# ExcavatorCovid: Extracting Events and Relations from Text Corpora for
Temporal and Causal Analysis for COVID-19

Bonan Min, Benjamin Rozonoyer, Haoling Qiu, Alexander Zamanian, Jessica
MacBride
Raytheon BBN Technologies, Cambridge, Massachusetts
<EMAIL_ADDRESS>
###### Abstract

Timely responses from policy makers to mitigate the impact of the COVID-19
pandemic rely on a comprehensive grasp of events, their causes, and their
impacts. These events are reported at such a speed and scale as to be
overwhelming. In this paper, we present ExcavatorCovid, a machine reading
system that ingests open-source text documents (e.g., news and scientific
publications), extracts COVID-19 related events and relations between them,
and builds a Temporal and Causal Analysis Graph (TCAG). Excavator will help
government agencies alleviate the information overload, understand likely
downstream effects of political and economic decisions and events related to
the pandemic, and respond in a timely manner to mitigate the impact of
COVID-19. We expect the utility of Excavator to outlive the COVID-19 pandemic:
analysts and decision makers will be empowered by Excavator to better
understand and solve complex problems in the future. An interactive TCAG
visualization is available at http://afrl402.bbn.com:5050/index.html. We also
released a demonstration video at https://vimeo.com/528619007.

##  1 Introduction

Timely responses from policy makers to mitigate the impact of the COVID-19
pandemic rely on a comprehensive grasp of events, their causes, and their
impacts. Since the beginning of the COVID-19 pandemic, an enormous amount of
articles are being published every day, that report many events 111We define
an event as any occurrence, action, process or state of affairs, following
O’Gorman et al. (2016). related to COVID as well as studies related to COVID.
It is very difficult, if not impossible, to keep track of these developing
events or to get a comprehensive overview of the temporal and causal dynamics
underlying these events.

To aid the policy makers in overcoming the information overload, we developed
ExcavatorCovid (or Excavator for short), a system that will ingest open-source
text sources (e.g., news articles and scientific publications), extract
COVID-19 related events and relations between them, and build a Temporal and
Causal Analysis Graph (TCAG). Excavator combines the following NLP techniques:

  * •

Extracting events (§3) for types in our comprehensive COVID-19 event taxonomy
(§2). Each event will have time and location if available in text, allowing
analyses targeted at specific times or geographic regions of interest.

  * •

Extracting three types of temporal and causal relations (§4) between pairs of
events.

  * •

Constructing a TCAG (§5) by assembling all events and relations, to provide a
comprehensive overview of the events related to COVID-19 as well as their
causes and impacts.

  * •

Supporting trend and correlation analysis of events, via visualizing event
popularity time series (§ 6) in the TCAG visualization.

Excavator produces a TCAG that is in a machine-readable JSON format and is
also human-understandable (visualized via a web-based interactive User
Interface), to support varied analytical and decision making needs. We hope
that Excavator will aid government agencies in efforts to understand likely
downstream effects of political and economic decisions and events related to
the pandemic, and respond in a timely manner to mitigate the impact of
COVID-19. The benefit of Excavator is realized through a comprehensive
visualization of events and how they affect each other. We expect the utility
of Excavator to outlive the COVID-19 pandemic: analysts and decision makers
will be empowered by Excavator to better understand and solve complex problems
in the future.

Figure 1: An partial illustration of the COVID-19 event taxonomy.

We first present our COVID-19 event taxonomy, and then we present details
about event extraction, causal and temporal relation extraction, measuring
event popularity using news text as “quantitative data”, and the approach for
constructing a TCAG. We then describe the system demonstration, present a
quantitative analysis of the extractions, and conclude with recommended use
cases.

##  2 Building a COVID-19 Event Taxonomy

COVID-19 affects many aspects of our political, economic, and personal lives.
A comprehensive analysis requires an event taxonomy that categorizes the
events related to COVID-19 in many sectors and domains. We developed a
COVID-19 event taxonomy using a hybrid approach of manual curation with
automated support: first, we run Stanza Qi et al. (2020) on a large sample
(10%) of the Aylien coronavirus news dataset (§ 7) to tag verb and noun
phrases that are likely to trigger events. Second, we represent each phrase as
the average of the BERT Devlin et al. (2019) contextualized embedding vectors
of the subwords within each phrase, and then run committee-based clustering
Pantel and Lin (2002) over the vector representations of the phrases to
discover salient clusters. Finally, we review the frequently appearing
clusters and define event types related to COVID-19.

The event taxonomy includes 76 event types and a short description of each
type. Figure 1 illustrates several branches of the event taxonomy (the
complete taxonomy will be publicly available via github.com). The events come
from a wide range of domains. We also manually added the hyponymy relation via
is_a links (e.g., COVID-19 is_a {Virus, Disease}) between pairs of event
types.

##  3 Extracting Events

We developed a neural network model for extracting events defined in the
COVID-19 event taxonomy (the event classification stage) and extracting the
location and time arguments (the event argument extraction stage), if they are
mentioned in text, for each event mention. The structured representation
(events with location and/or time) enables analyses of events targeting a
specific time or location. Both stages use a BERT-based sequence tagging
model. Figure 2(a) shows the model architecture. Given a sequence of tokens as
input, the model extracts a sequence of tags, one per each token. We use the
commonly used Begin-Inside-Outside (BIO) tags for both event types and event
argument role types for the event classification and argument attachment tasks
respectively.

Figure 2: The BERT-based sequence tagging model for event classification and
argument extraction. Figure (a) shows the architecture of the model, which
takes a sequence of words $x_{1},x_{2},...,x_{n}$ as input and outputs a
sequence of tags $y_{1},y_{2},...,y_{n}$. Figure (b) and (c) shows an example
for each of the two stages. “PolicyInt” is short for “PolicyIntervention”.

### Event classification:

a sequence tagging model is trained to predict BIO tags of event types such
that it identifies the event trigger span as well as the event type. Figure
2(b) shows an example.

### Event argument extraction:

similarly, another sequence tagging model is trained to predict BIO tags of
argument role types, such that it identifies token spans of event arguments as
well as their argument role types, with respect to a trigger has already been
identified in the event classification stage and marked in the input sentence
in “$<t>...</t>$”. Figure 2(c) shows an example.

We run these two models in a pipeline: the event classification model is
applied first to find event triggers and classify their types, then the event
argument extraction model is applied to find location and time arguments for
each event mention.

### Training data curation.

We apply our prior work on rapid customization for event extraction Chan et
al. (2019) to curate a dataset for training the event classification model.
Our developer spent about 13 minutes per event type to find, expand, and
filter potential event triggers in a held-out 10% of the Aylien coronavirus
news corpus. The statistics of the curated data set are shown in Table 1 (we
only show the top-10 most frequent event types for brevity). In total, there
are 11814 mentions in 7159 sentences. We plan to make this dataset available
via github.com.

To train the argument extraction model, we use the related event-argument
annotation from the ACE 2005 dataset Doddington et al. (2004). We focus on
location and time arguments 222For example, Place and Time event argument
roles in ACE can be used to train an argument-role model to extract location
and time arguments, respectively. and ignore other roles. At decoding time,
after extracting the argument mentions for events, we apply the AWAKE Boschee
et al. (2014) entity linking system to resolve each location argument to a
canonical geolocation, and use SERIF Boschee et al. (2005) to resolve each
time argument to a canonical time and then convert it to the month level. This
allows us to perform analyses of events targeting a specific geolocation or
month of interest.

Type | Counts | Type | Counts
---|---|---|---
COVID-19 | 2114 | SocialDistancingMeasures | 412
Virus | 1028 | TravelRestrictions | 403
Pandemic | 596 | Disease | 378
Unemployment | 506 | Death | 355
Shortage | 502 | Lockdown | 321

Table 1: Top-10 frequent events in the training dataset.

##  4 Extracting Temporal and Causal Relations

We develop two approaches for extracting temporal and causal relations: a
pattern-based approach and a neural network model. We take the union of the
outputs from both approaches to maximize recall. The list of causal and
temporal relations extracted by the systems is shown in Table 2. Our
extractors extract relations at the subtype level. However, we decided to
merge the subtypes into types because (a) a user survey shows that users
prefer to have a simplified definition of causality that only includes “event
X causes (positively impacts) event Y” and “X mitigates (reduces/prevents) Y”,
because finer-grained distinctions at sub-type level are difficult and less
useful, and (b) merging the subtypes into types improves accuracy to near or
above 0.8 as shown in Table 4, comparing to 0.7 at the sub-type level due to
extraction approaches struggling to differentiate between the sub-types.

Type | Subtype | Definition
---|---|---
Causes | Cause | Y happens because of X.
| Catalyst | If X, intensity of Y increases.
| Precondition | X must have occured for Y to happen.
Mitigates | Mitigation | If X, intensity of Y decreases.
| Preventative | If X happens, Y can’t happen.
Before | Before/after | X happens before/after Y.

Table 2: Causal and temporal relations between event X and Y.

### Pattern-based relation extraction.

We applied the temporal and causal relation extraction patterns from LearnIt
Min et al. (2020). A pattern is either a lexical pattern, which is a sequence
of words between a pair of events, e.g.,“X leads to Y” 333X and Y refer to the
left and right arguments of a relation., or a proposition pattern, which is
the (nested) predicate-argument structure that connects the pair of events.
For example, “verb:cause[subject=X] [object=Y]” is the proposition counterpart
of the lexical pattern “X causes Y”.

### Neural relation extraction.

We developed a mention pooling Baldini Soares et al. (2019) neural model for
causal and temporal relation extraction. Figure 3 shows the model
architecture. Taking a sentence in which a pair of event mention spans are
marked as input, the model first encodes the sentence with BERT Devlin et al.
(2019) 444The BERT-Base model is used.. For each of the left and right event
mentions, it then uses average pooling over the BERT contextualized vectors of
the words in the span to obtain fixed-dimension vectors $V_{1}$ and $V_{2}$ as
the span representations. It then concatenates the input embeddings $V_{1}$
and $V_{2}$ with the element-wise difference $|V_{1}-V_{2}|$ to generate the
pair representation $V=(V_{1},V_{2},|V_{1}-V_{2}|)$. $V$ is passed into a
linear layer followed by a softmax layer to make the relation prediction. The
model is trained with a blended dataset consisting of the Entities, Events,
Simple and Complex Cause Assertion Annotation datasets 555The catalog IDs of
the LDC datasets are LDC2019E48, LDC2019E61, LDC2019E70, LDC2019E82,
LDC2019E83. released by LDC 666www.ldc.upenn.edu, and 1.5K temporal relation
instances generated by applying the LearnIt temporal relation extraction
patterns to 10,000 sampled Gigaword Parker et al. (2011) articles.

Figure 3: The neural model for causal and temporal relation extraction.

##  5 Constructing a TCAG

We aggregate all extracted events and causal and temporal relations across the
corpus to construct a TCAG. The TCAG is visualized in the interactive
visualization, in which each node is an event type and each edge is a causal
or temporal relation 777is_a relations are also added as dashed edges in the
TCAG..

We use a simple approach to aggregate events: by default, all event mentions
sharing the same type are grouped into a single node named by the type; we
resort to the UI to allow the user to selectively focus on a specific location
and/or time, such that the UI will only show a TCAG involving event mentions
and causal relations between pairs of events for the location and/or time of
interest.

##  6 Measuring Event Popularity through Time

The TCAG only provides a qualitative analysis of the temporal and causal
relations between the COVID-related events. It will be more informative if we
can measure the popularity of events through time to enable trend analysis
(e.g., does lockdown go up or down between January and May, 2020?) and
correlation analysis (e.g., will a stricter lockdown improve or deteriorate
the economy?).

In order to support these analyses, we produce a timeseries of a popularity
score for each event type over time (a.k.a., event timeline). Extending our
prior work Min and Zhao (2019), we define the popularity score for event type
$e$ at time $t$ as:

$Popularity(e)_{t}=\frac{1}{T}\sum_{t^{\prime}\in[t-\frac{T}{2},t+\frac{T}{2}]}\frac{N_{e,t^{\prime}}}{M_{t^{\prime}}}\vspace{-3mm}$

in which $N_{e,t}$ is the frequency of event $e$ at month $t$. We calculate
the moving average centered at each $t$ with a sliding window of $T=3$ months
to reduce noise. $M_{t}$ is $1/500$ of the total number of articles published
in month $t$. The raw event frequency counts can be inflated due to the
increasing level of media activity. Therefore, we divide the raw counts by
$M_{t}$ to normalize the counts so that they are comparable across different
months.

##  7 System Demonstration

### Datasets.

We run Excavator on the following two corpora to produce a TCAG for COVID-19:
the first corpus is 1.2 million articles 888These articles do not overlap with
the held-out set for training data curation. from the Aylien Coronavirus News
Dataset 999https://aylien.com/blog/free-coronavirus-news-dataset, which
contains 1.6 million COVID-related articles published between November 2019
and July 2020 that are from $\sim$440 news sources. We only kept the articles
that are published between January and May 2020, since the corpus contains
fewer articles in other months. The second corpus is the COVID-19 Open
Research Dataset Wang et al. (2020). It contains coronavirus-related research
from PubMed’s PMC corpus, a corpus maintained by the WHO, and bioRxiv and
medRxiv pre-prints. As of 11/08/2020, it contains over 300,000 scholarly
articles.

We combine these two corpora because news and research articles are
complementary: news are rich in real-world events and are up to date, while
analytical articles contain more causal relationships. Therefore, combining
them is likely to lead to a more comprehensive analysis and new insights.

### Overall statistics of extractions.

Excavator extracted 6.2 million event mentions of 59 types. Table 3 shows the
event types that appear more than 50,000 times. We randomly sampled 100 event
mentions, manually reviewed them, and found that the extracted events are 83%
accurate. Excavator extracted 226,176 causal and temporal relations from the
two corpora. A summary of the extracted relations and their precision
101010Estimated by manually reviewing 40 instances per type are shown in Table
4.

Type | Counts | Type | Counts
---|---|---|---
COVID-19 | 2772.3 | Travel | 111.8
Death | 730.0 | FearOrPanic | 94.6
Pandemic | 689.2 | Closures | 92.3
Lockdown | 417.2 | TravelRestrictions | 76.9
Isolation* | 195.4 | Shortage | 68.3
DiseaseSpread | 145.4 | Conflict | 55.5
Testing | 130.7 | Virus | 54.8
Treatment | 112.8 | Symptom | 54.0

Table 3: Frequent events extracted from the corpora (ranked by frequency
reversely; numbers are in thousands). *Isolation refers to
IsolationOrConfinement.

Type | Count | Precision
---|---|---
Causes | 193,694 | 0.78
Mitigates | 30,452 | 0.87
Before | 2,030 | 0.81

Table 4: Causal and temporal relations extracted.

### TCAG Visualization

We developed an interactive visualization of the TCAG. Figure 4 shows a small
part of the TCAG centered on the event Lockdown. Each node represents an event
type in our COVID event taxonomy for which Excavator is able to extract events
and track their popularity scores (§ 6) through time. The three types of
relational edges (Causes, Mitigates and Before) are shown in different colors.
The size of the nodes and the thickness of the edges indicate the relative
frequency of the event types or relations in the log scale, respectively. For
example, Figure 4 shows that Death is mentioned more frequently than Lockdown,
and the causal relation {Lockdown, Causes, EconomicCrisis} appears more
frequently than {Lockdown, Mitigates (“reduces”), AccessToHealthcare}. To
support analysis focusing on a single event, we color the focused event in
blue, events that cause or precede the focused event in orange, and events
that the focused event causes or precedes in green.

### Event popularity timeseries visualization

For each node (event) in the TCAG visualization, we show its event popularity
timeseries visualization on the side. Figure 5 shows 3 screenshots of the
event popularity timeseries (§ 6) visualization between January and May 2020
for Lockdown, EconomicCrisis and COVID-19 respectively.

Figure 4: A screenshot of a partial TCAG centered on Lockdown. Green, pink,
and purple edges shows Cause, Mitigate and Before relations, respectively.
Blue, orange and green nodes show the focused node and nodes with incoming and
outgoing edges (with respect to the focused node), respectively.

##  8 Recommended Use Cases

We describe 3 recommended use cases below. More details are in our
demonstration video.

### Use case 1: causal and temporal analysis.

We can get a panoramic view of the underlying casual and temporal dynamics
between events related to COVID from the overall TCAG. We can start by
analyzing the causal or temporal relations centered at an event of interest.
For example, Figure 4 shows a diverse range of effects and consequences of
Lockdown, such as EconomicCrisis (economic), Shortage (supply-chain),
FearOrPanic (mental), etc. Interestingly, the graph also reveals surprises
such as {Lockdown, Causes, Death}: the UI shows supporting evidence such as
“lockdown exacerbates deaths and chronic health problems associated with
poverty, …”. Furthermore, the TCAG shows that Lockdown mitigates DiseaseSpread
but it also has a negative impact on the Economy, which will inform the
decision makers that they will need to understand the economic trade-offs when
implementing the Lockdown policy.

We can also analyze longer-distance causal pathways consisting of two or more
causal/temporal edges. For example, our demo video shows that COVID-19 causes
or precedes (Before) Lockdown, and that Lockdown causes or precedes
EconomicCrisis. This helps us understand details about how COVID causes
EconomicCrisis.

Figure 5: Event popularity timeseries (timeline) between 01-05/2020 for
Lockdown, EconomicCrisis and COVID-19. X-axis shows months between January and
May 2020. Y-axis shows event popularity scores.

### Use case 2: trend and correlation analysis.

We can inspect the event timeline for a node or an edge to perform a trend
analysis and a correlation analysis, respectively. Figure 5 shows screenshots
of the event popularity timeseries between January and May 2020 for Lockdown,
EconomicCrisis and COVID-19. First, the user can click on a single event to
perform a trend analysis: the popularity of Lockdown goes up continuously,
indicating an upward trend in implementing lockdown policies in more
geographic regions. The user can also click on a edge to perform a correlation
analysis for a pair of events: when the user clicks on the edge {Lockdown,
Causes, EconomicCrisis}, the UI shows a strong correlation between the two
upward curves. For another edge “Lockdown mitigates COVID-19”, the UI shows a
negative correlation near the end: as Lockdown rises, COVID-19 slightly falls
towards the end.

### Use case 3: analyses targeted at geolocations.

The event timeline visualization also allows the user to see the timeline for
geolocations such as each U.S. state individually, instead of the aggregate
for the entire U.S.. Figure 6 is a screenshot showing the 10 timelines for
Lockdown for the top-10 most frequently mentioned U.S. states. The screenshot
shows that the curves for California and New York go much higher than other
states. This roughly matches the stricter lockdown policies implemented in the
two states during this time period, comparing to other states. Such targeted
analysis is made possible because our events have location and time arguments.
We can also make the TCAG only show events and relations for a specific state,
if a user selects a state of interest in the UI.

Figure 6: Event popularity timeseries in 01-05/2020 for Lockdown, for top-10
frequently mentioned US states.

##  9 Related Work

### Extracting events.

Event extraction has been studied using feature-based approaches Huang and
Riloff (2012); Ji and Grishman (2008), or neural networks Chen et al. (2015);
Nguyen et al. (2016a); Wadden et al. (2019); Liu et al. (2020). GDELT Leetaru
and Schrodt (2013) creates an event database for the conflict and mediation
domain. It has very few event types related to COVID-19. To adapt event
extraction to new domains, Chen et al. Chan et al. (2019) developed a user-in-
the-loop rapid event customization system. Nguyen et al. Nguyen et al. (2016b)
proposed a neural model for event type extension given seed examples. Peng et
al. Peng et al. (2016) developed a minimally supervised approach using
triggers gathered from ACE annotation guideline.

### Extracting causal and temporal relations.

There are a lot of work in temporal D’Souza and Ng (2013); Chambers et al.
(2014); Ning et al. (2018b); Meng and Rumshisky (2018); Han et al. (2019);
Vashishtha et al. (2020); Wright-Bettner et al. (2020) and causal Bethard and
Martin (2008); Do et al. (2011); Riaz and Girju (2013); Roemmele and Gordon
(2018); Hashimoto (2019) relation extraction. Mirza and Tonelli Mirza and
Tonelli (2016) and Ning et al. Ning et al. (2018a) extract both in a single
framework.

### Constructing Causal Graphs from Text.

Eidos Sharp et al. (2019) uses a rule-based approach to extract causal
relations to build a causal analysis graph, that has limited coverage on
events related to COVID-19. LearnIt Min et al. (2020) enables rapid
customization of causal relation extractors. LearnIt does not focus on causal
relations involving COVID-related events. This work also differs from these
two in that we extract event arguments and temporal relations, and track event
popularity.

##  10 Conclusion

We present the Excavator system, a web-based TCAG visualization, and a video
demonstration.

## Acknowledgments

This research is based upon work supported in part by the Office of the
Director of National Intelligence (ODNI), Intelligence Advanced Research
Projects Activity (IARPA), via Contract No.: 2021-20102700002. The views and
conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies, either
expressed or implied, of ODNI, IARPA, or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for governmental
purposes not withstanding any copyright annotation therein.

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# Local coefficients and Gelfand-Graev representations for non-split covers on
SL$(2)$

Yeongseong Jo  Department of Mathematics Education, Ewha Womans University,
Seoul 03760, Republic of Korea
jo.59<EMAIL_ADDRESS>
###### Abstract.

A purely local approach has been developed by Krishnamurthy and Kutzko to
compute Langlands-Shahidi local coefficient for ${\rm SL}(2)$ via types and
covers à la Bushnell-Kutzko. In this paper, we extend their method to the non-
split case and complete their project. We also study the algebraic structure
of Gelfand-Graev representations, which generalizes the results of Chan-Savin
and Mishra-Pattanayak to ${\rm SL}(2)$ over non-archimidean local fields
without any restriction on the characteristic.

###### Key words and phrases:

Bushnell–Kutzko’s types and covers, Gelfand-Graev representations, Langlands-
Shahidi local coefficients

###### 2020 Mathematics Subject Classification:

Primary 11F70; Secondary 22E50

##  1\. Introduction

Throughout this paper $F$ will denote a non-arhimedean local field and we let
$G={\rm SL}_{2}(F)$. The purpose of this paper is to illustrate an algebraic
method of achieving an explicit formula for the Langlands-Shahidi local
coefficients and computing the isotypic subspace of Gelfand-Graev
representations of $G$ by means of types and non-split covers à la Bushnell-
Kutzko. At a first glance, the results seem to be rather tangential to each
other, but two topics are commonly concerned with what is known as Whittaker
models, which play an important role in the automorphic forms and
representation theory of linear reductive groups due to the nature of its
relation to $L$-functions, notably in the Langlands-Shahidi method and Rankin-
Selberg integrals as well. Specifically, local coefficient by definition is a
constant of proportionality arising from the uniqueness of Whittaker models
[Rod72], whereas the Gelfand-Graev representation is given by the dual of the
Whittaker space. In the meantime, the theory of types and covers provides a
systematic way of analyzing the structure of smooth representations of linear
reductive $p$-adic groups via the representation theory of compact open
subgroups (Cf. Section 2.1).

This paper is compromised of two parts. Our first main result of this paper is
a complete expression of local coefficients via types and covers, restricting
to the case of $F^{\times}$ as a Levi subgroup of $G$ but where the character
of $F^{\times}$ is quadratic and ramified (“non-split”), which is more
difficult than other situations. We resolve the only remaining case of non-
split cover, which is addressed in the work by Krishnamurthy and Kutzko [KK,
§3.4 Remark 2].

######  Theorem A (Theorem 3.1).

Let $\psi$ be a fixed additive character of level $0$. Let $\widetilde{\eta}$
be a non-trivial ramified quadratic character of $F^{\times}$ of level
$n_{\eta}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. Then we have

$C_{\psi}(s,\widetilde{\eta},{\rm
w}_{0})=\widetilde{\eta}(-\varpi_{F}^{n_{\eta}})\tau(\eta,\psi,\varpi_{F}^{-n_{\eta}})q_{F}^{-n_{\eta}(s-1)}.$

The local coefficient has a great influence on the development of contemporary
number theory, and is closed related to the theory of local factors à la the
Langlands-Shahidi method [Sha78, Sha81, Sha84]. Shahidi subsequently defined
the so-called Langlands-Shahidi $\gamma$-factors inductively so that the local
coefficient factorizes as a product of such $\gamma$-factors. Another briefly
aforementioned method for constructing local exterior square factors via the
theory of integral representations is developed by Bump and Friedberg in late
1980’s [BF89]. By definition, it is a proportionality factor between an
integral and its dual integral related to each other through the theory of
either Fourier transforms or intertwining operators [Mat15]. On the
perspective of the local Langlands correspondence, the resulting local factors
ought to be the same, probably up to some normalizations of certain Haar
measures. In practice, this flavor of comparisons is surely non-trivial to
answer, and is known for only a handful amount of cases; for example, we refer
the reader to [Kap15, Sha84] for Rankin-Selberg local factors, and [AKMSS21,
BP21] for Asai local factors. In this regard, Bump and Friedberg [BF89,
Conjecture 4] predicted that the equality should be valid in the context of
local exterior square factors attached to irreducible admissible
representation of ${\rm GL}_{n}(F)$. Our A confirms that the equality between
local factors [BF89, Conjecture 4] obtains from the Langlands-Shahidi method
and those obtained via Bump-Friedberg integrals holds unconditionally at least
for ${\rm GL}_{1}(F)\cong F^{\times}$.

Our strategy to tackle the problem of computing local coefficient on
$F^{\times}$ inside $G$ is going back to at least the pioneering work of
Casselman [Cas80], where local coefficients in the context of unramified
principal series representations are explicitly computed. The computation
there relies on finding the effect of the intertwining operator on the
subspace of vectors fixed by the Iwahori subgroup. The role played by the
trivial representation of the Iwahori subgroup in Casselman’s trick can be
regarded as an extreme incident of the theory of types and covers. Previously,
types and covers are adapted by Krishnamurthy and Kutzko [KK] for split covers
of $G$, or in other words, the case where the character is not an unramified
twist of a quadratic character. Afterwords, the author and Krishnamurthy
[JK21] dealt with local coefficients associated to covers of a homogeneous
pair of irreducible supercuspidal representations $(\pi_{1},\pi_{2})$ of the
Levi subgroup ${\rm GL}_{n}(F)\times{\rm GL}_{n}(F)$ embedded in ${\rm
GL}_{2n}(F)$.

Our next aim of this paper is to express the principal series block of the
Gelfand-Graev representation as a cyclic module over the Iwahori-Hecke
algebra.

######  Theorem B (Theorem 3.6).

Let $\psi$ be a fixed additive character of level $0$. Let $\widetilde{\eta}$
be a non-trivial ramified quadratic character of $F^{\times}$ of level
$n_{\eta}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. As
$\mathcal{H}(G,\lambda_{\eta})-$modules, we obtain isomorphisms

$({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}\cong\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}.$

The structure of Gelfand-Graev representation was originally treated by Chan
and Savin for unramified principal series blocks of split reductive groups
[CS18], and was further refined by Mishra and Pattanaya for principal series
blocks for connected reductive groups over $F$ whose residue characteristic is
large enough [MP21]. Shortly after, Gao, Gurevich, and Karasiewicz extended
Chan and Savin’s result for linear groups [CS18] to the Iwahori fixed vectors
in the Gelfan-Graev representation of covering groups as a module over the
Iwahori-Hecke algebra [GGK22].

A part of reasons why Mishra and Pattanaya [MP21] primarily considered limited
characteristics of $F$ is that they made a reduction to depth-zero cases and
then to finite group cases. In doing so, they was able to determine the
generator of isotypical space for the cover of Gelfan-Graev representations
over finite fields, which in turn admits the $1$-dimensional sign
representation. On the other hand, our proof takes elements from Chan and
Savin’s [CS18] unconditional and direct argument. Similarly to Chan and Savin,
we laboriously compute the outcome of Hecke operators on test functions from
which we find the generator possessing the $1$-dimensional sign representation
(Cf. Proposition 3.5).

In principle, all the computations boil down to understanding the behavior of
the intertwining or Hecke operators on certain test functions (Cf. Proposition
2.4,Proposition 3.5). We expect that those test functions that naturally occur
in the theory of types and cover open a new chapter for other situation, and
consider the present paper a groundwork for that direction. Benefiting from
[JK21], it is also our belief that the structure of the isotypic component of
the space of compactly supported Whittaker functions as modules over Iwahori-
Hecke algebra, is responsible for computing exterior square local coefficients
in more general settings, remarkably, Siegel Levi subgroups isomorphic to
${\rm GL}_{n}(F)$ lying inside ${\rm Sp}_{2n}(F)$. It will be therefore very
interesting to see if our robust arguement can be carried out to parabolically
induced representations of symplectic groups in $2n$-variables. The author
plans to investigate them in a future study.

##  2\. Non-split Covers and Intertwining Operators

###  2.1. Types and Covers

Let $F$ be a non-archimedean local field with its residual finite field
$\mathbb{F}_{q}$ and denote by $p$ the characteristic of $\mathbb{F}_{q}$. The
base field $F$ is a finite extension of $\mathbb{Q}_{p}$ or
$\mathbb{F}_{p}((t))$, called a $p$-adic field in characteristic 0, or a local
function field in characteristic $p>0$. Let $\mathfrak{o}_{F}$ be its ring of
integers, $\mathfrak{p}_{F}$ its maximal ideal. We fix a generator
$\varpi_{F}$ of $\mathfrak{p}_{F}$ and normalize the absolute value $|\cdot|$
of $F$ so that $|\varpi_{F}|=q_{F}^{-1}=q^{-1}$. When there is no possibility
of confusion, we sometimes drop the subscripts, while working over a fixed
$F$. We let $G={\rm SL}_{2}(F)$. Let $B$ be the subgroup of $F$-points of the
Borel subgroup of upper triangular matrices. Then $B=TU$, where

$T=\left\\{\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\,\middle|\,a\in
F^{\times}\right\\};\quad U=\left\\{u(x):=\begin{pmatrix}1&x\\\
0&1\end{pmatrix}\,\middle|\,x\in F\right\\}.$

To be specific, we identify $T$ with $F^{\times}$ when no confusion can arise.
Let $\overline{B}=T\overline{U}$ denote the opposite Borel subgroup of lower
triangular matrices, where

$\overline{U}=\left\\{\overline{u}(x):=\begin{pmatrix}1&0\\\
x&1\end{pmatrix}\,\middle|\,x\in F\right\\}.$

Let $K=G(\mathfrak{o}_{F})$ be a maximal compact subgroup and let ${\rm\bf
W}=N_{G}(T)/T=\\{{\rm I}_{2},{\rm w}_{0}\\}$, where

${\rm I}_{2}=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\quad\text{and}\quad{\rm
w}_{0}=\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}.$

We let $\ell$ denotes the length function on ${\rm\bf W}$ defined by
$\ell({\rm I}_{2})=0$ and $\ell({\rm w}_{2})=1$. The character $\nu_{s}$ is
given by the formula

$\nu_{s}\left(\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\right)=|a|^{s}.$

We may view it as a character of $B$ by extending $\nu_{s}$ trivially to $U$.
Let $\delta$ denote the modulus character of $B$: explicitly,

$\delta\left(\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}\begin{pmatrix}a&0\\\
0&a^{-1}\end{pmatrix}\right)=|a|^{2}.$

For any topological group $H$, we write $\widehat{H}$ to denote the group of
continuous homomorphism from $H$ to $\mathbb{C}^{\times}$. Particularly, we
are interested in the character group $\widehat{\mathfrak{o}}_{F}^{\times}$.
Let $T(\mathfrak{o}_{F}):=T\cap K$. Specifically,
$T(\mathfrak{o}_{F})\cong\mathfrak{o}_{F}^{\times}$. Then
$T/T(\mathfrak{o}_{F})$ is a free abelian group of rank $1$. Let $X(T)$ denote
the group of continuous homomorphisms of $T$ into $\mathbb{C}^{\times}$ which
are trivial on $T(\mathfrak{o}_{F})$ – called the group of unramified
characters of $T$. Moreover $X(T)$ is equipped with the structure of a complex
variety whose ring of regular functions is
$\mathbb{C}[T/T(\mathfrak{o}_{F})]\cong\mathbb{C}[t,t^{-1}]$.

By a a cuspidal pair in $G$, we mean a pair $(M,\sigma)$ in $G$, where $M$ is
either $T$ or $G$, and $\sigma$ is a supercuspidal representation of $M$. Two
such pairs $(M_{i},\sigma_{i})$, $i=1,2$, are said to be inertially equivalent
if there exist $g\in G$ and an unramified character $\mu$ of $G$ such that
$M_{2}=M_{1}^{g}=g^{-1}M_{1}g$, and $\sigma_{2}$ is equivalent to the
representation $\sigma_{1}^{g}\otimes\mu:x\mapsto\sigma_{1}(gxg^{-1})\mu(x)$
of $M_{2}$. We denote by $[(M,\sigma)]_{G}$ the $G$-inertial equivalence class
of a cuspidal pair $(M,\sigma)$ in $G$. Let $\mathfrak{B}(G)$ denote the set
of inertial equivalence classes of cuspidal pairs in $G$. In particular, we
let $\widetilde{\eta}$ be a character of $F^{\times}$ and we put
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. Let
$\widetilde{\eta}_{i}$, $i=1,2$, be characters of $T$. Then
$\widetilde{\eta}_{2}$ is $G$-inertially equivalent to $\widetilde{\eta}_{1}$
if and only if there exists $s\in\mathbb{C}$ such that
$\widetilde{\eta}_{2}=\widetilde{\eta}_{1}^{\pm 1}\nu_{s}$.

It is a fundamental result of Bernstein (cf. [Kut04, §1.4]) that
$\mathfrak{R}(G)$ of smooth complex representations of $G$ decomposes into a
product of full subcategories

$\mathfrak{R}(G)\cong\prod_{\mathfrak{s}\in\mathfrak{B}(G)}\mathfrak{R}^{\mathfrak{s}}(G).$

We divide the equivalence classes into so-called

  1. $(\mathrm{i})$

supercuspidal blocks;

  2. $(\mathrm{ii})$

principal series blocks.

$(\mathrm{i})$ For each irreducible supercuspidal representation $\sigma$ of
$G$, we write $\mathfrak{s}(\sigma)$ for the equivalence class of $\sigma$ in
$\mathfrak{R}(G)$. We let $\mathfrak{R}^{\mathfrak{s}(\sigma)}(G)$ be the full
subcategory of $\mathfrak{R}(G)$ whose objects are isomorphic to sums of
copies of $\sigma$. $(\mathrm{ii})$ Let $\chi$ be a character of $B$, and
$\iota^{G}_{B}$ the functor of normalized parabolic induction. We let
$\mathcal{F}_{B}(\chi)$ denote the space of $\iota^{G}_{B}(\chi)$. We denote
$G$-intertially equivalent class by $\mathfrak{s}_{\eta}$. We let
$\mathfrak{R}^{\mathfrak{s}_{\eta}}(G)$ be the full subcategory of
$\mathfrak{R}(G)$ whose irreducible objects are exactly those that occur as a
subquotient of some $\mathcal{F}_{Q}(\widetilde{\eta}\otimes\nu_{s})$, where
$Q$ is either $B$ or $\overline{B}$ [KK, §3.2, P.228].

With $\widetilde{\eta}_{i}$, $i=1,2$, as above, $\widetilde{\eta}_{1}$ is
$T$-inertially equivalent to $\widetilde{\eta}_{2}$ if and only if there exist
$s\in\mathbb{C}$ such that $\widetilde{\eta}_{2}=\widetilde{\eta}_{1}\nu_{s}$.
Let $\mathfrak{t}_{\eta}$ be the corresponding $T$-inertially equivalent
class. Let $\mathfrak{R}^{\mathfrak{t}_{\eta}}(T)$ be the full subcategory of
$\mathfrak{R}(T)$ whose object $(\pi,V)$ has the property that
$\pi(x)v=\eta(x)v$ for all $x\in T(\mathfrak{o}_{F})$ and $v\in V$. Then the
category $\mathfrak{R}(T)$ similarly decomposes as a product of its
subcategories $\mathfrak{R}^{\mathfrak{t}_{\eta}}(T)$:

$\mathfrak{R}(T)\cong\prod_{\eta\in\widehat{\mathfrak{o}}_{F}^{\times}}\mathfrak{R}^{\mathfrak{t}_{\eta}}(T).$

Let $J$ be a compact open subgroup of $G$, let $(\lambda,W)$ be a smooth
irreducible representation of $J$, and write $(\check{\lambda},\check{W})$ for
the contragredient representation. Then $\mathcal{H}(G,\lambda)$ is the space
of compactly supported function
$f:G\rightarrow\mathrm{End}_{\mathbb{C}}(\check{W})$ that satisfy

$f(hxk)=\check{\lambda}(h)f(x)\check{\lambda}(k),\quad x\in G,\;\;h,k\in J.$

It is a unital (associative) algebra with respect to the standard convolution
operation

$f_{1}\star f_{2}(y)=\int_{G}f_{1}(x)f_{2}(x^{-1}y)\,dx,\quad
f_{1},f_{2}\in\mathcal{H}(G,\lambda),\;\;y\in G,$

where we normalized the Haar measure on $G$ such that ${\rm vol}(J)=1$. An
element $x\in G$ is called intertwine if there is a non-zero $J\cap
x^{-1}Jx$-homomorphism between $\lambda$ and the conjugate representation
$\lambda^{x}$. It is equivalent to saying that the double coset $JxJ$ supports
a non-zero function in $\mathcal{H}(G,\lambda)$ [Kut04, §2.2]. Specifically,
when $\lambda$ is $1$-dimensional, $\lambda(xkx^{-1})=\lambda(k)$ for any
$k\in J\cap x^{-1}Jx$. A pair $(J,\lambda)$ is said to be a type for
$\mathfrak{s}\in\mathfrak{B}(G)$, or simply a $\mathfrak{s}$-type if for every
irreducible object $(\pi,V)\in\mathfrak{R}(G)$, we have
$(\pi,V)\in\mathfrak{R}^{\mathfrak{s}}(G)$ if and only if $\pi$ contains
$\lambda$, that is to say, the space of $\lambda$-covariants
$V_{\lambda}:=\mathrm{Hom}_{J}(W,V)$ is non-trivial. For
$a\in\mathrm{End}_{\mathbb{C}}(\check{W})$, $a^{\vee}$ denote the transpose of
$a$ with respect to the canonical pairing between $W$ and $\check{W}$. We
similarly define the Hecke algebra $\mathcal{H}(G,\check{\lambda})$. There is
a canonical anti-isomorphism $f\mapsto\check{f}$ from
$\mathcal{H}(G,\lambda)\rightarrow\mathcal{H}(G,\check{\lambda})$ given by
$\check{f}(g)=(f(g^{-1}))^{\vee}$. There is a natural left
$\mathcal{H}(G,\lambda)-{\rm Mod}$ structure (also denoted as $\pi$) given by

$(f\star\phi)(w):=(\pi(f)\phi)(w)=\int_{G}\pi(g)\phi(f(g)^{\vee}w)\,dg$

for $\phi\in V_{\lambda}$, $w\in W$, and $f\in\mathcal{H}(G,\lambda)$. For
$\mathfrak{s}$-type $(J,\lambda)$, the map $V\mapsto V_{\lambda}$ induces an
equivalence of categories
$\mathfrak{R}^{\mathfrak{s}}(G)\cong\mathcal{H}(G,\lambda)-{\rm Mod}$.

We fix a non-trivial ramified quadratic character $\widetilde{\eta}$ of
$F^{\times}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$ so that
$\widetilde{\eta}^{2}=1$. We set $\epsilon=\widetilde{\eta}(-1)$. We let
$n_{\eta}$ to be the smallest number $n$ so that
$1+\mathfrak{p}_{F}^{n}\subset\ker\eta$. The compact open subgroup is given by

$J_{\eta}=\left\\{\begin{pmatrix}c_{11}&c_{12}\\\
c_{21}&c_{22}\end{pmatrix}\in
G\,\middle|\,c_{11},c_{22}\in\mathfrak{o}_{F}^{\times},c_{12}\in\mathfrak{o},c_{21}\in\mathfrak{p}_{F}^{n_{\eta}}\right\\},$

and $\lambda_{\eta}$ is a function on $J_{\eta}$ given by

$\lambda_{\eta}\left(\begin{pmatrix}c_{11}&c_{12}\\\
c_{21}&c_{22}\end{pmatrix}\right)=\eta(c_{11}).$

It follows from [Kut04] that the pair $(J_{\eta},\lambda_{\eta})$ is a
$G$-cover for $(T(\mathfrak{o}_{F}),\eta)$. We recall certain crucial
properties of a $G$-cover $(J_{\eta},\lambda_{\eta})$:

  1. $(\mathrm{a})$

(Iwahori Factorization)
$J_{\eta}=(J_{\eta}\cap\overline{U})T(\mathfrak{o}_{F})(J_{\eta}\cap U)$.

  2. $(\mathrm{b})$

The representation $\lambda_{\eta}$ is trivial on $J_{\eta}\cap\overline{U}$
and $J_{\eta}\cap U$, while
$\lambda_{\eta}\restriction_{T(\mathfrak{o}_{F})}=\eta$. The pair
$(J_{\eta},\lambda_{\eta})$ is a type for $\mathfrak{s}_{\eta}$.

  3. $(\mathrm{c})$

There is a support preserving injective algebra map
$t_{Q}:\mathcal{H}(T,\eta)\rightarrow\mathcal{H}(G,\lambda_{\eta})$ that
realizes the parabolic induction functor $\iota_{Q}^{G}$ at the level of Hecke
algebras. It means that the following diagram commutes:

$\textstyle{\mathfrak{R}^{\mathfrak{t}_{\eta}}(T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota_{Q}^{G}}$$\scriptstyle{\cong}$$\textstyle{{\mathcal{H}}(T,\eta){\rm-
Mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(t_{Q})_{\ast}}$$\textstyle{\mathfrak{R}^{\mathfrak{s}_{\eta}}(G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{{\mathcal{H}}(G,\lambda_{\eta}){\rm-
Mod}}$

where $(t_{Q})_{\ast}$ is a right adjoint of the restriction functor
$t_{Q}^{\ast}:{\mathcal{H}}(G,\lambda_{\eta}){\rm-
Mod}\rightarrow{\mathcal{H}}(T,\eta){\rm-Mod}$.

We oftentimes identify ${\mathcal{H}}(T,\eta)$ as a sub-algebra of
$\mathcal{H}(G,\lambda_{\eta})$ using the embedding $t_{Q}$.

###  2.2. Intertwining Operators

We write $\mathcal{F}(s,\chi)$ to denote the induced space
$\mathcal{F}_{B}(\chi\otimes\nu_{s})$. We define $\mathbb{C}_{\lambda_{\eta}}$
to be the space of complex numbers for which $J_{\eta}$ acts on
$\mathbb{C}_{\lambda_{\eta}}$ by $j\cdot w=\lambda_{\eta}(j)w$. Throughout the
rest of the paper, we put $W=\mathbb{C}_{\lambda_{\eta}}$. As $\lambda_{\eta}$
is one-dimensional, it is straightforward from [BK98, (2.13)] that there is a
natural isomorphism $V_{\lambda_{\eta}}\cong
V_{\lambda_{\eta}}\otimes_{\mathbb{C}}W\cong V^{\lambda_{\eta}}$ given by
$\phi\otimes w\mapsto\phi(w)$. For $w\in W$, the functions $f_{{\rm
I}_{2},\widetilde{\eta}}$ and $f_{{\rm w}_{0},\widetilde{\eta}}$ in
$\mathcal{F}(s,\widetilde{\eta})^{\lambda_{\eta}}$ are given by

$f_{{\rm
I}_{2},\widetilde{\eta}}(w)(g)=\begin{cases}\widetilde{\eta}\nu_{s}\delta^{1/2}(b)\lambda_{\eta}(j)&\text{if}\quad
g=bj\in BJ_{\eta}\\\ 0&\text{otherwise,}\end{cases}$

and

$f_{{\rm
w}_{0},\widetilde{\eta}}(w)(g)=\begin{cases}\widetilde{\eta}\nu_{s}\delta^{1/2}(b)\lambda_{\eta}(j)&\text{if}\quad
g=b{\rm w}_{0}j\in B{\rm w}_{0}J_{\eta}\\\ 0&\text{otherwise.}\end{cases}$

For ${\rm w}\in{\rm\bf W}$, let $T_{\rm w}\in\mathcal{H}(G,\lambda_{\eta})$
denote the function supported on the double coset $J_{\eta}{\rm w}J_{\eta}$
and given by the formula $T_{\rm w}(h{\rm
w}k)=\check{\lambda}(h)\check{\lambda}(k)$, $h,k\in J_{\eta}$. Let
$\mathcal{H}(K,\lambda_{\eta})$ be the sub-algebra in
$\mathcal{H}(G,\lambda_{\eta})$ spanned by the function $T_{\rm w}$ for ${\rm
w}\in{\rm\bf W}$. Just as in [Kut04, §3.3], we let $T^{\ast}_{\rm w}$ be the
normalized Hecke operator so that

$T^{\ast}_{\rm w}=\epsilon^{\ell({\rm w})/2}{\rm vol}(J_{\eta}{\rm
w}J_{\eta})^{-1/2}T_{\rm w},\quad{\rm w}\in{\rm\bf W}.$

Since any element ${\rm w}$ in ${\rm\bf W}$ certainly intertwines the
representation $\lambda_{\eta}$, [Kut04, Lemma 2.3] (Cf. [BK98, (11.6)])
implies that $J_{\eta}{\rm\bf W}J_{\eta}=J_{\eta}KJ_{\eta}=K$ (cf. [MP21,
(5.1)]). The sub-algebra $\mathcal{H}(K,\lambda_{\eta})$ inherits a left
action on $\mathcal{H}(K,\check{\lambda}_{\eta})$ given by $C\bullet
D:=D\star\check{C}$ for $C\in\mathcal{H}(K,\lambda_{\eta})$ and
$D\in\mathcal{H}(K,\check{\lambda}_{\eta})$. We summarize the properties of
$\mathcal{H}(K,\lambda_{\eta})$ from [BK98, §11.5 and §11.6].

######  Lemma 2.1.

The set $\\{T_{{\rm I}_{2}},T_{{\rm w}_{0}}\\}$ is a $\mathbb{C}$-basis of
$\mathcal{H}(K,\lambda_{\eta})$. Specifically,
$\dim_{\mathbb{C}}(\mathcal{H}(K,\lambda_{\eta}))=2$.

We observe that

$\\{g\in G\,|\,\text{There is
$f\in\mathcal{F}(s,\widetilde{\eta})^{\lambda_{\eta}}$ with $f(g)\neq 0$}\\}$

is a subset of $G=BGJ_{\eta}=BKJ_{\eta}$, because $G=BK$. Hence, any
$f\in\mathcal{F}(s,\widetilde{\eta})^{\lambda_{\eta}}$ is determined by its
restriction to $K$. In this regard, we define

(2.1) $\iota:{\rm
Hom}_{J_{\eta}}(W,\mathcal{F}(s,\widetilde{\eta}))\rightarrow\mathcal{H}(K,\check{\lambda}_{\eta})$

by $\iota(\phi)(k)(w)=\phi(w)(k)$ for $\phi\in{\rm
Hom}_{J_{\eta}}(W,\mathcal{F}(s,\widetilde{\eta})),k\in J_{\eta}KJ_{\eta}$,
and $w\in W$.

######  Lemma 2.2.

$($Cf. [Kim16, Lemma 3.2.6]$)$ The map $\iota$ is well defined and is a
homomorphism of left $\mathcal{H}(K,\lambda_{\eta})-$modules.

###### Proof.

Let $j,j^{\prime}\in J_{\eta}$ and ${\rm w}\in{\rm\bf W}$. Thanks to
$(\mathrm{a})$, we decompose $j=j_{U}j_{T}j_{\overline{U}}$ with $j_{U}\in
J_{\eta}\cap U$, $j_{T}\in J_{\eta}\cap T$, and $j_{\overline{U}}\in
J_{\eta}\cap\overline{U}$. We see that elements ${\rm
w}^{-1}j_{\overline{U}}{\rm w}j^{\prime}$ intertwine the character
$\lambda_{\eta}$ and $\pi$ acts on $\mathcal{F}(s,\widetilde{\eta})$ by the
right translation. With these in hand, we have

$\iota(\phi)(j{\rm
w}j^{\prime})(w)=\lambda_{\eta}(j_{T})\phi(w)(j_{\overline{U}}{\rm
w}j^{\prime})=\lambda_{\eta}(j_{T})(\pi({\rm w}^{-1}j_{\overline{U}}{\rm
w}j^{\prime})\phi)(w)({\rm w})\\\
=\lambda_{\eta}(j_{T})\phi(\lambda_{\eta}({\rm w}^{-1}j_{\overline{U}}{\rm
w}j^{\prime})w)({\rm w}).$

Appealing to $(\mathrm{b})$, $\lambda_{\eta}({\rm w}^{-1}j_{\overline{U}}{\rm
w})$ is trivial, and it becomes

$\iota(\phi)(w)(j{\rm
w}j^{\prime})=\lambda_{\eta}(j_{T})\phi(\lambda_{\eta}(j^{\prime})w)({\rm
w})=\lambda_{\eta}(j_{T})\iota(\phi)({\rm
w})\lambda_{\eta}(j^{\prime})(w)=\lambda_{\eta}(j)\iota(\phi)({\rm
w})\lambda_{\eta}(j^{\prime})(w)$

having used the fact that $\lambda_{\eta}(j_{T})=\lambda_{\eta}(j)$. We
confirm that $\iota$ is well defined, that is to say, $\iota(\phi)$ belongs to
$\mathcal{H}(K,\check{\lambda}_{\eta})$.

It remains to show that $\iota$ is a homomorphism of left
$\mathcal{H}(K,\lambda_{\eta})-$modules. Upon making the change of variables
$g\mapsto k^{-1}g$, for $f\in\mathcal{H}(K,\lambda_{\eta})$,
$\iota(\pi(f)\phi)(k)(w)$ equals to

$(\pi(f)\phi)(w)(k)=\int_{G}\pi(g)\phi(f(g)^{\vee}w)(k)\,dg=\int_{G}\phi(f(g)^{\vee}w)(kg)\,dg=\int_{G}\phi(\check{f}(g^{-1})w)(kg)\,dg\\\
=\int_{G}\iota(\phi)(kg)(\check{f}(g^{-1})w)\,dg=\int_{G}\iota(\phi)(g)(\check{f}(g^{-1}k)w)\,dg=(\iota(\phi)\star\check{f})(k)(w)=(f\bullet\iota(\phi))(k)(w)$

from which the desired conclusion follows. It is noteworthy that the support
of all these integrals above is actually in $K$. ∎

The following proposition can be thought of as the ${\rm SL}_{2}(F)$-analogue
of [JK21, Proposition 5.9] and [Kim16, Lemma 3.2.9].

######  Proposition 2.3.

As left $\mathcal{H}(K,\lambda_{\eta})-$modules,
$\mathcal{H}(K,\lambda_{\eta})$ is isomorphic to
$\mathcal{F}(s,\widetilde{\eta})_{\lambda_{\eta}}$. Consequently, the
$\lambda_{\eta}$-isotypic subspace
$\mathcal{F}(s,\widetilde{\eta})^{\lambda_{\eta}}$ is two dimensional with a
$\mathbb{C}$-basis $\\{f_{{\rm I}_{2},\widetilde{\eta}},f_{{\rm
w}_{0},\widetilde{\eta}}\\}$.

###### Proof.

To verify that (2.1) is an isomorphism, it is sufficient to check that it
takes a basis of $\mathcal{H}(K,\lambda_{\eta})$ to a basis of
$\mathcal{F}(s,\widetilde{\eta})_{\lambda_{\eta}}$. The algebra
$\mathcal{H}(K,\check{\lambda}_{\eta})$ is isomorphic to the left regular
representation of $\mathcal{H}(K,\lambda_{\eta})$ via an isomorphism
$\iota^{\prime}:\mathcal{H}(K,\check{\lambda}_{\eta})\rightarrow\mathcal{H}(K,\lambda_{\eta})$
given by $\iota^{\prime}(D)=\check{D}$ for
$D\in\mathcal{H}(K,\check{\lambda}_{\eta})$. This is then immediate from Lemma
2.1, since $\iota(f_{{\rm w},\widetilde{\eta}})=\check{T}_{\rm w}$ for $\rm
w\in{\rm\bf W}$. ∎

Since $U,\overline{U}\cong F$, we may identify the measure on $U$ and
$\overline{U}$ with the additive measure $dx$ on $F$, and we take
$d^{\times}x$ to be $dx/|x|$ on $F^{\times}$. The Haar measure $dx$ is
normalized so that $J_{\eta}\cap U\cong\mathfrak{o}_{F}$ has volume one. We
define a standard $G$-intertwining operator
$A(s,\chi):\mathcal{F}(s,\chi)\rightarrow\mathcal{F}(-s,\chi^{-1})$ by

(2.2) $A(s,\chi,{\rm w}_{0})(f)(g)=\int_{U}f({\rm w}_{0}ug)\,du,$

for all $f\in\mathcal{F}(s,\chi)$. The integral converges absolutely for ${\rm
Re}(s)\gg 0$ and defines a rational function on a non-empty Zariski open
subset of the complex torus $X(T)$. The character $\chi$ is said to be regular
if $\chi\neq\chi^{-1}$. If $\chi\otimes\nu_{s}$ is regular,
$\iota_{B}^{G}(\chi\otimes\nu_{s})$ and
$\iota_{B}^{G}(\chi^{-1}\otimes\nu_{-s})$ are all irreducible so that every
$G$-morphism from $\mathcal{F}(s,\chi)$ to $\mathcal{F}(-s,\chi^{-1})$ is a
scalar multiple of $A(s,\chi,{\rm w}_{0})$. Our calculation is inspired from
the idea in [JK21].

######  Proposition 2.4.

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$ and assume that $\nu_{s}$ is regular. Then we have

$A(s,\widetilde{\eta},{\rm w}_{0})(f_{{\rm
I}_{2},\widetilde{\eta}})=\epsilon\cdot{\rm vol}(\overline{U}\cap
J_{\eta})f_{{\rm w}_{0},\widetilde{\eta}^{-1}}=\epsilon q^{-n_{\eta}}f_{{\rm
w}_{0},\widetilde{\eta}^{-1}}$

and

$A(s,\widetilde{\eta},{\rm w}^{-1}_{0})(f_{{\rm
w}_{0},\widetilde{\eta}})=\epsilon\cdot{\rm vol}(U\cap J_{\eta})f_{{\rm
I}_{2},\widetilde{\eta}^{-1}}=\epsilon f_{{\rm I}_{2},\widetilde{\eta}^{-1}}.$

###### Proof.

Since $\mathfrak{R}^{\mathfrak{s}_{\eta}}(G)$ is equivalent to
$\mathcal{H}(G,\lambda_{\eta})-$Mod, we have

${\rm
Hom}_{G}(\mathcal{F}(s,\widetilde{\eta}),\mathcal{F}(-s,\widetilde{\eta}^{-1}))\cong{\rm
Hom}_{\mathcal{H}(G,\lambda_{\eta})}(\mathcal{F}(s,\widetilde{\eta})_{\lambda_{\eta}},\mathcal{F}(-s,\widetilde{\eta}^{-1})_{\lambda_{\eta}}).$

Therefore, $A(s,\widetilde{\eta},{\rm w}_{0})$ induces an intertwining map
$\mathcal{F}(s,\widetilde{\eta})_{\lambda_{\eta}}\rightarrow\mathcal{F}(-s,\widetilde{\eta}^{-1})_{\lambda_{\eta}}$,
which we by abuse of notation we will again denote by
$A(s,\widetilde{\eta},{\rm w}_{0})$. We prove equalities when the integral in
(2.2) converges absolutely, that is to say, the real part of $s$ is
sufficiently large, and then extend meromorphically to the entire complex
plane.

Now it follows from Proposition 2.3 that $A(s,\widetilde{\eta},{\rm
w}_{0})(f_{{\rm I}_{2},\widetilde{\eta}})=a_{{\rm I}_{2}}f_{{\rm
I}_{2},\widetilde{\eta}^{-1}}+a_{{\rm w_{0}}}f_{{\rm
w_{0}},\widetilde{\eta}^{-1}}$. We may evaluate both sides of this equation at
${\rm w}_{0}$ to establish

$a_{{\rm w}_{0}}=\int_{U}f_{{\rm I}_{2},\widetilde{\eta}}({\rm w}_{0}u{\rm
w}_{0})\,du=\widetilde{\eta}(-1)\int_{\overline{U}}f_{{\rm
I}_{2},\widetilde{\eta}}(\overline{u})\,d\overline{u}=\epsilon\cdot{\rm
vol}(\overline{U}\cap J_{\eta}).$

For $x\neq 0$, we have the Iwasawa decomposition:

(2.3) ${\rm w}_{0}u(x)=\begin{pmatrix}x^{-1}&0\\\
0&x\end{pmatrix}\begin{pmatrix}1&-x\\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\\
x^{-1}&1\end{pmatrix}.$

In particular, for $x^{-1}\in\mathfrak{p}_{F}^{n_{\eta}}$, we have $f_{{\rm
I}_{2},\widetilde{\eta}}({\rm
w}_{0}u(x))=\widetilde{\eta}\nu_{s}\delta^{1/2}(x^{-1})$. We may rewrite $x\in
F^{\times}$ as $x=\varpi_{F}^{k}y$ with $k\in\mathbb{Z}$ and
$y\in\mathfrak{o}_{F}^{\times}$, and then use the relation of additive Haar
measure $d(cx)=|c|dx$ for $c\in F^{\times}$ to determine $a_{{\rm I}_{2}}$:

(2.4) $a_{{\rm I}_{2}}=\int_{U}f_{{\rm I}_{2},\widetilde{\eta}}({\rm
w}_{0}u)\,du=\int_{\\{x\,|\,x\neq
0,\;x^{-1}\in\mathfrak{p}_{F}^{n_{\eta}}\\}}\widetilde{\eta}(x^{-1})|x|^{-s-1}\,dx\\\
=\int_{|x|\geq
q_{F}^{n_{\eta}}}\widetilde{\eta}(x^{-1})|x|^{-s-1}\,dx=\sum_{n=-\infty}^{-n_{\eta}}\eta(\varpi_{F}^{-n})|\varpi_{F}^{-n}|^{-s}\int_{\mathfrak{o}_{F}^{\times}}\widetilde{\eta}(x)\,d^{\times}x.$

But $\widetilde{\eta}$ being ramified implies that the series of equality of
above integrals is zero. In other words, $a_{{\rm I}_{2}}=0$.

Let us turn our attention to $A(s,\widetilde{\eta},{\rm w}^{-1}_{0})(f_{{\rm
w}_{0},\widetilde{\eta}})$. Owing to Proposition 2.3, we know that

$A(s,\widetilde{\eta},{\rm w}^{-1}_{0})(f_{{\rm
w}_{0},\widetilde{\eta}})=b_{{\rm I}_{2}}f_{{\rm
I}_{2},\widetilde{\eta}^{-1}}+b_{{\rm w_{0}}}f_{{\rm
w_{0}},\widetilde{\eta}^{-1}}.$

Our aim is to determine unknown coefficients $b_{{\rm I}_{2}}$ and $b_{{\rm
w}_{0}}$ precisely. To this end, we evaluate both sides of the above equation
at ${\rm I}_{2}$. The test function $f_{{\rm w}_{0},\widetilde{\eta}}$ is
supported on $B{\rm w}_{0}J_{\eta}$ so that

$b_{{\rm I}_{2}}=\int_{U}f_{{\rm w}_{0},\widetilde{\eta}}({\rm
w}^{-1}_{0}u)\,du=\widetilde{\eta}(-1)\int_{U}f_{{\rm
w}_{0},\widetilde{\eta}}({\rm w}_{0}u)=\epsilon\cdot{\rm vol}(U\cap
J_{\eta}),$

thereby providing $b_{{\rm I}_{2}}=\widetilde{\eta}(-1)$. Just as in (2.4), we
use the Iwasawa decomposition (2.3) (Cf. (3.5)) to arrive at

$b_{{\rm w}_{0}}=\int_{U}f_{{\rm I}_{2},\widetilde{\eta}}({\rm
w}_{0}^{-1}u{\rm w}_{0})\,du=\int_{\\{x\,|\,x\neq
0,\;x^{-1}\in\mathfrak{o}_{F}\\}}\widetilde{\eta}(x^{-1})|x|^{-s-1}\,dx\\\
=\sum_{n\leq
0}\eta(\varpi_{F}^{-n})|\varpi_{F}^{-n}|^{-s}\int_{\mathfrak{o}_{F}^{\times}}\widetilde{\eta}(x)\,d^{\times}x.$

Once again, $\widetilde{\eta}$ being ramified forces that the innermost
integral over $\mathfrak{o}_{F}^{\times}$ is equal to zero, which yields
$b_{{\rm w}_{0}}=0$, as requested. ∎

The Plancherel constant is a scalar valued function
$\mu(s,\widetilde{\eta})\in\mathbb{C}$ attached to $\widetilde{\eta}$ is by
the defining relation [Sha84, §6]

$A(-s,\widetilde{\eta}^{-1},{\rm w}^{-1}_{0})\circ A(s,\widetilde{\eta},{\rm
w}_{0})=\mu(s,\widetilde{\eta})^{-1}\cdot
id_{\mathcal{F}(s,\widetilde{\eta})}$

on a Zariski open dense subset of $\mathbb{C}$. It is a rational function in
$q_{F}^{-s}$ and clearly depends on the measure defining intertwining
operators.

######  Theorem 2.5 (The Plancherel Constant I).

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$ of level $n_{\eta}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. Then we have

$\mu(s,\widetilde{\eta})={\rm vol}(\overline{U}\cap J_{\eta})^{-1}{\rm
vol}(U\cap J_{\eta})^{-1}=q_{F}^{n_{\eta}}.$

###### Proof.

We apply Proposition 2.4 twice. ∎

The Plancherel constant in Theorem 2.5 coincides with that in split cases [KK,
§3.3] on the common Zariski open dense subset of $\mathbb{C}$ (Refer to [KM09,
Theorem 4.5] for the precise description), and this recovers the work by
Kutzko and Morris [KM09, Theorem 4.5.(2)-(ii)].

##  3\. Local Coefficients and Gelfand-Graev Representations

###  3.1. The Local Coefficient

Let $\psi$ be a non-trivial additive character of $F$ trivial on
$\mathfrak{o}_{F}$ but not on $\mathfrak{p}_{F}^{-1}$, and $\chi$ a character
of $F^{\times}$. It is a theorem of Rodier [Rod72] that the dimension of the
space of $\psi$-Whittaker functionals on $\mathcal{F}(s,\chi)$ is one. We may
define a basis vector $\Omega_{s}$ for the $\psi$-Whittaker functionals on
$\mathcal{F}(s,\chi)$ by the formula

$\Omega_{s}(f)=\int_{U}f({\rm w}_{0}u)\psi^{-1}(u)\,du.$

This integral may not converge for all $f$ but can be extended to the whole
space as a principal value integral. We also have the following convenient
reinterpretation for $\Omega_{s}$ as a principal value integral; Given a
compact open subgroup $K_{\ast}$ of $G$, there exists a suitably large compact
open subgroup $U_{\ast}\subset U$ such that

$\Omega_{s}(f)=\int_{U_{\ast}}f({\rm w}_{0}u)\psi^{-1}(u)\,du$

for all $s$ and for all $f\in\mathcal{F}(s,\chi)^{K_{\ast}}$. We similarly
define $\Omega^{\prime}_{s}$ on $\mathcal{F}(-s,\chi^{-1})$ via

$\Omega^{\prime}_{s}(f)=\int_{U}f({\rm w}_{0}u)\psi^{-1}(u)\,du$

as a principal value integral in the above sense. Appealing to the
aforementioned result of Rodier [Rod72], there exists a non-zero constant
$C_{\psi}(s,\chi)$ called the Langlnads-Shahidi local coefficient satisfying

(3.1) $C_{\psi}(s,\chi)(\Omega^{\prime}_{s}\circ A(s,\chi,{\rm
w}_{0}))=\Omega_{s}.$

For $c\in F$, the Gauss sum attached to $\chi$ is defined by

$\tau(\chi,\psi,c)=\int_{\mathfrak{o}_{F}^{\times}}\chi(x^{-1})\overline{\psi(cx)}\,dx.$

######  Theorem 3.1 (The Local Coefficient).

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$ of level $n_{\eta}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. Then we have

$C_{\psi}(s,\widetilde{\eta})=\widetilde{\eta}(-\varpi_{F}^{n_{\eta}})\tau(\eta,\psi,\varpi_{F}^{-n_{\eta}})q_{F}^{-n_{\eta}(s-1)}.$

###### Proof.

Since $C_{\psi}(s,\widetilde{\eta})$ is a rational function in
$\mathbb{C}(q_{F}^{-s})$, it suffices to prove the assertion on a Zariski open
dense subset of $X(T)$. In particular, we impose the assumption that $\nu_{s}$
is regular so that $\mathcal{F}(s,\widetilde{\eta})$ and
$\mathcal{F}(-s,\widetilde{\eta}^{-1})$ are all irreducible. Our argument is
rather close in spirit to the work by Krishnamurthy and Kutzko [KK,
Proposition 3.1]. Evaluating one side of (3.1) at $f_{{\rm
I}_{2},\widetilde{\eta}}$ is straightforward, since the function $f_{{\rm
w}_{0},\widetilde{\eta}}$ is supported on $B{\rm w}_{0}J_{\eta}$. Namely,

$(\Omega^{\prime}_{s}\circ A(s,\widetilde{\eta},{\rm w}_{0}))(f_{{\rm
I}_{2},\widetilde{\eta}})=\widetilde{\eta}(-1){\rm vol}(\overline{U}\cap
J_{\eta})\Omega^{\prime}_{s}(f_{{\rm w}_{0},\widetilde{\eta}^{-1}})\\\
=\widetilde{\eta}(-1){\rm vol}(\overline{U}\cap J_{\eta})\int_{J_{\eta}\cap
U}f_{{\rm w}_{0},\widetilde{\eta}^{-1}}({\rm
w}_{0}u)\psi^{-1}(u)\,du=\widetilde{\eta}(-1){\rm vol}(\overline{U}\cap
J_{\eta})\int_{J_{\eta}\cap U}\psi^{-1}(u)\,du.$

We see that $\psi$ is trivial on $J_{\eta}\cap U$, and consequently,

(3.2) $(\Omega^{\prime}_{s}\circ A(s,\widetilde{\eta},{\rm w}_{0}))(f_{{\rm
I}_{2},\widetilde{\eta}})=\widetilde{\eta}(-1){\rm vol}(\overline{U}\cap
J_{\eta}){\rm vol}(U\cap J_{\eta})=\widetilde{\eta}(-1)q_{F}^{-n_{\eta}}.$

This brings us to the central issue of computing the other side
$\Omega_{s}(f_{{\rm I}_{2},\widetilde{\eta}})$. In contrast to $f_{{\rm
w}_{0},\widetilde{\eta}}$, this is not immediate, because $f_{{\rm
I}_{2},\widetilde{\eta}}$ is supported near the identity element. To this end,
we deduce from (2.3) that $f_{{\rm I}_{2},\widetilde{\eta}}({\rm w}_{0}u(x))$
is $\widetilde{\eta}\nu_{s}\delta^{1/2}(x^{-1})$, if
$x^{-1}\in\mathfrak{p}_{F}^{n_{\eta}}$, and $0$, otherwise. Then
$\Omega_{s}(f_{{\rm I}_{2},\widetilde{\eta}})$ equals to

$\int_{\\{\mathfrak{p}_{F}^{-m}-\\{0\\}\\}\cap\\{x\,|\,x^{-1}\in\mathfrak{p}_{F}^{n_{\eta}}\\}}\widetilde{\eta}(x^{-1})|x|^{-s-1}\psi^{-1}(x)\,dx$

for some large positive integer $m\gg 0$. For convenience, let $\mathcal{D}$
denote
$\\{\mathfrak{p}_{F}^{-m}-\\{0\\}\\}\cap\\{x\,|\,x^{-1}\in\mathfrak{p}_{F}^{n_{\eta}}\\}$.
For any integer $r$, let $\mathcal{D}_{r}$ denote the shell,
$\\{x\in\varpi_{F}^{r}\mathfrak{o}_{F}^{\times}\,|\,x^{-1}\in\mathfrak{p}_{F}^{n_{\eta}}\\}$.
Then our domain of the integration can be decomposed as shells
$\mathcal{D}=\cup_{-m\leq r\leq-n_{\eta}}\mathcal{D}_{r}$. The crux of the
proof of [KK, Proposition 3.1] is that only the last shell
$\mathcal{D}_{-n_{\eta}}=\varpi_{F}^{-n_{\eta}}\mathfrak{o}_{F}^{\times}$
contributes to $\Omega_{s}(f_{{\rm I}_{2},\widetilde{\eta}})$. Assembling all
of this information, we achieve

(3.3) $\Omega_{s}(f_{{\rm
I}_{2},\widetilde{\eta}})=\int_{\varpi_{F}^{-n_{\eta}}\mathfrak{o}_{F}^{\times}}\widetilde{\eta}(x^{-1})|x|^{-s}\psi^{-1}(x)\,d^{\times}x=\widetilde{\eta}(\varpi_{F}^{-n_{\eta}})q_{F}^{-n_{\eta}s}\int_{\mathfrak{o}_{F}^{\times}}\eta(x^{-1})\psi^{-1}(\varpi_{F}^{-n_{\eta}}x)d^{\times}x\\\
=\widetilde{\eta}(\varpi_{F}^{-n_{\eta}})q_{F}^{-n_{\eta}s}\tau(\eta,\psi,\varpi_{F}^{-n_{\eta}}).$

We can draw the conclusion from (3.2) combined with (3.3). ∎

This result should be compared with the work of Shahidi [Sha78, Lemma 4.4]. In
addition, the local coefficient coincides with the corresponding Hecke-Tate
local $\gamma$-factor [Tat77], which can be viewed as Bump and Friedberg
exterior square local factors for ${\rm GL}_{1}(F)\cong F^{\times}$ (cf.
[Mat15, Proof of Theorem 5.4]). Indeed, Theorem 3.1 confirms Conjecture 4 in
[BF89].

Let $P(X)\in\mathbb{C}[X]$ be the unique polynomial satisfying $P(0)=1$ such
that $P(q_{F}^{-s})$ is the numerator of $C_{\psi}(s,\chi)$. Whenever $\chi$
is unitary, the local $L$-factor is defined by

$L(s,\chi):=\frac{1}{P(q_{F}^{-s})}.$

The local $\varepsilon$-factor is defined to satisfy the relation:

$C_{\psi}(s,\chi)=\varepsilon(s,\chi,\psi)\frac{L(1-s,\chi^{-1})}{L(s,\chi)}.$

######  Corollary 3.2 (Local Factors).

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$. Then we have

$\varepsilon(s,\widetilde{\eta},\psi)=C_{\psi}(s,\widetilde{\eta})\quad\text{and}\quad
L(s,\widetilde{\eta})=1.$

Corollaries 3.3 and 3.4 match with the corresponding formulæ [Sha81,
Proposition 3.1.1] and [Sha84, §Introduction].

######  Corollary 3.3 (The Functional Equation).

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$. Then we have

$C_{\psi}(s,\widetilde{\eta})C_{\psi}(1-s,\widetilde{\eta}^{-1})=\epsilon.$

###### Proof.

The functional equation for the local constant
$\varepsilon(s,\widetilde{\eta},\psi)$ (Cf. [BH06, Corollary 23.4.2]) produces

$C_{\psi}(s,\widetilde{\eta})C_{\psi}(1-s,\widetilde{\eta}^{-1})=\varepsilon(1/2,\widetilde{\eta},\psi)\varepsilon(1/2,\widetilde{\eta}^{-1},\psi)=\widetilde{\eta}(-1).\qed$

The local coefficient is related to the Plancherel constant which is more or
less saying that “the square root of the local coefficient equals the
associated Plancherel constant” as given in Corollary 3.4.

######  Corollary 3.4 (The Plancherel Constant II).

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$. We assume that $\widetilde{\eta}$ is unitary. Then we have

$\mu(s,\widetilde{\eta})=C_{\psi}(s,\widetilde{\eta})C_{\psi^{-1}}(-s,\widetilde{\eta}^{-1})=|C_{\psi}(s,\widetilde{\eta})|^{2}.$

###### Proof.

We observe from [BH06, (23.6.3)] that

$\tau(\eta,\psi,\varpi_{F}^{-n_{\eta}})\tau(\eta^{-1},\psi^{-1},\varpi_{F}^{-n_{\eta}})=|\tau(\eta,\psi,\varpi_{F}^{-n_{\eta}})|^{2}=q_{F}^{-n_{\eta}}.\qed$

###  3.2. The Gelfand-Graev Representation

We take this occasion to explore the structure of the Gelfand-Graev space,
which extends the result of [CS18] and [MP21]. The Gelfand-Graev
representation ${\text{\rm c-ind}}_{U}^{G}{\psi}$ [CS18, §4.1] is provided by
the space of smooth functions $f:G\rightarrow\mathbb{C}$ which are compactly
supported modulo $U$. They also satisfy

(3.4) $f(ug)=\psi(u)f(g)\quad\text{for all $u\in U$ and $g\in G$.}$

Let ${\text{\rm Ind}}_{U}^{G}{\psi}$ denote the full space of smooth functions
$f:G\rightarrow\mathbb{C}$ which satisfy (3.4). As a potential application
(Cf. [JK21, §5.3.2 (5.25)]), the contragredient $({\text{\rm
c-ind}}_{U}^{G}\,{\overline{\psi}})^{\vee}\cong{\text{\rm Ind}}_{U}^{G}{\psi}$
of ${\text{\rm c-ind}}_{U}^{G}\,{\overline{\psi}}$ appears in the target space
of the Whittaker map
$\omega_{s}:\mathcal{F}(s,\widetilde{\eta})\rightarrow{\text{\rm
Ind}}_{U}^{G}{\psi}$ corresponding to $\Omega_{s}$ via Frobenius reciprocity.
For $w\in W$, we define the functions $\varphi^{\psi}_{{\rm I}_{2},\eta}$ and
$\varphi^{\psi}_{{\rm w}_{0},\eta}:G\rightarrow\mathbb{C}$ in the
$\lambda_{\eta}$-co-invariant Gelfand-Graev space $({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$ given by

$\varphi^{\psi}_{{\rm
I}_{2},\eta}(w)(g)=\begin{cases}\psi(u)\lambda_{\eta}(j),\ &\text{if}\quad
g=uj\in UJ_{\eta}\\\ 0,&\text{otherwise,}\end{cases}$

and

$\varphi^{\psi}_{{\rm
w}_{0},\eta}(w)(g)=\begin{cases}\psi(u)\lambda_{\eta}(j),&\text{if}\quad
g=u{\rm w}_{0}j\in U{\rm w}_{0}J_{\eta}\\\ 0,&\text{otherwise.}\end{cases}$

Given a smooth (complex) representation $(\pi,V)$ of $G$, let
$V_{\overline{U}}$ be the maximal quotient of $V$ on which $\overline{U}$ acts
trivially. According to [Bor76]*Lemma 4.7 [CS18]*Proposition 4.2 (Cf. [BK98,
Lemma 10.3]), the usual projection from ${\text{\rm c-ind}}_{U}^{G}{\psi}$
onto ${\text{\rm c-ind}}_{1}^{T}{\mathbb{C}}(\cong
C_{c}^{\infty}(T)\cong{\mathcal{H}}(T,{\textbf{1}}_{T}))$ defined by

$\mathcal{P}_{\delta}:f\mapsto
f_{\overline{U}}(t)=\delta^{1/2}(t)\int_{\overline{U}}f(t\overline{u})\,d\overline{u},\quad
t\in T$

descends to an isomorphism $\mathcal{S}_{\delta}:({\text{\rm
c-ind}}_{U}^{G}{\psi})_{\overline{U}}\rightarrow C_{c}^{\infty}(T)$ as
$C_{c}^{\infty}(T)$-modules. Let ${\rm
ch}^{\eta}_{T(\mathfrak{o}_{F})}\in(C_{c}^{\infty}(T))^{\eta}$ be a test
function supported on $T(\mathfrak{o}_{F})$ such that ${\rm
ch}^{\eta}_{T(\mathfrak{o}_{F})}(t)=\mathrm{vol}(\overline{U}\cap
J_{\eta})\eta(t)$ for all $t\in T(\mathfrak{o}_{F})$. Owing to [MP21, Theorem
1] (Cf. [CS18, The discussion preceding Lemma 4.3]), $\mathcal{P}_{\delta}$ in
turn endows the isomorphism as ${\mathcal{H}}(T,\eta)$-modules of $({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$ with
$(C_{c}^{\infty}(T))^{\eta}\cong{\mathcal{H}}(T,\eta)$, where ${\rm
ch}^{\eta}_{T(\mathfrak{o}_{F})}$ is a generator.

######  Proposition 3.5.

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. Then we have

  1. $(\mathrm{1})$

$\mathcal{S}_{\delta}(\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})={\rm
ch}^{\eta}_{T(\mathfrak{o}_{F})}$.

  2. $(\mathrm{2})$

$T^{\ast}_{\rm w}\star(\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})=(-1)^{\ell(({\rm
w})}(\varphi^{\psi}_{{\rm I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm
vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})$
for ${\rm w}\in{\rm\bf W}$.

###### Proof.

The first assertion $(\mathrm{1})$ is immediate from Claim located in the
halfway of the proof of [CS18, Lemma 4.1].

As for $(\mathrm{2})$, the function $T^{\ast}_{\rm
w}\star(\varphi^{\psi}_{{\rm I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm
vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})$ is
supported on $U{\rm w}J_{\eta}\cup U{\rm w}J_{\eta}{\rm w}_{0}J_{\eta}$. It is
a direct computation to check that ${\rm w}_{0}(J_{\eta}\cap\overline{U}){\rm
w}^{-1}_{0}\subseteq J_{\eta}\cap U$. Furthermore, via the Bruhat
decomposition (3.5), the element ${\rm w}_{0}u(x){\rm
w}^{-1}_{0}=\overline{u}(-x)$ with $x\in\mathfrak{o}_{F},x\neq 0$ lies in
$U{\rm w}_{0}J_{\eta}$. To see this, the diagonal part of $U{\rm
w}J_{\eta}\cup U{\rm w}J_{\eta}{\rm w}_{0}J_{\eta}$ belongs to
$T(\mathfrak{o}_{F})$. Altogether, the function $T^{\ast}_{\rm
w}\star(\varphi^{\psi}_{{\rm I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm
vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})$ is
compactly supported on $UJ_{\eta}\cup U{\rm w}_{0}J_{\eta}$. This permits us
to reduce the computation to specializing the values at ${\rm I}_{2}$ and
${\rm w}_{0}$.

We consider the case that ${\rm w}={\rm I}_{2}$. We know that $({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$ is a ${\mathcal{H}}(G,\eta)$-algebra,
while $T^{\ast}_{{\rm I}_{2}}$ is the identity in this Hecke algebra
${\mathcal{H}}(G,\eta)$ so must surely act as the identity on itself
$({\text{\rm c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$. Henceforth we conclude
that

$T^{\ast}_{{\rm I}_{2}}\star(\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})=\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}.$

We bring our attention to the effect of intertwiners $T^{\ast}_{{\rm w}_{0}}$
on the generator. We treat the term $T^{\ast}_{{\rm
w}_{0}}\star\varphi^{\psi}_{{\rm I}_{2},\eta}$ first. The value
$(T^{\ast}_{{\rm w}_{0}}\star\varphi^{\psi}_{{\rm I}_{2},\eta})({\rm I}_{2})$
is $0$, as supports $J_{\eta}{\rm w}_{0}J_{\eta}$ and $UJ_{\eta}$ of
$T^{\ast}_{{\rm w}_{0}}$ and $\varphi^{\psi}_{{\rm I}_{2},\eta}$,
respectively, are disjoint. Now thanks to the Iwahori factorization of
$J_{\eta}$, we may identify

$J_{\eta}/(J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}^{-1}_{0})=\begin{pmatrix}1&\mathfrak{o}_{F}/\mathfrak{p}_{F}^{n_{\eta}}\\\
&1\end{pmatrix}$

from which we deduce that $(\pi(T^{\ast}_{{\rm w}_{0}})\varphi^{\psi}_{{\rm
I}_{2},\eta})(w)({\rm w}_{0})$ is equal to

$\int_{J_{\eta}{\rm w}_{0}J_{\eta}}\varphi^{\psi}_{{\rm
I}_{2},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}(x^{-1})w)({\rm
w}_{0}x)\,dx=\sum_{j\in J_{\eta}/(J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}_{0}^{-1})}\int_{J_{\eta}}\varphi^{\psi}_{{\rm
I}_{2},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}({j^{\prime}}^{-1}{\rm
w}^{-1}_{0}j^{-1})w)({\rm w}_{0}j{\rm w}_{0}j^{\prime})dj^{\prime}\\\
=\widetilde{\eta}(-1)\sum_{j\in J_{\eta}/(J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}^{-1}_{0})}\varphi^{\psi}_{{\rm I}_{2},\eta}(\check{T}^{\ast}_{{\rm
w}_{0}}({\rm w}^{-1}_{0}j^{-1})w)({\rm w}_{0}j{\rm w}^{-1}_{0}).$

In this circumstance, we have employed the relation ${\rm w}_{0}^{2}=-1$ and
it is worthwhile to notice that the sign change occurs. Upon writing $j=u(x)$,
$\varphi^{\psi}_{{\rm I}_{2},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}({\rm
w}^{-1}_{0}j^{-1})w)({\rm w}_{0}j{\rm w}^{-1}_{0})\neq 0$ if and only if
$x\in\mathfrak{p}_{F}^{n_{\eta}}$. As a result, we end up at

$(\pi(T^{\ast}_{{\rm w}_{0}})\varphi^{\psi}_{{\rm I}_{2},\eta})(w)({\rm
w}_{0})=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm I}_{2},\eta}(w)({\rm
I}_{2})=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}.$

Let us focus our attention to the term $T^{\ast}_{{\rm
w}_{0}}\star\epsilon^{3/2}{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$. Then the support of
$\varphi^{\psi}_{{\rm w}_{0},\eta}$ tells us that

$(\pi(T^{\ast}_{{\rm w}_{0}})\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})(w)({\rm
I}_{2})=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\int_{J_{\eta}{\rm w}_{0}J_{\eta}}\varphi^{\psi}_{{\rm
w}_{0},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}(x^{-1})w)(x)\,dx\\\
=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\sum_{j\in
J_{\eta}/(J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}_{0}^{-1})}\int_{J_{\eta}}\varphi^{\psi}_{{\rm
w}_{0},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}({j^{\prime}}^{-1}{\rm
w}^{-1}_{0}j^{-1})w)(j{\rm w}_{0}j^{\prime})\,dj^{\prime}.$

In light of $[J_{\eta}:J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}_{0}^{-1}]=[J_{\eta}{\rm w}_{0}J_{\eta}:J_{\eta}]$, we are led to

$(\pi(T^{\ast}_{{\rm w}_{0}})\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})(w)({\rm I}_{2})\\\
=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\sum_{j\in
J_{\eta}/(J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}_{0}^{-1})}\varphi^{\psi}_{{\rm w}_{0},\eta}(\check{T}^{\ast}_{{\rm
w}_{0}}({\rm w}^{-1}_{0}j^{-1})w)(j{\rm w}_{0})\\\ ={\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1}[J_{\eta}:J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}_{0}^{-1}]\varphi^{\psi}_{{\rm w}_{0},\eta}(w)({\rm w}_{0})=1.$

Concerning the value $(\pi(T^{\ast}_{{\rm w}_{0}})\epsilon^{3/2}\cdot{\rm
vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm
w}_{0},\eta})(w)(\cdot)$ at ${\rm w}_{0}$, we use the fact that the integrand
below is supported on $J_{\eta}{\rm w}_{0}J_{\eta}$ and is right invariant
under $J_{\eta}$ coupled with the fact that the map $j\mapsto j{\rm w}_{0}$
induces a bijection $J_{\eta}/(J_{\eta}\cap{\rm w}_{0}J_{\eta}{\rm
w}_{0}^{-1})\rightarrow J_{\eta}{\rm w}_{0}J_{\eta}/J_{\eta}$ (Cf. [Kut04,
Lemma 3.2] to see that

$(\pi(T^{\ast}_{{\rm w}_{0}})\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})(w)({\rm
w}_{0})=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\int_{G}\varphi^{\psi}_{{\rm
w}_{0},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}(x^{-1})w)({\rm w}_{0}x)\,dx\\\
=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}[J_{\eta}{\rm
w}_{0}J_{\eta}:J_{\eta}]\int_{J_{\eta}}\varphi^{\psi}_{{\rm
w}_{0},\eta}(\check{T}^{\ast}_{{\rm w}_{0}}({\rm w}_{0}^{-1}j^{-1})w)({\rm
w}_{0}j{\rm w}_{0})\,dj.$

The integrand of the last integral is right invariant under
$J_{\eta}\cap\overline{B}$. In this way, we find that

$(\pi(T^{\ast}_{{\rm w}_{0}})\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})(w)({\rm
w}_{0})=\frac{{\rm vol}(J_{\eta})}{{\rm vol}(J_{\eta}\cap
U)}\int_{J_{\eta}\cap U}\varphi^{\psi}_{{\rm w}_{0},\eta}(w)({\rm w}_{0}u{\rm
w}_{0})\,du.$

The integrand is $0$, unless $x\notin\mathfrak{o}_{F}^{\times}$. Then for
$x\in\mathfrak{o}_{F}^{\times}$, we apply the Bruhat decomposition (Cf.
[Kut04, P. 606]);

(3.5) $\begin{pmatrix}1&0\\\ x&1\end{pmatrix}=-\begin{pmatrix}x^{-1}&0\\\
0&x\end{pmatrix}\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}{\rm
w}_{0}\begin{pmatrix}1&x^{-1}\\\ 0&1\end{pmatrix},$

whence $\varphi^{\psi}_{{\rm w}_{0},\eta}\left({\rm
w}_{0}\begin{pmatrix}1&x\\\ 0&1\end{pmatrix}{\rm
w}_{0}\right)=\eta(x^{-1})\psi(-x)$. Therefore it can be expressed in terms of
Gauss sums attached to the ramified character $\widetilde{\eta}$ and then
eventually this term vanishes

$(\pi(T^{\ast}_{{\rm w}_{0}})\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})(w)({\rm
w}_{0})=\int_{\mathfrak{o}_{F}^{\times}}\eta(x^{-1})\psi(x)\varphi^{\psi}_{{\rm
w}_{0},\eta}(w)({\rm w}_{0})dx=0,$

as was to be shown. In short, we accomplish that

$T^{\ast}_{{\rm w}_{0}}\star\varphi^{\psi}_{{\rm
I}_{2},\eta}=\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}\quad\text{and}\quad
T^{\ast}_{{\rm w}_{0}}\star\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}=\varphi^{\psi}_{{\rm
I}_{2},\eta}.$

and the outcome we seek for follows from this. ∎

It is noteworthy that by applying Proposition 3.5-$(\mathrm{2})$ repeatedly,
we get

$T^{\ast 2}_{{\rm w}_{0}}\star(\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta})=\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$

which is consistent with the consequence from $T^{\ast 2}_{{\rm
w}_{0}}=T^{\ast}_{{\rm I}_{2}}$ [Kut04, Proposition 3.3]. We denote by
$\mathbb{C}_{-1}$ the one-dimensional $\mathcal{H}(K,\lambda_{\eta})-$module
on which $T^{\ast}_{\rm w}$ acts as $(-1)^{\ell({\rm w})}$, for ${\rm
w}\in{\rm\bf W}$. We are now in a position to state our main theorem.

######  Theorem 3.6.

Let $\widetilde{\eta}$ be a non-trivial ramified quadratic character of
$F^{\times}$ with
$\eta=\widetilde{\eta}\restriction_{\mathfrak{o}^{\times}_{F}}$. As
$\mathcal{H}(G,\lambda_{\eta})$-modules, we have isomorphisms

$({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}\cong\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}.$

###### Proof.

In virtue of Proposition 3.5-$(\mathrm{2})$, we construct an element in ${\rm
Hom}_{{\mathcal{H}(K,\lambda_{\eta})}}(\mathbb{C}_{-1},({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}})$ given by
$1\mapsto\varphi^{\psi}_{{\rm I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm
vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$. We
attain the following Frobenius reciprocity

${\rm Hom}_{{\mathcal{H}(K,\lambda_{\eta})}}(\mathbb{C}_{-1},({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}})\cong{\rm
Hom}_{{\mathcal{H}(G,\lambda_{\eta})}}(\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1},({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}),$

where the element $1\mapsto\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$ corresponds to the
element $T^{\ast}_{{\rm I}_{2}}\otimes 1\mapsto\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$. In summary, we form
a non-zero map

$\vartheta:\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}\rightarrow({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$

of $\mathcal{H}(G,\lambda_{\eta})$-modules, which assigns $T^{\ast}_{{\rm
I}_{2}}\otimes 1$ to $\varphi^{\psi}_{{\rm
I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$.

It suffices to show that $\vartheta$ is an isomorphism of
$\mathcal{H}(T,\eta)$-modules. To verify it, Proposition 3.5-$(\mathrm{1})$
ensures that we find an isomorphism
${\mathcal{H}}(T,\eta)\rightarrow({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$ of ${\mathcal{H}}(T,\eta)$-modules
provided by ${\rm ch}^{\eta}_{T(\mathfrak{o}_{F})}$ mapping to
$\varphi^{\psi}_{{\rm I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm vol}(J_{\eta}{\rm
w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$. Upon invoking the
isomorphism
$\mathcal{H}(T,\eta)\otimes_{\mathbb{C}}\mathcal{H}(K,\lambda_{\eta})\cong\mathcal{H}(G,\lambda_{\eta})$
attributed to [Kut04, Proposition 3.1], we end up with a series of
isomorphisms as left $\mathcal{H}(T,\eta)-$modules

${\mathcal{H}(T,\eta)}\cong{\mathcal{H}(T,\eta)}\otimes_{\mathbb{C}}\mathbb{C}_{-1}\cong{\mathcal{H}(T,\eta)}\otimes_{\mathbb{C}}{\mathcal{H}(K,\lambda_{\eta})}\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}\cong\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1},$

which sends ${\rm ch}^{\eta}_{T(\mathfrak{o}_{F})}$ to $T^{\ast}_{{\rm
I}_{2}}\otimes 1$. Gathering all information, the conclusion thereby follows,
because $\vartheta$ is indeed $\mathcal{H}(T,\eta)$-modules. In other words,
$\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}$
and $({\text{\rm c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}$ are free
$\mathcal{H}(T,\eta)$-modules generated by $T^{\ast}_{{\rm I}_{2}}\otimes 1$
and $\varphi^{\psi}_{{\rm I}_{2},\eta}-\epsilon^{3/2}\cdot{\rm
vol}(J_{\eta}{\rm w}_{0}J_{\eta})^{-1/2}\varphi^{\psi}_{{\rm w}_{0},\eta}$
respectively. ∎

When the characteristic of the field is $0$ and that of the residual field is
greater than $3$, the above theorem has been settled in the work by Mishra and
Pattanayak [MP21, Theorem 3]. To the best of our knowledge, Theorem 3.6 is new
when $F$ has positive characteristics, and also when it has residual
characteristic $2$ or $3$.

######  Remark 3.7.

If $\eta^{2}\neq 1$, then the injection $t_{Q}$ in $(\mathrm{c})$ of §2.1
becomes an isomorphism [Kut04, Corollary 3.1] in which case the cover
$(J_{\eta},\lambda_{\eta})$ is said to be a split cover. Then Proposition
3.5-$(\mathrm{2})$ seems to be superfluous, because the map $t_{Q}$ already
gives rise to a series of
${\mathcal{H}}(T,\eta)\cong\mathcal{H}(G,\lambda_{\eta})$-module isomorphism

$({\text{\rm
c-ind}}_{U}^{G}{\psi})^{\lambda_{\eta}}\cong\mathcal{H}(G,\lambda_{\eta})\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}\cong{\mathcal{H}}(T,\eta)\otimes_{{\mathcal{H}(K,\lambda_{\eta})}}\mathbb{C}_{-1}\cong{\mathcal{H}}(T,\eta)\cong(C_{c}^{\infty}(T))^{\eta}.$

######  Acknowledgments.

I am deeply indebted to Muthu Krishnamurthy for introducing his unfinished
project [KK] to me. Without his suggestion and countless encouragement, this
paper never has come into existence. I also want to take this opportunity to
thank to Jack Buttcane, Brandon Hanson, and Andrew Knightly for fruitful
mathematical conversation and their feedbacks on some aspects of this
manuscript. I am grateful to the Department of Mathematics and Statistics at
University of Maine for their warm hospitality, while the paper was written.
The author would like to convey our appreciation to anonymous referees for
thoughtfully reading our paper and correcting many inaccuracies which
significantly improved the exposition of this manuscript. This work was
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (No. RS-2023-00209992).

######  Conflicts of interest.

The author states that there is no conflict of interest.

######  Data availability.

This manuscript has no associated data.

## References