religious beliefs, political opinions or union memberships, or locations;
financial or health data; biometric or genetic data; forms of government
identification, such as social security numbers; criminal history)? If so,
please provide a description.

     * •

Yes. DCLM-Pool is created from data that is available on the public internet.
Since people often debate their political views, sexual preferences, religious
beliefs and other such information, it is highly likely such information is
contained in the dataset. While such information is often published willingly
in the explicit intent that it will be publicly visible (see Q19), we do
encourage additional research on filtering such data both to preserve privacy
as well as to discard any potentially biased or toxic content from the
training data of the models.

  17. Q21

Any other comments?

     * •

DCLM-Pool is a research artifact, and we aim for it to be useful for those
studying ways to make internet-scale datasets safer.

####  Collection process

  1. Q22

How was the data associated with each instance acquired? Was the data directly
observable (e.g., raw text, movie ratings), reported by subjects (e.g., survey
responses), or indirectly inferred/derived from other data (e.g., part-of-
speech tags, model-based guesses for age or language)? If data was reported by
subjects or indirectly inferred/derived from other data, was the data
validated/verified? If so, please describe how.

     * •

Data is directly scraped from the public internet by Common Crawl.

  2. Q23

What mechanisms or procedures were used to collect the data (e.g., hardware
apparatus or sensor, manual human curation, software program, software API)?
How were these mechanisms or procedures validated?

     * •

We begin by downloading the entire Common Crawl data prior to 2023. We ran
Python-based processing scripts to parse these archives, filtering low-quality
or irrelevant content, deduplicate samples and in some cases decontaminate
against downstream tests sets, and compute various model-based features. We
ran processes on hundreds of AWS CPU nodes for Common Crawl parsing and data
downloading. Model-based features were run on GPU clusters. For software links
see Q37 and Appendix R or refer to https://datacomp.ai/dclm.

  3. Q24

If the dataset is a sample from a larger set, what was the sampling strategy
(e.g., deterministic, probabilistic with specific sampling probabilities)?

     * •

DCLM-Pool is not a probabilistic sample. As described in Q7, DCLM-Pool
contains all data from Common Crawl before 2023. Common Crawl is a sample of
the Web, and we refer to Common Crawl documentation for details of their
sampling process.

  4. Q25

Who was involved in the data collection process (e.g., students, crowdworkers,
contractors) and how were they compensated (e.g., how much were crowdworkers
paid)?

     * •

The authors participated in the data collection as part of an open-source
effort. No researchers received specific compensation for their contributions
to this project.

  5. Q26

Over what timeframe was the data collected? Does this timeframe match the
creation timeframe of the data associated with the instances (e.g., recent
crawl of old news articles)? If not, please describe the timeframe in which
the data associated with the instances was created.

     * •

The data was downloaded between January 2023 and May 2023. The urls are
collected from Common Crawl archives up to 2023. Common Crawl archives may
include URLs from the early days of the internet. Hence, the download /
collection timeframe does not match the creation timeframe. Additionally,
future users of DCLM-Pool and its subsets will have to download data
themselves using our tooling, though the snapshot should not be altered in any
way.

  6. Q27

Were any ethical review processes conducted (e.g., by an institutional review
board)? If so, please provide a description of these review processes,
including the outcomes, as well as a link or other access point to any
supporting documentation.

     * •

A formal ethics review / IRB has not been conducted to date because DCLM-Pool
contains only data that is already publicly available as part of Common Crawl.

  7. Q28

Does the dataset relate to people? If not, you may skip the remaining
questions in this section.

     * •

Yes. As described in Q17, people’s data may appear as part of the data
scraped.

  8. Q29

Did you collect the data from the individuals in question directly, or obtain
it via third parties or other sources (e.g., websites)?

     * •

The data was gathered from data scattered across the web.

  9. Q30

Were the individuals in question notified about the data collection? If so,
please describe (or show with screenshots or other information) how notice was
provided, and provide a link or other access point to, or otherwise reproduce,
the exact language of the notification itself.

     * •

Individuals were not notified about the data collection.

  10. Q31

Did the individuals in question consent to the collection and use of their
data? If so, please describe (or show with screenshots or other information)
how consent was requested and provided, and provide a link or other access
point to, or otherwise reproduce, the exact language to which the individuals
consented.

     * •

Following our usage of Common Crawl, we respect robots.txt files, which
specify parts of websites that a crawler may access. It is, however, possible
that some private content of people such as personal notes, medical
information or private correspondence were uploaded to the internet without a
person’s consent or under the assumption the host site is private. To mitigate
against such safety concerns we make an effort to exclude some malicious
domains and filter such content as low quality.

  11. Q32

If consent was obtained, were the consenting individuals provided with a
mechanism to revoke their consent in the future or for certain uses? If so,
please provide a description, as well as a link or other access point to the
mechanism (if appropriate).

     * •

While we have no control over the raw data scraped and hosted by Common Crawl,
we will make an effort to provide user a mechanism to request exclusion of
specific URLs, which can be filtered out of our DCLM-Pool and its derived
datasets.

  12. Q33

Has an analysis of the potential impact of the dataset and its use on data
subjects (e.g., a data protection impact analysis) been conducted? If so,
please provide a description of this analysis, including the outcomes, as well
as a link or other access point to any supporting documentation.

     * •

Luccioni & Viviano [102], Bender et al. [19] conducted such research that web-
based datasets still contain substantial amounts of hate speech and sexually
explicit content, even after filtering. Such content can propagate biases and
harmful stereotypes when used to train language models, resulting in outputs
that may be inappropriate or offensive in various contexts.

  13. Q34

Any other comments?

     * •

We anticipate and hope that future studies will leverage DCLM-Pool and DCLM-
baseline to investigate techniques for building better web-scale datasets.

####  Preprocessing, cleaning, and/or labeling

  1. Q35

Was any preprocessing/cleaning/labeling of the data done (e.g., discretization
or bucketing, tokenization, part-of-speech tagging, SIFT feature extraction,
removal of instances, processing of missing values)? If so, please provide a
description. If not, you may skip the remainder of the questions in this
section.

     * •

Yes. See Q7. For more details see Section 4 and Appendix E.

  2. Q36

Was the “raw” data saved in addition to the preprocessed/cleaned/labeled data
(e.g., to support unanticipated future uses)? If so, please provide a link or
other access point to the “raw” data.

     * •

The raw data is stored and accessible through Common Crawl. DCLM-Pool contains
raw text data after HTML extraction using resiliparse.

  3. Q37

Is the software used to preprocess/clean/label the instances available? If so,
please provide a link or other access point.

     * •

We use the following, open-source software to aid in data processing:

       * –

ChatNoir Resiliparse: https://github.com/chatnoir-eu/chatnoir-resiliparse

       * –

Ray: https://www.ray.io

       * –

BFF: https://github.com/allenai/bff

       * –

slurm: https://github.com/SchedMD/slurm

       * –

fastText: https://github.com/facebookresearch/fastText

       * –

nltk: https://github.com/nltk/nltk/blob/develop/LICENSE.txt

       * –

langdetect: https://github.com/Mimino666/langdetect

For a more complete list of software and associated licenses, please refer to
Appendix R.

  4. Q38

Any other comments?

     * •

The creation of DCLM-Pool, DCLM-baseline, the DCLM tooling and our trained
models relies heavily on tools developed by the open-source community and
would not have been possible without it.

####  Uses

  1. Q39

Has the dataset been used for any tasks already? If so, please provide a
description.

     * •

The full dataset (and subsets) have been used to train hundreds of language
models at various scales and compute budgets as presented in our main paper.
We evaluate these models on our testbed of 53 zero- and few-shot downstream
tasks. See Sections 4 and 3.5.

  2. Q40

Is there a repository that links to any or all papers or systems that use the
dataset? If so, please provide a link or other access point.

     * •

No. There is, however, a leaderboard connected to DCLM. Those interested can
review the submissions and examine publications that utilize our data. Refer
to: https://datacomp.ai/dclm/leaderboard.

  3. Q41

What (other) tasks could the dataset be used for?

     * •

Large language models are now widespread and used for an incredibly large
spectrum of tasks, ranging from spell-checking and translation to interactive
agents. The dataset could provide the necessary data to pretrain such models.
DCLM-Pool could also be used for sociological studies, such as examining
biases and trends in human communication, as well as studying human behavior
on the public internet.

  4. Q42

Is there anything about the composition of the dataset or the way it was
collected and preprocessed/cleaned/labeled that might impact future uses? For
example, is there anything that a future user might need to know to avoid uses
that could result in unfair treatment of individuals or groups (e.g.,
stereotyping, quality of service issues) or other undesirable harms (e.g.,
financial harms, legal risks) If so, please provide a description. Is there
anything a future user could do to mitigate these undesirable harms?

     * •

DCLM-Pool and its related datasets and models are not designed for use in
production systems, particularly those involving sensitive areas such as race,
gender identity or expression, ethnicity, sexual orientation, age,
socioeconomic status, disability, religion, national origin, or creed. DCLM-
Pool is unsuitable for applications that involve decision-making about
individuals. Since DCLM-Pool is sourced from the internet, it inherently
contains biases, unfairness, and stereotypes prevalent in society. It is
intended solely as a research tool to examine language-modeling dataset
curation on a large scale and to study the impact of various data curation
methods on downstream models.

  5. Q43

Are there tasks for which the dataset should not be used? If so, please
provide a description.

     * •

As mentioned in Q42, neither DCLM-Pool in its current state nor the subsets
included in this paper should be used in decision-making software involving
individuals. It is intended solely as a research tool for academic study.

  6. Q44

Any other comments?

     * •

Our aim with DCLM-Pool and DCLM was to establish a benchmark for the community
to measure dataset progress across various dimensions (e.g., model performance
on diverse tasks). We consider this essential for creating more effective and
safer datasets, minimizing redundant efforts, promoting knowledge sharing, and
making large language model research more accessible.

####  Distribution

  1. Q45

Will the dataset be distributed to third parties outside of the entity (e.g.,
company, institution, organization) on behalf of which the dataset was
created? If so, please provide a description.

     * •

Yes. We use HuggingFace datasets for public release.

  2. Q46

How will the dataset be distributed (e.g., tarball on website, API, GitHub)?
Does the dataset have a digital object identifier (DOI)?

     * •

The dataset will be distributed via HuggingFace.

  3. Q47

When will the dataset be distributed?

     * •

DCLM-Pool and DCLM-baseline will be available starting June 2024.

  4. Q48

Will the dataset be distributed under a copyright or other intellectual
property (IP) license, and/or under applicable terms of use (ToU)? If so,
please describe this license and/or ToU, and provide a link or other access
point to, or otherwise reproduce, any relevant licensing terms or ToU, as well
as any fees associated with these restrictions.

     * •

We distribute our datasets in full, including extracted page content and
associated metadata under a standard CC-BY-4.0 licence (see Appendix E). The
code associated with DCLM is released under the MIT license. We also note that
the use of this dataset is also subject to CommonCrawl’s Terms of Use as
described in https://commoncrawl.org/terms-of-use.

  5. Q49

Have any third parties imposed IP-based or other restrictions on the data
associated with the instances? If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any relevant
licensing terms, as well as any fees associated with these restrictions.

     * •

We do not copyright samples in the dataset.

  6. Q50

Do any export controls or other regulatory restrictions apply to the dataset
or to individual instances? If so, please describe these restrictions, and
provide a link or other access point to, or otherwise reproduce, any
supporting documentation.

     * •

No, the dataset is provided as individual samples with extracted content and
associated metadata based on the content in Common Crawl hosted data.

  7. Q51

Any other comments?

     * •

We provide several subsets of DCLM-Pool in different sizes, along with
extensive tooling to sample from it which makes it easy for any research
entity to download and experiment with the data at scale suited for them.

We release our code and dataset as open-source with permissive licenses as
described in Q48.

We, the authors, bear all responsibility for any violation of rights
associated with this dataset. While we have made maximal efforts to respect
all licenses of used assets and to mitigate any risks of causing harm, the
responsibility for any misuse of the dataset by others does not rest with us.
This dataset is intended solely for scientific research and not for use in
production systems. We strongly encourage all users to adhere to local and
national laws, respect privacy, and make every effort to avoid harming anyone
when using this dataset.

####  Maintenance

  1. Q52

Who will be supporting/hosting/maintaining the dataset?

     * •

HuggingFace currently hosts the datasets. The DCLM team will be responsible
for maintaining the dataset.

  2. Q53

How can the owner/curator/manager of the dataset be contacted (e.g., email
address)?

     * •

We can be contacted at<EMAIL_ADDRESS>
  3. Q54

Is there an erratum? If so, please provide a link or other access point.

     * •

There are no errata at this time. If any issues arise, we will inform the
public through our website at https://datacomp.ai/dclm.

  4. Q55

Will the dataset be updated (e.g., to correct labeling errors, add new
instances, delete instances)? If so, please describe how often, by whom, and
how updates will be communicated to users (e.g., mailing list, GitHub)?

     * •

Currently, there are no plans to update DCLM-Pool to maintain scientific
integrity and comparability among participants in the DCLM competition.
However, we will address user takedown requests (see Q56). DCLM-Pool is
inherently noisy, and its release aims to encourage researchers to study
dataset cleaning in the context of raw, web-crawled text samples.

  5. Q56

If the dataset relates to people, are there applicable limits on the retention
of the data associated with the instances (e.g., were individuals in question
told that their data would be retained for a fixed period of time and then
deleted)? If so, please describe these limits and explain how they will be
enforced.

     * •

Until we establish an automated method for takedown requests, users can
contact us through<EMAIL_ADDRESS>with takedown requests and specify the
offending URL.

  6. Q57

Will older versions of the dataset continue to be supported/hosted/maintained?
If so, please describe how. If not, please describe how its obsolescence will
be communicated to users.

     * •

This is the first version of DCLM-Pool and derivative DCLM-baseline dataset.
We do not intend to maintain deprecated version of DCLM-Pool. Any deprecation
or modification will be announced on our website at https://datacomp.ai/dclm.

  7. Q58

If others want to extend/augment/build on/contribute to the dataset, is there
a mechanism for them to do so? If so, please provide a description. Will these
contributions be validated/verified? If so, please describe how. If not, why
not? Is there a process for communicating/distributing these contributions to
other users? If so, please provide a description.

     * •

Each proposed modification to the dataset will be addressed individually.

  8. Q59

Any other comments?

     * •

We encourage community members to contact us at<EMAIL_ADDRESS>with any
suggestion or questions about dataset maintenance.


# Hecke algebras for the 1st congruence subgroup and bundles on
$\mathbb{P}^{1}$ I: the case of finite field

Alexander Braverman and David Kazhdan

###### Abstract.

Let $G$ be a split reductive group over a finite field $\mathbf{k}$ In this
note we study the space $V$ of finitely supported functions on the set
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}(\mathbf{k})$ of isomorphism
classes $G$-bundles on the projective line $\mathbb{P}^{1}$ endowed with a
trivialization at $0$ and $\infty$. We show that $V$ is naturally isomorphic
to the regular bimodule over the Hecke algebra $A$ of the group
$G(\mathbf{k}((t)))$ with respect to the first congruence subgroup. As a
byproduct we show that Hecke operators at points different from $0$ and
$\infty$ to generate the ”stable center” of $A$. We provide an expression of
the character of the lifting of an irreducible cuspidal representation of
$\operatorname{GL}(N,\mathbf{k})$ to
$\operatorname{GL}(N,\mathbf{k}^{\prime})$ where $\mathbf{k}^{\prime}$ is a
finite extension of $\mathbf{k}$ in terms of these generators.

In a subsequent publication we plan to develop analogous constructions in the
case when $\mathbf{k}$ is replaced by a local non-archimedian field.

##  1\. Introduction and setup

### 1.1.

Let $\mathbf{k}$ be the finite field $\mathbb{F}_{q}$ with a fixed algebraic
closure $\overline{\mathbf{k}}$. For every $i\geq 1$ let $\mathbf{k}_{i}$
denote the degree $i$ extension of $\mathbf{k}$ inside
$\overline{\mathbf{k}}$; we have the norm map
$\operatorname{Norm}_{i,j}:\mathbf{k}_{i}^{\times}\to\mathbf{k}_{j}^{\times}$
for every $i\geq j$. In particular the groups $\mathbf{k}_{i}^{\times}$ form a
projective system. Let also
$\mathbb{K}=\mathbf{k}((t)),{\mathbb{O}}=\mathbf{k}[[t]]$.

Let $G$ be a split connected reductive group over $\mathbf{k}$.

Denote by $K_{1}$ the first congruence subgroup of $G({\mathbb{O}})$
(consisting of elements equal to 1 modulo $t$). We denote by $A$ the Hecke
algebra of $G(\mathbb{K})$ with respect to $K_{1}$ with coefficients in an
algebraically closed field $F$ of characteristic $0$; this is an associative
algebra containing the group algebra of $G(\mathbf{k})$ as a subalgebra. We
denote by $G^{\vee}$ the Langlands dual group of $G$; unless otherwise stated
we shall think about it as a group over $F$.

Let $\mathcal{Z}(A)$ denote the center of $A$. It follows from [6] that
$\mathcal{Z}(A)$ contains the algebra $F[Z(G^{\vee},\mathbf{k})]^{G^{\vee}}$
of $G^{\vee}$-invariant functions on the scheme $Z_{G^{\vee},\mathbf{k}}$
consisting of pairs $(s,u)$ where

1) $s\in G^{\vee}$

2) $u:\underset{\leftarrow}{\lim}\ \mathbf{k}_{i}^{\times}\to G^{\vee}$

subject to the relation $sus^{-1}=u^{q}$ (it is easy to see that
$Z_{G^{\vee},\mathbf{k}}$ is indeed a scheme of finite type over $F$; this
follows from the fact that there exists some $i$ depending only on $G$ such
that for any pair $(s,u)$ as above the homomorphism $u$ factorizes through
$\mathbf{k}_{i}^{\times}$).

One of the purposes of this note is to give an alternative characterization of
the homomorphism $F[Z(G^{\vee},\mathbf{k})]^{G^{\vee}}\to\mathcal{Z}(A)$ (in
the future we shall call the image of this homomorphism the stable center of
$A$ and we shall denote it by $\mathcal{Z}_{st}(A)$). For this we are going to
use some global constructions discussed in the next subsection.

###  1.2. Bundles on $\mathbb{P}^{1}$ and $\mathcal{Z}(A)$

Let $\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ denote the moduli
stack of bundles on $\mathbb{P}^{1}$ (over $\mathbf{k}$) with trivializations
at $0$ and $\infty$. Let $V$ denote the space of $F$-valued functions on
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}(\mathbf{k})$ with finite
support. The space $V$ has two natural commuting actions of the algebra $A$
(by Hecke operators at $0$ and $\infty$). Our first statement is that $V$ is
essentially the regular bimodule over $A$. More precisely, we claim that there
exists an anti-involution $\iota:A\to A$ on $A$ such that if we use it to turn
$V$ into an $A$-bimodule – keeping the (left) action of $A$ at $0$ intact and
making the action of $A$ at $\infty$ into a right action by using $\iota$,
then we have

######  Theorem 1.1.

There is a natural isomorphism $V\simeq A$ of $A$-bimodules.

######  Corollary 1.2.

We have $\mathcal{Z}(A)=\operatorname{End}_{A\otimes A}(V)$.

We do not know any simple description of $\iota$ but it will satisfy the
following properties:

1) It preserves the subalgebra $F[G(\mathbf{k})]\subset A$, and on this
subalgebra it is induced by the map $g\mapsto g^{-1}$.

2) Let $e_{fin}$ be the idempotent $\frac{1}{\\#G(\mathbf{k})}\sum_{g\in
G(\mathbf{k})}g\in F[\Gamma(\mathbb{F}_{q})]\subset A$. Then the induced anti-
involution on $e_{fin}Ae_{fin}$ is equal to identity (note that
$e_{fin}Ae_{fin}$ is the spherical Hecke algebra of $G$ which is commutative).

Let us also note that $\iota$ is not completely local in the sense that it
depends on the choice of a generator of the maximal ideal of
$\mathbf{k}[[t]]$.

We shall explain the strategy of the proof of Theorem 1.1 in Subsection 1.5.

###  1.3. Hecke operators at other points and stable center

Let $D$ be a subset of $\overline{\mathbf{k}}^{\times}$ invariant under the
Galois group of $\overline{\mathbf{k}}$ over $\mathbf{k}$. Let also $f\in
F[G^{\vee}]^{G^{\vee}}$. To this data one can in the standard way associate
the Hecke operator $h_{D,f}\in\operatorname{End}_{A\otimes A}(V)$. According
to Corollary 1.2 we can regard $h_{D,f}$ as an element of $\mathcal{Z}(A)$.

######  Theorem 1.3.

  1. (i)

We have $h_{D,f}\in\mathcal{Z}_{st}$. In particular, $h_{D,f}$ corresponds to
a function $\phi_{D,f}$ on $Z(G^{\vee},\mathbf{k})$.

  2. (ii)

Assume that $D$ is the
$\operatorname{Gal}(\overline{\mathbf{k}}/\mathbf{k})$-orbit of some
$x\in\mathbf{k}_{i}^{\times}$ which does not belong to any $\mathbf{k}_{j}$
with $j<i$. Then

(1) $\phi_{D,f}(s,u)=f(s^{i}\,u(y))$

where $y\in\mathbf{k}_{n}^{\times}$ for some $n\geq i$ is such that $u$
factorizes through $\mathbf{k}_{n}^{\times}$, $n$ is divisible by $i$ and
$\operatorname{Norm}_{n,i}(y)=x$ (in particular, we claim that the RHS of (1)
does not depend on the choice of $n$ and $y$).

It is easy to see that Theorem 1.3 implies the following

######  Corollary 1.4.

$\mathcal{Z}_{st}(A)$ is the subalgebra of $\mathcal{Z}(A)$ generated by all
the $h_{D,f}$.

###  1.4. Action on cuspidal representations of $G(\mathbf{k})$

It is clear that stack $\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ is
acted on by $G\times G$ (by changing the trivializations at $0$ and $\infty$),
hence the group $G(\mathbf{k})\times G(\mathbf{k})$ acts on $V$. Let
$V_{\operatorname{cusp}}$ denote the maximal cuspidal $G(\mathbf{k})\times
G(\mathbf{k})$-subrepresentation of $V$ (it is easy to see that
$V_{\operatorname{cusp}}$ is also the maximal cuspidal
$G(\mathbf{k})$-subrepresentation of $V$ with respect to either action of
$G(\mathbf{k})$ – this follows immediately from Theorem 1.1). It is clear that
$V_{\operatorname{cusp}}$ is invariant under the action of $A\otimes A$ and
also under the operators $h_{D,f}$. s

The following lemma is easy:

######  Lemma 1.5.

Assume that $G$ is semi-simple. Then
$V_{\operatorname{cusp}}=F_{\operatorname{cusp}}(G(\mathbf{k}))$ (i.e. the
cuspidal part of the regular representation of $G(\mathbf{k})$.

In particular, the restriction of every $h_{D,f}$ to $V_{\operatorname{cusp}}$
defines a $G(\mathbf{k})\times G(\mathbf{k})$-endomorphism of
$F_{\operatorname{cusp}}(G(\mathbf{k}))$. Such a data is equivalent to a
function
$\eta_{D,f}:\operatorname{Irr}_{\operatorname{cusp}}(G(\mathbf{k}))\to F$
where $\operatorname{Irr}_{\operatorname{cusp}}(G(\mathbf{k}))$ denotes the
set of isomorphism classes of irreducible cuspidal representations of
$G(\mathbf{k})$. We  will now describe this function explicitly in the case
when $G=\operatorname{PGL}(N)$ and $f$ is the trace function.

First of all, for every $a\geq 1$ let us denote by $\mathcal{R}_{a}$ the
Grothedieck group of representations of
$\operatorname{PGL}(N,\mathbf{k}_{a})$; we shall write $\mathcal{R}$ instead
of $\mathcal{R}_{1}$. We have a natural map
$\operatorname{Lift}_{a}:\mathcal{R}\to\mathcal{R}_{a}$ defined as follows. It
is known (that $\mathcal{R}_{a}$ is freely spanned by the classes of Deligne-
Lusztig virtual representations $R(T,\theta)$ where $T$ is a maximal torus in
$\operatorname{PGL}(N)$ defined over $\mathbf{k}$ and and
$\theta:T(\mathbf{k})\to\mathbb{C}^{\times}$ (the representation $R(T,\theta)$
depends only on the pair $(T,\theta)$ up to conjugacy). Given $a\geq 1$ and a
pair $(T,\theta)$ we get by extension of scalars a torus $T_{a}$ in
$\operatorname{PGL}(N,\mathbf{k}_{a})$ endowed with the norm map
$T_{a}(\mathbf{k})\to T(\mathbf{k})$; composing this map with $\theta$ we get
a character $\theta_{a}$ of $T_{a}(\mathbf{k})$. The map
$\operatorname{Lift}_{a}$ is characterized by the condition that it sends the
class of $R(T,\theta)$ to the class of $R(T_{a},\theta_{a})$.

We shall also assume that $D$ is as in Theorem 1.3(2). Then we have the
following.

######  Theorem 1.6.

Under the above conditions on $D$ and $f$ we have:

  1. (i)

$\eta_{D,f}=0$ if $i$ is not divisible by $N$.

  2. (ii)

Assume that $i=N$. Any $x\in\mathbf{k}_{i}^{\times}=\mathbf{k}_{N}^{\times}$
can be thought of as an element of $\operatorname{GL}(N,\mathbf{k})$ defined
up to conjugacy; abusing the notation we denote its image in
$\operatorname{PGL}(N,\mathbf{k})$ by the same letter. Then for any
$\pi\in\operatorname{Irr}_{\operatorname{cusp}}(\operatorname{PGL}(N,\mathbf{k}))$
we have

(2) $\eta_{D,f}(\pi)=\chi_{\pi}(x).$

Here $\chi_{\pi}$ is the character of $\pi$.

  3. (iii)

Let now $i=aN$ with $a\geq 1$. Then

(3) $\eta_{D,f}(\pi)=\chi_{\operatorname{Lift}_{a}(\pi)}(x)$

Note that (3) provides an explicit description of the character of
$\operatorname{Lift}_{a}(\pi)$ on the set of elliptic elements in
$\operatorname{PGL}(N,\mathbf{k}_{a})$.

###  1.5. Sketch of the proof of Theorem 1.1

The proof of Theorem 1.1 will be divided into the following steps (we sketch
the proof here, since we think that it is of independent interest).

Step 1. Since the algebra $A$ naturally acts on $V$, in order to define a map
$\mathbf{act}:A\to V$ it is enough to choose an element $\delta\in V$ and set
$\mathbf{act}(a)=a\underset{0}{\star}\delta$ where $\underset{0}{\star}$
stands for the action of $A$ at $0$. We then need to show that $\mathbf{act}$
is an isomorphism – i.e. that $V$ is a free module over $A$ of rank 1 with
generator $\delta$. We shall choose $\delta$ to be the characteristic function
of the trivial $G$-bundle on $\mathbb{P}^{1}$ with the tautological
trivialization at $0$ and $\infty$.

On the other hand $V$ is endowed with a natural involution coming from the map
$z\to z^{-1}$ on $\mathbb{P}^{1}$ which fixes $\delta$. If we know that
$\mathbf{act}$ is an isomorphism, then we can transport this involution to a
map $\iota:A\to A$. Since $\delta$ is preserved by the above involution of $V$
we obviously have

$a\underset{0}{\star}\delta=\iota(a)\underset{\infty}{\star}\delta$

which immediately implies that $\iota(ab)=\iota(b)\iota(a)$, thus $\iota$ is
an anti-involution of $A$. The verification of properties 1) and 2) after
Corollary 1.2 is straightforward and we leave it to the reader.

Step 2. We now need to show that $\iota$ is an isomorphism. Let $\Lambda$
denote the coweight lattice of $G$ and let $\Lambda^{+}$ be the set of
dominant coweights. Both sets are endowed with a standard partial order
($\lambda>\mu$ if $\lambda-\mu$ is a sum of positive roots), and both $A$ and
$V$ acquire a natural filtration numbered by elements of $\Lambda^{+}$ and the
map $\mathbf{act}$ is compatible with the filtrations. We shall denote the
corresponding filtration components by $A_{\leq\lambda}$ and
$V_{\leq\lambda}$.

These filtrations are defined as follows. First, the double quotient
$K_{1}\backslash G(\mathbb{K})/K_{1}$ maps naturally to
$G({\mathbb{O}})\backslash G(\mathbb{K})/G({\mathbb{O}})$. It is known that
the latter quotient is naturally identified with $\Lambda^{+}$; we shall
denote by $(G({\mathbb{O}})\backslash
G(\mathbb{K})/G({\mathbb{O}}))_{\leq\lambda}$ the union of all double cosets
corresponding to $\mu\leq\lambda$. Similarly, we define $(K_{1}\backslash
G(\mathbb{K})/K_{1})_{\lambda}$ to be the preimage of
$(G({\mathbb{O}})\backslash G(\mathbb{K})/G({\mathbb{O}}))_{\leq\lambda}$ in
$(G({\mathbb{O}})\backslash G(\mathbb{K})/G({\mathbb{O}}))_{\leq\lambda}$. We
then define $A_{\leq\lambda}$ to be the subspace of $A$ consisting of
functions supported on $(G({\mathbb{O}})\backslash
G(\mathbb{K})/G({\mathbb{O}}))_{\leq\lambda}$. This is an algebra filtration
on $A$.

Similarly, the set of isomorphism classes of $G$-bundles on $\mathbb{P}^{1}$
is naturally identified with $\Lambda^{+}$; hence the stack
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ is stratified by elements
of $\Lambda^{+}$ and we define $V_{\leq\lambda}$ similarly to
$A_{\leq\lambda}$.

Step 3. Since the map $\mathbf{act}$ is compatible with the above filtrations,
in order to prove that $\mathbf{act}$ is an isomorphism, it is enough to show
that it induces an isomorphism on the accociatd graded spaces. So, we set
$A_{\lambda}=A_{\leq\lambda}/A_{<\lambda}$ and
$V_{\lambda}=V_{\leq\lambda}/V_{<\lambda}$. We also denote by
$\mathbf{act}_{\lambda}$ the corresponding map $A_{\lambda}\to V_{\lambda}$.

To each $\lambda$ we can naturally associate a parabolic subgroup
$P_{\lambda}$ of $G$ with a Levi subgroup $M_{\lambda}$. We also denote by
$P_{\lambda}^{-}$ the opposite parabolic (so $P_{\lambda}\cap
P_{\lambda}^{-}=M_{\lambda}$. The we prove the following

######  Theorem 1.7.

  1. (i)

We have natural isomorphisms

(4) $V_{\lambda}\simeq
F(((G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}))(\mathbf{k}))$

and

(5) $A_{\lambda}\simeq
F(((G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}^{-}))(\mathbf{k})).$

Here $\underset{M_{\lambda}}{\times}$ stands for the quotient of the product
by the diagonal action on $M_{\lambda}$.

  2. (ii)

Under the above isomorphisms the map $\mathbf{act}_{\lambda}$ is the standard
intertwining operator corresponding to the pair
$(P_{\lambda},P_{\lambda}^{-})$ acting on the 2nd factor (cf. Section 4 for
precise definitions).

Step 4. According to Theorem 1.7 it is enough to show that given two associate
parabolic subgroups $P$ and $Q$ of $G$ 111I.e. parabolic subgroups which have
a common Levi subgroup with unipotent radicals $U_{P},U_{Q}$ the standard
intertwining operator $I_{P,Q}:F(G/U_{P}(\mathbf{k}))\to
F(G/U_{Q}(\mathbf{k}))$ is an isomorphism. This is proven in [4] (cf. also
Theorem 2.4 in [7]).

###  1.6. Acknowledgements

We thank R. Bezrukavnikov and M. Finkelberg for very useful discussions.

##  2\. Bundles on $\mathbb{P}^{1}$ with trivializations and proof of (4)

###  2.1. The stack of $G$-bundles on $\mathbb{P}^{1}$

We denote by $\operatorname{Bun}_{G}(\mathbb{P}^{1})$ the moduli stack of
$G$-bundles on $\mathbb{P}^{1}$. It is well-known that
$\operatorname{Bun}_{G}(\mathbb{P}^{1})(\mathbf{k})$ naturally identifies with
$\Lambda^{+}$. This identification can be described as follows. Let
$\lambda\in\Lambda$; this is homomorphism from $\Gamma_{m}$ to the (chosen)
maximal torus $T$ in $G$; thus $\lambda$ defines a homomorphism $\Gamma_{m}\to
G$. Applying it to the line bundle $\mathcal{O}(1)$ on $\mathbb{P}^{1}$ we get
a $G$-bundle $\mathcal{O}(\lambda)$ whose isomorphism class only depends on
the orbit of $\lambda$ under the Weyl group $W$, and the corresponding map
$\Lambda^{+}\to\operatorname{Bun}_{G}(\mathbb{P}^{1})(\mathbf{k})$ is a
bijection. Note that for any $z\in\mathbb{P}^{1}(k)$ we can write canonically
$\mathcal{O}(1)=\mathcal{O}(z)$. Thus the bundle $\mathcal{O}(\lambda)$
acquires a canonical trivialization away from $z$. In what follows we shall
choose such a $z$ so that $z\in\mathbf{k}^{\times}$. Some of our construction
will slightly depend on this choice (in principle, one can choose $z=1$ but
there is nothing special about this choice).

Geometrically this defines a stratification of
$\operatorname{Bun}_{G}(\mathbb{P}^{1})$ – we denote by
$\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\lambda}$ the corresponding locally
closed substack of $\operatorname{Bun}_{G}(\mathbb{P}^{1})$. Denote by
$\overline{\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\lambda}}$ its closure.
Then it is easy to see that
$\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\mu}\subset\overline{\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\lambda}}$
if and only if $\lambda\leq\mu$. We set

$\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\leq\lambda}=\bigcup\limits_{\mu\leq\lambda}\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\mu}.$

This is an open substack of $\operatorname{Bun}_{G}(\mathbb{P}^{1})$.

###  2.2. Parabolic subgroups and Harder-Narasimhan filtrations

Recall that we have chosen a Cartan subgroup $T$ of $G$ and a Borel subgroup
$B$ containing it. Every $\lambda\in\Lambda$ defines a parabolic subgroup
$P_{\lambda}$ in $G$ (its unipotent radical is generated by root vectors
corresponding to roots $\alpha$ such that $\langle\lambda,\alpha\rangle>0$).
We have also the opposite parabolic $P_{\lambda^{-}}$ which have a common Levi
subgroup $M_{\lambda}$. We denote by $\pi_{\lambda}:P_{\lambda}\to
M_{\lambda}$, $i_{\lambda}:M_{\lambda}\to P_{\lambda}$ the natural maps
(similarly for $\pi_{\lambda}^{-}$ and $i_{\lambda}^{-}$).

By construction every $\mathcal{O}(\lambda)$ is endowed with a $T$-structure.
However, this structure is not canonical in the sense that it is not invariant
under the group of automorphisms of $\mathcal{O}(\lambda)$ – in particular, it
is not well-defined on a $G$-bundle abstractly isomorphic to
$\mathcal{O}(\lambda)$. However, it is easy to see that the induced
$P_{\lambda}$-structure is invariant under the automorphisms of
$\mathcal{O}(\lambda)$. For $G=\operatorname{GL}(N)$ this is exactly the
Harder-Narasimhan filtration, so for general $G$ we shall also sometimes use
this term for the $P_{\lambda}$ structure on $G$-bundle isomorphic to
$\mathcal{O}(\lambda)$.

###  2.3. The main stack

Our main geometric object in this Section is the stack
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ classifying $G$-bundles on
$\mathbb{P}^{1}$ with trivializations at $0$ and $\infty$. We denote by
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\leq\lambda}$ and
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}$ the
corresponding (open and locally closed) substacks of
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ which are preimages
respectively of $\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\leq\lambda}$ and of
$\operatorname{Bun}_{G}(\mathbb{P}^{1})^{\lambda}$. As before we set $V$ to be
the space of $F$-valued functions on
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}(\mathbf{k})$ with finite
support. We also set $V_{\leq\lambda}$ (resp. $V_{\lambda}$) to be the space
of $F$-valued functions on
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\leq\lambda}(\mathbf{k})$
(resp. on
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}(\mathbf{k})$).
Then $V_{\leq\lambda}$ define a filtration on $V$ with $V_{\lambda}$’s being
the corresponding associated graded quotients.

######  Theorem 2.1.

There exists a map
$\eta_{\lambda}:\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}\to
G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}$ such that:

  1. (i)

$\eta_{\lambda}$ is surjective on $\mathbf{k}$-points

  2. (ii)

The fibers of $\eta_{\lambda}$ are classifying stacks of some unipotent group.

In particular (i) and (ii) imply that $\eta_{\lambda}$ is an isomorphism on
the level of $\mathbf{k}$-points.

######  Corollary 2.2.

$V_{\lambda}\simeq
F(G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}(\mathbf{k}))$ (this
is the statement of (4)).

###### Proof.

Let us begin by observing that the variety
$G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}$ is the moduli space
of the following data: two $P_{\lambda}$-structures on the trivial $G$-torsor
together with an isomorphism between the corresponding $M_{\lambda}$-torsors.

Now any bundle on $\mathbb{P}^{1}$ with isomorphism type $\lambda$ is endowed
with a canonical $P_{\lambda}$-structure (for $G=GL(N)$ this is the Harder-
Narasimhan filtration). By looking at the image of this structure at $0$ and
$\infty$ we get a map from
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}$ to
$G/P_{\lambda}\times G/P_{\lambda}$. In addition, let us recall that we have
chosen some $z\in\mathbb{P}^{1}(\mathbf{k})\backslash\\{0,\infty\\}$. Then we
claim that the corresponding $M_{\lambda}$-torsors are canonically isomorphic.
Indeed, it is enough to check it for $G=\operatorname{GL}(N)$. In this case,
it follows from the fact the every graded piece of the associated graded of
any rank $N$ vector bundle with respect to its Harder-Narasimhan filtration is
isomorphic to $\mathcal{L}^{k}$ for some
$\mathcal{L}\in\operatorname{Pic}(\mathbb{P}^{1})$, $k\in\mathbb{Z}_{>0}$. Now
we have unique up to constant isomorphism $\mathcal{L}\simeq\mathcal{O}(d\cdot
z)$ where $d=\deg(\mathcal{L})$, which identifies the fibers of $\mathcal{L}$
at $0$ and $\infty$. Thus we get a map
$\eta_{\lambda}:\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}\to
G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}$.

We now need to prove (1) and (2). To prove (1) let us observe that
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}$ is acted on
transitively by the group $G\times G$, and the same is true for
$G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}(\mathbf{k})$;
moreover, this action is transitive on the latter.

To check (2) let us describe the stack
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}$ explicitly. Let
$\mathcal{F}$ be a $G$-bundle of type $\lambda$ with trivializations at $0$
and $\infty$. Let $\operatorname{Aut}(\mathcal{F})$ be the group of
automorphisms of $\mathcal{F}$ (just as a bundle – i.e. the automorphisms are
not supposed to preserve the trivializations). Looking at the action of
automorphisms at $0$ and $\infty$ we get a map
$\operatorname{Aut}(\mathcal{F})\to G\times G$. The stack
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}$ is equivalent to
$G\times G/\operatorname{Aut}(\mathcal{F})$.222The map
$\operatorname{Aut}(\mathcal{F})\to G\times G$ is in general not injective,
thus the quotient is indeed a stack. Let us determine the image of
$\operatorname{Aut}(\mathcal{F})$ in $G\times G$. For this it is enough to
take any $\mathcal{F}$ as above, and we take
$\mathcal{F}=\mathcal{O}(\lambda)$ with trivializations at $0$ and $\infty$
coming from the trivialization of $\mathcal{O}(\lambda)$ on
$\mathbb{P}^{1}\backslash\\{z\\}$. Then the image of
$\operatorname{Aut}(\mathcal{F})$ in $G\times G$ is easily seen to be the
subgroup of $P_{\lambda}\times P_{\lambda}$ consisting of pairs
$(p,p^{\prime})$ whose images in $M_{\lambda}$ coincide. Thus the quotient of
$G\times G$ by the image of $\operatorname{Aut}(\mathcal{F})$ is precisely
$G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}$. It remains to note
that the kernel of the map $\operatorname{Aut}(\mathcal{F})\to G\times G$ is
unipotent. ∎

##  3\. Hecke stack for the 1st congruence subgroup

###  3.1. The Hecke stack in the spherical case

Let $\mathcal{H}_{{\mathbb{O}}}$ be the stack classifying triples
$(\mathcal{F}_{1},\mathcal{F}_{2},\kappa)$

where

1) $\mathcal{F}_{i}$ is a $G$-bundle on the formal disc
$\mathcal{D}=\operatorname{Spec}({\mathbb{O}})$.

2) $\kappa$ is an isomorphism
$\mathcal{F}_{1}|_{\mathcal{D}^{*}}\to\mathcal{F}_{2}|_{\mathcal{D}^{*}}$
where $\mathcal{D}^{*}=\operatorname{Spec}(\mathbb{K})$.

This stack can be identified with the double quotient
$G({\mathbb{O}})\backslash G(\mathbb{K})/G({\mathbb{O}})$. As before we have
the natural bijection $\mathcal{H}_{{\mathbb{O}}}(\mathbf{k})=\Lambda^{+}$.
Here every $\lambda\in\Lambda^{+}$ corresponds to the element
$t^{\lambda}=\lambda(t)\in T(\mathbb{K})\subset G(\mathbb{K})$. In fact, the
stack $\mathcal{H}_{{\mathbb{O}}}$ endowed with a natural stratification

$\mathcal{H}_{{\mathbb{O}}}=\bigsqcup\limits_{\lambda\in\Lambda^{+}}\mathcal{H}^{\lambda},$

where $\mathcal{H}_{{\mathbb{O}}}^{\lambda}$ is the (reduced) locally closed
substack of $\mathcal{H}_{{\mathbb{O}}}$ whose only $\mathbf{k}$-point is
given by the above construction. The closure
$\mathcal{H}^{\leq\lambda}_{{\mathbb{O}}}$ of
$\mathcal{H}_{{\mathbb{O}}}^{\lambda}$ is equal to the union of all
$\mathcal{H}_{{\mathbb{O}}}^{\mu}$ with $\mu\leq\lambda$.

###  3.2. The Hecke stack for the 1st congruence subgroup

Let $\mathcal{H}$ denote the stack of 5-tuples
$(\mathcal{F}_{1},\mathcal{F}_{2},\phi_{1},\phi_{2},\kappa)$ where

1) $\mathcal{F}_{i}$ is a $G$-bundle on the formal disc
$\mathcal{D}=\operatorname{Spec}({\mathbb{O}})$ where
${\mathbb{O}}=\mathbf{k}[[t]]$.

2) $\phi_{i}$ is the trivialization of the fiber of $\mathcal{F}_{i}$ at $0$.

3) $\kappa$ is an isomorphism
$\mathcal{F}_{1}|_{\mathcal{D}^{*}}\to\mathcal{F}_{2}|_{\mathcal{D}^{*}}$
where $\mathcal{D}^{*}=\operatorname{Spec}(\mathbb{K})$ where
$\mathbb{K}=\mathbf{k}((t))$.

Let $\mathcal{H}^{\lambda}$ denote the preimage of
$\mathcal{H}^{\lambda}_{{\mathbb{O}}}$ under the natural map
$\mathcal{H}\to\mathcal{H}_{{\mathbb{O}}}$; this is a locally closed substack
of $\mathcal{H}$. Similarly, we denote by $\mathcal{H}^{\leq\lambda}$ the
preimage of $\mathcal{H}^{\leq\lambda}_{{\mathbb{O}}}$.

###  3.3. The algebra $A$ and its filtration

Let $A$ denote the space of finitely supported $F$-valued functions on
$\mathcal{H}(\mathbf{k})$. This is an algebra with respect to convolution. We
let $A_{\leq\lambda}$ be the space of functions on
$\mathcal{H}^{\leq\lambda}$. It is easy to see that $A_{\lambda}\cdot
A_{\mu}\subset A_{\leq\lambda+\mu}$; thus $A_{\leq\lambda}$ is a filtration on
$A$ by dominant coweights. The associated graded algebra is the direct sum of
all the $A_{\lambda}$ where $A_{\lambda}$ is the space of $F$-valued functions
on $\mathcal{H}^{\lambda}$.

We would like to describe the space $A_{\lambda}$. For this we are going to
need the following geometric description of the stack $\mathcal{H}^{\lambda}$.

######  Theorem 3.1.

There is a map $\mathcal{H}^{\lambda}\to
G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}^{-}$ which is
surjective on $\mathbf{k}$-points and whose fibers are classifying spaces of
some pro-unipotent groups (in particular, this map is bijective of
$\mathbf{k}$-points).

To construct this map we need a digression on Jantzen filtrations.

###  3.4. Jantzen filtrations

Let $\mathcal{E},\mathcal{E}^{\prime}$ be two finitely generated torsion-free
${\mathbb{O}}$-modules endowed with an isomorphism
$\kappa:\mathcal{E}\underset{{\mathbb{O}}}{\otimes}\mathbb{K}\to\mathcal{E}^{\prime}\underset{{\mathbb{O}}}{\otimes}\mathbb{K}$.
Let $E,E^{\prime}$ be the fibers of $\mathcal{E},\mathcal{E}^{\prime}$ at $0$.

Set $E_{i}$ be the set of all $v\in E$ such that for some (equivalently for
any) $s\in\mathcal{E}$ with $s(0)=v$ we have
$t^{-i}\kappa(s)\in\mathcal{E}^{\prime}$. Similarly we define $E^{\prime}_{i}$
by replacing $\kappa$ by $\kappa^{-1}$. Then it is clear that

(1) $E_{i}$ is an increasing filtration on $E$, $E_{i}^{\prime}$ is an
increasing filtration on $E^{\prime}$ (we are going to call them Jantzen
filtrations).

(2) $\kappa$ establishes a canonical isomorphism between
$\operatorname{gr}_{i}(E)$ and $\operatorname{gr}_{-i}(E^{\prime})$.

Let $N=\dim(E)=\dim(E^{\prime})$. Then we can view $\mathcal{E}$ and
$\mathcal{E}^{\prime}$ as rank $N$ vector bundles on $\mathcal{D}$, and
$\kappa$ defines a point of the double quotient stack
$\operatorname{GL}(N,{\mathbb{O}})\backslash\operatorname{GL}(N,\mathbb{K})/\operatorname{GL}(N,{\mathbb{O}})$.
On the other hand, the above Jantzen fitrations define a reduction of $V$ and
$V^{\prime}$ to some parabolic subgroups $P,P^{\prime}$ of
$\operatorname{GL}(N)$. Moreover, (2) above implies that the type of
$P^{\prime}$ is opposite to that of $P$.

Recall that $\mathbf{k}$-points of
$\operatorname{GL}(N,{\mathbb{O}})\backslash\operatorname{GL}(N,\mathbb{K})/\operatorname{GL}(N,{\mathbb{O}})$
are in one-to-one correspondence with $\lambda\in\Lambda^{+}$ (where
$\Lambda^{+}$ denotes the set of dominant coweights of
$\operatorname{GL}(N)$).

######  Lemma 3.2.

Assume that $(\mathcal{E},\mathcal{E}^{\prime},\kappa)$ has type $\lambda$.
Then the Jantzen filtration on $E$ defines a reduction of $E$ to
$P_{\lambda}$. Similarly, the Jantzen filtration on $E^{\prime}$ defines a
reduction of $E^{\prime}$ to $P_{\lambda}^{-}$.

As before, to prove this lemma, it is enough to look at the case
$\mathcal{E}=\mathcal{E}^{\prime}={\mathbb{O}}^{N}$ and $\kappa$ being the
diagonal matrix with entries $(t^{n_{1}},\cdots,t^{n_{N}})$ where
$\lambda=(n_{1},\cdots,n_{N})$, in which case the above lemma is obvious.

Since Jantzen filtrations are completely canonical, they are functorial and
are compatible with tensor products. As a corollary, we get the following

######  Lemma 3.3.

Let $G$ be as before, and let $(\mathcal{F}_{1},\mathcal{F}_{2},\kappa)$ be a
$\mathbf{k}$-point of the stack $G({\mathbb{O}})\backslash
G(\mathbb{K})/G({\mathbb{O}})$ of type $\lambda$. Let $F_{i}$ be the fiber of
$\mathcal{F}_{i}$ at $0$. Then $F_{1}$ is endowed with a
$P_{\lambda}$-structure, $F_{2}$ is endowed with a $P_{\lambda}^{-}$-structure
and the corresponding $M_{\lambda}$-bundles are canonically isomorphic.

###  3.5. End of the proof of Theorem 3.1

It is clear that
$G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}^{-}$ is the moduli
space of the following data:

1) A reduction of the trivial $G$-torsor to $P_{\lambda}$.

2) A reduction of the trivial $G$-torsor to $P_{\lambda}^{-}$.

3) An isomorphism between the corresponding $M_{\lambda}$-torsors.

Thus using Lemma 3.3 we immediately get a map $\mathcal{H}^{\lambda}\to
G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}^{-}$. We now need to
prove that it is surjective on $\mathbf{k}$-points and that the fibers are
classifying stacks of some pro-unipotent group. The former follows again from
the fact that $G(\mathbf{k})\times G(\mathbf{k})$ acts transitively on
$G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}^{-}(\mathbf{k})$. To
prove the latter as before we note that $\mathcal{H}^{\lambda}$ is equivalent
to $G\times G/A_{\lambda}$ where $A_{\lambda}$ is the group of automorphisms
of the triple $(\mathcal{F}_{1},\mathcal{F}_{2},\kappa)$ where
$\mathcal{F}_{i}$ are trivial bundles and $\kappa=t^{\lambda}$ (it maps to
$G\times G$ by looking at its actions on the fibers of $\mathcal{F}_{i}$ at
$0$). In other words, $A_{\lambda}$ is the subgroup of $G({\mathbb{O}})$
consisting of elements $g\in G({\mathbb{O}})$ such that $\kappa
g\kappa^{-1}\in G({\mathbb{O}})$ and the map to $G\times G$ is given by
$(g\mod t,\kappa g\kappa^{-1}\mod t)$. It is easy to see that the kernel of
the map $A_{\lambda}\to G\times G$ is pro-unipotent and the image of
$A_{\lambda}$ in $G\times G$ is equal to

$\\{(p,p^{\prime})\in P_{\lambda}\times P_{\lambda}^{-}|\
\pi_{\lambda}(p)=\pi_{\lambda}^{-}(p^{\prime})\\}.$

which finishes the proof.

##  4\. Hecke correspondences and proof of Theorem 1.7

The purpose of this Section is to prove Theorem 1.7. As was explained in the
Introduction this implies Theorem 1.1.

###  4.1. Intertwining operators

We begin by recalling the definition and properties of the standrd
intertwining operators for induced representations of the group
$G(\mathbf{k})$.

Let $P$ and $P^{-}$ be any two opposite parabolics in $G$ – i.e. two
parabolics whose intersection $M$ is a Levi subgroup in both. Let $\gamma$
(resp. $\gamma_{-}$) be the natural map $G\to G/U$ (resp. $G\to G/U_{-}$).
Then we can view $G$ as a correspondence between $G/U_{-}$ and $G/U$ and we
define $\Phi_{P_{-},P}:F(G/U^{-}(\mathbf{k}))\to F(G/U(\mathbf{k}))$ be the
corresponding intertwining operator (a.k.a. ”Radon transform”). This operators
commutes with the natural action of $G(\mathbf{k})\times M(\mathbf{k})$ on
both sides. The following result is proven in [4] (cf. also [7]):

######  Theorem 4.1.

The operator $\Phi_{P_{-},P}$ is an isomorphism.

###  4.2. The map $\mathbf{act}$ and the stack $\mathcal{C}$

The algebra $A\otimes A$ acts naturally on the space $V$ by convolution at $0$
and $\infty$; as in subsection 1.5 we denote these actions by
$a\underset{0}{\star}v$ and $a\underset{\infty}{\star}v$.

Let $\delta\in V$ be the characteristic function of the $\mathbf{k}$-point of
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ corresponding to the
trivial bundle with tautological trivializations at $0$ and $\infty$. We
define the map $\mathbf{act}:A\to V$ by setting

$\mathbf{act}(a)=a\underset{0}{\star}\delta.$

It is easy to see that $\mathbf{act}(a)$ is also equal to
$\iota(a)\underset{\infty}{(}\delta)$ where $\iota$ is defined in subsection
1.2. Indeed

###  4.3. The space $\mathcal{C}^{\lambda}$

Let $\mathcal{C}^{\lambda}$ denote the moduli space of the following data:

(1) A $G$-bundle on $\mathcal{F}$ on $\mathbb{P}^{1}$ of isomorphism class
$\lambda$.

(2) Trivialization of the fiber of $\mathcal{F}$ at $0$.

(3) A trivialization of $\mathcal{F}|_{\mathbb{P}^{1}\backslash\\{0\\}}$ which
has type $\lambda$ near $0$.

We have obvious maps

$\begin{CD}\mathcal{C}^{\lambda}@>{}>{}>\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}\\\
@V{}V{}V\\\ \mathcal{H}^{\lambda}\end{CD}$

Here the horizontal arrow is obtained by looking at (1),(2) and (3) above and
using only the image of (3) at $\infty$, and the vertical arrow is obtained by
restricting all of the data to the formal disc near $0$ (and by observing that
the trivial bundle is endowed with a canonical trivialization of its fiber at
$0$). Thus in view of the previous sections we get natural maps

$\begin{CD}\mathcal{C}^{\lambda}@>{\beta}>{}>G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}\\\
@V{\alpha}V{}V\\\
G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U^{-}_{\lambda}.\end{CD}$

Note that

$G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}=G\times
G/(U_{\lambda}\times U_{\lambda})\cdot\Delta M_{\lambda}$

and

$(G/U_{\lambda}\times G/U^{-}_{\lambda})/M_{\lambda}=G\times
G/(U_{\lambda}\times U_{\lambda}^{-})\cdot\Delta M_{\lambda}.$

We now want to describe $\mathcal{C}^{\lambda}$ explicity. First, consider the
embedding $P_{\lambda}\hookrightarrow G\times G$ which sends $p$ to
$(p,i_{\lambda}\circ\pi_{\lambda}(p))$. Let $P_{\lambda}^{\prime}$ denote the
image of $P_{\lambda}$ under this embedding. Note that $P_{\lambda}$ is
contained in both $(U_{\lambda}\times U_{\lambda})\cdot\Delta M_{\lambda}$ and
$(U_{\lambda}\times U_{\lambda}^{-})\cdot\Delta M_{\lambda}$. Note that
$P_{\lambda}^{\prime}$ is contained in both $(U_{\lambda}\times
U_{\lambda})\cdot\Delta M_{\lambda}$ and in $(U_{\lambda}\times
U_{\lambda}^{-})\cdot\Delta M_{\lambda}$. Thus we get natural maps

$\begin{CD}(G\times
G)/P_{\lambda}^{\prime}@>{\delta}>{}>G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U_{\lambda}\\\
@V{\gamma}V{}V\\\
(G/U_{\lambda}\underset{M_{\lambda}}{\times}G/U^{-}_{\lambda}.\end{CD}$

######  Theorem 4.2.

There is a $G\times G$-equivariant isomorphism
$\mathcal{C}^{\lambda}\simeq(G\times G)/P_{\lambda}^{\prime}$ under which
$\alpha$ goes to $\gamma$ and $\beta$ goes to $\delta$.

###### Proof.

Since $(G\times G)/P_{\lambda}^{\prime}$ is acted on transitively by $G\times
G$ in order to construct the above isomorphism it is enough to check the
following things:

(a) The two maps $\mathcal{C}^{\lambda}\to G/P_{\lambda}$ given by composing
either $\alpha$ or $\beta$ with the natural (first) projection to
$G/P_{\lambda}$ coincide

(b) The fibers of these maps are $G$-torsors with respect to the action of the
2nd copy of $G$ (corresponding to changing the trivialization of $\mathcal{F}$
on $\mathbb{P}^{1}\backslash\\{0\\}$ by a constant map
$\mathbb{P}^{1}\backslash\\{0\\}\to G$) are $G$-torsors. Moreover, the fiber
over 1 is canonically the trivial torsor (i.e. it is equal to $G$)

(c) Any $p\in P_{\lambda}$ acts on the fiber over $1\in G/P_{\lambda}$ (which
is naturally isomorphic to $G$ by the last sentence of (b)) by the right
multiplication by $i_{\lambda}\pi_{\lambda}(p^{-1})$.

First, we claim that $\mathcal{C}^{\lambda}$ is acted on transitively by
$G\times G$. Indeed, for this it is enough to check that a trivialization of
$\mathcal{O}(\lambda)$ on $\mathbb{P}^{1}\backslash\\{0\\}$ of type $\lambda$
is unique up to constant change of trivialization. This is a special case of
the modular description of transversal slices in the to
$G({\mathbb{O}})$-orbits in the affine Grassmannian (cf. the proof of Theorem
2.8 in [2]).

Now (a) follows from the fact that the fiber of the Harder-Narasimhan flag of
$\mathcal{F}$ at $0$ is equal to the Jantzen flag at $0$ w.r.to the
trivialization on $\mathbb{P}^{1}\backslash\\{0\\}$. In view of the above
transitivity in order to check this it is enough to assume that
$\mathcal{F}=\mathcal{O}(\lambda)$ with the standard trivialization; in this
case it is a straightforward calculation.

Let us check (b). Again, it follows from the proof of Theorem 2.8 in [2] that
the space $\overline{\mathcal{C}}^{\lambda}$ consisting just of data (1) and
(3) from Subsection 4.3 is isomorphic to $G/P_{\lambda}$, where the action of
$G$ is given by changing the trivialization by a constant map. More precisely,
the claim is that $G$ acts transitively on such data and given any
$(\mathcal{F},\kappa)\in\overline{\mathcal{C}}^{\lambda}$ the group of all
$g\in\operatorname{Aut}(\mathcal{F})$ which commute with $\kappa$ maps
injectively to the group $\operatorname{Aut}(\mathcal{F}_{0})$ (automorphisms
of the fiber of $\mathcal{F}$ at $0$) and its image consists of those
automorphisms of $\mathcal{F}_{0}$ which preserve the restriction of the
Harder-Narasimhan flag of $\mathcal{F}$ to $0$. This implies that if we add to
$(\mathcal{F},\kappa)$ a trivialization of $\mathcal{F}_{0}$ such that the
restriction of the Harder-Narasimhan flag of $\mathcal{F}$ to $0$ gives a
fixed $P_{\lambda}$-structure on the corresponding trivial $G$-torsor, then
$G$ still acts transitively but with trivial stabilizer. This is precisely the
first assertion of (b). For the 2nd assertion we just need to say that we have
the bundle $\mathcal{O}(\lambda)$ with a natural trivilization on
$\mathbb{P}^{1}\backslash\\{0\\}$ and on $\mathbb{P}^{1}\backslash\\{z\\}$
which gives rise to a point of $\mathcal{C}^{\lambda}$ which sits over $1\in
G/P_{\lambda}$, which trivializes the corresponding torsor. Part (c) is then
straightforward. ∎

######  Corollary 4.3.

Let $\mathbf{k}=\mathbb{F}_{q}$. Then $\mathcal{C}^{\lambda}$ (viewed as a
correspondence) defines an isomorphism between
$\mathbb{C}[\mathcal{H}^{\lambda}(\mathbf{k})]$ and
$\mathbb{C}[\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}^{\lambda}(\mathbf{k})]$.

##  5\. Proof of Theorems 1.3 and 1.6

###  5.1. Proof of Theorem 1.3

First of all, it is clear that assertion (i) of Theorem 1.3 follows from
assertion (ii). Also it is enough to prove (ii) in the case
$F=\overline{\mathbb{Q}}_{\ell}$ (here $\ell$ is prime to the characteristic
of $\mathbf{k}$). In this case (ii) follows from the (weak form of the) global
Langlands correspondence for functional fields proved by L. Lafforgue. Namely,
the following statement is contained in [8].

Let $\pi=\otimes^{\prime}\pi_{\nu}$ be an irreducible automorphic
representation $G$ for the global field $E=\mathbf{k}(\mathbb{P}^{1})$ (here
$\nu$ runs over places of $E$) such that

a) $\pi_{\nu}$ is unramfied by $\nu\neq 0,\infty$ (i.e.
$\pi_{\nu}^{G({\mathbb{O}}_{E_{\nu}})}\neq 0$ for $\nu\neq 0,\infty$);

b) $\pi_{\nu}^{K_{1},\nu}\neq 0$ for $\nu=0,\infty$ where $K_{1,\nu}$ denotes
the 1st congruence subgroup of $G(E_{\nu})$.

Let $\pi_{1}(\mathbb{P}^{1})_{0,\infty}$ denote the fundamental group of
$\mathbb{P}^{1}\backslash\\{0,\infty\\}$ tamely ramified at $0$ and $\infty$.
Then there exists a homomorphism $\rho:\pi_{1}(\mathbb{P}^{1})_{0,\infty}\to
G^{\vee}(\overline{\mathbb{Q}}_{\ell})$ such that the Satake parameter of
$\pi_{\nu}$ for $\nu\neq 0,\infty$ (which a priori is a semi-simple conjugacy
class in $G^{\vee}(\overline{\mathbb{Q}}_{\ell})$) is equal to
$\rho(\operatorname{Fr}_{\nu})$ (where $\operatorname{Fr}_{\nu}$ is the
Frobenius element at the place $\nu$ which is well defined as an element of
$\pi_{1}(\mathbb{P}^{1})_{0,\infty}$ up to conjugacy).

The group $\Gamma=\pi_{1}(\mathbb{P}^{1})_{0,\infty}$ is isomorphic to the
semidirect product of $\widehat{\mathbb{Z}}$ (with generator
$\operatorname{Fr}$, corresponding to the Frobenius element at the point
$1\in\mathbf{k}^{\times}$) and the group
$\Gamma_{0}=\underset{\leftarrow}{\lim}\ \mathbf{k}_{i}^{\times}$ (this is the
corresponding geometric fundamental group), such that
$\operatorname{Fr}\gamma\operatorname{Fr}^{-1}=\gamma^{q}$ for
$\gamma\in\Gamma_{0}$. Thus a homomorphism $\Gamma\to
G^{\vee}(\overline{\mathbb{Q}}_{\ell})$ is given by a pair $(s,u)$ as in
Subsection 1.1. The proof then follows from the following two observations:

1) If $x\in\mathbf{k}_{i}^{\times}$ and $x$ does not belong to
$\mathbf{k}_{j}$ for any $j<i$ then $\rho(\operatorname{Fr}_{x})$ is conjugate
to $s^{i}u(y)$ where $y$ as in Theorem 1.3.

2) Every irreducible representation of $A$ does appear as $\pi_{0}$ for some
$\pi$ as above (this follows immediately from Theorem 1.1).

###  5.2. Proof of Lemma 1.5

Assume that $G$ is semi-simple. Then it is clear that $P_{\lambda}\neq G$ for
any $\lambda\neq 0$. Hence Lemma 1.5 follows from (4).

###  5.3. Proof of Theorem 1.6(i)

For the remainder of this Section we set $G=\operatorname{PGL}(N)$ and we use
freely the notation from Subsection 1.4. The set of connected components of
the stack $\operatorname{Bun}_{G}(\mathbb{P}^{1})$ (and thus of
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$) is naturally identified
$\mathbb{Z}/N\mathbb{Z}$. Thus the space $V$ acquires a natural
$\mathbb{Z}/N\mathbb{Z}$ grading $V=\oplus_{d\in\mathbb{Z}/N\mathbb{Z}}V_{d}$.
We claim that $(V_{d})_{\operatorname{cusp}}=0$ unless $d=0$ – this again
follows immediately from (4). On the other hand, the operator $h_{D,f}$ maps
$V_{d}$ to $V_{d+\overline{i}}$ where $\overline{i}$ is the image of $i$ in
$\mathbb{Z}/N\mathbb{Z}$, which shows that $h_{D,f}$ acts by $0$ on
$V_{\operatorname{cusp}}$ unless $\overline{i}=0$.

###  5.4. Proof of Theorem 1.6(ii)

Here we give a direct geometric proof of Theorem 1.6(ii). In the next
subsection we shall give a proof of Theorem 1.6(iii) which contains Theorem
1.6(ii) as a special case – but that proof will be indirect (it will use
Theorem 1.3 (and thus it will rely on our knowledge of the global Langlands
correspondence) and the formula for the character of Deligne-Lusztig
representation from [5]).

First of all, let us recall the precise definition of the operator $h_{D,f}$
in our case. Let $D$ for the moment be any subset of
$\overline{\mathbf{k}}^{\times}$ invariant under the Galois group of
$\overline{\mathbf{k}}$ over $\mathbf{k}$. Let us first work with
$G=\operatorname{GL}(N)$ rather than $\operatorname{PGL}(N)$. Then we can
consider triples
$(\widetilde{\mathcal{F}_{1}},\widetilde{\mathcal{F}_{2}},\widetilde{\kappa})$
where $\widetilde{\mathcal{F}_{i}}$ is a locally free sheaf on
$\mathbb{P}^{1}$ of rank $N$ trivialized at $0$ and $\infty$ and
$\widetilde{\kappa}:\widetilde{\mathcal{F}_{1}}\to\widetilde{\mathcal{F}_{2}}$
is an injective map such that
$\widetilde{\mathcal{F}_{2}}/\widetilde{\mathcal{F}_{1}}=\mathcal{O}_{D}$ (the
structure sheaf of $D$). For $\operatorname{PGL}(N)$ we consider the quotient
of this stack by $\operatorname{Pic}(\mathbb{P}^{1})$ acting diagonally. We
denote the resulting stack by $\mathcal{H}_{D,N}$ and we denote its typical
point by $(\mathcal{F}_{1},\mathcal{F}_{2},\kappa)$ (here each
$\mathcal{F}_{i}$ is a $\operatorname{PGL}(N)$-bundle on $\mathbb{P}^{1}$
trivialized at $0$ and $\infty$ and $\kappa$ is an isomorphism between
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ away from $D$ (the isomorphism is not
required to respect the trivializations at $0$ and $\infty$)). It can be
viewed as a correspondence from
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$ to
$\operatorname{Bun}_{G}(\mathbb{P}^{1})_{0,\infty}$. We also denote by
$\mathcal{H}_{D,N}^{0}$ the locus where both $\mathcal{F}_{i}$ are trivial.
This is a scheme (rather than a stack). It can be viewed as a correspondence
from $G$ to $G$.

Then $h_{D,f}$ is the operator induced by the above correspondence multiplied
by $(-q^{-1/2})^{(N-1)\deg(D)}$. We now specialize to the case when $D$ is the
Frobenius orbit of some $x\in\mathbf{k}_{i}^{\times}$ (which doesn’t belong to
any $\mathbf{k}_{j}$ with $j<i$). In this case $\deg(D)=i$. In view of the
previous subsection we only care about the case when $i$ is divisible by $N$.
In this case the number $i(N-1)$ is even and hence

$(-q^{-1/2})^{(N-1)\deg(D)}=q^{-\frac{iN}{2}}.$

To prove Theorem 1.6(ii) we need the following

######  Lemma 5.1.

Let $D$ be as above. Assume in addition that $i=N$. Then $x$ defines a
conjugacy class in $G=\operatorname{PGL}(N)$; we shall denote this class by
$\Omega_{x}\subset G$. Then as a correspondence from $G$ to $G$ the scheme
$\mathcal{H}_{D,N}$ is isomorphic to the subset of $G\times G$ consisting of
elements $(g_{1},g_{2})$ such that $g_{1}g_{2}^{-1}\in\Omega_{x}$.

###### Proof.

First, let us formulate the analogous statement over $\overline{\mathbf{k}}$
and also replace $\operatorname{PGL}(N)$ by $\operatorname{GL}(N)$. Then we
geet the following statement. Let
$(a_{1},\cdots,a_{N})\in\overline{\mathbf{k}}^{\times}$ be such that
$a_{j}\neq a_{j}$ for $i\neq j$. Let
$\Omega_{a_{1},\cdots,a_{N}}\subset\operatorname{GL}(N,\mathbf{k})$ be the
orbits of the diagonal matrix $(-a_{1},\cdots,-a_{N})$ in the group
$\operatorname{GL}(N,\mathbf{k})$. Let $Z$ denote the space of injective maps
$\phi:\mathcal{O}^{N}\to\mathcal{O}(1)^{N}$ such that

$\mathcal{O}(1)^{N}/\phi(\mathcal{O}^{N})=\bigoplus\limits_{i=0}^{N}\mathcal{O}_{a_{i}}.$

We can think about $\phi$ as an $N\times N$ matrix with entries in
$\mathbf{k}[t]$ which is invertible away from $a_{1},\cdots,a_{N}$ and such
that $\phi(t)/t$ is regular at $\infty$ and its value at $\infty$ is
invertible. Then $Z$ is isomorphic to the subset of
$\operatorname{GL}(N)\times\operatorname{GL}(N)$ consisting of pairs
$(g_{1},g_{2})$ such that $g_{1}g_{2}^{-1}\in\Omega_{a_{1},\cdots,a_{N}}$,
where the map to $G\times G$ is given by
$(\phi(0),\frac{\phi(t)}{t}(\infty))$. The proof is as follows. The space of
subsheaves $\mathcal{E}\subset\mathcal{O}(1)^{N}$ such that
$\mathcal{O}(1)^{N}/\mathcal{E}=\bigoplus\limits_{i=0}^{N}\mathcal{O}_{a_{i}}$
is naturally isomorphic to $(\mathbb{P}^{N-1})^{N}$. The open subset
corresponding to the condition that $\mathcal{E}$ is trivial is equal to the
complement to all diagonals. Thus the group $\operatorname{GL}(N)$ acts
transitively on it (the action is by automorphisms of $\mathcal{O}(1)^{N}$).
Thus it follows that if in addition we choose a trivialization of
$\mathcal{E}$ (which is what we need to get precisely the scheme $Z$) we get a
scheme isomorphic to $\operatorname{GL}(N)\times\Omega$ where $\Omega$ is some
adjoint orbit in $\operatorname{GL}(N)$. It remains to remark that the
diagonal matrix $\phi(t)=(t-a_{1},\cdots,t-a_{N})$ lies in $Z$ and in this
case $\phi(0)=(-a_{1},\cdots,-a_{N})$ and $\frac{\phi(t)}{t}(\infty)$ is the
identity matrix.

∎

Let finish the proof of Theorem 1.6(ii). If $\pi$ is an irreducible
representation of $G(\mathbf{k})$ then the element
$\sum\limits_{g\in\Omega_{x}}g\in F[G(\mathbf{k})]$ acts in $\pi$ by by a
scalar $c(\pi,x)$. Lemma 5.1 implies that if $\pi$ is cuspidal then
$\eta_{D,f}(\pi)=q^{-\frac{N(N-1)}{2}}c(\pi,x)$. On the other hand, it is
clear (by computing the trace of $\sum_{g\in\Omega_{x}}g$ in $\pi$ that

$|\Omega_{x}(\mathbf{k})|\chi_{\pi}(x)=\dim(\pi)\cdot c(\pi,x).$

But it is easy to see that

$|\Omega_{x}(\mathbf{k})|=|\operatorname{GL}(N,\mathbf{k})|/|\mathbf{k}_{N}^{\times}|=\prod\limits_{j=1}^{N-1}(q^{N}-q^{j})$

On the other hand, it follows from [5] that if $\pi$ is cuspidal and
irreducible then

$\dim(\pi)=\prod\limits_{j=1}^{N-1}(q^{j}-1)=q^{-\frac{N(N-1)}{2}}\prod\limits_{j=1}^{N-1}(q^{N}-q^{j})=q^{-\frac{N(N-1)}{2}}|\Omega_{x}(\mathbf{k})|.$

Hence

$\eta_{D,f}(\pi)=q^{-\frac{N(N-1)}{2}}c(\pi,x)=\eta_{D,f}(\pi)=q^{-\frac{N(N-1)}{2}}\chi_{\pi}(x)\frac{|\Omega_{x}(\mathbf{k})|}{\dim(\pi)}=\chi_{\pi}(x)$

which finishes the proof.

###  5.5. Proof of Theorem 1.6(iii)

Assume first that $x\in\mathbf{k}_{N}^{\times}$ and let
$[\pi]=(-1)^{N-1}R(T,\theta)$ where
$T(\mathbf{k})=\mathbf{k}_{N}^{\times}/\mathbf{k}^{\times}$ and
$\theta:T(\mathbf{k})\to F^{\times}$ is a character such that
$\theta^{q^{j}}\neq\theta$ for all $0<j<N$ (the latter condition is equivalent
to cuspidality of $\pi$)333The sign $(-1)^{N-1}$ is needed in order to turn
$R(T,\theta)$ into actual representation (as opposed to a virtual
representation). Let us view $\pi$ as a representation of $G({\mathbb{O}})$
via the evaluation map $G({\mathbb{O}})\to G(\mathbf{k})$. Then
$\operatorname{ind}_{G({\mathbb{O}})}^{G(\mathbb{K})}(\pi)$ (here ind stands
for compact induction) is irreducible and thus
$(\operatorname{ind}_{G({\mathbb{O}})}^{G(\mathbb{K})}(\pi))^{K_{1}}$ is an
irreducible (obviously non-zero) representation of $A$. The corresponding
parameters $(s,u)$ are described as follows.

######  Lemma 5.2.

Under the above conditions $u:\Gamma_{0}=\underset{\leftarrow}{\lim}\
\mathbf{k}_{i}^{\times}\to G^{\vee}(F)=\operatorname{SL}(N,F)$ factorizes
through $\mathbf{k}_{N}^{\times}$ and it is equal to the diagonal matrix
$(-1)^{N-1}(\theta,\theta^{q},\theta^{q^{2}},\cdots,\theta^{q^{N-1}})$. Also
$s$ is given by the cyclic permutation $(1\to 2\to\cdots\to N\to 1)$.

The proof easily follows from the suitable generalization of [3]
(specifically, the arguments from the proof Theorem 2.2.8. in loc. cit.) to
the ramified setting. This is relatively straightforward, but we postpone it
for another publication.

So, $s^{N}=1$ and the RHS of (1) is equal to

$(-1)^{N-1}\sum\limits_{i=0}^{N-1}\theta(x)^{q^{i}}.$

This is precisely the value of the character of $(-1)^{N-1}R(T,\theta)$ at
$x$.

We now proceed to the case of general $x$; i.e. we assume that
$x\in\mathbf{k}_{i}^{\times}$ with $i=aN$ and that $x$ does not belong to any
$\mathbf{k}_{j}$ with $j<i$. Then again $s^{i}=1$ and since
$\mathbf{k}_{N}\subset\mathbf{k}_{i}$ the RHS of (1) becomes just equal to

$(-1)^{N-1}\sum\limits_{i=0}^{N-1}\theta(\operatorname{Norm}_{i,N}(x))^{q^{i}},$

which is again the character of the above representation $\pi$.

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