# The COVID-19 Infodemic: Twitter versus Facebook

Kai-Cheng Yang,1 Francesco Pierri,2 Pik-Mai Hui,1 David Axelrod,1
Christopher Torres-Lugo,1 John Bryden,1 and Filippo Menczer1
1Observatory on Social Media, Indiana University, Bloomington, USA
2Dipartimento di Elettronica, Informatica e Bioingegneria
Politecnico di Milano, Italy

###### Abstract

The global spread of the novel coronavirus is affected by the spread of
related misinformation — the so-called COVID-19 Infodemic — that makes
populations more vulnerable to the disease through resistance to mitigation
efforts. Here we analyze the prevalence and diffusion of links to low-
credibility content about the pandemic across two major social media
platforms, Twitter and Facebook. We characterize cross-platform similarities
and differences in popular sources, diffusion patterns, influencers,
coordination, and automation. Comparing the two platforms, we find divergence
among the prevalence of popular low-credibility sources and suspicious videos.
A minority of accounts and pages exert a strong influence on each platform.
These misinformation “superspreaders” are often associated with the low-
credibility sources and tend to be verified by the platforms. On both
platforms, there is evidence of coordinated sharing of Infodemic content. The
overt nature of this manipulation points to the need for societal-level rather
than in-house mitigation strategies. However, we highlight limits imposed by
inconsistent data-access policies on our capability to study harmful
manipulations of information ecosystems.

##  1 Introduction

The impact of the COVID-19 pandemic has been felt globally, with almost 70
million detected cases and 1.5 million deaths as of December 2020
(coronavirus.jhu.edu/map.html). Epidemiological strategies to combat the virus
require collective behavioral changes. To this end, it is important that
people receive coherent and accurate information from media sources that they
trust. Within this context, the spread of false narratives in our information
environment can have acutely negative repercussions on public health and
safety. For example, misinformation about masks greatly contributed to low
adoption rates and increased disease transmission [19]. The problem is not
going away any time soon: false vaccine narratives [18] will drive hesitancy,
making it difficult to reach herd immunity and prevent future outbreaks.

It is concerning that many people believe, and many more have been exposed to,
misinformation about the pandemic [22, 35, 23, 34]. The spread of this
misinformation has been termed the Infodemic [44]. Social media play a strong
role in propagating misinformation because of peer-to-peer transmission [41].
There is also evidence that social media is manipulated [36, 40] and used to
spread COVID-19 misinformation [12]. It is therefore important to better
understand how users disseminate misinformation across social media networks.

A number of studies have used a variety of perspectives to assess the extent
to which the public has been exposed to COVID-19 misinformation on social
media. These include quantitative studies of the prevalence of links to low-
credibility websites [5, 6, 14, 38] as well as qualitative analyses of posts
on Twitter [33, 1, 20]. These different approaches have yielded disparate
estimates of misinformation prevalence levels — between 1% and 70%. Meanwhile,
surveys of the general public have found exposure levels to be around 30–50%
[22, 23, 34]. These widely varying statistics indicate that different
approaches to experimental design, including uneven access to data on
different platforms and inconsistent definitions of misinformation, can
generate inconclusive or misleading results. We follow the reasoning from [14]
that it is better to clearly define a misinformation metric, and then use it
in a comparative way — to look at how misinformation varies over time or is
influenced by other factors.

A common limitation with current studies is that they only focus on a single
social media platform [5, 14, 38, 33, 1, 20]. However, the modern information
environment is an ecosystem consisting of different platforms over which
information propagates concurrently and in diverse ways. Each platform can
have different vulnerabilities [2]. A key goal of the present work is to
compare and contrast the extent to which the Infodemic has spread on Twitter
and Facebook.

A second gap in our understanding of COVID-19 misinformation is in the
patterns of diffusion within social media. It is important to understand how
certain user accounts, or groups of accounts, can play a disproportionate role
in amplifying the spread of misinformation. Inauthentic social media accounts,
known as social bots and trolls, can play an important role in amplifying the
spread of low-credibility content in general [13, 40, 36], and COVID-19
misinformation in particular [11], on Twitter. The picture on Facebook is less
clear, as there is little access to data that would enable a determination of
social bot activity. It is, however, possible to look for evidence of
manipulation in how multiple accounts can be coordinated with one another,
potentially controlled by a single entity. For example, accounts may exhibit
suspiciously similar sharing behaviors [27].

We extract website links from social media posts that include COVID-19 related
keywords. We identify a link with low-credibility content in one of two ways.
First, we follow the convention of classifying misinformation at the source
rather than the article level [17]. We do this by matching links to an
independently-generated corpus of low-credibility website domains (or
sources). Second, in the case of links to YouTube, we flag videos as
suspicious if they have been banned by the site, or are otherwise unavailable
to the public. This enables us to quantitatively study the prevalence of
likely misinformation about COVID-19 and the different ways in which it has
been shared on the two commonly used social media platforms.

The main contributions of this study stem from exploring three sets of
research questions:

  * •

What is the prevalence of low-credibility content on Twitter and Facebook? Are
there similarities in how sources are shared over time? How does this activity
compare to that of popular high-credibility sources? Are the same suspicious
sources and YouTube videos shared in similar volumes across the two platforms?

  * •

Is the sharing of misinformation concentrated around a few active accounts? Do
a few influential accounts dominate the resharing of popular misinformation?
What is the role of verified accounts and those associated with the low-
credibility sources on the two platforms?

  * •

Is there evidence of inauthentic coordinated behavior in sharing low-
credibility content? Can we identify clusters of users, pages, or groups with
suspiciously similar sharing patterns? Is low-credibility content amplified by
Twitter bots more prevalent on Twitter as compared to Facebook?

In the next section we describe the methodology and data employed in this
paper. The following three sections present results to answer the above
research questions. Finally, we discuss the limitations of our analyses and
implications of our findings for mitigation strategies.

##  2 Methods

###  2.1 Identification of low-credibility information

We focus on news articles linked in social media posts and identify those
pertaining to low-credibility domains by matching the URLs to sources,
following a corpus of literature [17, 36, 16, 29, 3]. We define our list of
low-credibility domains based on information provided by the Media Bias/Fact
Check website (MBFC, mediabiasfactcheck.com), an independent organization that
reviews and rates the reliability of news sources. We gather the sources
labeled by MBFC as having a “Very Low” or “Low” factual-reporting level. We
then add “Questionable” or “Conspiracy-Pseudoscience” sources and we leave out
those with factual-reporting levels of “Mostly-Factual,” “High,” or “Very
High.” We remark that although many websites exhibit specific political
leanings, these do not affect inclusion in the list. The list has 674 low-
credibility domains and we have shared it separately [26].

###  2.2 High-credibility sources

Table 1:  List of high-credibility domains. The political leanings are fetched from Allsides.com (www.allsides.com/media-bias/media-bias-ratings, accessed Dec. 1, 2020.)  Domain | Political leaning
---|---
huffpost.com | left
newyorker.com | left
msnbc.com | left
newsweek.com | left
cnn.com | center left
nytimes.com | center left
economist.com | center left
time.com | center left
washingtonpost.com | center left
reuters.com | center
apnews.com | center
npr.org | center
usatoday.com | center
wsj.com | center
foxnews.com | center right
marketwatch.com | center right
nypost.com | center right
dailycaller.com | right
theblaze.com | right
dailywire.com | right
cdc.gov | N/A
who.int | N/A

As a benchmark for interpreting the prevalence of low-credibility content, we
also curate a list of 20 high-credibility information sources. We start from
the list provided in a recent Pew Research Center report [21] and used in a
few studies on online disinformation [31, 32], and we select popular news
outlets that cover the full U.S. political spectrum. In addition, we include
the websites of two organizations that acted as authoritative sources of
COVID-19 related information, namely the Centers for Disease Control and
Prevention (CDC) and World Health Organization (WHO). The full list can be
found in Table 1.

###  2.3 Data collection

We collect data related to COVID-19 from both Twitter and Facebook. To provide
a general and unbiased view of the discussion, we chose the following generic
terms as keywords: coronavirus, covid (to capture variations like covid19 and
covid-19), and sars (to capture sars-cov-2 and related variations).

####  2.3.1 Twitter data.

Figure 1:  Structure of the data collected from Twitter and Facebook. On
Twitter, we have the information about original tweets, retweets, and all the
accounts involved. On Facebook, we have information about original posts and
public groups/pages that posted them. For each post, we also have aggregate
numbers of reshares, comments, and reactions, with no information about the
users responsible for those interactions.

Our Twitter data was collected using an API from the Observatory on Social
Media [8], which allows to search tweets from the Decahose, a 10% random
sample of public tweets. We searched for English tweets containing the
keywords between Jan. 1 and Oct. 31, 2020, resulting in over 53M tweets posted
by about 12M users. Note that since the Decahose samples tweets and not users,
the sample of users in our Twitter dataset is biased toward more active users.

Our collection contains two types of tweets, namely original tweets and
retweets. The content of original tweets is published by users directly, while
retweets are generally used to endorse/amplify original tweets by others (no
quoted tweets are included). We refer to authors of original tweets as “root”
users, and to authors of retweets as “leaf” users (see Fig. 1).

####  2.3.2 Facebook data.

We used the posts/search endpoint of the CrowdTangle API [7] to collect data
from Facebook. We filtered the entire set of English posts published by public
pages and groups in the period from Jan. 1 to Oct. 31, 2020 using the above
list of keywords, resulting in over 37M posts from over 140k public
pages/groups.

Our Facebook data collection is limited by the coverage of pages and groups in
CrowdTangle, a public tool owned and operated by Facebook. CrowdTangle
includes over 6M public Facebook pages and groups with at least 100k
followers/members (public groups with at least 2k members if based in the
U.S.), and a very small subset of verified profiles that can be followed like
public pages. We include these public accounts among pages and groups. In
addition, some pages and groups with fewer followers and members are also
included by CrowdTangle upon request from users. This might bias the dataset
in ways that are hard to gauge. For example, requests from researchers
interested in monitoring low-credibility pages and groups might lead to over-
representation of such content.

Figure 2:  Pearson correlation coefficients between Facebook metrics
aggregated at the domain level for low-credibility domains. The reaction
consists of “like,” “love,” “wow,” “haha,” “sad,” “angry,” and “care.” All
correlations are significant ($p<0.01$).

As shown in Fig. 1, the collected data contains information about original
Facebook posts and the pages/groups that published these posts. For each post,
we also have access to aggregated statistics such as the number of reshares,
comments, and reactions (e.g., “like”) by Facebook users. The numbers of
comments and reactions are highly correlated with reshares (Fig. 2), so we
focus on reshares in this study.

Similarly to Twitter, Facebook pages and groups that publish posts are
referred to as “roots” and users who reshare them are “leaves”. However,
differently from Twitter, we don’t have access to any information about leaf
users on Facebook. We refer generically to Twitter users and Facebook pages
and groups as “accounts.”

To compare Facebook and Twitter in a meaningful way, we compare root users
with root pages/groups, original tweets with original posts, and retweet
counts with reshare counts. We define prevalence as the sum of original tweets
and retweets on Twitter, and as the sum of original posts and reshares on
Facebook.

####  2.3.3 YouTube data.

We observed a high prevalence of links pointing to youtube.com on both
platforms — over 64k videos on Twitter and 204k on Facebook. Therefore, we
also provide an analysis of popular videos published on Facebook and Twitter.
Specifically, we focus on popular YouTube videos that are likely to contain
low-credibility content. To approach this in a systematic manner, we assume
that videos removed by YouTube are more likely to include low-credibility
information. In addition, we also consider private videos to be suspicious,
since this can be used as a tactic to evade the platform’s sanctions when
violating terms of service.

To identify the most popular and suspicious YouTube content, we first select
the 16,669 videos shared at least once on both platforms. We focus on the
1,828 (11%) of these videos that had been removed or made private by querying
the YouTube API Videos:list endpoint to collect metadata. We rank these videos
by their prevalence on each platform. As validation for using unavailability
to flag suspicious videos, we manually inspected a small sample (about 1%) of
unavailable videos and confirmed their low-credibility through cross-
referencing with other YouTube content (e.g., by identifying other low-
credibility videos that describe or link to them).

###  2.4 Link extraction

Table 2:  List of URL shortening services.  bit.ly | dlvr.it | liicr.nl | tinyurl.com
---|---|---|---
goo.gl | ift.tt | ow.ly | fxn.ws
buff.ly | back.ly | amzn.to | nyti.ms
nyp.st | dailysign.al | j.mp | wapo.st
reut.rs | drudge.tw | shar.es | sumo.ly
rebrand.ly | covfefe.bz | trib.al | yhoo.it
t.co | shr.lc | po.st | dld.bz
bitly.com | crfrm.us | flip.it | mf.tt
wp.me | voat.co | zurl.co | fw.to
mol.im | read.bi | disq.us | tmsnrt.rs
usat.ly | aje.io | sc.mp | gop.cm
crwd.fr | zpr.io | scq.io | trib.in
owl.li |  |  |

Estimating the prevalence of low-credibility information requires matching
URLs, extracted from tweets and Facebook metadata, against our lists of low-
and high-credibility websites. As shortened links are very common, we also
identified 49 link shortening services that appear at least 50 times in our
datasets (Table 2) and expanded shortened URLs referring to these services
through HTTP requests to obtain the actual domains. We finally match the
extracted and expanded links against the lists of low- and high-credibility
domains. A breakdown of matched posts/tweets is shown in Table 3. For low-
credibility content, the ratio of retweets to tweets is 2.7:1, while the ratio
of reshares to posts is 68:1. This large discrepancy is due to various
factors: the difference in traffic on the two platforms, the fact that we only
have a 10% sample of tweets, and the bias toward popular pages and groups on
Facebook.

Table 3: Breakdown of Facebook and Twitter posts/tweets matched to low- and high-credibility domains. | Low-credibility | High-credibility
---|---|---
Facebook |  |
Original posts | 303,119 | 1,194,634
Reshares | 20,462,035 | 98,415,973
Twitter |  |
Original tweets | 245,620 | 734,409
Retweets | 653,415 | 2,184,050

##  3 Infodemic prevalence

In this section, we provide results about the prevalence of links to low-
credibility domains on the two platforms. As described in the Methods section,
we sum tweets and retweets for Twitter, and original posts and reshares for
Facebook. Note that deleted content is not included in our data. Therefore,
our estimations should be considered as lower bounds for the prevalence of
low-credibility information on both platforms.

###  3.1 Prevalence trends

Figure 3: Infodemic content surge on both platforms around the COVID-19
pandemic waves, from Jan. 1 to Oct. 31, 2020. All curves are smoothed via
7-day moving averages. (a) Daily volume of posts/tweets linking to low-
credibility domains on Twitter and Facebook. Left and right axes have
different scales and correspond to Twitter and Facebook, respectively. (b)
Overall daily volume of pandemic-related tweets and worldwide COVID-19
hospitalization rates (data source: Johns Hopkins University). (c) Daily ratio
of volume of low-credibility links to volume of high-credibility links on
Twitter and Facebook. The noise fluctuations in early January are due to low
volume. The horizontal lines indicate averages across the period starting Feb.
1.

We plot the daily prevalence of links to low-credibility sources on Twitter
and Facebook in Fig. 3(a). The two time series are strongly correlated
(Pearson $r=0.87$, $p<0.01$). They both experience a drastic growth during
March, when the number of COVID-19 cases was growing worldwide. Towards
summer, the prevalence of low-credibility information decreases to a
relatively low level and then becomes more stable.

To analyze the Infodemic surge with respect to the pandemic’s development and
public awareness, Fig. 3(b) shows the worldwide hospitalization rate and the
overall volume of tweets in our collection. The Infodemic surge roughly
coincides with the general attention given to the pandemic, captured by the
overall Twitter volume. The peak in hospitalizations trails by a few weeks. A
similar delay was recently reported between peaks of exposure to Infodemic
tweets and of COVID-19 cases in different countries [14]. The present findings
indicate that this delay is related to general attention toward the pandemic
rather than specifically toward misinformation.

To further explore whether the decrease of low-credibility information is
organic or due to platform interventions, we also compare the prevalence of
low-credibility content to that of links to credible sources. As shown in Fig.
3(c), the ratios are relatively stable across the observation period. These
results suggest that the prevalence of low-credibility content is mostly
driven by the public attention to the pandemic in general, which progressively
decreases after the initial outbreak. We finally observe that Twitter exhibits
a higher ratio of low-credibility information than Facebook (32% vs. 21% on
average). However, this difference may be due to biases in the data collection
protocols, as discussed in Methods.

###  3.2 Prevalence of specific domains

Figure 4:  Total prevalence of links to low- and high-credibility domains on
both (a) Facebook and (b) Twitter. Due to space limitation, we only show the
40 most frequent domains on the two platforms. The high-credibility domains
are all within the top 40. We also show low-credibility information as a whole
(cf. “low cred combined”).

We use the high-credibility domains as a benchmark to assess the prevalence of
low-credibility domains on each platform. As shown in Fig. 4, we notice that
the low-credibility sources exhibit disparate levels of prevalence. We also
see that low-credibility content as a whole reaches considerable volume on
both platforms, with prevalence surpassing every single high-credibility
domain considered in this study. On the other hand, low-credibility domains
generally exhibit much lower prevalence compared to high-credibility ones,
(with a few exceptions, notably thegatewaypundit.com and breitbart.com).

###  3.3 Source popularity comparison

Figure 5:  (a) Rank comparison of low-credibility sources on Facebook and
Twitter. Each dot in the figure represents a low-credibility domain. The most
popular domain ranks first. Domains close to the vertical line have similar
ranks on the two platforms. Domains close to the edges are much more popular
on one platform or the other. We annotate a few selected domains that exhibit
high rank discrepancy. (b) A zoom-in on the sources ranked among the top 50 on
both platforms (highlighted square in (a)).

As shown in Fig. 4, we observed that low-credibility websites may have
different prevalence on the two platforms. To further contrast the most
popular sources on Twitter and Facebook, we measure the popularity of websites
on each platform by ranking them by prevalence, and then compare the resulting
ranks in Fig. 5. The ranks on the two platforms are not strongly correlated
(Spearman $r=0.57$, $p<0.01$). A few domains are much more popular or only
appear on one of the platforms (see annotations in Fig. 5(a)). We also show
the domains that are very popular on both platforms in Fig. 5(b). They are
dominated by right-wing and state sources, such as breitbart.com,
washingtontimes.com, thegatewaypundit.com, oann.com, and rt.com.

###  3.4 YouTube Infodemic content

Thus far, we examined the prevalence of links to low-credibility sources.
However, a significant portion of the links shared on Twitter and Facebook
point to YouTube videos, which can also carry COVID-19 misinformation.
Previous work has shown that bad actors utilize YouTube in this manner for
their campaigns [42]. Specifically, anti-scientific narratives on YouTube
about vaccines, Idiopathic Pulmonary Fibrosis, and the COVID-19 pandemic have
been documented [9, 15, 10].

Figure 6:  Rank comparison of top 500 suspicious YouTube videos on Facebook
and Twitter. The most popular domain ranks first. Each dot in the figure
represents a suspicious video. Videos close to the vertical line have similar
ranks on both platforms. Videos close to the edges are more popular on one
platform.

To measure the prevalence of Infodemic content from YouTube, we consider the
unavailability of videos as an indicator of suspicious content, as explained
in the Methods section. Fig. 6 compares the prevalence rankings on Twitter and
Facebook for unavailable videos ranked within the top 500 on both platforms.
These videos are linked between 6 and 980 times on Twitter and 39 and 64,257
times on Facebook. While we cannot rigorously measure rank correlation due to
the exclusion of low-prevalence videos, we do not observe a correlation in the
cross-platform popularity of suspicious content from a qualitative inspection
of the figure. A caveat to this analysis is that the same low-credibility
content can be found within many videos, each having a unique video ID. Some
of these videos are promptly removed while others are not. Therefore, the lack
of correlation could be driven by YouTube’s efforts to remove Infodemic
content in conjunction with attempts by uploaders to counter those efforts.

Figure 7:  Percentages of suspicious YouTube videos against their percent rank
among all videos linked from pandemic-related tweets/posts on both Twitter and
Facebook.

Having looked at the prevalence of suspicious content from YouTube, we wish to
explore the question from another angle: are popular videos more likely to be
suspicious? Fig. 7 shows this to be the case on both platforms: a larger
portion of videos with higher prevalence are unavailable, but the trend is
stronger on Twitter than on Facebook. The overall trend suggests that YouTube
may have a bias toward moderating videos that attract more attention. For
example, an Infodemic video that is spreading virally on Twitter may receive
many abuse reports on YouTube.

##  4 Infodemic spreaders

Links to low-credibility sources are published on social media through
original tweets and posts first, then retweeted and reshared by leaf users. In
this section, we study this dissemination process on Twitter and Facebook with
a focus on the top spreaders, or “superspreaders”: those responsible for the
original publication of the most popular Infodemic content.

###  4.1 Concentration of influence

We wish to measure whether low-credibility content originates from a wide
range of users, or can be attributed to a few influential actors. For example,
a source published 100 times could owe its popularity to 100 distinct users,
or to a single root whose post is republished by 99 leaf users. To quantify
the concentration of original posts/tweets or reshares/retweets for a source
$s$, we use the inverse normalized entropy [24], defined as:

$\mathcal{C}_{s}=1+\sum_{r=1}^{N_{s}}\frac{P_{r}(s)\log{P_{r}(s)}}{\log{N_{s}}}\text{,}$

where $r$ represents a root user/group/page linking to $s$, $P_{r}(s)$ stands
for the fraction of posts/tweets/reshares/retweets linking to $s$ and
associated with $r$, and $N_{s}$ is the total number of roots linking to $s$.
Entropy measures how evenly quantities of content are distributed across
roots; it is normalized to account for the varying numbers of roots in
different cases. Its inverse captures concentration, and is defined in the
unit interval. It is maximized at 1 when the content originates from a single
root user/group/page, and minimized at 0 when each root makes an equal
contribution. We set $\mathcal{C}_{s}=1$ when $N_{s}=1$.

Figure 8:  Evidence of Infodemic superspreaders. Boxplots show the median
(white line), 25th–75th percentiles (boxes), 5th–95th percentiles (whiskers),
and outliers (dots). Significance of statistical tests is indicated by ***
($p<0.001$). (a) Distributions of the concentration of original tweets,
retweets, original posts, and reshares linking to low-credibility domains
around root accounts. Each domain corresponds to one observation. (b)
Distributions of the total number of retweets and reshares of low-credibility
content posted by verified and unverified accounts. Each account corresponds
to one observation. (c) Fractions of original tweets, retweets, original
posts, and reshares by verified accounts.

Let us gauge the concentration of activity around root accounts through their
numbers of original tweets/posts for each source. Similarly, we calculate the
concentration of popularity around the root accounts using their numbers of
retweets/reshares for each source. We show the distributions of these
concentration variables in Fig. 8(a). On both platforms, we find that
popularity is significantly more concentrated around root accounts compared to
their activity ($p<0.001$ for paired sample t-tests). This suggests the
existence of superspreaders: despite the diversity of root accounts publishing
links to low-credibility content on both platforms, only messages from a small
group of influential accounts are shared extensively.

###  4.2 Who are the Infodemic superspreaders?

Both Twitter and Facebook provide verification of accounts and embed such
information in the metadata. Although the verification processes differ, we
wish to explore the hypothesis that verified accounts on either platform play
an important role as top spreaders of low-credibility content. Fig. 8(b)
compares the popularity of verified accounts to unverified ones on a per-
account basis. We find that verified accounts tend to receive a significantly
higher number of retweets/reshares on both platforms ($p<0.001$ for Mann-
Whitney U-tests).

We further computed the proportion of original tweets/posts and
retweets/reshares that correspond to verified accounts on both platforms.
Verified accounts are a small minority compared to unverified ones, i.e., 1.9%
on Twitter and 4.5% on Facebook, among root accounts involved in publishing
low-credibility content. Despite this, Fig. 8(c) shows that verified accounts
yield almost 40% of the total number of retweets on Twitter and almost 70% of
reshares on Facebook.

These results suggest that verified accounts play an outsize role in the
spread of Infodemic content. Are superspreaders all verified? To answer this
question, let us analyse superspreader accounts separately for each low-
credibility source. We extract the top user/page/group (i.e., the account with
most retweets/reshares) for each source, and find that 19% and 21% of them are
verified on Twitter and Facebook, respectively. While these values are much
higher than the percentages of verified accounts among all roots, they show
that not all superspreaders are verified.

Table 4:  List of social media handles for the 23 top low-credibility sources
from Fig. 5(b). Accounts with a checkmark (✓) are verified. Accounts with an
asterisk (*) are the top spreaders for the corresponding domains.

Domain | Twitter handle | Facebook page/group
---|---|---
thegatewaypundit.com | @gatewaypundit ✓* | @gatewaypundit ✓*
breitbart.com | @BreitbartNews ✓* | @Breitbart ✓*
zerohedge.com | @zerohedge * | @ZeroHedge *
washingtontimes.com | @WashTimes ✓* | @TheWashingtonTimes ✓
rt.com | @RT_com ✓* | @RTnews ✓*
swarajyamag.com | @SwarajyaMag ✓* | @swarajyamag ✓*
pjmedia.com | @PJMedia_com * | @PJMedia ✓*
waynedupree.com | @WayneDupreeShow ✓* | @WayneDupreeShow ✓*
bongino.com | @dbongino ✓* | @dan.bongino ✓*
trendingpolitics.com | – | @trendingpoliticsdotcom
oann.com | @OANN ✓* | @OneAmericaNewsNetworkD ✓*
wnd.com | @worldnetdaily ✓* | @WNDNews *
sputniknews.com | @SputnikInt ✓* | @SputnikNews ✓*
dailystar.co.uk | @dailystar ✓* | @thedailystar ✓*
politicalflare.com | @nicolejames | @nicolejameswriter
thenationalpulse.com | @RaheemKassam ✓* | @thenationalpulse *
americanthinker.com | @AmericanThinker | @AmericanThinker *
gellerreport.com | @PamelaGeller ✓ | @pamelageller ✓
cbn.com | @CBNOnline ✓ | @cbnonline ✓
100percentfedup.com | @100PercFEDUP * | @100PercentFEDUp *
newspunch.com | – | @thepeoplesvoicetv *
thepoliticalinsider.com | @TPInsidr | @ThePoliticalInsider ✓*
hannity.com | @seanhannity ✓* | @SeanHannity ✓*

Who are the top spreaders of Infodemic content? Table 4 answers this question
for the 23 top low-credibility sources in Fig. 5(b). We find that the top
spreader for each source tends to be the corresponding official account. For
instance, about 20% of the retweets containing links to thegatewaypundit.com
pertain to @gatewaypundit, the official handle associated with The Gateway
Pundit website, on Twitter. The remaining retweets have 10,410 different root
users. Similarly, on Facebook, among all 2,821 pages/groups that post links to
thegatewaypundit.com, the official page @gatewaypundit accounts for 68% of the
reshares. We observe in Table 4 that most of the top low-credibility sources
have official accounts on both Twitter and Facebook, which tend to be verified
(15 out of 21 on Twitter and 15 out of 23 on Facebook). They are also the top
spreaders of those domains in 16 out of 21 cases on Twitter and 18 out of 23
on Facebook.

##  5 Infodemic manipulation

Here we consider two types of inauthentic behaviors that can be used to spread
and amplify COVID-19 misinformation: coordinated networks and automated
accounts.

###  5.1 Coordinated amplification of low-credibility content

Social media accounts can act in a coordinated fashion (possibly controlled by
a single entity) to increase influence and evade detection [27, 25, 37]. We
apply the framework proposed by [27] to identify coordinated efforts in
promoting low-credibility information, both on Twitter and Facebook.

The idea is to build a network of accounts where the weights of edges
represent how often two accounts link to the same domains. A high weight on an
edge means that there is an unusually high number of domains shared by the two
accounts. We first construct a bipartite graph between accounts and low-
credibility domains linked in tweets/posts. The edges of the bipartite graph
are weighted using TF-IDF [39] to discount the contributions of popular
sources. Each account is therefore represented as a TF-IDF vector of domains.
A projected co-domain network is finally constructed, with edges weighted by
the cosine similarity between the account vectors.

We apply two filters to focus on active accounts/pages/groups and highly
similar pairs. On Twitter, the accounts must have at least 10 tweets
containing links, and we retain edges with similarity above 0.99. On Facebook,
the pages/groups must have at least 5 posts containing links, and we retain
edges with similarity above 0.95. These thresholds are selected by manually
inspecting the outputs.

Figure 9: Networks showing clusters that share suspiciously similar sets of
sources on (top) Twitter and (bottom) Facebook. Nodes represent Twitter
accounts or Facebook pages/groups. Edges are drawn between pairs of nodes that
share an unlikely high number of the same website domains. The most shared
sources are annotated for some of the clusters. Facebook pages associated with
Salem Media Group radio stations are highlighted by a dashed box.

Fig. 9 shows dense connected components in the co-domain networks for Twitter
and Facebook. These clusters of accounts share suspiciously similar sets of
sources. They likely act in coordinated fashion to amplify Infodemic messages,
and are possibly controlled by the same entity or organization. We highlight
the fact that using a more stringent threshold on the Twitter dataset yields a
higher number of clusters than a more lax threshold on the Facebook dataset.
However, this does not necessarily imply a higher level of abuse on Twitter;
it could be due to the difference in the units of analysis. On Facebook, we
only have access to public groups and pages with a bias toward high
popularity, and not to all accounts as on Twitter.

An examination of the sources shared by the suspicious clusters on both
platforms shows that they are predominantly right-leaning and mostly
U.S.-centric. The list of domains shared by likely coordinated accounts on
Twitter is mostly concentrated on the leading low-credibility sources, such as
breitbart.com and thegatewaypundit.com, while likely coordinated groups and
pages on Facebook link to a more varied list of sources. Some of the amplified
websites feature polarized rhetoric, such as defending against “attack by
enemies” (see www.frontpagemag.com/about) or accusations of “liberal bias”
(cnsnews.com/about-us), among others. Additionally, there are clusters on both
platforms that share Russian state-affiliated media such as rt.com and an
Indian right-wing magazine (swarajyamag.com).

In terms of the composition of the clusters, they mostly consist of users,
pages, and groups that mention President Trump or his campaign slogans. Some
of the Facebook clusters are notable because they consist of groups or pages
that are owned by organizations with a wider reach beyond the platform, or
that are given an appearance of credibility by being verified. Examples of the
former are the pages associated with The Answer radio stations (highlighted in
Fig. 9). These are among 100 stations owned by the publicly traded Salem Media
Group, which also airs content on 3,100 affiliate stations. Examples of
verified pages engaged in likely coordinated behavior are those affiliated
with the non-profit Media Research Center, some of which have millions of
followers. On Twitter, some of the clusters include accounts with tens of
thousands of followers. Many of the suspicious accounts in Fig. 9 no longer
exist.

###  5.2 Role of social bots

We are interested in revealing the role of inauthentic actors in spreading
low-credibility information on social media. One type of inauthentic behavior
stems from accounts controlled in part by algorithms, known as social bots
[13]. Malicious bots are known to spread low-credibility information [36] and
in particular create confusion in the online debate about health-related
topics like vaccination [4].

We adopt BotometerLite [43], a bot detection model that allows efficient bot
detection on Twitter. BotometerLite generates a bot score between 0 and 1 for
each Twitter account; higher scores indicate bot-like profiles. To the best of
our knowledge, there are no similar techniques designed for Facebook, because
insufficient training data is available. Therefore we limit this analysis to
Twitter.

Figure 10:  Total number of tweets with links posted by likely humans vs.
likely bots for each low-credibility source. The slope of the fitted line is
1.04. The color of each source represents the difference between its
popularity rank on the two platforms. Red means more popular on Facebook, blue
more popular on Twitter.

When applying BotometerLite to our Twitter dataset, we use 0.5 as a threshold
to categorize accounts as likely humans or likely bots. For each domain, we
calculate the total number of original tweets plus retweets authored by likely
humans ($n_{h}$) and bots ($n_{b}$). We plot the relationship between the two
in Fig. 10. The linear trend on the log-log plot signifies a power law
$n_{h}\sim n_{b}^{\gamma}$ with exponent $\gamma\approx 1.04$, suggesting a
weak level of bot amplification (4%) [36].

While we are unable to perform automation detection on Facebook groups and
pages, the ranks of the low-credibility sources on both platforms allow us to
investigate whether sources with more Twitter bot activity are more prevalent
on Twitter or Facebook. For each domain, we calculate the difference of its
ranks on Twitter and Facebook and use the value of the difference to color the
dots in Fig. 10. The results show that sources with more bot activity on
Twitter are equally shared on both platforms.

##  6 Discussion

In this paper, we provide the first comparison between the prevalence of low-
credibility content related to the COVID-19 pandemic on two major social media
platforms, namely Twitter and Facebook. Our results indicate that the primary
drivers of low-credibility information tend to be high-profile, official, and
verified accounts. We also find evidence of coordination among accounts
spreading Infodemic content on both platforms, including many controlled by
influential organizations. Since automated accounts do not appear to play a
strong role in amplifying content, these results indicate that the COVID-19
Infodemic is an overt, rather than a covert, phenomenon.

We find that, taken as a whole, low-credibility content has higher prevalence
than content from any single high-credibility source. However, there is
evidence of differences in the misinformation ecosystems of the two platforms,
with many low-credibility websites and suspicious YouTube videos at higher
prevalence on one platform when compared to the other. Such a discrepancy
might be due to different user demographics on the two platforms [30] or to
the asymmetric presence of official accounts associated with specific low-
credibility websites.

During the first months of the pandemic, we observe similar surges of low-
credibility content on both platforms. The strong correlation between the
timelines of low- and high-credibility content volume reveals that these peaks
were likely driven by public attention to the crisis rather than by bursts of
malicious content.

Our results provide us with a way to assess evidence on how effective the two
platforms have been at combating the Infodemic. The ratio of low- to high-
credibility information on Facebook is lower than on Twitter, indicating that
perhaps Facebook was more effective. On the other hand, we also find that
verified accounts played a stronger role on Facebook than Twitter in spreading
low-credibility content. However, these comparisons are qualified by the
different data collection biases. In general, the unavailable videos shared
from YouTube tended to have asymmetric prevalence on Facebook and Twitter. In
addition to the aforementioned reasons also applicable to websites, a
speculative interpretation of this finding is that the videos may have been
quickly removed by YouTube before they had a chance to migrate across
platforms. Subsequent duplicates of removed content may also play a role, with
one version becoming popular on Facebook and another on Twitter.

There are a number of limitations to our work. As we have remarked throughout
the paper, both inherent differences between the platforms and differences in
data sampling and selection biases make direct and fair comparisons impossible
in many cases. The content collected from the Twitter Decahose is biased
toward active users due to being sampled on a per-tweet basis. The Facebook
accounts provided by CrowdTangle are biased toward popular pages and public
groups, and data availability is also based upon requests made by other
researchers. The small set of keywords driving our data collection pipeline
may have introduced additional biases in the analyses. This is an inevitable
limitation of any collection system, including the Twitter COVID-19 stream
(developer.twitter.com/en/docs/labs/covid19-stream/filtering-rules). The use
of source-level rather than article-level labels for selecting low-credibility
content is necessary [17], but not ideal; some links from low-credibility
sources may point to credible information. In addition, the list of low-
credibility sources was not specifically tailored to our subject of inquiry.
Finally, we do not have access to many deleted Twitter and Facebook posts,
which may lead to an underestimation of the Infodemic’s prevalence. All of
these limitations highlight the need for cross-platform, privacy-sensitive
protocols for sharing data with researchers [28].

Low-credibility information on the pandemic is an ongoing concern for society.
Our study raises a number of questions about how social media platforms are
handling the flow of information and are allowing likely dangerous content to
spread. Regrettably, since we find that high-status accounts play an important
role, addressing this problem will probably prove difficult. As Twitter and
Facebook have increased their moderation of COVID-19 misinformation, they have
been accused of political bias. While there are many moral and ethical
considerations around free speech and censorship, our work suggests that these
questions cannot be avoided and are an important part of the debate around how
we can improve our information ecosystem.

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# On generalization of a theorem of Harish-Chandra

Taiwang DENG  Yau Mathematical Sciences Center, Tsinghua University, Haidian
District, Beijing, 100084, China<EMAIL_ADDRESS>
###### Abstract.

In this paper under some conditions we generalize a theorem of Harish-Chandra
concerning representability of Fourier transforms of orbital integrals.

###### Key words and phrases:

Fourier transform, Orbital integrals, Orbital Gauss sum, Harish-Chandra
theorem

##  1\. Introduction

In this short paper we consider the question of representability of Fourier
transform of orbital integrals for general representations of a reductive
groups $\displaystyle H$. In the case
$\displaystyle\rho:H\rightarrow\mathfrak{h}$ is the adjoint representation,
let

$\displaystyle\widehat{\mathop{\mathrm{Orb}}\nolimits}(X,f):=\mathop{\mathrm{Orb}}\nolimits(X,\widehat{f}),f\in{\mathcal{C}}_{c}^{\infty}(\mathfrak{h}),X\in\mathfrak{h},$

where $\displaystyle\mathop{\mathrm{Orb}}\nolimits(X,f)$ is the orbital
integral of $\displaystyle f$ along the $\displaystyle H$-orbit of
$\displaystyle X$, then Harish-Chandra [2, Theorem 1.1] shows that the
distribution is representable by a kernel function
$\displaystyle\kappa(X,\cdot)$. He approaches the problem by local trace
formula and a finiteness theorem due to Howe. In [7, §4], Zhang raises the
question whether analogues theorem holds for general representations of
$\displaystyle H$. Note that Howe’s finiteness theorem is known to fail in
general(see loc.cit. for counter examples). In this paper, we answer this
question affirmatively under some mild conditions. Our main result is Theorem
2.5. After we finish the proof, we notice that our proof is analogues to that
of [3].

Acknowledgement. The work is done when the author is a postdoc at Yau
Mathematical Sciences Center of Tsinghua University. He wants to thank their
hospitality. He also would like to thank Bin Xu for a careful reading of the
first version of the paper.

##  2\. Fourier Transform of Orbital integrals

Let $\displaystyle F$ be a finite extension of $\displaystyle\mathbb{Q}_{p}$.
Let $\displaystyle\mathcal{O}_{F}$ be the ring of integers of $\displaystyle
F$ and $\displaystyle\pi_{F}$ be its uniformizer. Let
$\displaystyle\mathrm{k}_{F}$ be its residue field. We fix a valuation
$\displaystyle\nu_{\pi_{F}}$ on $\displaystyle F$ with
$\displaystyle\nu_{\pi_{F}}(\pi_{F})=1$. Let us also fix an additive character
$\displaystyle\psi$ of $\displaystyle F$ satisfying

$\displaystyle\psi|_{\mathcal{O}_{F}}=1,\quad\sum_{a\in\pi_{F}^{-1}\mathcal{O}_{F}/\mathcal{O}_{F}}\psi(a)=1.$

Assume that $\displaystyle H$ is a quasi-split reductive group defined over
$\displaystyle\mathcal{O}_{F}$. In general, $\displaystyle H\otimes F$ may
have different models over $\displaystyle\mathcal{O}_{F}$, and the reductive
models depend on a choice of a special vertex on the Bruhat-Tits building of
$\displaystyle H\otimes F$. Note that it follows from our assumption that the
group $\displaystyle H\otimes\mathrm{k}_{F}$ is also reductive. Finally we
denote by $\displaystyle\mathfrak{h}$ the Lie algebra of $\displaystyle H$.

Let us consider the following setup. Assume that $\displaystyle(\rho,V)$ is an
algebraic representation of a reductive group $\displaystyle H$ over
$\displaystyle\mathcal{O}_{F}$. We assume further that there is an central
embedding $\displaystyle\iota:\mathbb{G}_{m}\rightarrow H$ such that the
composition with $\displaystyle\rho$ gives
$\displaystyle\rho\circ\iota=\mathop{\mathrm{Id}}\nolimits:\mathbb{G}_{m}\rightarrow\mathbb{G}_{m}$,
where the target is identified with the center of
$\displaystyle\mathop{\mathrm{Aut}}\nolimits(V)$.

According to [5, §3 and §7], there exists an involution $\displaystyle\theta$
of $\displaystyle H$ defined over $\displaystyle F$(called Chevalley
involution) which interchanges an irreducible and its contragredient.

The existence of the Chevalley involution allows us to construct a non-
degenerate symmetric pairing on $\displaystyle V_{F}=V\otimes F$ (without
confusion we identify $\displaystyle V_{F}$ with its $\displaystyle
F$-rational points $\displaystyle V(F)$)

$\displaystyle\langle,\rangle:V_{F}\otimes V_{F}\rightarrow F$

satisfying

$\displaystyle\langle hX,\theta(h)Y\rangle=\langle X,Y\rangle,\quad\forall
X,Y\in V(F),h\in H(F).$

######  Definition 2.1.

We fix a basis $\displaystyle\\{e_{1},\cdots,e_{d}\\}$ of $\displaystyle
V(\mathcal{O}_{F})$. And define a valuation $\displaystyle\nu_{\pi_{F}}$ on
$\displaystyle V$ by letting

$\displaystyle\nu_{\pi_{F}}(\sum_{i}x_{i}e_{i})=\min\\{\nu_{\pi_{F}}(x_{i}):i=1,\cdots,d\\}.$

We also define the norm
$\displaystyle|Y|=|\mathrm{k}_{F}|^{-\nu_{\pi_{F}}(Y)}$. And for any subspace
$\displaystyle V_{1}$ of $\displaystyle V(F)$, define

$\displaystyle d_{\langle,\rangle}(V_{1})=\min\\{\nu_{\pi_{F}}(Y)|Y\in
V_{1},\langle Z,Y\rangle\in\mathcal{O}_{F},\forall Z\in
V(\mathcal{O}_{F})\\}.$

Finally, define the depth of $\displaystyle\langle,\rangle$ by

$\displaystyle
d_{\langle,\rangle}:=\max\\{d_{\langle,\rangle}(V_{1})|V_{1}\subseteq V\\}.$

It follows immediately that

######  Lemma 2.2.

The valuation $\displaystyle\nu_{\pi_{F}}$ is independent of the choice of the
basis of $\displaystyle V(\mathcal{O}_{F})$.

######  Lemma 2.3.

For any $\displaystyle Y\in V(F)$ with $\displaystyle\nu_{\pi_{F}}(Y)\geq
d_{\langle,\rangle}$, we have

$\displaystyle\langle Z,Y\rangle\in\mathcal{O}_{F},\forall Z\in
V(\mathcal{O}_{F}).$

###### Proof.

In fact, let $\displaystyle Y\in V(F)$ with $\displaystyle\nu_{\pi_{F}}(Y)\geq
d_{\langle,\rangle}$ and $\displaystyle Z\in V(\mathcal{O}_{F})$ such that

$\displaystyle\langle Z,Y\rangle\notin\mathcal{O}_{F}.$

Then take $\displaystyle V_{1}$ to be the subspace generated by $\displaystyle
Y$. Then $\displaystyle d_{\langle,\rangle}(V_{1})\geq\nu_{\pi_{F}}(Y)\geq
d_{\langle,\rangle}$, which is absurd. ∎

######  Assumption 2.4.

We assume that our choice of $\displaystyle\langle,\rangle$ is of depth zero.

Let $\displaystyle{\mathcal{C}}_{c}^{\infty}(V)$ be the space of locally
constant functions on $\displaystyle V(F)$ with compact support. For
$\displaystyle Y\in V(F)$ and $\displaystyle f\in{\mathcal{S}}(V)$, we define
the Fourier transform of $\displaystyle f$ by

$\displaystyle\widehat{f}(Y)=\int_{V(F)}f(X)\psi(\langle X,Y\rangle)dX.$

We define the orbital integrals of $\displaystyle f$ on $\displaystyle V(F)$
to be a linear functional on $\displaystyle{\mathcal{C}}_{c}^{\infty}(V)$:

$\displaystyle\mathop{\mathrm{Orb}}\nolimits(Y,f):=\int_{{\mathscr{O}}_{H}(Y)}f(X)dX$

And we denote by $\displaystyle{\mathscr{S}}(V)$ the linear functional on
$\displaystyle{\mathcal{C}}_{c}^{\infty}(V)$ generated by the orbital
integrals. And we are interested in the question of representability of the
orbital integral $\displaystyle\mathop{\mathrm{Orb}}\nolimits(Y,\widehat{f})$,
cf. [7, §3].

Let us make the assumption that $\displaystyle P(X)$ is a quasi-invariant in
the polynomial ring $\displaystyle F[V]$ such that the set

$\displaystyle\\{X\in V(F)|P(X)\neq 0\\}$

is open and dense in $\displaystyle V(F)$( in the p-adic topology). By
assumption, we have

$\displaystyle(g.P)(X)=P(g^{-1}X)=\nu^{-1}(g)P(X),\forall g\in H,$

for some algebraic character $\displaystyle\nu:H\rightarrow\mathbb{G}_{m}$.
Let $\displaystyle\nu_{s}(g)=|\nu(g)|^{s}$. We require that $\displaystyle\nu$
is nontrivial.

We can modify our definition of orbital integral to be

$\displaystyle\mathop{\mathrm{Orb}}\nolimits(Y,f,\nu_{s}):=\int_{{\mathscr{O}}_{H}(Y)}f(X)|P(X)|^{s}dX,$

where $\displaystyle dX$ is certain quasi-invariant measure on
$\displaystyle{\mathscr{O}}_{H}(Y)$ which will be discussed in detailed later.
Note that by fixing a base point $\displaystyle Y_{0}\in{\mathscr{O}}_{H}(Y)$,
we can write

$\displaystyle\mathop{\mathrm{Orb}}\nolimits(Y_{0},f,\nu_{s})=|P(Y_{0})|^{s}\int_{H/H_{Y_{0}}}f(hY_{0})\nu_{s}(h)d\overline{h}.$

In particular, by letting $\displaystyle s=0$ we fall back to our previous
definition. Our main result in this section is the following

######  Theorem 2.5.

There exists a locally constant function
$\displaystyle\kappa(\nu_{s},X,\cdot)$ well defined outside a measure zero
subset on $\displaystyle V(F)$ given by

$\displaystyle\kappa(\nu_{s},X,Y)=\int_{{\mathscr{O}}_{H}(X)}|P(Z)|^{s}\psi(\langle
Z,Y\rangle)dZ,$

such that for $\displaystyle f\in{\mathcal{C}}_{c}^{\infty}(V)$,

(2.0.1)
$\displaystyle\displaystyle\mathop{\mathrm{Orb}}\nolimits(X,\widehat{f},\nu_{s})=\int_{V(F)}f(Y)\kappa(\nu_{s},X,Y)dY.$

######  Remark 2.6.

When $\displaystyle V_{F}$ contains only finitely many open $\displaystyle
H$-orbits $\displaystyle\\{{\mathscr{O}}_{1},\cdots,{\mathscr{O}}_{r}\\}$,
then the above theorem gives

$\displaystyle\mathop{\mathrm{Orb}}\nolimits({\mathscr{O}}_{i},\widehat{f},\nu_{s})=\sum_{j}\Gamma_{ij}(\nu_{s})\mathop{\mathrm{Orb}}\nolimits({\mathscr{O}}_{j},f,\theta(\delta^{-1}\nu_{s}))$

where $\displaystyle\delta:H(F)\rightarrow\mathbb{C}^{\times}$ via
$\displaystyle d(h_{1}\overline{h})=\delta(h_{1})d\overline{h}$ and
$\displaystyle\theta(\delta^{-1}\nu_{s}))(h)=\delta^{-1}\nu_{s}(\theta(h))$,
this is the local functional equation for local Igusa Zeta function [4].

We need several lemmas.

######  Lemma 2.7.

For $\displaystyle Z_{0}\in V(F)\backslash\\{0\\}$, let $\displaystyle V_{0}$
be the $\displaystyle\mathcal{O}_{F}$-module generated by $\displaystyle
H(\mathcal{O}_{F}).Z_{0}$. Let

$\displaystyle d_{V_{0}}=\min\\{\nu_{\pi_{F}}(Y)|Y\in V(F),\langle
Z,Y\rangle\in\mathcal{O}_{F},\forall Z\in V_{0}\\}.$

Let

$\displaystyle U_{Z_{0}}=\\{Y\in
V(F)|\psi(\frac{\langle\cdot,\pi_{F}^{d_{V_{0}}-\nu_{\pi_{F}}(Z_{0})-\nu_{\pi_{F}}(Y)}Y\rangle}{\pi_{F}}):V_{0}\otimes\mathrm{k}_{F}\rightarrow\mathbb{C}^{\times}\text{
is nontrivial}\\}.$

Then $\displaystyle U_{Z_{0}}$ is an non-empty open subset of $\displaystyle
V(F)$ whose complement is closed of measure zero.

###### Proof of Lemma 2.7.

The non-emptiness of $\displaystyle U_{Z_{0}}$ follows from the definition of
$\displaystyle d_{V_{0}}$. By definition, if $\displaystyle Y\notin
U_{Z_{0}}$, then $\displaystyle\langle
Z,\pi_{F}^{d_{V_{0}}-\nu_{\pi_{F}}(Y)-\nu_{\pi_{F}}(Z)}Y\rangle\in\pi_{F}\mathcal{O}_{F}$
for all $\displaystyle Z\in V_{0}$. Let $\displaystyle U^{c}$ be the
complement of $\displaystyle U_{Z_{0}}$ in $\displaystyle V(F)$ and
$\displaystyle U^{c}_{n}=U^{c}\cap\pi_{F}^{n}\mathcal{O}_{F}$. It suffices to
show that for any $\displaystyle n>0$, the image of $\displaystyle U_{n}^{c}$
in $\displaystyle\pi_{F}^{n}\mathcal{O}_{F}/\pi_{F}^{n+1}\mathcal{O}_{F}$ is
not surjective. In fact, let $\displaystyle Y\in U_{n}^{c}\backslash
U_{n+1}^{c}$, then $\displaystyle\nu_{\pi_{F}}(Y)=n$, but by definition
$\displaystyle Y_{0}=\pi_{F}^{-n}Y\in U_{0}^{c}$, hence we only need to show
that the image of $\displaystyle U_{0}^{c}$ in $\displaystyle
V(\mathrm{k}_{F})$ is not surjective. But it readily follows from the
definition there exists $\displaystyle Y\in V(F),Z\in V_{0}$ such that
$\displaystyle\nu_{\pi_{F}}(Y)=0$ and $\displaystyle\langle
Z,\pi_{F}^{d_{V_{0}}-\nu_{\pi_{F}}(Z_{0})}Y\rangle\in\mathcal{O}_{F}\backslash\pi_{F}\mathcal{O}_{F}$.
Hence the projection of $\displaystyle U_{0}^{c}$ is not surjective to
$\displaystyle V(\mathrm{k}_{F})$. ∎

######  Lemma 2.8.

We keep the notations in Lemma 2.7. Then for $\displaystyle Y\in U_{Z_{0}}$,
we have

$\displaystyle\sum_{Z\in V_{0}\otimes\mathrm{k}_{F}}\psi(\frac{\langle
Z,\pi_{F}^{d_{V_{0}}-\nu_{\pi_{F}}(Z_{0})-\nu_{\pi_{F}}(Y)}Y\rangle}{\pi_{F}})=0.$

###### Proof of Lemma 2.8.

Note that $\displaystyle\psi(\frac{\langle
Z,\pi_{F}^{d_{V_{0}}-\nu_{\pi_{F}}(Z_{0})-\nu_{\pi_{F}}(Y)}Y\rangle}{\pi_{F}})$
is a non-trivial character on $\displaystyle V_{0}\otimes\mathrm{k}_{F}$. But
for any nontrivial character
$\displaystyle\chi:V_{0}\otimes\mathrm{k}_{F}\rightarrow\mathbb{C}^{\times}$,
assume that $\displaystyle\chi(Z)\neq 1$ and $\displaystyle
V_{0}\otimes\mathrm{k}_{F}=V^{\prime}\oplus\mathrm{k}_{F}Z$, then

$\displaystyle\sum_{X\in
V_{0}\otimes\mathrm{k}_{F}}\chi(X)=\sum_{Z^{\prime}\in
V^{\prime}}\chi(Z^{\prime})\sum_{a\in\mathrm{k}_{F}}\chi(Z)^{a}=0.$

∎

######  Lemma 2.9.

We consider the sequence of functions

$\displaystyle f_{n}=1_{\pi_{F}^{-n}V(\mathcal{O}_{F})},\quad n\in\mathbb{Z}.$

And denote

$\displaystyle
I_{n}(Y)=\int_{{\mathscr{O}}_{H}(X)}f_{n}(Z)|P(Z)|^{s}\psi(\langle
Z,Y\rangle)dZ=\int_{\pi_{F}^{-n}V(\mathcal{O}_{F})\cap{\mathscr{O}}_{H}(X)}|P(Z)|^{s}\psi(\langle
Z,Y\rangle)dZ.$

Then there exists an open dense subset $\displaystyle U$ of $\displaystyle
V(F)$ such that $\displaystyle V(F)\backslash U$ is of measure zero and for
any $\displaystyle Y\in U$, the sequence
$\displaystyle\\{I_{n}(Y)|n\in\mathbb{Z}\\}$ stabilizes as $\displaystyle
n\rightarrow\infty$. Consequently, the sequence
$\displaystyle\\{I_{n}(Y)|n\in\mathbb{Z}\\}$ convereges to a locally constant
function $\displaystyle I_{\infty}(Y)$ on $\displaystyle U$ and the function
$\displaystyle I_{\infty}(Y)$ is of polynomial growth in
$\displaystyle|\nu(\pi_{F})|^{-\nu_{\pi_{F}}(Y)s}$ as $\displaystyle Y$
approaches the complement of $\displaystyle U$.

Remark: The example below with $\displaystyle
H=\mathop{\mathrm{GL}}\nolimits_{n}$ shows that the restriction to
$\displaystyle U$ is necessary.

###### Proof of Lemma 2.9.

By our assumption at the beginning of the section, the image of $\displaystyle
H$ contains the center of $\displaystyle\mathop{\mathrm{Aut}}\nolimits(V)$.
Let $\displaystyle
W_{n}=\pi_{F}^{-n}V(\mathcal{O}_{F})\cap{\mathscr{O}}_{H}(X)$, hence

$\displaystyle\displaystyle I_{n}(Y)$
$\displaystyle\displaystyle=\int_{W_{n}}|P(Z)|^{s}\psi(\langle Z,Y\rangle)dZ$
$\displaystyle\displaystyle=\int_{W_{0}}|P(\frac{Z}{\pi_{F}^{n}})|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})d(\frac{Z}{\pi_{F}^{n}})$
$\displaystyle\displaystyle=(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{0}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ$
$\displaystyle\displaystyle=(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{0}\backslash
W_{-1}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ+(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{-1}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ.$

And we observe that

$\displaystyle\displaystyle(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{-1}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ$
$\displaystyle\displaystyle=(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{0}}|P(\pi_{F}Z)|^{s}\psi(\frac{\langle\pi_{F}Z,Y\rangle}{\pi_{F}^{n}})d(\pi_{F}Z)$
$\displaystyle\displaystyle=(\nu_{s}\delta)^{-n+1}(\pi_{F})\int_{W_{0}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n-1}})d(Z)$ $\displaystyle\displaystyle=I_{n-1}(Y).$

Therefore, we have

$\displaystyle
I_{n}(Y)-I_{n-1}(Y)=(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{0}\backslash
W_{-1}}|P(Z)|^{s}\psi(\frac{\langle Z,Y\rangle}{\pi_{F}^{n}})dZ.$

We need to show that for $\displaystyle n>>0$,

$\displaystyle\int_{W_{0}\backslash W_{-1}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ=0.$

In fact by assumption $\displaystyle H$ is defined over
$\displaystyle\mathcal{O}_{F}$, therefore $\displaystyle W_{0}\backslash
W_{-1}$ will decompose into $\displaystyle H(\mathcal{O}_{F})$ orbits:

$\displaystyle W_{0}\backslash
W_{-1}=\coprod_{i=1}^{r}H(\mathcal{O}_{F}).Z_{i}$

with $\displaystyle D=\\{Z_{1},\cdots,Z_{r}\\}$ a set of representatives.
Hence it suffices to show that

$\displaystyle\int_{H(\mathcal{O}_{F}).Z_{i}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ=0,\text{ for }n>>0.$

Note that for $\displaystyle g\in H(\mathcal{O}_{F})$, we have

$\displaystyle|P(gZ_{i})|=|\nu(g)||P(Z_{i})|=|P(Z_{i})|$

since $\displaystyle\nu$ is an algebraic character. Hence

$\displaystyle\displaystyle\int_{H(\mathcal{O}_{F}).Z_{i}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ$ $\displaystyle\displaystyle=c_{i}\sum_{h\in
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})}\psi(\frac{\langle
hZ_{i},Y\rangle}{\pi_{F}^{n}})$

with $\displaystyle
c_{i}=\frac{\mathop{\mathrm{vol}}\nolimits(\mathop{\mathrm{ker}}\nolimits(H(\mathcal{O}_{F})\rightarrow
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})))|P(Z_{i})|^{s}}{|\mathop{\mathrm{Im}}\nolimits\\{H^{i}\rightarrow
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})\\}|}$, where $\displaystyle
H^{i}$ is the stabilizer of $\displaystyle Z_{i}$ in $\displaystyle
H(\mathcal{O}_{F})$. Assume that $\displaystyle c\neq 0$ otherwise we are
done.

Now let $\displaystyle Y_{1}=\pi_{F}^{-\nu_{\pi_{F}}(Y)}Y$. And

$\displaystyle\sum_{h\in
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})}\psi(\frac{\langle
hZ_{i},Y\rangle}{\pi_{F}^{n}})=\sum_{h\in
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})}\psi(\frac{\langle
hZ_{i},Y_{1}\rangle}{\pi^{n-\nu_{\pi_{F}}(Y)}}).$

We claim that

(2.0.2) $\sum_{h\in
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})}\psi(\frac{\langle
hZ_{i},Y\rangle}{\pi_{F}^{n}})=0,\quad\forall n\geq\nu_{\pi_{F}}(Y)+3.$

Upon replacing $\displaystyle Y$ by $\displaystyle Y_{1}$ above, we only need
to show that for $\displaystyle\nu_{\pi_{F}}(Y)=0$,

$\displaystyle\sum_{h\in
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})}\psi(\frac{\langle
hZ_{i},Y\rangle}{\pi_{F}^{n}})=0,\quad\forall n\geq 3.$

Recall that by our assumption, for any $\displaystyle Y\in
V(\mathcal{O}_{F})$, $\displaystyle\langle Z,Y\rangle\in\mathcal{O}_{F}$ holds
for any $\displaystyle Z\in V(\mathcal{O}_{F})$(cf. Lemma 2.3). Let
$\displaystyle
H_{n}=\mathop{\mathrm{ker}}\nolimits(H(\mathcal{O}_{F})\rightarrow
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F}))$ and $\displaystyle H_{n,n-1}$
the image of $\displaystyle H_{n-1}$ in $\displaystyle
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})$ under the natural projection.
The key observation is that $\displaystyle H_{n,n-1}$ is a
$\displaystyle\mathrm{k}_{F}$-vector space which is isomorphic to the Lie
algebra $\displaystyle\mathfrak{h}(\mathrm{k}_{F})$ via
$\displaystyle\mathop{\mathrm{Id}}\nolimits+\pi_{F}^{n-1}h\mapsto h$. Note
that the action of $\displaystyle H$ on $\displaystyle V$ induces an action of
$\displaystyle\mathfrak{h}$ on $\displaystyle V$, by which we also denote
$\displaystyle(h,X)\mapsto hX$ for $\displaystyle h\in\mathfrak{h},X\in V(F)$.
Now

$\displaystyle\displaystyle\sum_{h\in
H(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})}\psi(\frac{\langle
hZ_{i},Y\rangle}{\pi_{F}^{n}})$ $\displaystyle\displaystyle=\sum_{h_{1}\in
H(\mathcal{O}_{F}/\pi_{F}^{n-1}\mathcal{O}_{F})}\sum_{\mathop{\mathrm{Id}}\nolimits+\pi_{F}^{n-1}h_{2}\in
H_{n,n-1}}\psi(\frac{\langle
h_{1}Z_{i},Y\rangle}{\pi_{F}^{n}})\psi(\frac{\langle
h_{2}Z_{i},Y\rangle}{\pi_{F}})$ $\displaystyle\displaystyle=\sum_{h_{1}\in
H(\mathcal{O}_{F}/\pi_{F}^{n-1}\mathcal{O}_{F})}\psi(\frac{\langle
h_{1}Z_{i},Y\rangle}{\pi_{F}^{n}})\sum_{h_{2}\in\mathfrak{h}(\mathrm{k}_{F})}\psi(\frac{\langle
h_{2}Z_{i},Y\rangle}{\pi_{F}}).$

Now apply Lemma2.8, we know that there is an open dense subset $\displaystyle
U_{Z_{i}}$ of $\displaystyle V(F)$ such that $\displaystyle V(F)\backslash
U_{Z_{i}}$ is of measure zero and
$\displaystyle\psi(\frac{\langle\cdot,Y\rangle}{\pi_{F}})$ is non-trivial as a
character of $\displaystyle\mathfrak{h}(\mathrm{k}_{F})Z_{i}$, moreover, we
have

$\displaystyle\sum_{h\in\mathfrak{h}(\mathrm{k}_{F})}\psi(\frac{\langle
hZ_{i},Y\rangle}{\pi_{F}})=0.$

Here there is a discrepancy: we are summing over
$\displaystyle\mathfrak{h}(\mathrm{k}_{F})$ instead of
$\displaystyle\mathfrak{h}(\mathrm{k}_{F})Z_{i}$ but since the argument is the
same we omit it. Let $\displaystyle U=\cap U_{Z_{i}}$, then for $\displaystyle
Y\in U$, we have

$\displaystyle\int_{W_{0}\backslash W_{-1}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ=0.$

which finishes the proof of the claim (2.0.2). Now for $\displaystyle Y\in U$,
we have

$\displaystyle
I_{\infty}(Y)=\sum_{n=\nu_{\pi_{F}}(Y)+1}^{\nu_{\pi_{F}}(Y)+2}(\nu_{s}\delta)^{-n}(\pi_{F})\int_{W_{0}\backslash
W_{-1}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{n}})dZ+I_{\nu_{\pi_{F}}(Y)}(Y).$

Furthermore,

$\displaystyle\displaystyle I_{\nu_{\pi_{F}}(Y)}(Y)$
$\displaystyle\displaystyle=(\nu_{s}\delta)^{-\nu_{\pi_{F}}(Y)}(\pi_{F})\int_{W_{0}}|P(Z)|^{s}\psi(\frac{\langle
Z,Y\rangle}{\pi_{F}^{\nu_{\pi_{F}}(Y)}})dZ$
$\displaystyle\displaystyle=(\nu_{s}\delta)^{-\nu_{\pi_{F}}(Y)}(\pi_{F})\int_{W_{0}}|P(Z)|^{s}dZ,$

here we use the assumption that $\displaystyle\langle,\rangle$ is of depth
zero. Now a theorem of Denef states that the integral

$\displaystyle\int_{W_{0}}|P(Z)|^{s}dZ$

is a rational function in $\displaystyle p^{-s}$ and the order of poles of
this rational function are at most
$\displaystyle\mathop{\mathrm{dim}}\nolimits(V)[F:\mathbb{Q}_{p}]$. The last
part of the lemma follows. ∎

Remark: For $\displaystyle H=\mathop{\mathrm{SL}}\nolimits_{2}$ and
$\displaystyle V=\mathop{\mathrm{sl}}\nolimits_{2}$ the adjoint
representation, U. Everling prove that the functions
$\displaystyle\\{f_{n}(Y)\\}$ stabilize as $\displaystyle n\rightarrow\infty$
by employing similar arguments from which he deduces the existence of
$\displaystyle\kappa(X,Y)$(cf. [3, Proposition 4]) in this case.

Let $\displaystyle D$ be a subset of $\displaystyle
V(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})(n>0)$. Let
$\displaystyle\Phi_{D}(X)$ be the characteristic function of the preimage of
$\displaystyle D$ under the projection
$\displaystyle\varphi_{n}:V(\mathcal{O}_{F})\rightarrow
V(\mathcal{O}_{F}/\pi_{F}^{n}\mathcal{O}_{F})$. Let us determine its Fourier
transform. If there is no confusion we will also regard
$\displaystyle\Phi_{D}(X)$ as a function on $\displaystyle
V(\mathcal{O}_{F}/\pi_{F}^{d}\mathcal{O}_{F})$ for $\displaystyle d\geq n$. As
for the Fourier transform, we have

(2.0.3)
$\displaystyle\displaystyle\widehat{\Phi}_{D}(Y)=\int_{V(F)}\Phi(X)\psi(\langle
X,Y\rangle)dX=c_{d}\sum_{X\in
V(\mathcal{O}_{F}/\pi_{F}^{d}\mathcal{O}_{F})}\Phi_{D}(X)\psi(\langle
X,Y\rangle)$

with $\displaystyle
c_{d}=\mathop{\mathrm{vol}}\nolimits(\pi_{F}^{d}V(\mathcal{O}_{F}))$ and
$\displaystyle d\geq\max\\{-\nu_{\pi_{F}}(Y),n\\}$.

###### Proof of Theorem 2.5.

It is clear from the Lemmas before that the function
$\displaystyle\kappa(\nu_{s},X,Y)$ as well as the right hand side of (2.0.1)
is well defined. It remains to show that the two sides of (2.0.1) is equal.

Let $\displaystyle d$ be an integer. Let $\displaystyle
W_{d}=\pi_{F}^{-d}V(\mathcal{O}_{F})\cap{\mathscr{O}}_{H}(X)$. Since
$\displaystyle\widehat{\Phi}_{D}(Y)$ is compactly supported, we have for
$\displaystyle d>>0$,

$\displaystyle\displaystyle\mathop{\mathrm{Orb}}\nolimits(X,\widehat{f},\nu_{s})$
$\displaystyle\displaystyle=\int_{W_{d}}\widehat{\Phi}(Z)|P(Z)|^{s}dZ$
$\displaystyle\displaystyle=c_{d}\int_{W_{d}}(\sum_{Y\in
V(\mathcal{O}_{F}/\pi_{F}^{d}\mathcal{O}_{F})}\Phi_{D}(Y)\psi(\langle
Y,Z\rangle)|P(Z)|^{s}dZ$ $\displaystyle\displaystyle=c_{d}\sum_{Y\in
V(\mathcal{O}_{F}/\pi_{F}^{d}\mathcal{O}_{F})}\Phi_{D}(Y)\int_{W_{d}}\psi(\langle
Y,Z\rangle)|P(Z)|^{s}dZ$ $\displaystyle\displaystyle=c_{d}\sum_{Y\in
V(\mathcal{O}_{F}/\pi_{F}^{d}\mathcal{O}_{F})}\Phi_{D}(Y)I_{d}(Y).$

Let $\displaystyle D=\\{a_{1},\cdots,a_{r}\\}$ in $\displaystyle
V(\mathcal{O}_{F}/\pi^{n}\mathcal{O}_{F})$. If $\displaystyle D$ does not
contain 0, then $\displaystyle\Phi_{D}(Y)\neq 0$ imply that $\displaystyle
1\leq\nu_{\pi_{F}}(Y)\leq n$. Following the proof of Lemma 2.9, take
$\displaystyle d$ large enough we have

$\displaystyle I_{\infty}(Y)=I_{d}(Y)=\kappa(\nu_{s},X,Y).$

Hence

$\displaystyle\mathop{\mathrm{Orb}}\nolimits(X,\widehat{f},,\nu_{s})=c_{d}\sum_{Y\in
V(\mathcal{O}_{F}/\pi_{F}^{d}\mathcal{O}_{F})}\Phi_{D}(Y)I_{\infty}(Y)=\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY.$

It remains to treat the case where $\displaystyle D=\\{0\\}$, i.e.,

$\displaystyle\Phi_{D}(X)=1_{\pi_{F}^{n}V(\mathcal{O}_{F})}(X).$

Let $\displaystyle\delta_{1}$ be the modulus character of $\displaystyle H$
with respect to the Haar measure on $\displaystyle V$. In this case, we have

$\displaystyle\displaystyle\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY$
$\displaystyle\displaystyle=\int_{\pi^{n}V(\mathcal{O}_{F})}\kappa(\nu_{s},X,Y)dY$
$\displaystyle\displaystyle=\delta_{1}(\pi_{F})^{n}\int_{V(\mathcal{O}_{F})}\kappa(\nu_{s},X,\pi_{F}^{n}Y)dY$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\kappa(\nu_{s},X,\pi_{F}^{n+j}Y)dY.$

And

$\displaystyle\displaystyle\kappa(\nu_{s},X,\pi_{F}^{n+j}Y)=\int_{{\mathscr{O}}_{H}(X)}|P(Z)|^{s}\psi(\langle
Z,\pi_{F}^{n+j}Y\rangle)dZ=(\delta\nu_{s}(\pi_{F}))^{-n-j}\int_{{\mathscr{O}}_{H}(X)}|P(Z)|^{s}\psi(\langle
Z,Y\rangle)dZ.$

Therefore,

$\displaystyle\displaystyle\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY=\sum_{j=0}^{\infty}(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-n-j}\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\kappa(\nu_{s},X,Y)dY.$

This sequence( with $\displaystyle s$ as a complex variable) converges on
certain half plane of the complex numbers depending on
$\displaystyle|\nu(\pi_{F})|$.

For $\displaystyle Y\in
V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})$, we have
$\displaystyle\nu_{\pi_{F}}(Y)=0$, therefore for $\displaystyle d\geq 3$ we
have

$\displaystyle\kappa(\nu_{s},X,Y)=I_{\infty}(Y)=I_{d}(Y)=\int_{W_{d}}|P(Z)|^{s}\psi(\langle
Z,Y\rangle)dZ.$

Now it comes to the point that we are allowed to interchange the two integrals
$\displaystyle\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\int_{W_{d}}$
and obtain

$\displaystyle\displaystyle\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY=\sum_{j=0}^{\infty}(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-n-j}\int_{W_{d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle
Z,Y\rangle)dY.$

Now reverse the above process, we obtain

$\displaystyle\displaystyle\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-n-j}\int_{W_{d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle
Z,Y\rangle)dY$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\int_{W_{n+j+d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle\pi_{F}^{n+j}Z,Y\rangle)dY.$

We want to take take limit with respect to $\displaystyle d$ but need some
justification. Let

$\displaystyle
S_{d}=\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\int_{W_{n+j+d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle\pi_{F}^{n+j}Z,Y\rangle)dY.$

then

$\displaystyle\displaystyle S_{\infty}-S_{d}$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\int_{{\mathscr{O}}_{H}(X)\backslash
W_{n+j+d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle\pi_{F}^{n+j}Z,Y\rangle)dY$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\sum_{k=d}^{\infty}\int_{W_{n+j+k+1}\backslash
W_{n+j+k}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle\pi_{F}^{n+j}Z,Y\rangle)dY$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-n-j}\sum_{k=d}^{\infty}\int_{W_{k+1}\backslash
W_{k}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle
Z,Y\rangle)dY$
$\displaystyle\displaystyle=\sum_{j=0}^{\infty}(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-n-j}\int_{{\mathscr{O}}_{H}(X)\backslash
W_{d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle
Z,Y\rangle)dY$
$\displaystyle\displaystyle=\frac{(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-n}}{1-(\delta\delta_{1}^{-1}\nu_{s}(\pi_{F}))^{-1}}\int_{{\mathscr{O}}_{H}(X)\backslash
W_{d}}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle
Z,Y\rangle)dY$

From this we deduce that for $\displaystyle d>>0$,

$\displaystyle|S_{\infty}-S_{d}|<\epsilon,\forall\epsilon>0.$

Hence by letting $\displaystyle d\rightarrow\infty$ we obtain

$\displaystyle\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY=\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\int_{{\mathscr{O}}_{H}(X)}|P(Z)|^{s}dZ\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle\pi_{F}^{n+j}Z,Y\rangle)dY.$

Finally since
$\displaystyle\sum_{j=0}^{\infty}\delta_{1}(\pi_{F})^{n+j}\int_{V(\mathcal{O}_{F})\backslash\pi_{F}V(\mathcal{O}_{F})}\psi(\langle\pi_{F}^{n+j}Z,Y\rangle)dY$
converges absolutely to
$\displaystyle\int_{\pi_{F}^{n}V(\mathcal{O}_{F})}\psi(\langle
Z,Y\rangle)dY$(note that $\displaystyle\delta_{1}(\pi_{F})<1$), we obtain

$\displaystyle\int_{V(F)}\Phi_{D}(Y)\kappa(\nu_{s},X,Y)dY=\int_{{\mathscr{O}}_{H}(X)}|P(Z)|^{s}\int_{\pi^{n}\mathcal{O}_{F}}\psi(\langle
Z,Y\rangle)dY.$

This proves Theorem 2.5 for the functions $\displaystyle\Phi_{D}$. Observe
that the functions $\displaystyle\Phi_{D}$ and their translations under
$\displaystyle H(F)$ form a basis of the space of compactly supported locally
constant functions $\displaystyle{\mathcal{C}}_{c}^{\infty}(V)$, we observe
that the equality (2.0.1) is $\displaystyle H$-equivariant, hence we are done.

∎

######  Definition 2.10.

We will call the function $\displaystyle\kappa(\nu_{s},X,Y)$ and orbital Gauss
function.

Remark: When $\displaystyle\nu=1$ is trivial, we will omit it and simply write
$\displaystyle\kappa(X,Y)$.

In special cases, this function has been studied by T. Taniguchi and F. Thorne
[6].

Let us consider some examples. For $\displaystyle H=GL_{1}$ and $\displaystyle
V=F$. Then $\displaystyle H$ acts on $\displaystyle V$ by scalar
multiplication. Let $\displaystyle P(X)=X$. There are two orbits in this case:
$\displaystyle V_{0}=\\{0\\}$ and $\displaystyle V_{1}=F^{\times}$. The second
orbit is open.

Now let us consider the sequence of functions $\displaystyle
f_{n}=1_{p^{-n}\mathbb{Z}_{p}}$ then the sequence
$\displaystyle\\{f_{n}\\}_{n=1}^{\infty}$ converge to $\displaystyle 1$ in the
space of distributions $\displaystyle{\mathcal{D}}(V)$ on $\displaystyle V$.
Now for $\displaystyle X\in V_{1}$, in $\displaystyle{\mathcal{D}}(V)$ we have

(2.0.4)
$\displaystyle\displaystyle\kappa(\nu_{s},X,Y)=\int_{F^{\times}}|Z|^{s}\psi(ZY)dZ=\lim_{n}\int_{F^{\times}}|Z|^{s}f_{n}(Z)\psi(ZY)dZ=|Y|^{-s-1}\gamma_{p}(-s)$

where

$\displaystyle\gamma_{p}(s)=\frac{L_{p}(1-s)}{L_{p}(s)}=\frac{1-p^{-s}}{1-p^{s-1}}.$

Similar idea was used by Beineke and Bump in computing the Fourier transform
of $\displaystyle g\mapsto|\det(g)|^{s}$([1, §6] ). This also follows from the
functional equation for $\displaystyle H=\mathop{\mathrm{GL}}\nolimits_{1}$.
In fact consider the zeta integral for the trivial representation we have

$\displaystyle Z(\Phi,s)=\int_{F^{\times}}\Phi(X)|X|^{s-1}dX,$

now local function equation tells us

$\displaystyle Z(\widehat{\Phi},1-s)=\gamma(s)Z(\Phi,s).$

Note that the definition of $\displaystyle\kappa(\nu_{s},X,Y)$ extends to
$\displaystyle Y=0$. Note that in this case if $\displaystyle s=0$ the
function $\displaystyle\kappa(\nu_{s},X,Y)=0$, hence our orbital integral will
vanish identically(in distributional sense).

Generally, for $\displaystyle H=GL_{n}\times GL_{n}$ and $\displaystyle
V=\mathop{\mathrm{gl}}\nolimits_{n}$ with the action

$\displaystyle((g_{1},g_{2}),X)\mapsto g_{1}Xg_{2}^{-1}.$

We take $\displaystyle P(X)=\det(X)$. We have finitely many orbits, indexing
by the rank of $\displaystyle X$. Let us consider $\displaystyle
X=\mathop{\mathrm{Id}}\nolimits$, a point on the open orbit. As before let us
take $\displaystyle
f_{n}=1_{p^{-n}\mathop{\mathrm{gl}}\nolimits_{n}(\mathbb{Z}_{p})}$, then
$\displaystyle f_{n}\rightarrow 1(n\rightarrow\infty)$ in distribution sense.
Let us consider the function

$\displaystyle\kappa(\nu_{s},X,Y)=\int_{GL_{n}(F)}|\det(Z)|^{s}\psi(\mathop{\mathrm{Tr}}\nolimits(ZY))dZ=\lim_{n}\int_{GL_{n}(F)}f_{n}(X)|\det(X)|^{s}\psi(\mathop{\mathrm{Tr}}\nolimits(ZY))dZ.$

As before, our computation shows that

$\displaystyle\kappa(\nu_{s},X,Y)=|\det(Y)|^{-s-n}\gamma(-s),\quad\gamma(s)=\frac{L_{p}(n-s)}{L_{p}(s)}.$

As before, this also follows from the local functional equation for the
trivial representation of $\displaystyle\mathop{\mathrm{GL}}\nolimits_{n}$.

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