  13.30 |  1.75 (0.84) |  3.22 (2.64)
RSU |  - |  1.88 |  3.40
RU |  - |  2.27 |  3.59
nl |  SS |  38.04 |  1.23 (0.32) |  2.21 (1.29)
RSS |  12.90 |  1.64 (0.82) |  2.95 (2.44)
RSU |  - |  1.71 |  2.86
|  RU |  - |  1.81 |  3.1

##  4 Results and Discussion

###  4.1 Baseline multilingual ASR

Monolingual (mono) baseline systems are the language dependent acoustic,
pronunciation and language models which are trained on a language specific
data set. The train sets of all the languages are then mixed to train
multilingual (multi) acoustic and language models. The multilingual acoustic
model is then used with monolingual Language Model (LM) of the target language
which is termed as mono-lm in the reported results. The results of the
baseline systems for all the languages are given in Table 2 in terms of WER
and PER. Since only the language model is changed for mono-lm, thus the PER
remains unchanged when compared with that of multilingual system. The results
show that the error for all the languages increases in multilingual setup in
spite of the balanced data duration for each language. This indicates that the
reason of performance degradation in multilingual setups can not only be
attributed to rich resources of these languages or unbalanced data sampling.

Figure 2: % Relative improvement in PER per shared phoneme compared with
monolingual ASR for en target language

According to the assumption of multilingual systems [17] discussed earlier in
section 2, if the articulatory representations of phonemes are considered
language independent units then the performance of shared phonemes should
improve with the more training data in the multilingual systems. The
languages, being studied here, have an overlapping set of 24 phonemes. As a
case study of en as the target language, the relative improvement in PER of
shared phonemes is analysed with gradually increasing the languages in the
training data. It is evident from the Fig. 2 that even the performance of
shared phonemes is degraded in bilingual and multilingual setups, though nl is
less detrimental than de for the en language.

Table 4: Posterior distribution similarity for the nl test set AM |  Biphones subset |  % Correct SAMC |  KL-Div (SAMC) |  Entropy (SAMC)
---|---|---|---|---
en |  SS |  46.51 |  1.33 (0.37) |  1.75 (0.95)
RSS |  15.43 |  1.87 (1.26) |  2.36 (1.77)
RSU |  - |  2.113 |  2.66
RU |  - |  2.173 |  2.69
de |  SS |  40.90 |  1.44 (0.46) |  1.89 (1.19)
RSS |  17.30 |  1.88 (1.15) |  2.33 (1.81)
RSU |  - |  2.04 |  2.43
|  RU |  - |  2.13 |  2.58
Table 5: Posterior distribution similarity for the de test set AM |  Biphones subset |  % Correct SAMC |  KL-Div (SAMC) |  Entropy (SAMC)
---|---|---|---|---
en |  SS |  43.56 |  0.83 (0.22) |  1.76 (1.07)
RSS |  14.90 |  1.21 (0.74) |  2.52 (2.12)
RSU |  - |  1.15 |  2.21
RU |  - |  1.27 |  2.54
nl |  SS |  36.37 |  1.05 (0.31) |  1.95 (1.24)
RSS |  13.68 |  1.38 (0.82) |  2.5 (2.09)
RSU |  - |  1.54 |  2.67
|  RU |  - |  1.41 |  2.56

###  4.2 Similarity analysis

Table 3 shows the similarity measure for English (en) test set when passed
through German (de) and Dutch (nl) acoustic models. In clustering, shared seen
biphones (SS) may share the same cluster with unseen (SU) or unshared biphones
(U) and vice versa. So for insightful observations, analysis is restricted to
the clusters which have only one biphone (shown with ``R" prefix). It is clear
from the results that KL divergence increases from shared seen biphones
towards unshared biphones.

(a) de to en mapping network

(b) nl to en mapping network

Figure 3: Posteriorgram and entropy plot of $N_{de-en}$ and $N_{nl-en}$
mapping networks with one-hot vectors as input. Behaviour for only hundred
source biphone classes is shown. Posteriorgram shows sorted probabilities of
top ten mapped classes. Each box on horizontal axis is a one-hot vector of
source language and on vertical axis is the probability of a mapped output
class

For shared seen biphone sets (SS and RSS), being present in both (source and
the target) languages, mapping network should learn one-to-one mapping to same
biphone class if the multilingual assumption holds true. This measure is
calculated as percentage of these biphones recognised correctly by source
acoustic model and reported as `correct Source Acoustic Model Class (SAMC)' in
the results. The results (low percentage of ``Correct SAMC" in tables) show
that the source acoustic models are not very good at recognising these
biphones but the lower values of KL divergence (KL-Div in parenthesis)
indicate that it is easier for mapping network to learn a one-to-one mapping
in these cases. It implies that the source ASR has a pattern in errors of
these biphones sets which mapping network could learn easily. On analysis, it
appears that the source acoustic models confuse these biphones with several
close biphones. For example, a biphone \i\,\z\from English test set is
frequently confused with \i\,\s\and \E\,\s\by German acoustic model.

To study the confidence of the mapping network in the mapped classes, it is
sequentially fed the one-hot vectors as input. As a case study, the
posteriorgram and entropy of mapping networks from source languages to en is
analysed. Behaviour for the top n source classes, which are mapped to the
target classes with minimum entropy (more confidently), is observed. Entropy
and posteriorgram from both networks, $N_{de-en}$ and $N_{nl-en}$, for same n
($n=100$) is visualised in Fig. 3. Only top ten most probable mapped output
classes are shown in the sorted posteriorgram. It can be seen that for the
same value of $n$, the entropy range is lower for nl-en mapping network than
the range for de-en network. It evidences that nl-en mapping network could
learn better mappings and nl phonemes are more amenable to transfer to en
phonemes. Furthermore, the entropy for each biphones subset is also measured
which also shows the same trend as the KL divergence. It implies that the
language similarities can be estimated through the entropy of the mapping
network only without measuring KL-div between target and mapped posteriors.

Table 6: Mean KL-divergence (as the cross-lingual similarity measure) and bilingual ASR performance Target Language | Source Languages(KL-Div/% WER)
---|---
en | de | nl
en | 0/43.84 | 1.56/46.35 |  1.44/44.46
de |  1.00/38.94 | 0/37.77 | 1.18/39.46
nl |  1.60/42.32 | 1.61/44.03 | 0/37.94

In the case of unseen (RSU) and unshared (RU) biphones, the source model tries
to map them to the nearest biphone clusters. However, the performance of
unshared biphones mapping goes further down in most of the cases. Almost the
same pattern is observed for the remaining two languages (Table 4 and 5). The
average cross-lingual KL divergence along with the WER of bilingual ASRs is
tabulated in Table 6. Diagonal entries are the monolingual ASRs. These results
can be seen in comparison with cross-lingual phoneme sharing of Table 1. For
example, Table 1 infers that de shares more phonemes with nl compared to en,
but the mean KL divergence is smaller for en (in Table 6). It implies that en
is more closer to de than nl and this claim is corroborated by % WER of
bilingual ASRs in Table 6. So, the phoneme sharing statistics are not a very
informative metric to measure languages closeness and thus lead to degradation
in the multilingual setups.

Table 7: Performance of model fusion in % WER Language | mono | mono-lm | mf | Rel. imp
---|---|---|---|---
English (en) | 43.84 | 47.48 | 43.43 | 0.94%
German (de) | 37.77 | 40.81 | 36.16 | 4.26%
Dutch (nl) | 37.94 | 58.84 | 34.94 | 7.91%

###  4.3 Model fusion for multilingual ASR

A multilingual acoustic model is imitated by fusing the target language and
the mapped source language posteriors. The fusion is the linear weighted sum
of all of these posterior distributions. In Table 7, the results of proposed
model fusion (mf) approach is compared with mono and mono-lm of Table 2. The
setup of fusion technique makes results directly comparable to those of mono-
lm but the fusion outperforms even the mono ASR. So, the reported relative
improvement here is in comparison with the mono ASR. However, fused posteriors
give a relative gain of 6.4% to 30% when compared with mono-lm.

##  5 Conclusion

In this work, cross-lingual acoustic-phonetic similarities are estimated by
comparing the posterior distributions from source and the target acoustic
models. A regression neural network is trained to map source languages
posteriors to the target language posteriors. This study reveals that the
behaviour of different languages for multilingual ASRs is more complex than
predicting from cross-lingual phoneme sharing perspective. The languages which
share more phonemes, does not guarantee performance gain in multilingual
setups. The analysis observes that the phonemes with identical representations
across languages are not acoustically identical. Finally, the mapped
posteriors are fused for decoding of a target language. A maximum gain of
$\sim$8% in relative improvement over the monolingual system is achieved. The
relative gain increases to 30% when compared with the fusion model's
counterpart.

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# Asymptotics of G-equivariant Szegő kernels

Rung-Tzung Huang  Department of Mathematics, National Central University,
Chung-Li, Taoyuan 32001, Taiwan<EMAIL_ADDRESS>and  Guokuan Shao
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082,
Guangdong, China<EMAIL_ADDRESS>
###### Abstract.

Let $(X,T^{1,0}X)$ be a compact connected orientable CR manifold of dimension
$2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected
compact Lie group $G$ action. Under certain natural assumptions about the
group $G$ action, we define $G$-equivariant Szegő kernels and establish the
associated Boutet de Monvel-Sjöstrand type theorems. When $X$ admits also a
transversal CR $S^{1}$ action, we study the asymptotics of Fourier components
of $G$-equivariant Szegő kernels with respect to the $S^{1}$ action.

###### Key words and phrases:

equivariant Szegő kernel, moment map, CR manifold

###### 2010 Mathematics Subject Classification:

Primary: 58J52, 58J28; Secondary: 57Q10

The first author was supported by Taiwan Ministry of Science and Technology
projects 107-2115-M-008-007-MY2 and 109-2115-M-008-007-MY2.

##  1\. Introduction and statement of the main results

Let $(X,T^{1,0}X)$ be a CR manifold of dimension $2n+1$, $n\geq 1$. Let
$\Box^{(q)}_{b}$ be the Kohn Lalpacian acting on $(0,q)$ forms. The orthogonal
projection $S^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{(q)}_{b}$ onto
${\rm Ker\,}\Box^{(q)}_{b}$ is called the Szegő projection. The Szegő kernel
is its distribution kernel $S^{(q)}(x,y)$. The study of the Szegő projection
and kernels is a classical subject in several complex variables and CR
geometry. When $X$ is the boundary of a strictly pseudoconvex domain, Boutet
de Monvel-Sjöstrand [1] showed that $S^{(0)}(x,y)$ is a complex Fourier
integral operator.

The Boutet de Monvel-Sjöstrand theorem had a profound impact in many research
areas, especially through [2]: several complex variables, symplectic and
contact geometry, geometric quantization, Kähler geometry, semiclassical
analysis, quantum chaos, etc. cf. [3, 4, 6, 15, 16, 17, 18, 23, 25]. Recently,
Hsiao-Huang [10] obtained $G$-invariant Boutet de Monvel-Sjöstrand type
theorems and Hsiao-Ma-Marinescu [11] established geometric quantization on CR
manifolds by using $G$-invariant Szegő kernels asymptotic expansions.

In this paper, we study $G$-equivariant Szegő kernels with respect to all
equivalent classes of irreducible unitary representations of $G$. We establish
$G$-equivariant Boutet de Monvel-Sjöstrand type theorems. When $X$ admits also
a transversal CR $S^{1}$ action, we derive the asymptotic expansion of Fourier
components of $G$-equivariant Szegő kernels with respect to the $S^{1}$
action.

We now formulate the main results. We refer to Section 2 for some notations
and terminology used here. Let $(X,T^{1,0}X)$ be a compact connected
orientable CR manifold of dimension $2n+1$, $n\geq 1$, where $T^{1,0}X$
denotes the CR structure of $X$. Fix a global non-vanishing real $1$-form
$\omega_{0}\in C^{\infty}(X,T^{*}X)$ such that
$\langle\,\omega_{0}\,,\,u\,\rangle=0$, for every $u\in T^{1,0}X\oplus
T^{0,1}X$. The Levi form of $X$ at $x\in X$ is the Hermitian quadratic form on
$T^{1,0}_{x}X$ given by

$\mathcal{L}_{x}(U,\overline{V})=-\frac{1}{2i}\langle\,d\omega_{0}(x)\,,\,U\wedge\overline{V}\,\rangle,\
\text{for all}\ U,V\in T^{1,0}_{x}X.$

In this paper, we assume that

######  Assumption 1.1.

The Levi form is non-degenerate of constant signature $(n_{-},n_{+})$ on $X$.
That is, the Levi form has exactly $n_{-}$ negative and $n_{+}$ positive
eigenvalues at each point of $X$, where $n_{-}+n_{+}=n$.

Let $HX=\left\\{{\rm Re\,}u;\,u\in T^{1,0}X\right\\}$ and let $J:HX\rightarrow
HX$ be the complex structure map given by
$J(u+\overline{u})=iu-i\overline{u}$, for every $u\in T^{1,0}X$. In this
paper, we assume that $X$ admits a $d$-dimensional connected compact Lie group
$G$ action. Let $\mathfrak{g}$ denote the Lie algebra of $G$. For any
$\xi\in\mathfrak{g}$, we write $\xi_{X}$ to denote the vector field on $X$
induced by $\xi$. That is, $(\xi_{X}u)(x)=\frac{\partial}{\partial
t}\left(u(\exp(t\xi)\circ x)\right)|_{t=0}$, for any $u\in C^{\infty}(X)$. Let
$\underline{\mathfrak{g}}={\rm Span\,}(\xi_{X};\,\xi\in\mathfrak{g})$. We
assume throughout that

######  Assumption 1.2.

The Lie group $G$ action is CR and preserves $\omega_{0}$ and $J$.

We recall that the Lie group $G$ action preserves $\omega_{0}$ and $J$ means
that $g^{\ast}\omega_{0}=\omega_{0}$ on $X$ and $g_{\ast}J=Jg_{\ast}$ on $HX$,
for every $g\in G$, where $g^{*}$ and $g_{*}$ denote the pull-back map and
push-forward map of $G$, respectively. The $G$ action is CR means that for
every $\xi_{X}\in\underline{\mathfrak{g}}$,

$[\xi_{X},C^{\infty}(X,T^{1,0}X)]\subset C^{\infty}(X,T^{1,0}X).$

######  Definition 1.3.

The moment map associated to the form $\omega_{0}$ is the map
$\mu:X\to\mathfrak{g}^{*}$ such that, for all $x\in X$ and
$\xi\in\mathfrak{g}$, we have

$\langle\mu(x),\xi\rangle=\omega_{0}(\xi_{X}(x)).$ (1.1)

We assume also that

######  Assumption 1.4.

$0$ is a regular value of $\mu$ and $G$ acts locally free near $\mu^{-1}(0)$.

By Assumption 1.4, $\mu^{-1}(0)$ is a $d$-codimensional orbifold of $X$. Note
that if $G$ acts freely near $\mu^{-1}(0)$ and the Levi form is positive at
$\mu^{-1}(0)$, it is known that $\mu^{-1}(0)/G$ is a CR manifold with natural
CR structure induced by $T^{1,0}X$ of dimension $2n-2d+1$, (see [10]).

Let $R=\\{R_{1},R_{2},...\\}$ be the collection of all irreducible unitary
representations of $G$, including only one representation from each equivalent
class (see Section 2.4). Write

$\begin{split}R_{k}:G&\to GL(\mathbb{C}^{d_{k}}),\ \ d_{k}<\infty,\\\
g&\to(R_{k,j,l}(g))_{j,l=1}^{d_{k}},\end{split}$ (1.2)

where $d_{k}$ is the dimension of the representation $R_{k}$. Denote by
$\chi_{k}(g):=\text{Tr}R_{k}(g)$ the trace of the matrix $R_{k}(g)$ (the
character of $R_{k}$). Let $u\in\Omega^{0,q}(X)$. For every $k=1,2,\cdots,$ we
define

$u_{k}(x)=d_{k}\int_{G}(g^{\star}u)(x)\overline{\chi_{k}(g)}d\mu(g),$ (1.3)

where $d\mu(g)$ is the probability Haar measure on $G$. We will show that
$u=\sum_{k=0}^{\infty}u_{k}$. By Assumption 1.2, $u_{k}\in\Omega^{0,q}(X)$ if
$u\in\Omega^{0,q}(X)$. Set

$\Omega^{0,q}(X)_{k}:=\\{u(x)\in\Omega^{0,q}(X)|u(x)=u_{k}(x)\\}.$ (1.4)

Denote by $L^{2}_{(0,q)}(X)_{k}$ the completion of $\Omega^{0,q}(X)_{k}$ with
respect to the inner product $(\,\cdot\,|\,\cdot\,)$.

######  Definition 1.5.

The $G$-equivariant Szegő projection is the orthogonal projection

$S^{(q)}_{k}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{(q)}_{b}\bigcap
L^{2}_{(0,q)}(X)_{k}$

with respect to $(\,\cdot\,|\,\cdot\,)$. Denote by $S^{(q)}_{k}(x,y)\in
D^{\prime}(X\times X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ the distribution
kernel of $S^{(q)}_{k}$.

The first main result of this work is the following

######  Theorem 1.6.

With the assumptions and notations above, suppose that $\Box^{(q)}_{b}:{\rm
Dom\,}\Box^{(q)}_{b}\rightarrow L^{2}_{(0,q)}(X)$ has closed range. If
$q\notin\left\\{n_{-},n_{+}\right\\}$, then $S^{(q)}_{k}\equiv 0$ on $X$.

Suppose $q\in\left\\{n_{-},n_{+}\right\\}$. Let $D$ be an open neighborhood of
$X$ with $D\bigcap\mu^{-1}(0)=\emptyset$. Then, $S^{(q)}_{k}\equiv 0$ on $D$.

Let $p\in\mu^{-1}(0)$ and let $U$ be an open neighborhood of $p$ and let
$x=(x_{1},\ldots,x_{2n+1})$ be local coordinates defined in $U$. Let
$N_{p}=\\{g\in G:g\circ p=p\\}=\\{g_{1}=e_{0},g_{2}...,g_{r}\\}$. Then, there
exist continuous operators
$\hat{S}_{k,-},\hat{S}_{k,+}:\Omega^{0,q}_{c}(U)\rightarrow\Omega^{0,q}(U)$
such that

$S^{(q)}_{k}\equiv\hat{S}_{k,-}+\hat{S}_{k,+}\ \ \mbox{on $U$},$ (1.5)

and $\hat{S}_{k,-}(x,y)$, $\hat{S}_{k,+}(x,y)$ satisfy

$\begin{split}&\hat{S}_{k,-}(x,y)\equiv\sum_{\alpha=1}^{r}\int^{\infty}_{0}e^{i\Phi_{k,-}(g_{\alpha}\circ
x,y)t}a_{k,\alpha,-}(x,y,t)dt\ \ \mbox{on $U$},\\\
&\hat{S}_{k,+}(x,y)\equiv\sum_{\alpha=1}^{r}\int^{\infty}_{0}e^{i\Phi_{k,+}(g_{\alpha}\circ
x,y)t}a_{k,\alpha,+}(x,y,t)dt\ \ \mbox{on $U$},\end{split}$ (1.6)

with

$\begin{split}&a_{k,\alpha,+}(x,y,t),a_{k,\alpha,-}(x,y,t)\in
S^{n-\frac{d}{2}}_{{\rm cl\,}}(U\times
U\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&a_{k,\alpha,-}(x,y,t)=0\ \ \mbox{if $q\neq n_{-}$},\ \
a_{k,\alpha,+}(x,y,t)=0\ \ \mbox{if $q\neq n_{+}$},\\\
&a^{0}_{k,\alpha,-}(x,x)\neq 0,\ \ \forall x\in U,\ \ \mbox{if $q=n_{-}$},\ \
a^{0}_{k,\alpha,+}(x,x)\neq 0,\ \ \forall x\in U,\ \ \mbox{if
$q=n_{+}$},\end{split}$ (1.7)

where $a^{0}_{k,\alpha,-}(x,x)$ and $a^{0}_{k,\alpha,+}(x,x)$,
$x\in\mu^{-1}(0)\bigcap U$, are the leading terms of the asymptotic expansion
of $a_{k,\alpha,-}(x,x)$ and $a_{k,\alpha,+}(x,x)$ in
$S^{n-\frac{d}{2}}_{1,0}(U\times
U\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, respectively, and
$\Phi_{k,-}(x,y)\in C^{\infty}(U\times U)$,

$\begin{split}&{\rm Im\,}\Phi_{k,-}(x,y)\geq 0,\\\
&d_{x}\Phi_{k,-}(x,x)=-d_{y}\Phi_{k,-}(x,x)=-\omega_{0}(x),\ \ \forall x\in
U\bigcap\mu^{-1}(0),\\\ \end{split}$ (1.8)

and $-\bar{\Phi}_{k,+}(x,y)$ satisfies (1.8).

We refer the readers to the discussion before (2.1) and Definition 3.1 for the
precise meanings of $A\equiv B$ and the symbol space $S^{n-\frac{d}{2}}_{{\rm
cl\,}}$, respectively.

Assume that $G$ acts freely on $\mu^{-1}(0)$ for a moment. To state the
formulas for $a^{0}_{k,-}(x,x)$ and $a^{0}_{k,+}(x,x)$, we introduce some
notations. For a given point $x_{0}\in X$, let $\\{W_{j}\\}_{j=1}^{n}$ be an
orthonormal frame of $(T^{1,0}X,\langle\,\cdot\,|\,\cdot\,\rangle)$ near
$x_{0}$, for which the Levi form is diagonal at $x_{0}$. Put

$\mathcal{L}_{x_{0}}(W_{j},\overline{W}_{\ell})=\mu_{j}(x_{0})\delta_{j\ell}\,,\;\;j,\ell=1,\ldots,n\,.$

We will denote by

$\det\mathcal{L}_{x_{0}}=\prod_{j=1}^{n}\mu_{j}(x_{0})\,.$ (1.9)

Let $\\{T_{j}\\}_{j=1}^{n}$ denote the basis of $T^{*0,1}X$, dual to
$\\{\overline{W}_{j}\\}^{n}_{j=1}$. We assume that $\mu_{j}(x_{0})<0$ if
$1\leq j\leq n_{-}$ and $\mu_{j}(x_{0})>0$ if $n_{-}+1\leq j\leq n$. Put

$\begin{array}[]{ll}&\mathcal{N}(x_{0},n_{-}):=\left\\{cT_{1}(x_{0})\wedge\ldots\wedge
T_{n_{-}}(x_{0});\,c\in\mathbb{C}\right\\},\\\
&\mathcal{N}(x_{0},n_{+}):=\left\\{cT_{n_{-}+1}(x_{0})\wedge\ldots\wedge
T_{n}(x_{0});\,c\in\mathbb{C}\right\\}\end{array}$

and let

$\tau_{n_{-}}=\tau_{x_{0},n_{-}}:T^{*0,q}_{x_{0}}X\rightarrow\mathcal{N}(x_{0},n_{-})\,,\quad\tau_{n_{+}}=\tau_{x_{0},n_{+}}:T^{*0,q}_{x_{0}}X\rightarrow\mathcal{N}(x_{0},n_{+})\,,$
(1.10)

be the orthogonal projections onto $\mathcal{N}(x_{0},n_{-})$ and
$\mathcal{N}(x_{0},n_{+})$ with respect to
$\langle\,\cdot\,|\,\cdot\,\rangle$, respectively.

Fix $x\in\mu^{-1}(0)$, consider the linear map

$\begin{array}[]{rll}R_{x}:\underline{\mathfrak{g}}_{x}&\rightarrow&\underline{\mathfrak{g}}_{x},\\\
u&\rightarrow&R_{x}u,\ \
\langle\,R_{x}u\,|\,v\,\rangle=\langle\,d\omega_{0}(x)\,,\,Ju\wedge
v\,\rangle.\end{array}$

Let $\det R_{x}=\lambda_{1}(x)\cdots\lambda_{d}(x)$, where $\lambda_{j}(x)$,
$j=1,2,\ldots,d$, are the eigenvalues of $R_{x}$.

Fix $x\in\mu^{-1}(0)$, put $Y_{x}=\left\\{g\circ x;\,g\in G\right\\}$. $Y_{x}$
is a $d$-dimensional submanifold of $X$. The $G$-invariant Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ induces a volume form $dv_{Y_{x}}$ on
$Y_{x}$. Put

$V_{{\rm eff\,}}(x):=\int_{Y_{x}}dv_{Y_{x}}.$

Note that the function $V_{{\rm eff\,}}(x)$ was already appeared in Ma-Zhang
[19, (0,10)] as exactly the role in the expansion, cf. [19, (0.14)].

######  Theorem 1.7.

With the notations used above, if $G$ acts freely on $\mu^{-1}(0)$, then for
$a^{0}_{k,-}(x,y)$ and $a^{0}_{k,+}(x,y)$ in (1.7), we have

$\begin{split}a^{0}_{k,-}(x,x)=2^{d-1}\frac{d^{2}_{k}}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{-}},\ \
\forall x\in\mu^{-1}(0)\\\ a^{0}_{k,+}(x,x)=2^{d-1}\frac{d^{2}_{k}}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{+}},\ \
\forall x\in\mu^{-1}(0).\end{split}$ (1.11)

Assume that $X$ admits an $S^{1}$ action $e^{i\theta}$: $S^{1}\times
X\rightarrow X$. Let $T\in C^{\infty}(X,TX)$ be the global real vector field
induced by the $S^{1}$ action given by
$(Tu)(x)=\frac{\partial}{\partial\theta}\left(u(e^{i\theta}\circ
x)\right)|_{\theta=0}$, $u\in C^{\infty}(X)$. Let the $S^{1}$ action
$e^{i\theta}$ be CR and transversal (see Definition 4.1). We assume throughout
that

######  Assumption 1.8.

$\mbox{$T$ is transversal to the space $\underline{\mathfrak{g}}$ at every
point $p\in\mu^{-1}(0)$},$ (1.12) $e^{i\theta}\circ g\circ x=g\circ
e^{i\theta}\circ x,\ \ \forall x\in X,\ \ \forall\theta\in[0,2\pi[,\ \ \forall
g\in G,$ (1.13)

and

$\mbox{$G\times S^{1}$ acts locally freely near $\mu^{-1}(0)$}.$ (1.14)

Let $u\in\Omega^{0,q}(X)$. Define

$Tu:=\frac{\partial}{\partial\theta}\bigr{(}(e^{i\theta})^{*}u\bigr{)}|_{\theta=0}\in\Omega^{0,q}(X).$
(1.15)

For every $m\in\mathbb{Z}$, let

$\begin{split}&\Omega^{0,q}_{m}(X):=\left\\{u\in\Omega^{0,q}(X);\,Tu=imu\right\\},\
\ q=0,1,2,\ldots,n,\\\
&\Omega^{0,q}_{m}(X)_{k}=\left\\{u\in\Omega^{0,q}(X)_{k};\,Tu=imu\right\\},\ \
q=0,1,2,\ldots,n.\end{split}$ (1.16)

Denote by $C^{\infty}_{m}(X):=\Omega^{0,0}_{m}(X)$,
$C^{\infty}_{m}(X)_{k}:=\Omega^{0,0}_{m}(X)_{k}$. From the CR property of the
$S^{1}$ action and (1.13), we have

$Tg^{*}\overline{\partial}_{b}=g^{*}T\overline{\partial}_{b}=\overline{\partial}_{b}g^{*}T=\overline{\partial}_{b}Tg^{*}\
\ \mbox{on $\Omega^{0,q}(X)$},\ \ \forall g\in G.$

Hence,

$\overline{\partial}_{b}:\Omega^{0,q}_{m}(X)_{k}\rightarrow\Omega^{0,q+1}_{m}(X)_{k},\
\ \forall m\in\mathbb{Z}.$ (1.17)

Assume that the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$\mathbb{C}TX$ is $G\times S^{1}$ invariant. Then the $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle$ is $G\times S^{1}$-invariant. We then have

$\begin{split}&Tg^{*}\overline{\partial}^{*}_{b}=g^{*}T\overline{\partial}^{*}_{b}=\overline{\partial}^{*}_{b}g^{*}T=\overline{\partial}^{*}_{b}Tg^{*}\
\ \mbox{on $\Omega^{0,q}(X)$},\ \ \forall g\in G,\\\
&Tg^{*}\Box^{(q)}_{b}=g^{*}T\Box^{(q)}_{b}=\Box^{(q)}_{b}g^{*}T=\Box^{(q)}_{b}Tg^{*}\
\ \mbox{on $\Omega^{0,q}(X)$},\ \ \forall g\in G,\end{split}$

where $\overline{\partial}^{*}_{b}$ is the $L^{2}$ adjoint of
$\overline{\partial}_{b}$ with respect to $(\,\cdot\,|\,\cdot\,)$.

Let $L^{2}_{(0,q),m}(X)_{k}$ be the completion of $\Omega_{m}^{0,q}(X)_{k}$
with respect to $(\,\cdot\,|\,\cdot\,)$. Write
$L^{2}_{m}(X)_{k}:=L^{2}_{(0,0),m}(X)_{k}$. Put

$H^{q}_{b,m}(X)_{k}:=({\rm Ker\,}\Box^{(q)}_{b})\bigcap
L^{2}_{(0,q),m}(X)_{k}.$

Since $\Box^{(q)}_{b}-T^{2}$ is elliptic, we have for every $m\in\mathbb{Z}$,
$H^{q}_{b,m}(X)_{k}\subset\Omega^{0,q}_{m}(X)_{k}$ and ${\rm
dim\,}H^{q}_{b,m}(X)_{k}<\infty$.

######  Definition 1.9.

The $m$-th $G$-equivariant Szegő projection is the orthogonal projection

$S^{(q)}_{k,m}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{(q)}_{b})\bigcap
L^{2}_{(0,q),m}(X)_{k}$

with respect to $(\,\cdot\,|\,\cdot\,)$. Let $S^{(q)}_{k,m}(x,y)\in
C^{\infty}(X\times X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution
kernel of $S^{(q)}_{k,m}$.

The second main result of this work is the following

######  Theorem 1.10.

With the assumptions and notations used above, if $q\notin n_{-}$, then, as
$m\rightarrow+\infty$, $S^{(q)}_{k,m}=O(m^{-\infty})$ on $X$.

Suppose $q=n_{-}$. Let $D$ be an open neighborhood of $X$ with
$D\bigcap\mu^{-1}(0)=\emptyset$. Then, as $m\rightarrow+\infty$,

$S^{(q)}_{k,m}=O(m^{-\infty})\ \ \mbox{on $D$}.$

Let $p\in\mu^{-1}(0)$ and let $N_{p}=\\{g\in G:g\circ
p=p\\}=\\{g_{1}=e_{0},g_{2}...,g_{r}\\}$. Let $U$ be an open neighborhood of
$p$ and let $x=(x_{1},\ldots,x_{2n+1})$ be local coordinates defined in $U$.
Then, as $m\rightarrow+\infty$,

$\begin{split}&S^{(q)}_{k,m}(x,y)\equiv\sum_{\alpha=1}^{r}e^{im\Psi_{k}(g_{\alpha}\circ
x,y)}b_{k,\alpha}(x,y,m),\\\ &b_{k,\alpha}(x,y,m)\in S^{n-\frac{d}{2}}_{{\rm
loc\,}}(1;U\times U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&\mbox{$b_{k,\alpha}(x,y,m)\sim\sum^{\infty}_{j=0}m^{n-\frac{d}{2}-j}b_{k,\alpha}^{j}(x,y)$
in $S^{n-\frac{d}{2}}_{{\rm loc\,}}(1;U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\ &b_{k,\alpha}^{j}(x,y)\in
C^{\infty}(U\times U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \
j=0,1,2,\ldots,\end{split}$ (1.18)

$\Psi_{k}(x,y)\in C^{\infty}(U\times U)$,
$d_{x}\Psi_{k}(x,x)=-d_{y}\Psi_{k}(x,x)=-\omega_{0}(x)$, for every
$x\in\mu^{-1}(0)$, $\Psi_{k}(x,y)=0$ if and only if $x=y\in\mu^{-1}(0)$.

In particular, if $G\times S^{1}$ acts freely near $\mu^{-1}(0)$, then

$b_{k}^{0}(x,x)=2^{d-1}\frac{d^{2}_{k}}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{-}},\ \
\forall x\in\mu^{-1}(0),$ (1.19)

where $\tau_{x,n_{-}}$ is given by (1.10).

We provide a special case when $G=T^{d}$ on an irregular Sasakian manifold,
where $T^{d}$ denotes the $d$-dimensional torus. Recall that a CR manifold $X$
is irregular Sasakian if it admits a CR transversal $\mathbb{R}$-action, which
does not come from any circle action. The $\mathbb{R}$-action can be
interpreted as a CR torus action $T^{d+1}=T^{d}\times S^{1}$ (cf. Herrmann-
Hsiao-Li [7]). Fix $(p_{1},...,p_{d})\in\mathbb{Z}^{d}$, we define a
$T^{d}$-action as follows:

$\begin{split}T^{d}\times X&\rightarrow X\\\
\bigl{(}(e^{i\theta_{1}},...,e^{i\theta_{d}}),x\bigr{)}&\mapsto(e^{i\theta_{1}},...,e^{i\theta_{d}},e^{-ip_{1}\theta_{1}-\cdot\cdot\cdot-
ip_{d}\theta_{d}})\circ x.\end{split}$ (1.20)

The $T^{d}$-action satisfies Assumption 1.8. All irreducible unitary
representations of $T^{d}$ are
$\\{R_{p_{1},...,p_{d}}:(e^{i\theta_{1}},...,e^{i\theta_{d}})\mapsto
e^{ip_{1}\theta_{1}+\cdot\cdot\cdot+ip_{1}\theta_{d}}\\}$. Let $T_{j}$ be the
induced vector fields of the $T^{d}$-action. That is,
$T_{j}u(x):=\frac{\partial}{\partial\theta_{j}}|_{\theta_{j}=0}u\bigl{(}(1,...,e^{i\theta_{j}},...,1)\circ
x\bigr{)}$, for $j=1,...,d$ and $u\in C^{\infty}(X)$. Then

$H_{b,mp_{1},...,mp_{d},m}^{q}(X)=\\{u\in
H_{b}^{q}(X):Tu=imu,T_{j}u=imp_{j}u,j=1,...,d\\}.$ (1.21)

The $m$-th equivariant Szegő kernel is the distributional kernel of the
orthogonal projection $S_{mp_{1},...,mp_{d},m}^{q}:L^{2}_{(0,q)}\rightarrow
H_{b,mp_{1},...,mp_{d},m}^{q}(X)$.

Assume that $(-p_{1},...,-p_{d})\in\mathbb{Z}^{d}$ is a regular value of the
torus invariant CR moment map

$\mu_{0}:X\rightarrow\mathbb{R}^{d},\mu_{0}(x):=\bigl{(}\langle\omega_{0}(x),T_{1}(x)\rangle,...,\langle\omega_{0}(x),T_{d}(x)\rangle\bigr{)}.$
(1.22)

The CR moment map of $T^{d}$-action defined in (1.20) is

$\mu:X\rightarrow\mathbb{R}^{d},\mu(x):=\bigl{(}\langle\omega_{0}(x),-p_{1}T+T_{1}(x)\rangle,...,\langle\omega_{0}(x),-p_{d}T+T_{d}(x)\rangle\bigr{)}.$
(1.23)

For $x\in\mu_{0}^{-1}(-p_{1},...,-p_{d})$, $\omega_{0}(-p_{j}T+T_{j})=0$. Then
$0$ is the regular value of $\mu$. By Theorem 1.10, we deduce the following
which covers Shen’s result [22] when $X$ is strongly pseudoconvex.

######  Corollary 1.11.

Fix $(-p_{1},...,-p_{d})\in\mathbb{Z}^{d}$, the action of $T^{d}$ on the
irregular Sasakian manifold $X$ is defined in (1.20). With the assumptions and
notations used above, If $q\notin n_{-}$, then, as $m\rightarrow+\infty$,
$S^{(q)}_{mp_{1},...,mp_{d},m}=O(m^{-\infty})$ on $X$.

Suppose $q=n_{-}$. Let $D$ be an open neighborhood of $X$ with
$D\bigcap\mu^{-1}(0)=\emptyset$. Then, as $m\rightarrow+\infty$,

$S^{(q)}_{mp_{1},...,mp_{d},m}=O(m^{-\infty})\ \ \mbox{on $D$}.$

Let $p\in\mu^{-1}(0)$ and let $N_{p}=\\{g\in G:g\circ
p=p\\}=\\{g_{1}=e_{0},g_{2}...,g_{r}\\}$. Let $U$ be an open neighborhood of
$p$ and let $x=(x_{1},\ldots,x_{2n+1})$ be local coordinates defined in $U$.
Then, as $m\rightarrow+\infty$,

$\begin{split}&S^{(q)}_{mp_{1},...,mp_{d},m}(x,y)\equiv\sum_{\alpha=1}^{r}e^{im\Psi(g_{\alpha}\circ
x,y)}b_{\alpha}(x,y,m),\\\ &b_{\alpha}(x,y,m)\in S^{n-\frac{d}{2}}_{{\rm
loc\,}}(1;U\times U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&\mbox{$b_{\alpha}(x,y,m)\sim\sum^{\infty}_{j=0}m^{n-\frac{d}{2}-j}b_{\alpha}^{j}(x,y)$
in $S^{n-\frac{d}{2}}_{{\rm loc\,}}(1;U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\ &b_{\alpha}^{j}(x,y)\in
C^{\infty}(U\times U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \
j=0,1,2,\ldots,\end{split}$ (1.24)

$\Psi(x,y)\in C^{\infty}(U\times U)$,
$d_{x}\Psi(x,x)=-d_{y}\Psi(x,x)=-\omega_{0}(x)$, for every $x\in\mu^{-1}(0)$,
$\Psi(x,y)=0$ if and only if $x=y\in\mu^{-1}(0)$.

In particular, if $T^{d}\times S^{1}$ acts freely near $\mu^{-1}(0)$, then

$b^{0}(x,x)=\frac{2^{d-1}}{V_{{\rm eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{-}},\ \
\forall x\in\mu^{-1}(0),$ (1.25)

where $\tau_{x,n_{-}}$ is given by (1.10).

##  2\. Preliminaries

###  2.1. Standard notations

Let $M$ be a $C^{\infty}$ paracompact manifold. We let $TM$ and $T^{*}M$
denote the tangent bundle of $M$ and the cotangent bundle of $M$,
respectively. The complexified tangent bundle of $M$ and the complexified
cotangent bundle of $M$ will be denoted by $\mathbb{C}TM$ and
$\mathbb{C}T^{*}M$, respectively. Write $\langle\,\cdot\,,\cdot\,\rangle$ to
denote the pointwise duality between $TM$ and $T^{*}M$. We extend
$\langle\,\cdot\,,\cdot\,\rangle$ bilinearly to
$\mathbb{C}TM\times\mathbb{C}T^{*}M$.

Let $F$ be a $C^{\infty}$ vector bundle over $M$. The fiber of $F$ at $x\in M$
will be denoted by $F_{x}$. Let $E$ be a vector bundle over a $C^{\infty}$
paracompact manifold $M_{1}$. We write $F\boxtimes E^{*}$ to denote the vector
bundle over $M\times M_{1}$ with fiber over $(x,y)\in M\times M_{1}$
consisting of the linear maps from $E_{y}$ to $F_{x}$. Let $Y\subset M$ be an
open set. From now on, the spaces of distribution sections of $F$ over $Y$ and
smooth sections of $F$ over $Y$ will be denoted by $D^{\prime}(Y,F)$ and
$C^{\infty}(Y,F)$, respectively. Let $E^{\prime}(Y,F)$ be the subspace of
$D^{\prime}(Y,F)$ whose elements have compact support in $Y$. Put
$C^{\infty}_{c}(M,F):=C^{\infty}(M,F)\cap E^{\prime}(M,F)$.

We recall the Schwartz kernel theorem [8, Theorems 5.2.1, 5.2.6], [16, Thorem
B.2.7]. Let $F$ and $E$ be $C^{\infty}$ vector bundles over paracompact
orientable $C^{\infty}$ manifolds $M$ and $M_{1}$, respectively, equipped with
smooth densities of integration. If $A:C^{\infty}_{c}(M_{1},E)\rightarrow
D^{\prime}(M,F)$ is continuous, we write $K_{A}(x,y)$ or $A(x,y)$ to denote
the distribution kernel of $A$. The following two statements are equivalent

  1. (1)

$A$ is continuous: $E^{\prime}(M_{1},E)\rightarrow C^{\infty}(M,F)$,

  2. (2)

$K_{A}\in C^{\infty}(M\times M_{1},F\boxtimes E^{*})$.

If $A$ satisfies (1) or (2), we say that $A$ is smoothing on $M\times M_{1}$.
Let $A,\hat{A}:C^{\infty}_{0}(M_{1},E)\rightarrow D^{\prime}(M,F)$ be
continuous operators. We write

$A\equiv\hat{A}$ (on $M\times M_{1}$) (2.1)

if $A-\hat{A}$ is a smoothing operator. If $M=M_{1}$, we simply write “on
$M$”.

Let $H(x,y)\in D^{\prime}(M\times M_{1},F\boxtimes E^{*})$. We write $H$ to
denote the unique continuous operator $C^{\infty}_{c}(M_{1},E)\rightarrow
D^{\prime}(M,F)$ with distribution kernel $H(x,y)$. In this work, we identify
$H$ with $H(x,y)$.

###  2.2. Some standard notations in semi-classical analysis

Let $W_{1}$ be an open set in $\mathbb{R}^{N_{1}}$ and let $W_{2}$ be an open
set in $\mathbb{R}^{N_{2}}$. Let $E$ and $F$ be vector bundles over $W_{1}$
and $W_{2}$, respectively. An $m$-dependent continuous operator
$A_{m}:C^{\infty}_{c}(W_{2},F)\rightarrow D^{\prime}(W_{1},E)$ is called
$m$-negligible on $W_{1}\times W_{2}$ if, for $m$ large enough, $A_{m}$ is
smoothing and, for any $K\Subset W_{1}\times W_{2}$, any multi-indices
$\alpha$, $\beta$ and any $N\in\mathbb{N}$, there exists
$C_{K,\alpha,\beta,N}>0$ such that

$\lvert\partial^{\alpha}_{x}\partial^{\beta}_{y}A_{m}(x,y)\rvert\leq
C_{K,\alpha,\beta,N}m^{-N}\>\>\text{on $K$},\ \ \forall m\gg 1.$ (2.2)

In that case we write

$A_{m}(x,y)=O(m^{-\infty})\>\>\text{on $W_{1}\times W_{2}$,}$

or

$A_{m}=O(m^{-\infty})\>\>\text{on $W_{1}\times W_{2}$.}$

If $A_{m},B_{m}:C^{\infty}_{c}(W_{2},F)\rightarrow D^{\prime}(W_{1},E)$ are
$m$-dependent continuous operators, we write $A_{m}=B_{m}+O(m^{-\infty})$ on
$W_{1}\times W_{2}$ or $A_{m}(x,y)=B_{m}(x,y)+O(m^{-\infty})$ on $W_{1}\times
W_{2}$ if $A_{m}-B_{m}=O(m^{-\infty})$ on $W_{1}\times W_{2}$. When
$W=W_{1}=W_{2}$, we sometime write “on $W$”.

Let $X$ and $M$ be smooth manifolds and let $E$ and $F$ be vector bundles over
$X$ and $M$, respectively. Let $A_{m},B_{m}:C^{\infty}(M,F)\rightarrow
C^{\infty}(X,E)$ be $m$-dependent smoothing operators. We write
$A_{m}=B_{m}+O(m^{-\infty})$ on $X\times M$ if on every local coordinate patch
$D$ of $X$ and local coordinate patch $D_{1}$ of $M$,
$A_{m}=B_{m}+O(m^{-\infty})$ on $D\times D_{1}$. When $X=M$, we sometime write
on $X$.

We recall the definition of the semi-classical symbol spaces

######  Definition 2.1.

Let $W$ be an open set in $\mathbb{R}^{N}$. Let

$\displaystyle S(1;W):=\Big{\\{}a\in
C^{\infty}(W)\,|\,\forall\alpha\in\mathbb{N}^{N}_{0}:\sup_{x\in
W}\lvert\partial^{\alpha}a(x)\rvert<\infty\Big{\\}},$ $\displaystyle
S^{0}_{{\rm
loc\,}}(1;W):=\Big{\\{}(a(\cdot,m))_{m\in\mathbb{R}}\,|\,\forall\alpha\in\mathbb{N}^{N}_{0},\forall\chi\in
C^{\infty}_{0}(W)\,:\>\sup_{m\in\mathbb{R},m\geq 1}\sup_{x\in
W}\lvert\partial^{\alpha}(\chi a(x,m))\rvert<\infty\Big{\\}}\,.$

For $k\in\mathbb{R}$, let

$S^{k}_{{\rm loc}}(1):=S^{k}_{{\rm
loc}}(1;W)=\Big{\\{}(a(\cdot,m))_{m\in\mathbb{R}}\,|\,(m^{-k}a(\cdot,m))\in
S^{0}_{{\rm loc\,}}(1;W)\Big{\\}}\,.$

Hence $a(\cdot,m)\in S^{k}_{{\rm loc}}(1;W)$ if for every
$\alpha\in\mathbb{N}^{N}_{0}$ and $\chi\in C^{\infty}_{0}(W)$, there exists
$C_{\alpha}>0$ independent of $m$, such that $\lvert\partial^{\alpha}(\chi
a(\cdot,m))\rvert\leq C_{\alpha}m^{k}$ holds on $W$.

Consider a sequence $a_{j}\in S^{k_{j}}_{{\rm loc\,}}(1)$,
$j\in\mathbb{N}_{0}$, where $k_{j}\searrow-\infty$, and let $a\in
S^{k_{0}}_{{\rm loc\,}}(1)$. We say

$a(\cdot,m)\sim\sum\limits^{\infty}_{j=0}a_{j}(\cdot,m)\>\>\text{in
$S^{k_{0}}_{{\rm loc\,}}(1)$},$

if, for every $\ell\in\mathbb{N}_{0}$, we have $a-\sum^{\ell}_{j=0}a_{j}\in
S^{k_{\ell+1}}_{{\rm loc\,}}(1)$ . For a given sequence $a_{j}$ as above, we
can always find such an asymptotic sum $a$, which is unique up to an element
in $S^{-\infty}_{{\rm loc\,}}(1)=S^{-\infty}_{{\rm
loc\,}}(1;W):=\cap_{k}S^{k}_{{\rm loc\,}}(1)$.

Similarly, we can define $S^{k}_{{\rm loc\,}}(1;Y,E)$ in the standard way,
where $Y$ is a smooth manifold and $E$ is a vector bundle over $Y$.

###  2.3. CR manifolds

Let $(X,T^{1,0}X)$ be a compact, connected and orientable CR manifold of
dimension $2n+1$, $n\geq 1$, where $T^{1,0}X$ is a CR structure of $X$, that
is, $T^{1,0}X$ is a subbundle of rank $n$ of the complexified tangent bundle
$\mathbb{C}TX$, satisfying $T^{1,0}X\cap T^{0,1}X=\\{0\\}$, where
$T^{0,1}X=\overline{T^{1,0}X}$, and
$[\mathcal{V},\mathcal{V}]\subset\mathcal{V}$, where
$\mathcal{V}=C^{\infty}(X,T^{1,0}X)$. There is a unique subbundle $HX$ of $TX$
such that $\mathbb{C}HX=T^{1,0}X\oplus T^{0,1}X$, i.e. $HX$ is the real part
of $T^{1,0}X\oplus T^{0,1}X$. Let $J:HX\rightarrow HX$ be the complex
structure map given by $J(u+\overline{u})=iu-i\overline{u}$, for every $u\in
T^{1,0}X$. By complex linear extension of $J$ to $\mathbb{C}TX$, the
$i$-eigenspace of $J$ is
$T^{1,0}X\,=\,\left\\{V\in\mathbb{C}HX\,:\,JV\,=\,\sqrt{-1}V\right\\}.$ We
shall also write $(X,HX,J)$ to denote a compact CR manifold.

We fix a real non-vanishing $1$ form $\omega_{0}\in C(X,T^{*}X)$ so that
$\langle\,\omega_{0}(x)\,,\,u\,\rangle=0$, for every $u\in H_{x}X$, for every
$x\in X$. For each $x\in X$, we define a quadratic form on $HX$ by

$\mathcal{L}_{x}(U,V)=\frac{1}{2}d\omega_{0}(JU,V),\forall\ U,V\in H_{x}X.$
(2.3)

We extend $\mathcal{L}$ to $\mathbb{C}HX$ by complex linear extension. Then,
for $U,V\in T^{1,0}_{x}X$,

$\mathcal{L}_{x}(U,\overline{V})=\frac{1}{2}d\omega_{0}(JU,\overline{V})=-\frac{1}{2i}d\omega_{0}(U,\overline{V}).$
(2.4)

The Hermitian quadratic form $\mathcal{L}_{x}$ on $T^{1,0}_{x}X$ is called
Levi form at $x$. Let $T\in C^{\infty}(X,TX)$ be the non-vanishing vector
field determined by

$\begin{split}&\omega_{0}(T)=-1,\\\ &d\omega_{0}(T,\cdot)\equiv 0\ \ \mbox{on
$TX$}.\end{split}$ (2.5)

Note that $X$ is a contact manifold with contact form $\omega_{0}$, contact
plane $HX$ and $T$ is the Reeb vector field.

Fix a smooth Hermitian metric $\langle\cdot\mid\cdot\rangle$ on $\mathbb{C}TX$
so that $T^{1,0}X$ is orthogonal to $T^{0,1}X$, $\langle u\mid v\rangle$ is
real if $u,v$ are real tangent vectors, $\langle\,T\,|\,T\,\rangle=1$ and $T$
is orthogonal to $T^{1,0}X\oplus T^{0,1}X$. For $u\in\mathbb{C}TX$, we write
$|u|^{2}:=\langle u|u\rangle$. Denote by $T^{*1,0}X$ and $T^{*0,1}X$ the dual
bundles $T^{1,0}X$ and $T^{0,1}X$, respectively. They can be identified with
subbundles of the complexified cotangent bundle $\mathbb{C}T^{*}X$. Define the
vector bundle of $(0,q)$-forms by $T^{*0,q}X:=\wedge^{q}T^{*0,1}X$. The
Hermitian metric $\langle\cdot|\cdot\rangle$ on $\mathbb{C}TX$ induces, by
duality, a Hermitian metric on $\mathbb{C}T^{*}X$ and also on the bundles of
$(0,q)$ forms $T^{*0,q}X,q=0,1,\cdots,n$. We shall also denote all these
induced metrics by $\langle\cdot|\cdot\rangle$. Note that we have the
pointwise orthogonal decompositions:

$\begin{array}[]{c}\mathbb{C}T^{*}X=T^{*1,0}X\oplus
T^{*0,1}X\oplus\left\\{\lambda\omega_{0}:\lambda\in\mathbb{C}\right\\},\\\
\mathbb{C}TX=T^{1,0}X\oplus T^{0,1}X\oplus\left\\{\lambda
T:\lambda\in\mathbb{C}\right\\}.\end{array}$ (2.6)

For $x,y\in X$, let $d(x,y)$ denote the distance between $x$ and $y$ induced
by the Hermitian metric $\langle\cdot\mid\cdot\rangle$. Let $A$ be a subset of
$X$. For every $x\in X$, let $d(x,A):=\inf\left\\{d(x,y);\,y\in A\right\\}$.

Let $D$ be an open set of $X$. Let $\Omega^{0,q}(D)$ denote the space of
smooth sections of $T^{*0,q}X$ over $D$ and let $\Omega^{0,q}_{0}(D)$ be the
subspace of $\Omega^{0,q}(D)$ whose elements have compact support in $D$.

###  2.4. Fourier analysis on compact Lie groups

Let $\rho:G\to GL(\mathbb{C}^{d})$ be a representation of $G$, where $d$ is
the dimension of the representation $\rho$. Two representations $\rho_{1}$ and
$\rho_{2}$ are equivalent if they have the same dimension and there is an
invertible matrix $A$ such that $\rho_{1}(g)=A\rho_{2}(g)A^{-1}$ for all $g\in
G$. Let

$R=\\{R_{1},R_{2},...\\}$

be the collection of all irreducible unitary representations of $G$, where
each $R_{k}$ comes from exactly only one equivalent class. For each $R_{k}$,
let $(R_{k,j,l})_{j,l=1}^{d_{k}}$ be its matrix, where $d_{k}$ is the
dimension of $R_{k}$. Let $d\mu(g)$ be the probability Haar measure on $G$.
Let $(\cdot\mid\cdot)_{G}$ be the natural inner product on $C^{\infty}(G)$
induced by $d\mu(g)$. Let $L^{2}(G)$ be the completion of $C^{\infty}(G)$ with
respect to $(\cdot\mid\cdot)_{G}$. By the orthogonality relations for compact
Lie groups and the Peter-Weyl theorem [24], we have

######  Theorem 2.2.

The set $\\{\sqrt{d_{k}}R_{k,j,l};j,l=1,...,d_{k},k=1,2,...\\}$ form an
orthonormal basis of $L^{2}(G)$.

For a function $f\in C^{\infty}(G)$, the Fourier component of $f$ with respect
to $\sqrt{d_{k}}R_{k,j,l}$ is

$f_{k,j,l}:=d_{k}R_{k,j,l}(g)\int_{G}f(h)\overline{R_{k,j,l}(h)}d\mu(h)\in
C^{\infty}(G).$ (2.7)

The smooth version of the Peter-Weyl theorem on compact Lie groups is the
following [24]

######  Theorem 2.3.

Let $f\in C^{\infty}(G)$. For every $t\in\mathbb{N}$ and every
$\varepsilon>0$, there exists a $N_{0}\in\mathbb{N}$ such that for every
$N\geq N_{0}$, we have

$\bigl{\|}f-\sum_{k=1}^{N}\sum_{j,l=1}^{d_{k}}f_{k,j,l}\bigr{\|}_{C^{t}(G)}\leq\varepsilon.$
(2.8)

We put

$\chi_{m}(g):=\text{Tr}R_{k}(g)=\sum_{j=1}^{d_{k}}R_{k,j,j}(g).$

######  Definition 2.4.

The $k$-th Fourier component of $u\in\Omega^{0,q}(X)$ is defined as

$u_{k}(x)=d_{k}\int_{G}(g^{\star}u)(x)\overline{\chi_{k}(g)}d\mu(g)\in\Omega^{0,q}(X).$

We have the following theorem about Fourier components. For the readers’
convenience, we present the proof, see also [5].

######  Theorem 2.5.

Let $u\in\Omega^{0,q}(X)$. Then

$\lim_{N\to\infty}\sum_{k=1}^{N}u_{k}(x)=u(x),\forall x\in X,$ (2.9) $\langle
u_{k}(x)|u_{t}(x)\rangle=0,\ \ \text{if}\ \ k\neq t,\forall x\in X,$ (2.10)
$\sum_{k=1}^{N}\|u_{k}\|^{2}\leq\|u\|^{2},\forall N\in\mathbb{N}.$ (2.11)

###### Proof.

We fix $x\in X$ and consider a smooth function $f:g\in G\to(g^{\star}u)(x)$.
Then

$f_{k,j,l}(g)=d_{k}R_{k,j,l}(g)\int_{G}(h^{\star}u)(x)\overline{R_{k,j,l}(h)}d\mu(h).$
(2.12)

By Theorem 2.3, for every $\varepsilon>0$, there exists a $N_{0}\in\mathbb{N}$
such that for every $N\geq N_{0}$, we have

$\bigl{|}(g^{\star}u)(x)-\sum_{k=1}^{N}\sum_{j,l=1}^{d_{k}}f_{k,j,l}(g)\bigr{|}\leq\varepsilon,\forall
g\in G.$ (2.13)

Take $g=e_{0}$, where $e_{0}$ is the identity element of $g$, we obtain that
for every $N\geq N_{0}$,

$\bigl{|}u(x)-\sum_{k=1}^{N}\sum_{j,l=1}^{d_{k}}f_{k,j,l}(e_{0})\bigr{|}\leq\varepsilon.$
(2.14)

Note that by (2.12),

$f_{k,j,l}(e_{0})=d_{k}\delta_{j,l}\int_{G}(h^{\star}u)(x)\overline{R_{k,j,l}(h)}d\mu(h).$

Then

$\sum_{k=1}^{N}\sum_{j,l=1}^{d_{k}}f_{k,j,l}(e_{0})=\sum_{k=1}^{N}u_{k}(x).$
(2.15)

Hence (2.9) is true by (2.14) and (2.15).

By Theorem 2.2 and (2.12), we have

$\begin{split}&\sum_{k=1}^{\infty}\sum_{j,l=1}^{d_{k}}\int_{G}|f_{k,j,l}(g)|^{2}d\mu(g)\\\
&=\sum_{k=1}^{\infty}\sum_{j,l=1}^{d_{k}}d_{k}\bigl{|}\int_{G}(h^{\star}u)(x)\overline{R_{k,j,l}(h)}d\mu(h)\bigr{|}^{2}\\\
&=\int_{G}|(h^{\star}u)(x)|^{2}d\mu(h),\forall x\in X.\end{split}$ (2.16)

Since the metric on $X$ is $G$-invariant, we have

$\langle p|q\rangle=\langle h^{\star}p|h^{\star}q\rangle,\forall
p,q\in\Omega^{0,q}(X),\forall h\in G.$

Then for every $k,t\in\mathbb{N}$,

$\langle u_{k}|u_{t}\rangle=\int_{G}\langle
h^{\star}u_{k}|h^{\star}u_{t}\rangle d\mu(h).$ (2.17)

For every $h\in G$,

$\begin{split}h^{\star}u_{k}&=d_{k}\int_{G}(h^{\star}g^{\star}u)(x)\overline{\chi_{k}(g)}d\mu(g)\\\
&=d_{k}\int_{G}((g\circ h)^{\star}u)(x)\overline{\chi_{k}(g)}d\mu(g)\\\
&=d_{k}\int_{G}(g^{\star}u)(x)\overline{\chi_{k}(g\circ
h^{-1})}d\mu(g).\end{split}$ (2.18)

It is easy to see that

$\overline{\chi_{k}(g\circ
h^{-1})}=\sum_{j=1}^{d_{k}}\sum_{l=1}^{d_{k}}\overline{R_{k,j,l}(g)}R_{k,j,l}(h).$
(2.19)

Hence

$h^{\star}u_{k}=d_{k}\int_{G}(g^{\star}u)(x)(\sum_{j=1}^{d_{k}}\sum_{l=1}^{d_{k}}\overline{R_{k,j,l}(g)}R_{k,j,l}(h))d\mu(g).$
(2.20)

Similarly,

$h^{\star}u_{t}=d_{t}\int_{G}(g^{\star}u)(x)(\sum_{j=1}^{d_{t}}\sum_{l=1}^{d_{t}}\overline{R_{t,j,l}(g)}R_{t,j,l}(h))d\mu(g).$
(2.21)

From Theorem 2.2, (2.20) and (2.21), we have

$\int_{G}\langle h^{\star}u_{k}(x)|h^{\star}u_{t}(x)\rangle d\mu(h)=0,\forall
k\neq t,\forall x\in X.$ (2.22)

Then we deduce (2.10) from (2.17) and (2.22).

For $k=t$, we have

$\int_{G}\langle h^{\star}u_{k}(x)|h^{\star}u_{k}(x)\rangle
d\mu(h)=\sum_{j,l=1}^{d_{k}}\bigl{|}\int_{G}(g^{\star}u)(x)\overline{R_{k,j,l}(g)}d\mu(g)\bigr{|}^{2}.$
(2.23)

With (2.16), we deduce that for every $N\in\mathbb{N}$ and every $x\in X$,

$\sum_{k=1}^{N}\int_{G}\langle h^{\star}u_{k}(x)|h^{\star}u_{k}(x)\rangle
d\mu(h)\leq\int_{G}|(g^{\star}u)(x)|^{2}d\mu(g).$ (2.24)

Then for every $N\in\mathbb{N}$,

$\begin{split}\sum_{k=1}^{N}\|u_{k}\|^{2}&=\int_{X}\sum_{k=1}^{N}\bigl{(}\int_{G}\langle
h^{\star}u_{k}(x)|h^{\star}u_{k}(x)\rangle d\mu(h)\bigr{)}dv_{X}(x)\\\
&\leq\int_{X}\int_{G}|(g^{\star}u)(x)|^{2}d\mu(g)dv_{X}(x)=\|u\|^{2}.\end{split}$

The proof is completed. ∎

We can also prove the following, see [5, Theorem 3.5].

######  Theorem 2.6.

With the notations as above,

$\lim_{N\to\infty}\sum_{k=1}^{N}u_{k}(x)=u(x)$ (2.25)

in $C^{\infty}$-topology.

Moreover, we have (see [5]),

######  Proposition 2.7.

Let $u\in\Omega^{0,q}(X)$, then $u\in\Omega_{k}^{0,q}(X)$ if and only if
$u=u_{k}$ on $X$.

##  3\. $G$-equivariant Szegő kernel asymptotics

In this section, we establish asymptotic expansions of the $G$-equivariant
Szegő kernels. We first review some known results for Szegő kernels mainly
based on [9] and [13].

###  3.1. Szegő kernel asymptotics

Fix a Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on $\mathbb{C}TX$
which induces a Hermitian metric on the bundles of $(0,q)$ forms $T^{*0,q}X$,
$q=0,1,\ldots,n$. Let $D\subset X$ be an open set. Let $\Omega^{0,q}(D)$
denote the space of smooth sections of $T^{*0,q}X$ over $D$.

Let

$\overline{\partial}_{b}:\Omega^{0,q}(X)\rightarrow\Omega^{0,q+1}(X)$ (3.1)

be the tangential Cauchy-Riemann operator. The natural global $L^{2}$ inner
product $(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by $dv(x)$ and
$\langle\,\cdot\,|\,\cdot\,\rangle$ is given by

$(\,u\,|\,v\,):=\int_{X}\langle\,u(x)\,|\,v(x)\,\rangle\,dv(x)\,,\quad
u,v\in\Omega^{0,q}(X)\,.$ (3.2)

We denote by $L^{2}_{(0,q)}(X)$ the completion of $\Omega^{0,q}(X)$ with
respect to $(\,\cdot\,|\,\cdot\,)$. Write $L^{2}(X):=L^{2}_{(0,0)}(X)$. We
extend $\overline{\partial}_{b}$ to $L^{2}_{(0,r)}(X)$, $r=0,1,\ldots,n$, by

$\overline{\partial}_{b}:{\rm Dom\,}\overline{\partial}_{b}\subset
L^{2}_{(0,r)}(X)\rightarrow L^{2}_{(0,r+1)}(X)\,,$ (3.3)

where ${\rm Dom\,}\overline{\partial}_{b}:=\\{u\in
L^{2}_{(0,r)}(X);\,\overline{\partial}_{b}u\in L^{2}_{(0,r+1)}(X)\\}$ and, for
any $u\in L^{2}_{(0,r)}(X)$, $\overline{\partial}_{b}u$ is defined in the
sense of distributions. We also write

$\overline{\partial}^{*}_{b}:{\rm Dom\,}\overline{\partial}^{*}_{b}\subset
L^{2}_{(0,r+1)}(X)\rightarrow L^{2}_{(0,r)}(X)$ (3.4)

to denote the Hilbert adjoint of $\overline{\partial}_{b}$ in the $L^{2}$
space with respect to $(\,\cdot\,|\,\cdot\,)$. Let $\Box^{(q)}_{b}$ denote the
(Gaffney extension) of the Kohn Laplacian given by

$\begin{split}{\rm Dom\,}\Box^{(q)}_{b}=\Big{\\{}s\in
L^{2}_{(0,q)}(X);&\,s\in{\rm Dom\,}\overline{\partial}_{b}\cap{\rm
Dom\,}\overline{\partial}^{*}_{b},\,\overline{\partial}_{b}s\in{\rm
Dom\,}\overline{\partial}^{*}_{b},\overline{\partial}^{*}_{b}s\in{\rm
Dom\,}\overline{\partial}_{b}\Big{\\}}\,,\\\
\Box^{(q)}_{b}s&=\overline{\partial}_{b}\overline{\partial}^{*}_{b}s+\overline{\partial}^{*}_{b}\overline{\partial}_{b}s\>\>\text{for
$s\in{\rm Dom\,}\Box^{(q)}_{b}$}\,.\end{split}$ (3.5)

By a result of Gaffney, for every $q=0,1,\ldots,n$, $\Box^{(q)}_{b}$ is a
positive self-adjoint operator (see [16, Proposition 3.1.2]). That is,
$\Box^{(q)}_{b}$ is self-adjoint and the spectrum of $\Box^{(q)}_{b}$ is
contained in $\overline{\mathbb{R}}_{+}$, $q=0,1,\ldots,n$. Let

$S^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{(q)}_{b}$ (3.6)

be the orthogonal projections with respect to the $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$ and let

$S^{(q)}(x,y)\in D^{\prime}(X\times X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$
(3.7)

denote the distribution kernel of $S^{(q)}$.

We recall Hörmander symbol spaces. Let $D\subset X$ be a local coordinate
patch with local coordinates $x=(x_{1},\ldots,x_{2n+1})$.

######  Definition 3.1.

For $m\in\mathbb{R}$, $S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ is the space of all
$a(x,y,t)\in C^{\infty}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ such that, for all
compact $K\Subset D\times D$ and all $\alpha,\beta\in\mathbb{N}^{2n+1}_{0}$,
$\gamma\in\mathbb{N}_{0}$, there is a constant $C_{\alpha,\beta,\gamma}>0$
such that

$\lvert\partial^{\alpha}_{x}\partial^{\beta}_{y}\partial^{\gamma}_{t}a(x,y,t)\rvert\leq
C_{\alpha,\beta,\gamma}(1+\lvert t\rvert)^{m-\gamma},\ \ \forall(x,y,t)\in
K\times\mathbb{R}_{+},\ \ t\geq 1.$

Put

$\begin{split}&S^{-\infty}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})\\\
&:=\bigcap_{m\in\mathbb{R}}S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).\end{split}$

Let $a_{j}\in S^{m_{j}}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, $j=0,1,2,\ldots$
with $m_{j}\rightarrow-\infty$, as $j\rightarrow\infty$. Then there exists
$a\in S^{m_{0}}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ unique modulo
$S^{-\infty}$, such that $a-\sum^{k-1}_{j=0}a_{j}\in S^{m_{k}}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$ for
$k=0,1,2,\ldots$.

If $a$ and $a_{j}$ have the properties above, we write
$a\sim\sum^{\infty}_{j=0}a_{j}$ in $S^{m_{0}}_{1,0}\big{(}D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$. We write

$s(x,y,t)\in S^{m}_{{\rm cl\,}}\big{(}D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$ (3.8)

if $s(x,y,t)\in S^{m}_{1,0}\big{(}D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$ and

$\begin{split}&s(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}(x,y)t^{m-j}\text{ in
}S^{m}_{1,0}\big{(}D\times
D\times\mathbb{R}_{+}\,,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}\,,\\\
&s^{j}(x,y)\in C^{\infty}\big{(}D\times
D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)},\ j\in\mathbb{N}_{0}.\end{split}$
(3.9)

The following was proved in Theorem 4.8 in [13]

######  Theorem 3.2.

Given $q=0,1,2,\ldots,n$. Assume that $q\notin\left\\{n_{-},n_{+}\right\\}$.
Then, $S^{(q)}\equiv 0$ on $X$.

We have the following (see Theorem 1.2 in [9], Theorem 4.7 in [13] and see
also [1] for $q=0$)

######  Theorem 3.3.

Let $q=n_{-}$ or $n_{+}$. Suppose that $\Box^{(q)}_{b}$ has $L^{2}$ closed
range. Then, $S^{(q)}(x,y)\in C^{\infty}(X\times X\setminus{{\rm
diag\,}(X\times X)},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$. Let $D\subset X$ be
any local coordinate patch with local coordinates $x=(x_{1},\ldots,x_{2n+1})$.
Then, there exist continuous operators
$S_{-},S_{+}:\Omega^{0,q}_{0}(D)\rightarrow D^{\prime}(D,T^{*0,q}X)$ such that

$S^{(q)}\equiv S_{-}+S_{+}\ \ \mbox{on $D$},$ (3.10)

and $S_{-}(x,y)$, $S_{+}(x,y)$ satisfy

$\begin{split}&S_{-}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{-}(x,y)t}s_{-}(x,y,t)dt\
\ \mbox{on $D$},\\\
&S_{+}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{+}(x,y)t}s_{+}(x,y,t)dt\ \
\mbox{on $D$},\end{split}$

with

$\begin{split}&s_{-}(x,y,t),s_{+}(x,y,t)\in S^{n}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\ &s_{-}(x,y,t)=0\
\ \mbox{if $q\neq n_{-}$},\ \ s_{+}(x,y,t)=0\ \ \mbox{if $q\neq n_{+}$},\\\
&s_{-}(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}_{-}(x,y)t^{n-j}\quad\text{ in
}S^{n}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&s_{+}(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}_{+}(x,y)t^{n-j}\quad\text{ in
}S^{n}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&s^{j}_{+}(x,y),s^{j}_{-}(x,y)\in C^{\infty}(D\times
D,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j=0,1,2,3,\ldots,\\\
&s^{0}_{-}(x,x)\neq 0,\ \ \forall x\in D,\ \ s^{0}_{+}(x,x)\neq 0,\ \ \forall
x\in D,\end{split}$ (3.11)

and the phase functions $\varphi_{-}$, $\varphi_{+}$ satisfy

$\begin{split}&\varphi_{+}(x,y),\varphi_{-}\in C^{\infty}(D\times D),\ \ {\rm
Im\,}\varphi_{-}(x,y)\geq 0,\\\ &\varphi_{-}(x,x)=0,\ \ \varphi_{-}(x,y)\neq
0\ \ \mbox{if}\ \ x\neq y,\\\
&d_{x}\varphi_{-}(x,y)\big{|}_{x=y}=-\omega_{0}(x),\ \
d_{y}\varphi_{-}(x,y)\big{|}_{x=y}=\omega_{0}(x),\\\
&\varphi_{-}(x,y)=-\overline{\varphi}_{-}(y,x),\\\
&-\overline{\varphi}_{+}(x,y)=\varphi_{-}(x,y).\end{split}$ (3.12)

######  Remark 3.4.

Note that for a strictly pseudoconvec CR manifold of dimension $3$,
$\Box^{(0)}_{b}$ does not have $L^{2}$ closed range in general (see [21]).
Kohn [14] proved that if $q=n_{-}=n_{+}$ or $\lvert n_{-}-n_{+}\rvert>1$ then
$\Box^{(q)}_{b}$ has $L^{2}$ closed range.

###  3.2. $G$-equivariant Szegő kernel

Since $G$ preserves $J$ and $(\,\cdot\,|\,\cdot\,)$ is $G$-invariant, it is
straightforward to see that for all $g\in G$

$\begin{split}&g^{*}\overline{\partial}_{b}=\overline{\partial}_{b}g^{*}\ \
\mbox{on $\Omega^{0,q}(X)$},\\\
&g^{*}\overline{\partial}^{*}_{b}=\overline{\partial}^{*}_{b}g^{*}\ \ \mbox{on
$\Omega^{0,q}(X)$},\\\ &g^{*}\Box^{(q)}_{b}=\Box^{(q)}_{b}g^{*}\ \ \mbox{on
$\Omega^{0,q}(X)$}.\end{split}$ (3.13)

Denote by $\overline{\partial}_{b,k}$ (resp. $\Box_{b,k}^{(q)}$) the
restriction of $\overline{\partial}_{b}$ (resp. $\Box_{b}^{(q)}$) on
$\Omega_{k}^{0,q}(X)$. By Definition 2.4, we have

$\begin{split}&\overline{\partial}_{b,k}:\Omega_{k}^{0,q}(X)\to\Omega_{k}^{0,q+1}(X),\\\
&\Box_{b,k}^{(q)}:\Omega_{k}^{0,q}(X)\to\Omega_{k}^{0,q}(X).\end{split}$
(3.14)

The $G$-equivariant Szegő projection is the orthogonal projection

$S^{(q)}_{k}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{(q)}_{b}\bigcap
L^{2}_{(0,q),k}(X)$

with respect to $(\,\cdot\,|\,\cdot\,)$. Let $S^{(q)}_{k}(x,y)\in
D^{\prime}(X\times X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be its distribution
kernel.

######  Lemma 3.5.

$S^{(q)}_{k}(x,y)=d_{k}\int_{G}S^{(q)}(g\circ
x,y)\overline{\chi_{k}(g)}d\mu(g).$ (3.15)

###### Proof.

Let

$\begin{split}Q_{k}:&L^{2}_{(0,q)}(X)\to L^{2}_{(0,q),k}(X)\\\ &u\to
u_{k}=d_{k}\int_{G}(g^{\star}u)(x)\overline{\chi_{k}(g)}d\mu(g).\end{split}$
(3.16)

Then $S^{(q)}_{k}=Q_{k}\circ S^{(q)}$. For $u\in L^{2}_{(0,q)}(X)$, we have

$\begin{split}S^{(q)}_{k}u&=Q_{k}\circ S^{(q)}u\\\ &=Q_{k}\int
S^{(q)}(x,y)u(y)dy\\\ &=d_{k}\int_{G}g^{\star}(\int
S^{(q)}(x,y)u(y)dy)\overline{\chi_{k}(g)}d\mu(g)\\\ &=d_{k}\int_{G}(\int
S^{(q)}(g\circ x,y)u(y)dy)\overline{\chi_{k}(g)}d\mu(g)\\\
&=\int(d_{k}\int_{G}S^{(q)}(g\circ
x,y)\overline{\chi_{k}(g)}d\mu(g))u(y)dy.\end{split}$ (3.17)

Then the proof is completed. ∎

Note that

$\begin{split}S^{(q)}_{k}(h\circ x,y)&=d_{k}\int_{G}S^{(q)}(g\circ h\circ
x,y)\overline{\chi_{k}(g)}d\mu(g)\\\ &=d_{k}\int_{G}S^{(q)}(g\circ
x,y)\overline{\chi_{k}(g\circ h^{-1})}d\mu(g)\\\ &=d_{k}\int_{G}S^{(q)}(g\circ
x,y)\sum_{j=1}^{d_{k}}\sum_{l=1}^{d_{k}}\overline{R_{k,j,l}(g)}R_{k,j,l}(h)d\mu(g).\end{split}$
(3.18)

So $S^{(q)}_{k}(x,y)$ is not $G$-invariant.

###  3.3. $G$-equivariant Szegő kernels near $\mu^{-1}(0)$

In this subsection, we will study $G$-equivariant Szegő kernel near
$\mu^{-1}(0)$. Let $e_{0}\in G$ be the identity element. Let
$v=(v_{1},\ldots,v_{d})$ be the local coordinates of $G$ defined in a
neighborhood $V$ of $e_{0}$ with $v(e_{0})=(0,\ldots,0)$. From now on, we will
identify the element $e\in V$ with $v(e)$. We recall the following on group
actions in local coordinates, see [10, Theorem 3.6].

######  Theorem 3.6.

Let $p\in\mu^{-1}(0)$. There exist local coordinates $v=(v_{1},\ldots,v_{d})$
of $G$ defined in a neighborhood $V$ of $e_{0}$ with $v(e_{0})=(0,\ldots,0)$,
local coordinates $x=(x_{1},\ldots,x_{2n+1})$ of $X$ defined in a neighborhood
$U=U_{1}\times U_{2}$ of $p$ with $0\leftrightarrow p$, where
$U_{1}\subset\mathbb{R}^{d}$ is an open set of $0\in\mathbb{R}^{d}$,
$U_{2}\subset\mathbb{R}^{2n+1-d}$ is an open set of $0\in\mathbb{R}^{2n+1-d}$
and a smooth function $\gamma=(\gamma_{1},\ldots,\gamma_{d})\in
C^{\infty}(U_{2},U_{1})$ with $\gamma(0)=0\in\mathbb{R}^{d}$ such that

$\begin{split}&(v_{1},\ldots,v_{d})\circ(\gamma(x_{d+1},\ldots,x_{2n+1}),x_{d+1},\ldots,x_{2n+1})\\\
&=(v_{1}+\gamma_{1}(x_{d+1},\ldots,x_{2n+1}),\ldots,v_{d}+\gamma_{d}(x_{d+1},\ldots,x_{2n+1}),x_{d+1},\ldots,x_{2n+1}),\\\
&\forall(v_{1},\ldots,v_{d})\in V,\ \ \forall(x_{d+1},\ldots,x_{2n+1})\in
U_{2},\end{split}$ (3.19) $\begin{split}&\underline{\mathfrak{g}}={\rm
span\,}\left\\{\frac{\partial}{\partial x_{1}},\ldots,\frac{\partial}{\partial
x_{d}}\right\\},\\\ &\mu^{-1}(0)\bigcap
U=\left\\{x_{d+1}=\cdots=x_{2d}=0\right\\},\\\ &\mbox{On $\mu^{-1}(0)\bigcap
U$, we have $J(\frac{\partial}{\partial x_{j}})=\frac{\partial}{\partial
x_{d+j}}+a_{j}(x)\frac{\partial}{\partial x_{2n+1}}$,
$j=1,2,\ldots,d$},\end{split}$ (3.20)

where $a_{j}(x)$ is a smooth function on $\mu^{-1}(0)\bigcap U$, independent
of $x_{1},\ldots,x_{2d}$, $x_{2n+1}$ and $a_{j}(0)=0$, $j=1,\ldots,d$,

$\begin{split}&T^{1,0}_{p}X={\rm
span\,}\left\\{Z_{1},\ldots,Z_{n}\right\\},\\\
&Z_{j}=\frac{1}{2}(\frac{\partial}{\partial x_{j}}-i\frac{\partial}{\partial
x_{d+j}})(p),\ \ j=1,\ldots,d,\\\ &Z_{j}=\frac{1}{2}(\frac{\partial}{\partial
x_{2j-1}}-i\frac{\partial}{\partial x_{2j}})(p),\ \ j=d+1,\ldots,n,\\\
&\langle\,Z_{j}\,|\,Z_{l}\,\rangle=\delta_{j,l},\ \ j,l=1,2,\ldots,n,\\\
&\mathcal{L}_{p}(Z_{j},\overline{Z}_{l})=\mu_{j}\delta_{j,l},\ \
j,l=1,2,\ldots,n\end{split}$ (3.21)

and

$\begin{split}\omega_{0}(x)&=(1+O(\lvert
x\rvert))dx_{2n+1}+\sum^{d}_{j=1}4\mu_{j}x_{d+j}dx_{j}\\\
&\quad+\sum^{n}_{j=d+1}2\mu_{j}x_{2j}dx_{2j-1}-\sum^{n}_{j=d+1}2\mu_{j}x_{2j-1}dx_{2j}+\sum^{2n}_{j=d+1}b_{j}x_{2n+1}dx_{j}+O(\lvert
x\rvert^{2}),\end{split}$ (3.22)

where $b_{d+1}\in\mathbb{R},\ldots,b_{2n}\in\mathbb{R}$.

The following describes the phase function in Theorem 3.3, see [10, Theorem
3.7].

######  Theorem 3.7.

Let $p\in\mu^{-1}(0)$ and take local coordinates $x=(x_{1},\ldots,x_{2n+1})$
of $X$ defined in an open set $U$of $p$ with $0\leftrightarrow p$ such that
(3.20), (3.21) and (3.22) hold. Let $\varphi_{-}(x,y)\in C^{\infty}(U\times
U)$ be as in Theorem 3.3. Then,

$\begin{split}\varphi_{-}(x,y)&=-x_{2n+1}+y_{2n+1}-2\sum^{d}_{j=1}\mu_{j}x_{j}x_{d+j}+2\sum^{d}_{j=1}\mu_{j}y_{j}y_{d+j}+i\sum^{n}_{j=1}\lvert\mu_{j}\rvert\lvert
z_{j}-w_{j}\rvert^{2}\\\
&+\sum^{n}_{j=1}i\mu_{j}(\overline{z}_{j}w_{j}-z_{j}\overline{w}_{j})+\sum^{d}_{j=1}(-\frac{i}{2}b_{d+j})(-z_{j}x_{2n+1}+w_{j}y_{2n+1})\\\
&+\sum^{d}_{j=1}(\frac{i}{2}b_{d+j})(-\overline{z}_{j}x_{2n+1}+\overline{w}_{j}y_{2n+1})+\sum^{n}_{j=d+1}\frac{1}{2}(b_{2j-1}-ib_{2j})(-z_{j}x_{2n+1}+w_{j}y_{2n+1})\\\
&+\sum^{n}_{j=d+1}\frac{1}{2}(b_{2j-1}+ib_{2j})(-\overline{z}_{j}x_{2n+1}+\overline{w}_{j}y_{2n+1})+(x_{2n+1}-y_{2n+1})f(x,y)+O(\lvert(x,y)\rvert^{3}),\end{split}$
(3.23)

where $z_{j}=x_{j}+ix_{d+j}$, $j=1,\ldots,d$, $z_{j}=x_{2j-1}+ix_{2j}$,
$j=2d+1,\ldots,2n$, $\mu_{j}$, $j=1,\ldots,n$, and
$b_{d+1}\in\mathbb{R},\ldots,b_{2n}\in\mathbb{R}$ are as in (3.22) and $f$ is
smooth and satisfies $f(0,0)=0$, $f(x,y)=\overline{f}(y,x)$.

We now study $S^{(q)}_{k}(x,y)$. From Theorem 3.2 and Lemma 3.5, we get

######  Theorem 3.8.

Assume that $q\notin\left\\{n_{-},n_{+}\right\\}$.Then, $S^{(q)}_{k}\equiv 0$
on $X$.

Assume that $q=n_{-}$ and $\Box^{(q)}_{b}$ has $L^{2}$ closed range. Fix
$p\in\mu^{-1}(0)$ and let $v=(v_{1},\ldots,v_{d})$ and
$x=(x_{1},\ldots,x_{2n+1})$ be the local coordinates of $G$ and $X$ as in
Theorem 3.6. Assume that $d\mu=m(v)dv=m(v_{1},\ldots,v_{d})dv_{1}\cdots
dv_{d}$ on $V$, where $V$ is an open neighborhood of $e_{0}\in G$ as in
Theorem 3.6. From Lemma 3.5, we have

$S^{(q)}_{k}(x,y)=d_{k}\int_{G}\chi(g)\overline{\chi_{k}(g)}S^{(q)}(g\circ
x,y)d\mu(g)+d_{k}\int_{G}(1-\chi(g))\overline{\chi_{k}(g)}S^{(q)}(g\circ
x,y)d\mu(g),$ (3.24)

where $\chi\in C^{\infty}_{0}(V)$, $\chi=1$ near $e_{0}$.

Assume first $G$ is globally free on $\mu^{-1}(0)$, if $U$ and $V$ are small,
there is a constant $c>0$ such that

$d(g\circ x,y)\geq c,\ \ \forall x,y\in U,g\in{\rm Supp\,}(1-\chi),$ (3.25)

where $U$ is an open set of $p\in\mu^{-1}(0)$ as in Theorem 3.6. From now on,
we take $U$ and $V$ small enough so that (3.25) holds. By Theorem 3.3,
$S^{(q)}(x,y)$ is smoothing away from diagonal. From this observation and
(3.25), we have $\int_{G}(1-\chi(g))\overline{\chi_{k}(g)}S^{(q)}(g\circ
x,y)d\mu(g)\equiv 0$ on $U$ and hence

$\mbox{$S^{(q)}_{k}(x,y)\equiv
d_{k}\int_{G}\chi(g)\overline{\chi_{k}(g)}S^{(q)}(g\circ x,y)d\mu(g)$ on
$U$}.$ (3.26)

From Theorem 3.3 and (3.26), we have

$\begin{split}&\mbox{$S^{(q)}_{k}(x,y)\equiv\hat{S}_{k,-}(x,y)+\hat{S}_{k,+}(x,y)$
on $U$},\\\
&\hat{S}_{k,-}(x,y)=d_{k}\int_{G}\chi(g)\overline{\chi_{k}(g)}S_{-}(g\circ
x,y)d\mu(g),\\\
&\hat{S}_{k,+}(x,y)=d_{k}\int_{G}\chi(g)\overline{\chi_{k}(g)}S_{+}(g\circ
x,y)d\mu(g).\end{split}$ (3.27)

More precisely,

$\hat{S}_{k,-}(x,y)\equiv d_{k}\int_{0}^{\infty}\int_{V}e^{i\phi_{-}(v\circ
x,y)t}\chi(v)\overline{\chi_{k}(v)}s_{-}(v\circ x,y,t)m(v)dvdt.$ (3.28)

By using stationary phase formula of Melin-Sjöstrand [20], it follows from the
arguments in [10, Section 3.3] that

$\hat{S}_{k,-}(x,y)\equiv\int e^{i\Phi_{k,-}(x,y)t}a_{k,-}(x,y,t)dt\ \ \
\mbox{on $U$},$ (3.29)

where
$a_{k,-}(x,y,t)\sim\sum^{\infty}_{j=0}t^{n-\frac{d}{2}-j}a^{j}_{k,-}(x,y)$ in
$S^{n-\frac{d}{2}}_{1,0}(U\times
U\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,

$a^{j}_{k,-}(x,y)\in C^{\infty}(U\times U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\
\ j=0,1,\ldots,$

In this work, $G$ acts locally free on $\mu^{-1}(0)$ under Assumption 1.4. Let
$N_{p}=\\{g\in G:g\circ p=p\\}=\\{g_{1}=e_{0},g_{2}...,g_{r}\\}$. Similarly to
(3.25), we can choose $U$ and $V$ to be small such that the subsets
$\\{g_{j}V\\}_{\alpha=1}^{r}$ are mutually disjoint and there is a constant
$c>0$ satisfying

$d(h\circ x,y)\geq c,\ \ \forall x,y\in U,h\in{\rm
Supp\,}(1-\sum_{\alpha=1}^{r}\chi(gg_{\alpha}^{-1})).$ (3.30)

Then on $U$

$\begin{split}S^{(q)}_{k}(x,y)&\equiv
d_{k}\sum_{\alpha=1}^{r}\int_{G}\chi(gg_{\alpha}^{-1})\overline{\chi_{k}(g)}S^{(q)}(g\circ
x,y)d\mu(g)\\\
&=d_{k}\sum_{\alpha=1}^{r}\int_{G}\chi(g)\overline{\chi_{k}(gg_{\alpha})}S^{(q)}(gg_{\alpha}\circ
x,y)d\mu(g).\end{split}$ (3.31) $\hat{S}_{k,-}(x,y)\equiv
d_{k}\sum_{\alpha=1}^{r}\int_{0}^{\infty}\int_{V}e^{i\phi_{-}(vv_{\alpha}\circ
x,y)t}\chi(v)\overline{\chi_{k}(vv_{\alpha})}s_{-}(vv_{\alpha}\circ
x,y,t)m(v)dvdt,$ (3.32)

where $v_{\alpha}$ is the coordinate of $g_{\alpha}$. By using stationary
phase formula of Melin-Sjöstrand [20], it follows from the above argument that

$\hat{S}_{k,-}(x,y)\equiv\sum_{\alpha=1}^{r}\int
e^{i\Phi_{k,-}(v_{\alpha}\circ x,y)t}a_{k,\alpha,-}(x,y,t)dt\ \ \ \mbox{on
$U$},$ (3.33)

where
$a_{k,\alpha,-}(x,y,t)\sim\sum^{\infty}_{j=0}t^{n-\frac{d}{2}-j}a^{j}_{k,\alpha,-}(x,y)$
in $S^{n-\frac{d}{2}}_{1,0}(U\times
U\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$,

$a^{j}_{k,\alpha,-}(x,y)\in C^{\infty}(U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \ j=0,1,\ldots,$

In particular, it follows from the arguments in [10, Subsection 3.3] with
minor modification, if $G$ acts freely on $\mu^{-1}(0)$, then for
$a^{0}_{k,-}(x,y)$ and $a^{0}_{k,+}(x,y)$ in (1.7), we have

$\begin{split}a^{0}_{k,-}(x,x)=2^{d-1}\frac{d^{2}_{k}}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{-}},\ \
\forall x\in\mu^{-1}(0)\\\ a^{0}_{k,+}(x,x)=2^{d-1}\frac{d^{2}_{k}}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{+}},\ \
\forall x\in\mu^{-1}(0).\end{split}$

###  3.4. $G$-equivariant Szegő kernel asymptotics away $\mu^{-1}(0)$

The goal of this section is to prove the following

######  Theorem 3.9.

Let $D$ be an open neighborhood of $X$ with $D\bigcap\mu^{-1}(0)=\emptyset$.
Then,

$S^{(q)}_{k}\equiv 0\ \ \mbox{on $D$}.$

We first need

######  Lemma 3.10.

Let $p\notin\mu^{-1}(0)$. Then, there are open neighborhoods $U$ of $p$ and
$V$ of $e\in G$ such that for any $\chi\in C^{\infty}_{0}(V)$, we have for
every $k$,

$\int_{G}S^{(q)}(x,g\circ y)\chi(g)\overline{\chi_{k}(g)}d\mu(g)\equiv 0\ \
\mbox{on $U$}.$ (3.34)

The proof of the above lemma follows from [10, Lemma 3.14] by adding
$\overline{\chi_{k}(g)}$.

######  Lemma 3.11.

Let $p\notin\mu^{-1}(0)$ and let $h\in G$. We can find open neighborhoods $U$
of $p$ and $V$ of $h$ such that for every $\chi\in C^{\infty}_{0}(V)$, we have
for every $k$,

$\int_{G}S^{(q)}(g\circ x,y)\chi(g)\overline{\chi_{k}(g)}d\mu(g)\equiv 0\ \
\mbox{on $U$}.$

###### Proof.

Let $U$ and $V$ be open sets as in Lemma 3.10. Let $\hat{V}=hV$. Then,
$\hat{V}$ is an open set of $G$. Let $\hat{\chi}\in C^{\infty}_{0}(\hat{V})$.
We have

$\begin{split}\int_{G}S^{(q)}(g\circ
x,y)\hat{\chi}(g)\overline{\chi_{k}(g)}d\mu(g)&=\int_{G}S^{(q)}(h\circ g\circ
x,y)\hat{\chi}(h\circ g)\overline{\chi_{k}(h\circ g)}d\mu(g)\\\
&=\int_{G}S^{(q)}(h\circ g\circ x,y)\chi(g)d\mu(g),\end{split}$ (3.35)

where $\chi(g):=\hat{\chi}(h\circ g)\overline{\chi_{k}(h\circ g)}\in
C^{\infty}_{0}(V)$. From (3.35) and Lemma 3.10, we deduce that

$\int_{G}S^{(q)}(g\circ x,y)\hat{\chi}(g)\overline{\chi_{k}(g)}d\mu(g)\equiv
0\ \ \mbox{on $U$}.$

The lemma follows. ∎

###### Proof of Theorem 3.9.

Fix $p\in D$. We need to show that $S^{(q)}_{k}$ is smoothing near $p$. Let
$h\in G$. By Lemma 3.11, we can find open sets $U_{h}$ of $p$ and $V_{h}$ of
$h$ such that for every $\chi\in C^{\infty}_{0}(V_{h})$, we have

$\int_{G}S^{(q)}(g\circ x,y)\chi(g)\overline{\chi_{k}(g)}d\mu(g)\equiv 0\ \
\mbox{on $U_{h}$}.$ (3.36)

Since $G$ is compact, we can find open sets $U_{h_{j}}$ and $V_{h_{j}}$,
$j=1,\ldots,N$, such that $G=\bigcup^{N}_{j=1}V_{h_{j}}$. Let
$U=D\bigcap\Bigr{(}\bigcap^{N}_{j=1}U_{h_{j}}\Bigr{)}$ and let
$\tilde{\chi}_{j}\in C^{\infty}_{0}(V_{h_{j}})$, $j=1,\ldots,N$, with
$\sum^{N}_{j=1}\tilde{\chi}_{j}=1$ on $G$. From (3.36), we have

$\begin{split}S^{(q)}_{k}(x,y)&=d_{k}\int_{G}S^{(q)}(g\circ
x,y)\overline{\chi_{k}(g)}d\mu(g)\\\
&=d_{k}\sum^{N}_{j=1}\int_{G}S^{(q)}(g\circ
x,y)\tilde{\chi}_{j}(g)\overline{\chi_{k}(g)}d\mu(g)\equiv 0\ \ \mbox{on
$U$}.\end{split}$ (3.37)

The theorem follows. ∎

From Section 3.3 and Section 3.4, we get Theorem 1.6.

##  4\. $G$-equivariant Szegő kernel asymptotics on CR manifolds with $S^{1}$
action

Let $X$ admit an $S^{1}$ action $e^{i\theta}$: $S^{1}\times X\rightarrow X$.
Let $T\in C^{\infty}(X,TX)$ be the global real vector field induced by the
$S^{1}$ action given by
$(Tu)(x)=\frac{\partial}{\partial\theta}\left(u(e^{i\theta}\circ
x)\right)|_{\theta=0}$, $u\in C^{\infty}(X)$.

######  Definition 4.1.

The $S^{1}$ action $e^{i\theta}$ is CR if $[T,C^{\infty}(X,T^{1,0}X)]\subset
C^{\infty}(X,T^{1,0}X)$ and the $S^{1}$ action is transversal if for each
$x\in X$, $\mathbb{C}T(x)\oplus T_{x}^{1,0}X\oplus
T_{x}^{0,1}X=\mathbb{C}T_{x}X$. Moreover, the $S^{1}$ action is locally free
if $T\neq 0$ everywhere.

Note that transversality implies local freeness. Let $(X,T^{1,0}X)$ be a
compact connected CR manifold with a transversal CR $S^{1}$ action
$e^{i\theta}$ and $T$ be the global vector field induced by the $S^{1}$
action. Let $\omega_{0}\in C^{\infty}(X,T^{*}X)$ be the global real one form
determined by $\langle\,\omega_{0}\,,\,u\,\rangle=0$, for every $u\in
T^{1,0}X\oplus T^{0,1}X$, and $\langle\,\omega_{0}\,,\,T\,\rangle=-1$. Note
that $\omega_{0}$ and $T$ satisfy (2.5). Recall that we work with Assumption
1.8.

Assume that the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$\mathbb{C}TX$ is $G\times S^{1}$ invariant. Then the $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle$ is $G\times S^{1}$-invariant. We then have

$\begin{split}&Tg^{*}\overline{\partial}^{*}_{b}=g^{*}T\overline{\partial}^{*}_{b}=\overline{\partial}^{*}_{b}g^{*}T=\overline{\partial}^{*}_{b}Tg^{*}\
\ \mbox{on $\Omega^{0,q}(X)$},\ \ \forall g\in G,\\\
&Tg^{*}\Box^{(q)}_{b}=g^{*}T\Box^{(q)}_{b}=\Box^{(q)}_{b}g^{*}T=\Box^{(q)}_{b}Tg^{*}\
\ \mbox{on $\Omega^{0,q}(X)$},\ \ \forall g\in G.\end{split}$

Let $L^{2}_{(0,q),m}(X)_{k}$ be the completion of $\Omega^{0,q}_{m}(X)_{k}$
with respect to $(\,\cdot\,|\,\cdot\,)$. We write
$L^{2}_{m}(X)_{k}:=L^{2}_{(0,0),m}(X)_{k}$. Put

$H^{q}_{b,m}(X)_{k}:=({\rm Ker\,}\Box^{(q)}_{b})\bigcap
L^{2}_{(0,q),m}(X)_{k}.$

The $m$-th $G$-equivariant Szegő projection is the orthogonal projection

$S^{(q)}_{k,m}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{(q)}_{b})\bigcap
L^{2}_{(0,q),m}(X)_{k}$

with respect to $(\,\cdot\,|\,\cdot\,)$. Let $S^{(q)}_{k,m}(x,y)\in
C^{\infty}(X\times X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution
kernel of $S^{(q)}_{k,m}$. Then

$S^{(q)}_{k,m}(x,y)=\frac{1}{2\pi}\int^{\pi}_{-\pi}S^{(q)}_{k}(x,e^{i\theta}\circ
y)e^{im\theta}d\theta.$ (4.1)

The goal of this section is to study the asymptotics of $S^{(q)}_{k,m}$ as
$m\rightarrow+\infty$.

From Theorem 3.9, (4.1) and by using integration by parts several times, we
get

######  Theorem 4.2.

Let $D\subset X$ be an open set with $D\bigcap\mu^{-1}(0)=\emptyset$. Then,

$S^{(q)}_{k,m}=O(m^{-\infty})\ \ \mbox{on $D$}.$

We now study $S^{(q)}_{k,m}$ near $\mu^{-1}(0)$. We can repeat the proof of
Theorem 3.6 with minor change and get

######  Theorem 4.3.

Let $p\in\mu^{-1}(0)$. There exist local coordinates $v=(v_{1},\ldots,v_{d})$
of $G$ defined in a neighborhood $V$ of $e_{0}$ with $v(e_{0})=(0,\ldots,0)$,
local coordinates $x=(x_{1},\ldots,x_{2n+1})$ of $X$ defined in a neighborhood
$U=U_{1}\times(\hat{U}_{2}\times]-2\delta,2\delta[)$ of $p$ with
$0\leftrightarrow p$, where $U_{1}\subset\mathbb{R}^{d}$ is an open set of
$0\in\mathbb{R}^{d}$, $\hat{U}_{2}\subset\mathbb{R}^{2n-d}$ is an open set of
$0\in\mathbb{R}^{2n-d}$, $\delta>0$, and a smooth function
$\gamma=(\gamma_{1},\ldots,\gamma_{d})\in
C^{\infty}(\hat{U}_{2}\times]-2\delta,2\delta[,U_{1})$ with
$\gamma(0)=0\in\mathbb{R}^{d}$ such that

$\begin{split}&(v_{1},\ldots,v_{d})\circ(\gamma(x_{d+1},\ldots,x_{2n+1}),x_{d+1},\ldots,x_{2n+1})\\\
&=(v_{1}+\gamma_{1}(x_{d+1},\ldots,x_{2n+1}),\ldots,v_{d}+\gamma_{d}(x_{d+1},\ldots,x_{2n+1}),x_{d+1},\ldots,x_{2n+1}),\\\
&\forall(v_{1},\ldots,v_{d})\in V,\ \
\forall(x_{d+1},\ldots,x_{2n+1})\in\hat{U}_{2}\times]-2\delta,2\delta[,\end{split}$
(4.2) $\begin{split}&T=-\frac{\partial}{\partial x_{2n+1}},\\\
&\underline{\mathfrak{g}}={\rm span\,}\left\\{\frac{\partial}{\partial
x_{1}},\ldots,\frac{\partial}{\partial x_{d}}\right\\},\\\ &\mu^{-1}(0)\bigcap
U=\left\\{x_{d+1}=\cdots=x_{2d}=0\right\\},\\\ &\mbox{On $\mu^{-1}(0)\bigcap
U$, we have $J(\frac{\partial}{\partial x_{j}})=\frac{\partial}{\partial
x_{d+j}}+a_{j}(x)\frac{\partial}{\partial x_{2n+1}}$,
$j=1,2,\ldots,d$},\end{split}$ (4.3)

where $a_{j}(x)$ is a smooth function on $\mu^{-1}(0)\bigcap U$, independent
of $x_{1},\ldots,x_{2d}$, $x_{2n+1}$ and $a_{j}(0)=0$, $j=1,\ldots,d$,

$\begin{split}&T^{1,0}_{p}X={\rm
span\,}\left\\{Z_{1},\ldots,Z_{n}\right\\},\\\
&Z_{j}=\frac{1}{2}(\frac{\partial}{\partial x_{j}}-i\frac{\partial}{\partial
x_{d+j}})(p),\ \ j=1,\ldots,d,\\\ &Z_{j}=\frac{1}{2}(\frac{\partial}{\partial
x_{2j-1}}-i\frac{\partial}{\partial x_{2j}})(p),\ \ j=d+1,\ldots,n,\\\
&\langle\,Z_{j}\,|\,Z_{k}\,\rangle=\delta_{j,k},\ \ j,k=1,2,\ldots,n,\\\
&\mathcal{L}_{p}(Z_{j},\overline{Z}_{k})=\mu_{j}\delta_{j,k},\ \
j,k=1,2,\ldots,n\end{split}$ (4.4)

and

$\begin{split}\omega_{0}(x)&=(1+O(\lvert
x\rvert))dx_{2n+1}+\sum^{d}_{j=1}4\mu_{j}x_{d+j}dx_{j}\\\
&\quad+\sum^{n}_{j=d+1}2\mu_{j}x_{2j}dx_{2j-1}-\sum^{n}_{j=d+1}2\mu_{j}x_{2j-1}dx_{2j}+O(\lvert
x\rvert^{2}).\end{split}$ (4.5)

From Theorem 3.8, we get

######  Theorem 4.4.

Assume that $q\notin\left\\{n_{-},n_{+}\right\\}$. Then,
$S^{(q)}_{k,m}=O(m^{-\infty})$ on $X$.

###### Proof of Theorem 1.10.

It suffices to show the cases when $q=n_{-}$ and $q=n_{+}\neq n_{-}$. Assume
that $q=n_{-}$. It is well-known [13, Theorem 1.12] that when $X$ admits a
transversal $S^{1}$ action, then $\Box^{(q)}_{b}$ has $L^{2}$ closed range.
Fix $p\in\mu^{-1}(0)$. Let $N_{p}=\\{g\in G:g\circ
p=p\\}=\\{g_{1}=e_{0},g_{2}...,g_{r}\\}$. Let $v=(v_{1},\ldots,v_{d})$ and
$x=(x_{1},\ldots,x_{2n+1})$ be the local coordinates of $G$ and $X$ as in
Theorem 4.3 and let $U$ and $V$ be open sets as in Theorem 4.3. We take $U$
small enough so that there is a constant $c>0$ such that

$d(e^{i\theta}\circ g\circ x,y)\geq c,\ \ \forall(x,y)\in U\times U,\ \
\forall g\in G,\theta\in[-\pi,-\delta]\bigcup[\delta,\pi],$ (4.6)

where $\delta>0$ is as in Theorem 4.3. We repeat the same procedure in [10,
Section 4] as follows.

$\begin{split}S^{(q)}_{k,m}(x,y)&=\frac{1}{2\pi}\int^{\pi}_{-\pi}S^{(q)}_{k}(x,e^{i\theta}\circ
y)e^{im\theta}d\theta=\frac{1}{2\pi}\int^{\pi}_{-\pi}e^{-imx_{2n+1}+imy_{2n+1}}S^{(q)}_{k}(\mathring{x},e^{i\theta}\circ\mathring{y})e^{im\theta}d\theta\\\
&=I+II,\\\
&I=\frac{1}{2\pi}\int^{\pi}_{-\pi}e^{-imx_{2n+1}+imy_{2n+1}}\chi(\theta)S^{(q)}_{k}(\mathring{x},e^{i\theta}\circ\mathring{y})e^{im\theta}d\theta,\\\
&II=\frac{1}{2\pi}\int^{\pi}_{-\pi}e^{-imx_{2n+1}+imy_{2n+1}}(1-\chi(\theta))S^{(q)}_{k}(\mathring{x},e^{i\theta}\circ\mathring{y})e^{im\theta}d\theta,\end{split}$
(4.7)

where $\mathring{x}=(x_{1},\ldots,x_{2n},0)\in U$,
$\mathring{y}=(y_{1},\ldots,y_{2n},0)\in U$, $\chi\in
C^{\infty}_{0}(]-2\delta,2\delta[)$, $\chi=1$ on $[-\delta,\delta]$. It is
easy to check that

$II=O(m^{-\infty}).$ (4.8)

For $I$, we have

$\begin{split}&I=I_{0}+I_{1},\\\
&I_{0}=\frac{1}{2\pi}\sum_{\alpha=1}^{r}\int^{\infty}_{0}\int^{\pi}_{-\pi}e^{-imx_{2n+1}+imy_{2n+1}}\chi(\theta)e^{i(-\theta+\hat{\Phi}_{k,-}(g_{\alpha}\circ\mathring{x},\mathring{y}))t+im\theta}a_{k,\alpha,-}(\mathring{x},(\mathring{y},-\theta),t)dtd\theta,\\\
&I_{1}=\frac{1}{2\pi}\sum_{\alpha=1}^{r}\int^{\infty}_{0}\int^{\pi}_{-\pi}e^{-imx_{2n+1}+imy_{2n+1}}\chi(\theta)e^{i(\theta+\hat{\Phi}_{k,+}(g_{\alpha}\circ\mathring{x},\mathring{y}))t+im\theta}a_{k,\alpha,+}(\mathring{x},(\mathring{y},-\theta),t)dtd\theta.\end{split}$
(4.9)

From
$\frac{\partial}{\partial\theta}\Bigr{(}i(\theta+\hat{\Phi}_{k,+}(g_{\alpha}\circ\mathring{x},\mathring{y}))t+im\theta\Bigr{)}\neq
0$, we can integrate by parts with respect to $\theta$ several times and
deduce that

$I_{1}=O(m^{-\infty}).$ (4.10)

For $I_{0}$, we have

$I_{0}=\frac{1}{2\pi}\sum_{\alpha=1}^{r}\int^{\infty}_{0}\int^{\pi}_{-\pi}e^{-imx_{2n+1}+imy_{2n+1}}\chi(\theta)e^{im(-\theta
t+\hat{\Phi}_{k,-}(g_{\alpha}\circ\mathring{x},\mathring{y})t+\theta)}ma_{k,\alpha,-}(\mathring{x},(\mathring{y},-\theta),mt)dtd\theta.$
(4.11)

We apply the complex stationary phase formula of Melin-Sjöstrand [20, Theorem
2.3] to carry the $dtd\theta$ integration in (4.11). The calculation is
similar as in the proof of Theorem 3.17 in [12]. Then

$\begin{split}&I_{0}\equiv\sum_{\alpha=1}^{r}e^{im\Psi_{k}(g_{\alpha}\circ
x,y)}b_{k,\alpha}(x,y,m),\\\
&\Psi_{k}(x,y)=\hat{\Phi}_{k,-}(\mathring{x},\mathring{y})-x_{2n+1}+y_{2n+1},\\\
&b_{k,\alpha}(x,y,m)\in S^{n-\frac{d}{2}}_{{\rm loc\,}}(1;U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\\\
&\mbox{$b_{k,\alpha}(x,y,m)\sim\sum^{\infty}_{j=0}m^{n-\frac{d}{2}-j}b_{k,\alpha}^{j}(x,y)$
in $S^{n-\frac{d}{2}}_{{\rm loc\,}}(1;U\times
U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$},\\\ &b_{k,\alpha}^{j}(x,y)\in
C^{\infty}(U\times U,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}),\ \
j=0,1,2,\ldots,\end{split}$ (4.12)

Assume that $q=n_{+}\neq n_{-}$. If $m\rightarrow-\infty$, then the expansion
for $S^{(q)}_{k,m}(x,y)$ as $m\rightarrow-\infty$ is similar to $q=n_{-}$
case. When $m\rightarrow+\infty$, we can repeat the method above with minor
change and deduce that $S^{(q)}_{k,m}(x,y)=O(m^{-\infty})$ on $X$. In
particular, it follows from the argument in [10, Section 4] with minor
modification, if $G\times S^{1}$ acts freely near $\mu^{-1}(0)$, then

$b_{k}^{0}(x,x)=2^{d-1}\frac{d^{2}_{k}}{V_{{\rm
eff\,}}(x)}\pi^{-n-1+\frac{d}{2}}\lvert\det
R_{x}\rvert^{-\frac{1}{2}}\lvert\det\mathcal{L}_{x}\rvert\tau_{x,n_{-}},\ \
\forall x\in\mu^{-1}(0),$

where $\tau_{x,n_{-}}$ is given by (1.10). The proof is completed. ∎

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