# ArT: All-round Thinker for Unsupervised Commonsense
Question Answering

Jiawei Wang1,2, Hai Zhao1,2
1 Department of Computer Science and Engineering, Shanghai Jiao Tong
University
2 Key Laboratory of Shanghai Education Commission for Intelligent Interaction
and Cognitive Engineering, Shanghai Jiao Tong University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Corresponding author. This paper was partially supported by Key Projects of
National Natural Science Foundation of China (U1836222 and 61733011).

###### Abstract

Without labeled question-answer pairs for necessary training, unsupervised
commonsense question-answering (QA) appears to be extremely challenging due to
its indispensable unique prerequisite on commonsense source like knowledge
bases (KBs), which are usually highly resource consuming in construction.
Recently pre-trained language models (PLMs) show effectiveness as an
alternative for commonsense clues when they play a role of knowledge
generator. However, existing work either relies on large-scale in-domain or
out-of-domain labeled data, or fails to generate knowledge of high quality in
a general way. Motivated by human thinking experience, we propose an approach
of All-round Thinker (ArT) by fully taking association during knowledge
generating. In detail, our model first focuses on key parts in the given
context, and then generates highly related knowledge on such a basis in an
association way like human thinking. Besides, for causal reasoning, a reverse
thinking mechanism is especially added to further enhance bidirectional
inferring between cause and effect. ArT is totally unsupervised and KBs-free.
We evaluate it on three commonsense QA benchmarks: COPA, SocialIQA and SCT. On
all scales of PLM backbones, ArT shows its brilliant performance and
outperforms previous advanced unsupervised models. Our code is available at
https://github.com/WangJW424/commonsenseQA-ArT.

##  1 Introduction

Figure 1: An commonsense QA exmaple in COPA. Text with blue background is the
raw example and the gold answer is ticked. Text with green background is the
needed commonsense knowledge. Text with orange background is notes generated
by our ArT. Keyphrases are marked in red.

Commonsense question-answering (QA) has been a more challenging natural
language understanding (NLU) task than conventional QA tasks, for it requires
extra commonsense knowledge, which cannot be directly acquired from the given
context, to make an appropriate answer Niu et al. (2021). Figure 1 gives a
typical example. To select the correct cause of the fact “My body cast a
shadow over the grass”, models should know that “Light is obscured to produce
shadows” and “The sun emits light”. As human beings, we can immediately make
this association and judgment because above knowledge is deeply branded in our
mind due to daily seeing and hearing, and maybe long-term education, which is
so-called commonsense. However, it could be much difficult for models to be
equipped with such ability. Besides, commonsense QA usually takes an
unsupervised setting which makes this task even more difficult. This setting
means there is no labeled training data available as commonsense is too broad
to build a sufficient labeled training dataset Shwartz et al. (2020).

To deal with it, prior work focused on building large-scale knowledge bases
(KBs), also known as knowledge graphs (KGs), which usually contain millions of
nodes and edges to record the relation between entities as relation triple:
$<$$e_{1},r,e_{2}$$>$ Speer et al. (2017); Sap et al. (2019a). QA models then
can be injected with commonsense through retrieving over KBs Miller et al.
(2016). Though impressive improvements are gained, such method is resource
consuming in building or finding a good and suitable KB (e.g. Building
ConceptNet needs 30 GB of RAM111Declared by Speer et al. (2017) at
https://github.com/commonsense/conceptnet5). Recently, pre-trained language
models (PLMs) have been widely used and proved to be effective in commonsense
QA Niu et al. (2021); Xia et al. (2022). Thanks to the self-supervised pre-
training strategy on large-scale unlabeled text (e.g. WebText Radford et al.
(2019) and Wikipedia222https://www.english-corpora.org/wiki), PLMs are
competent for many tasks even under a zero-shot setting. And fine-tuning PLMs
on task-specific data in a supervised way can further produce even stronger
results Schick and Schütze (2021a); Gao et al. (2021); Schick and Schütze
(2021b). Since high-quality labeled datasets are rare, researching on
unsupervised commonsense QA is still of great significance. With the help of
PLMs, existing studies have explored some good solutions Niu et al. (2021);
Bosselut et al. (2021); Shwartz et al. (2020), however they either rely on
large-scale in-domain or out-of-domain labeled data, or need to be
specifically designed for different tasks. In this work, we focus on designing
a simple and general method to solve commonsense QA tasks in a strictly
unsupervised way.

Based on two empirical observations from human thinking,

  1. (1)

Given a question with context, people firstly tend to focus on several key
parts (as marked in red in Figure 1) and then make corresponding associations
to choose the right answer;

  2. (2)

For causal reasoning, people tends to carry out a bidirectional inferring to
assist answer selection or verify the answer correctness.

we propose All-round Thinker (ArT) for unsupervised commonsense QA, which
includes two principal methods: notes taking (NT) and reverse thinking (RT).
Specifically,

  1. (1)

NT extracts some keyphrases out of the context and then generates
corresponding notes, which will be added as extra knowledge in later
evaluation. Based on an unsupervised keyphrase extractor, we designed our
knowledge generation rule to be simple and general.

  2. (2)

RT converts the causal inferring question to two different forms: (cause $\to$
effect) and (effect $\to$ cause), and then integrates the decisions made from
the two reverse directions.

Our proposed model is strictly unsupervised and KBs-free for all it needs is
PLMs. We test ArT and validate its effectiveness on three commonsense QA
benchmark datasets: COPA Roemmele et al. (2011), SocialIQA Sap et al. (2019b)
and SCT Mostafazadeh et al. (2016). Our contribution is summarized as follows:

  * •

ArT can generate highly related knowledge through the imitation of human
behaviour and thought, which is qualified with inherent interpretablility.

  * •

Compared with existing work, ArT is simple and general, which totally gets rid
of the needs of any labeled data and the specific design on any specific task.

  * •

We conduct experiments on 3 commonsense QA benchmarks with 4 different scales
of PLMs, so as to reach solid and reproducible results. The results show that
ArT outperforms other advanced unsupervised models.

Figure 2: Overview of the proposed method All-round Thinker (ArT), which
contains two principal method: Notes Taking (within blue dashline box) and
Reverse Thinking (within green dashline box). The arrow denotes the flow of
data stream, with number/letter within circle marking its order in time.

##  2 Related Work

###  2.1 Building and Usage of Knowledge Bases

In order to equip QA models with commonsense reasoning ability, previous
researches were devoted to building and retrieving large-scale knowledge bases
(KBs). ConceptNet Speer et al. (2017) is one of the most famous traditional
KBs, which contains over 21 million edges (for relations) and over 8 million
notes (for entities). While ATOMIC Sap et al. (2019a) focuses more on if-then
relations between events.

Previous work applies a relatively standard routine to solve commonsense QA.
Given an existing KB, QA models can retrieve relation triples
$<$$e_{1},r,e_{2}$$>$ over it, which can be injected into models as word
embedding directly Wang et al. (2014); Paul and Frank (2019) or first
converted to a complete sentence according to preset templates (e.g. $<$bird,
CapableOf, fly$>$ $\to$ Bird can fly.) and then integrated with the raw input
text Ma et al. (2019); Mihaylov and Frank (2018); Bauer et al. (2018). This
type of methods may give remarkable performance for commonsense QA Weissenborn
et al. (2017). However, building and retrieving of KBs are both resource
consuming Bosselut et al. (2021).

Bosselut et al. (2019) claimed that commonsense knowledge does not cleanly fit
into a schema of comparing two entities with a known relation so that they
proposed COMET, which can generate rich and diverse commonsense descriptions
in natural language. However, COMET has to extract knowledge triples from
existing KBs as seed for PLM fine-tuning, while our method is totally KBs-
free.

###  2.2 PLMs in Unsupervised Commonsense QA

Due to excellent performance and versatility, PLMs now dominate the backbone
design of many NLP tasks, especially in QA Wang et al. (2021); Zhang et al.
(2021). Benefiting from the long-term pre-training on large-scale unlabeled
text, PLMs are equipped with implicit or explicit commonsense Davison et al.
(2019). As a result, there comes a tendency to rely on PLMs as the sole source
of world knowledge to solve commonsense QA in a zero-shot setting Shwartz et
al. (2020); Bosselut et al. (2021).

SEQA proposed by Niu et al. (2021) exploits generative PLMs to generate
hundreds of pseudo-answers, and then applies Sentence-BERT Reimers and
Gurevych (2019) to calculate semantic similarity between candidates and these
generated pseudo-answers. A voting mechanism is designed to make final
selection. Though SEQA shows impressive performance, it relies much on PLMs
fine-tuned on large-scale labeled NLI (Natural Language Inference) datasets.
Without such fine-tuning on such labeled datasets, SEQA’s effectiveness will
sharply decline. Instead, our method relies nothing but vanilla PLMs.

Shwartz et al. (2020) proposed Self-talk, which creatively applies PLMs as
commonsense generator. It uses preset question prefixes to firstly prompt the
PLM to generate information seeking questions (ISQs). ISQs will be put back to
secondly prompt the PLM to generate their answers as clarifications, which
will work as knowledge in later evaluation. Though this method is strictly
unsupervised, the question prefixes are not general and have to be carefully
designed according to different tasks. Besides, Self-talk fails to generate
knowledge of high quality since the generation of ISQs and clarifications are
both erratic. Inspired by Self-talk and towards its shortage, we designed a
general method to generate knowledge of high relevance.

###  2.3 Unsupervised Keyphrase Extraction

Keyphrase extraction (KE) is the task of selecting several words or phrases
that can summarize the main topic of the document Hasan and Ng (2014).
Unsupervised KE methods use different features of the document, such as word
frequency, position feature, relationship between words, etc Mihalcea and
Tarau (2004); Bougouin et al. (2013).

SIFRank Sun et al. (2020) is one of the most advanced unsupervised keyphrase
extractors. It leverages different features of words in document and evaluates
the weight of candidate phrases according to word embedding provided by PLMs.
SIFRank is originally designed to extract key noun phrases from given
documents, and we made slight modifications to adapt it to our model.

Key | Value | Replacing rule
---|---|---
“NP” | { “The definition of [NP] is”, “The main function of [NP] is”, “[NP] is a/an” } |  | directly replace
“VP” | { “[VP] means”, “After [VP], ”, “Before [VP], ” } |  | convert to gerund first
“PNP” | { “[PNP] is a/an”, “[PNP] felt”, “After this, [PNP]”, “[PNP] did this because” } |  | directly replace

Table 1: Note templates lookup table.

##  3 All-round Thinker

Inspired by the behaviour and thought of human beings during solving questions
requiring commonsense, we propose All-round Thinker (ArT). Figure 2 gives its
overview. We will firstly introduce the task definition and the basic
solution, and then describe the two principal methods of ArT in detail: Notes
Taking (NT) and Reverse Thinking (RT).

###  3.1 Task Definition and Basic Solution

This work focuses on unsupervised commonsense QA task in multi-choice style,
which consists of a $context$ ($C$) with related $question$ ($Q$), and asks
models to select a single answer from a given $option$ set:
$O=\\{O^{i}\\}_{i=1}^{|O|}$. Though there are some variants of the task form,
such as Piqa Bisk et al. (2020) and WinoGrande Sakaguchi et al. (2020), they
can be conveniently transformed into this normalized formulation: ($C$, $Q$,
$O$) Shwartz et al. (2020).

Following previous work, we adopt a basic solution which uses PLMs as scorer.
Based on the pre-training strategy: predicting the probability distribution of
token $n$ according to previous $(n-1)$-gram, e.g. OpenAI GPT Radford et al.
(2019), or bidirectional context, e.g. BERT Devlin et al. (2019), PLMs are
qualified for the role of $option$ scorer even under a zero-shot setting
Bosselut et al. (2021). Sentence likelihood is the commonly-used scoring
function for $O^{i}$:

$\begin{split}\bm{S}(O^{i}|C,Q)&=P_{LM}(O^{i}|C,Q)\\\
&=\frac{1}{|O^{i}|}\sum\limits_{t=1}^{|O^{i}|}\log
P_{LM}(O_{t}^{i}|C,Q,O_{<t}^{i})\end{split}$ (1)

where $P_{LM}$ refers to the probability function abstracted from PLMs. The
final predicted answer ($\hat{A}$) is selected as:

$\hat{A}=\operatorname*{argmax}\limits_{{O^{i}}}\bm{S}(O^{i}|C,Q)$ (2)

###  3.2 Notes Taking

To make full use of PLMs’ potential of commonsense reasoning and overcome the
shortage of Self-talk Shwartz et al. (2020), we propose notes taking (NT) to
generate commonsense descriptions in natural language (defined as “notes” in
our work) in a simple and general way. NT is composed of two phases: keyphrase
extraction and notes generation.

####  3.2.1 Keyphrase Extraction

Taking $O^{i}$ and its corresponding $C$ and $Q$ as input (as shown in left
bottom of Figure 2), ArT firstly rewrites the original interrogative $Q$ into
a declarative form (We followed the question rewriting method proposed by
Shwartz et al. (2020)) and then concatenate $<$$C$, $Q$, $O^{i}$$>$ as a
statement ($ST^{i}_{O}$). Then, we use an unsupervised keyphrase extractor to
extract keyphrases from $ST^{i}_{O}$. To be specific, we extract three types
of phrases: noun phrase (NP), verb phrase (VP) and person name phrase (PNP).

To implement this, for each type of phrase we designed a simple CFG (context-
free grammar Chomsky and Schützenberger (1959)) rule to extract it out of the
whole sentence, as following:

  * •

$NP\to(nn|adj)*+nn$

  * •

$VP\to vb+(pr)\\{0,1\\}+NP$

  * •

$PNP\to(pn)\\{1,2\\}$

where VP, NP and PNP are non-terminators; $vb$ (verb), $nn$ (noun), $adj$
(adjective), $pn$ (person name) and $pr$ (preposition) are terminators; ‘$+$’
means concatenation; $\\{a,b\\}$ means repetition times range from $a$ to $b$.
‘$*$’ is equivalent to $\\{0,\infty\\}$.

Then, we add these CFG rules to the RegexpPaser tool of NLTK Bird (2006). We
will extract top 5 most important keyphrases without any label or fine-tuned
model but word embeddings obtained from a PLM, i.e. ELMo Peters et al. (2018).

####  3.2.2 Notes Generation

Once keyphrases are obtained, we can retrieve a preset note templates set for
getting notes prefixes. Considering that: (1) For an object (NP), people will
think “what is it” and “what is it for”; (2) For a behavior (VP), people will
think “what does it mean” and “what is the cause/effect”; (3) For a person
(PNP), people will think “who is he/she” and “what is his/her
feeling/motivation/reaction”, our proposed general-purpose note templates set
is presented in Table 1.

We use the types of keyphrases as $keys$ and templates lists as $values$ to
build the note templates set as a lookup table. Given such a lookup table, we
can immediately retrieve the note templates for our extracted keyphrases and
simply replace the tag ([NP], [VP] and [PNP]) with these keyphrases to form
the note prefixes333In order to keep a correct grammar, for VP we will firstly
convert its verb into gerund form before replacing.. Though this lookup table
seems to be simple, it shows effectiveness and generality on different
benchmarks.

Next, for each note prefix, we will concatenate it to $C$ and input them into
a generative PLM to generate the complete note. Specifically, nucleus sampling
Holtzman et al. (2020) with $p=0.8$ is applied as the decoding strategy rather
than greedy/beam search to increase the diversity of generated text.
Meanwhile, to ensure the quality and scale the number of generated notes, we
sort all the notes according to their perplexity estimated by the PLM. We
denote this process as Rethinking. Finally, we will retain top $K$ notes to
construct the notes set: $\\{note^{k}\\}_{k=1}^{K}$, as shown in middle top of
Figure 2. Each $note^{k}$ will be inserted into $ST^{i}_{O}$ as extra
knowledge to assist later option scoring. The score of $O^{i}$ w.r.t
$note^{k}$ is calculated as:

$\begin{split}&score^{i,k}_{O}=P_{LM}(O^{i}|note^{k}+ST^{i}_{O}-O^{i})\\\
&=\frac{1}{|O^{i}|}\sum\limits_{t=1}^{|O^{i}|}\log
P_{LM}(O_{t}^{i}|note^{k}+ST^{i}_{O}-O^{i}+O_{<t}^{i})\end{split}$ (3)

in which ‘+’ means concatenation and ‘-’ means removing. Eventually, a voting
mechanism is applied to integrate the scores w.r.t all notes as:

$\begin{split}Score^{i}_{O}&=\frac{1}{K}\sum\limits_{k=1}^{K}score^{i,k}_{O}\end{split}$
(4)

It is worth noting that there is no need in training from any labeled data
throughout NT. And there is no need to modify the note templates according to
different tasks since our note templates is designed for general purpose.

###  3.3 Reverse Thinking

For causal reasoning questions, we additionally introduce reverse thinking
(RT) which conducts a bidirectional inferring between cause and effect. To
implement this, besides the ordered rewriting ($ST^{i}_{O}$) as mentioned in
Section 3.2.1, we also apply reverse rewriting that concatenates them in the
order of $<$$O^{i}$, $Q_{R}$, $C$$>$ (denoted as $ST^{i}_{R}$), as shown in
the right bottom of Figure 2. Note that $Q_{R}$ is the opposite question of
$Q$. To be specific, after question rewriting, “Because” and “Therefore” are
two opposite questions in causal reasoning tasks.

To conduct bidirectional inferring, except for $Score^{i}_{O}$ as introduced
in Section 3.2 for the ordered inferring: $C+Q\to O^{i}$, we set another
scoring function $Score^{i}_{R}$ for reverse inferring: $O^{i}+Q_{R}\to C$,
as:

$\begin{split}Score^{i}_{R}&=P_{LM}(C|ST^{i}_{R}-C)\\\
&=\frac{1}{|C|}\sum\limits_{t=1}^{|C|}\log
P_{LM}(C_{t}|ST^{i}_{R}-C+C_{<t})\end{split}$ (5)

To take advantage of bidirectional inferring, we design a mixed scoring
function by simply compute the average value of the above two:

$Score^{i}_{X}=\frac{1}{2}(Score^{i}_{O}+Score^{i}_{R})$ (6)

Finally, formula (2) is applied to select the answer by replacing $\bm{S}$
with $Score^{i}_{O}$ (default) or $Score^{i}_{X}$ (for causal reasoning). From
the perspective of model enhancing, averaging $Score^{i}_{O}$ and
$Score^{i}_{R}$ can be regarded as the assembly of two models (“cause” model
and “effect” model), which is a common method to enhance model performance and
robustness.

##  4 Experiment

###  4.1 Datasets

ArT is evaluated on three different commonsense QA benchmarks: COPARoemmele et
al. (2011), SocialIQASap et al. (2019b) and SCTMostafazadeh et al. (2016).
Here are the detailed information:

  * •

COPA444https://people.ict.usc.edu/~gordon/copa.html (Choice of Plausible
Alternatives): evaluates the ability of causal reasoning about a certain
event, which is described as a single sentence. Each question is accompanied
with two candidate options.

  * •

SocialIQA555https://leaderboard.allenai.org/socialiqa (Social Interaction
Question Answering): evaluates the reasoning ability on social interactions.
It has various questions, including the subject’s motivation, reaction,
personality, etc. Each question is accompanied with three candidate options.

  * •

SCT 666https://www.cs.rochester.edu/nlp/rocstories/ (Story Cloze Test):
requires models to select the right ending of the given short story from two
alternatives. Each story is composed of four sentences.

Since two test sets of the three datasets are hidden, we report all results on
dev sets. Note that the labels are kept invisible and only used for final
accuracy evaluating.

|  | Our (re-)running | Published
---|---|---|---
Dataset | Models | DistilGPT-2 | GPT-2${}_{\textup{medium}}$ | GPT-2${}_{\textup{large}}$ | GPT-2${}_{\textup{xlarge}}$ | GPT-2${}_{\textup{xlarge}}$
COPA | Baseline | 57.8 | 62.4 | 65.8 | 66.0 | –
SEQA | 51.4 (63.0) | 53.0 (68.4) | 53.8 (72.0) | 54.4 (75.4) | 79.4
Self-talk | 59.8 ($\uparrow$2.0) | 65.0 ($\uparrow$2.6) | 66.6 ($\uparrow$0.8) | 66.2 ($\uparrow$0.2) | 68.6
CGA | – | – | – | – | 72.2
ArT | 60.2 ($\uparrow$2.4) | 64.8 ($\uparrow$2.4) | 67.0 ($\uparrow$1.2) | 67.6 ($\uparrow$1.6) | –
| ArT ($Score_{X}^{i}$) |  61.0 ($\uparrow$3.2) |  65.6 ($\uparrow$3.2) |  69.4 ($\uparrow$3.6) |  69.8 ($\uparrow$3.8) | –
SocialIQA | Baseline | 41.3 | 44.3 | 45.5 | 45.9 | –
SEQA | 34.9 (43.9) | 35.9 (44.6) | 36.5 (46.6) | 36.6 (47.5) | 47.5
Self-talk | 40.5 ($\downarrow$0.8) | 44.8 ($\uparrow$0.5) | 46.1 ($\uparrow$0.6) | 47.2 ($\uparrow$1.3) | 47.5
CGA | – | – | – | – | 45.4
ArT |  42.0 ($\uparrow$0.7) |  45.6 ($\uparrow$1.3) |  47.6 ($\uparrow$2.1) |  47.3 ($\uparrow$1.4) | –
SCT | Baseline | 59.6 | 67.4 | 69.1 | 70.5 | –
SEQA | 50.7 (74.7) | 53.3 (80.5) | 54.2 (82.4) | 54.9 (83.2) | 83.2
Self-talk | 59.8 ($\uparrow$0.2) |  68.5 ($\uparrow$1.1) | 69.2 ($\uparrow$0.1) | 70.4 ($\downarrow$0.1) | 70.4
CGA | – | – | – | – | 71.5
ArT |  60.2 ($\uparrow$0.6) | 68.3 ($\uparrow$0.9) |  69.5 ($\uparrow$0.4) |  71.6 ($\uparrow$1.1) | –
Table 2: Accuracy (%) on COPA, SocialIQA and SCT. All results except last
column are run by ourselves. Best results are depicted in boldface (only
consider fully unsupervised models for fairness). $\uparrow$/$\downarrow$
refer to relative increase/decrease compared with baseline. For SEQA, we list
results of two settings: SRoBERTa${}_{\textup{large}}^{Origin}$ (before
brackets) and SRoBERTa${}_{\textup{large}}^{NLI}$ (in brackets).

###  4.2 Baseline and Contrastive models

Our baseline is constructed as only using PLMs as scorer without any explicit
knowledge injection. Formula (1) is applied as the scoring function as
described in Section 3.1. We also compare ArT with other advanced unsupervised
models:

  * •

Self-talkShwartz et al. (2020): It acquires knowledge through a two-stage
prompting of PLMs. Different question prefixes had to be specially designed
for different tasks.

  * •

SEQANiu et al. (2021): It applies PLMs to generate hundreds of pseudo-answers
and compares them with each $option$. However, its scorer relies on PLMs fine-
tuned on large-scale labeled NLI datasets, which is not strictly unsupervised.
For fair comparison, we design another setting that replaces the fine-tuned
PLM with the original one (only pre-trained on unlabeled text) .

  * •

CGABosselut et al. (2021): It employs a generative KB COMETBosselut et al.
(2019), which is trained on an existing seed KB (e.g. ConceptNet), to
construct context-relevant knowledge graphs to reason over.

###  4.3 Setup

Following previous work, we employ OpenAI GPTRadford et al. (2019) as the PLM
backbone. To reach solid and reproducible results, we conduct experiments on
GPT-2 of 4 different scales: distil, medium, large and xlarge. For ArT and
Self-talk, the same scale GPT is applied during both knowledge generating and
$option$ scoring. For SEQA, GPTs of different scales are used for pseudo-
answers generation and SRoBERTa${}_{\textup{large}}$Reimers and Gurevych
(2019) is used for semantic similarity calculation. To distinguish,
SRoBERTa${}_{\textup{large}}^{NLI}$ and SRoBERTa${}_{\textup{large}}^{Origin}$
refer to SRoBERTa${}_{\textup{large}}$ with and without further fine-tuning on
NLI datasets, respectively. For Self-talk777
https://github.com/vered1986/self_talk and SEQA888
https://github.com/heyLinsir/Semantic-based-QA, we re-run their codes with
their original settings and report both our re-running results999For each
setting except GPT-2${}_{\textup{xlarge}}$ (limited by the computational
power), we run 3 times and report the average number. and results coming from
their publications. For CGA, we report results provided by Niu et al. (2021).
For ArT, we modified the open source code of
SIFRank101010https://github.com/sunyilgdx/SIFRank to enable it to extract more
kinds of phrases rather than only noun phrase. The size of notes set ($K$) is
set as 32 as default. Except for $Score^{i}_{O}$, ArT takes another setting
$Score^{i}_{X}$ on COPA.

###  4.4 Results

Table 2 shows the results on three benchmarks. The results of our re-running
is highly consistent with those reported in their publications (last column).
Note that published results on COPA seem to have a deviation with our
reproduction. It because that they are on test set, while ours on dev set.
Shwartz et al. (2020) reported 66.0% on COPA in their paper, which is tested
on the dev set of a small version which contains 1/5 instances of that other
researches used. And ArT reaches 68.0% on that set under the same setting.

On all datasets, ArT obtains state-of-the-art performance among almost all
fully unsupervised models. Besides, it is noticed that on GPT of different
scales, ArT with NT stably brings positive improvement over baseline. On
causal reasoning task COPA, adding RT will bring further accuracy improvement.

In contrast with ArT, Self-talk fails to maintain effectiveness, whose
accuracy is even slightly lower than baseline from time to time, especially on
SCT. It indicates that the knowledge generated by Self-talk could be noisy and
as a result it misguides model evaluation.

With the help of SRoBERTa${}_{\textup{large}}^{NLI}$, SEQA can reach very
impressive results on all the datasets, especially SCT (exceed all models than
over 10%). However, when working in a strictly unsupervised mode, this method
quickly becomes invalid (close to random selection).

We also observe a common phenomenon on all our ArT, baseline and contrastive
models: when enlarging GPT-2 from large (750M parameters) to xlarge (1500M
parameters), we encounter no obvious model performance increasing and even
decreasing in some scenarios. It indicates that there could be a limit to only
using the method of increasing model parameters to improve the performance of
language models as sentence scorer or knowledge generator.

##  5 Analysis and Discussion

###  5.1 Effect of Different Modules

In order to determine the source of performance growth, we conduct ablation
study on ArT, as shown in Table 3. As expected, injecting knowledge with our
NT has positive effect on all three tasks. When removing the Rethinking
mechanism, all results slightly decrease, which indicates that Rethinking can
help increase the quality of generated knowledge. Besides, $Score^{i}_{X}$ can
bring further improvement whether or not NT is employed on COPA.

Models | COPA | SocialIQA | SCT
---|---|---|---
Baseline | 65.8 | 45.5 | 69.1
$+Score^{i}_{X}$ | 68.9 | – | –
$+$NT | 67.0 | 47.6 | 69.5
$+$NT$-\textit{Rethinking}$ | 66.2 | 46.7 | 69.4
$+$NT$+Score^{i}_{X}$ | 69.4 | – | –
Table 3: Ablation study for ArT modules on GPT-2${}_{\textup{large}}$.

###  5.2 Number of Notes and Keyphrases

Figure 3 shows the effect of notes set size $K$. On all the benchmarks, along
with the increasing of $K$, the accuracy curves basically show an upward
trend, which indicates that as the number of generated knowledge increases,
ArT will not accumulate too much noise as Niu et al. (2021) observed in Self-
talk. Therefore, compared with Self-talk, ArT tends to generate highly related
knowledge, which is contributed by our NT mechanism.

Figure 3: Accuracy curves of ArT on GPT-2${}_{\textup{medium}}$ w.r.t the size $K$ of notes set. $N$ | 1 | 3 | 5 | 7 | 9
---|---|---|---|---|---
baseline | 59.6
ArT | 59.6 | 60.2 | 60.2 | 60.0 | 60.0
Table 4: Accuracy (%) of ArT (GPT-2${}_{\textup{distil}}$) on SCT under
different settings of $N$.

In our experiments (Section 4), the number ($N$) of keyphrases to extract is
set as 5 as default. To show the effect of $N$, we conduct an ablation study
on SCT by setting $N\in\\{1,3,5,7,9\\}$. The PLM is selected as
GPT-2${}_{\textup{distil}}$. Table 4 shows the results. It is noticed that
extracting more keyphrases does not always results in a better performance.
But in general, the choice of $N$ does not have an obvious impact on the final
performance.

Dataset | Model | Rationality | Usefulness
---|---|---|---
COPA | Self-talk | 0.24 | 0.20
ArT | 0.32 | 0.28
SocialIQA | Self-talk | 0.17 | 0.16
ArT | 0.26 | 0.27
Table 5: Human evaluation on the rationality and usefulness of generated
knowledge.

###  5.3 Quality of Generated Knowledge

To compare the quality of generated knowledge (whether the knowledge is
reasonable enough to be a “fact” and correlative enough to be useful), we
conduct human evaluation on the rationality (-1: irrational, 0: meaningless,
1: rational) and usefulness (-1: negative, 0: neutral, 1: positive) of
knowledge generated by ArT and Self-talk. Two annotators are asked to
independently annotate 100 randomly selected knowledge for both COPA and
SocialIQA. The overall average scores of two annotators for each dataset are
shown in Table 5. It is noticed that ArT outperforms Self-talk in generating
knowledge with both rationality and usefulness.

Figure 4: Statistics on the quality of knowledge generated by Self-talk and
ArT on COPA.

We further make statistics on the kinds of all the knowledge generated by
Self-talk and ArT on COPA. We divide them into four classifications:

  * •

Positive: Baseline makes the right prediction. Adding the knowledge still
makes a right prediction.

  * •

Essential: Baseline makes the wrong prediction. Adding the knowledge helps
make a right prediction.

  * •

Invalid: Baseline makes the wrong prediction. Adding the knowledge still makes
a wrong prediction.

  * •

Negative: Baseline makes the right prediction. Adding the knowledge leads to a
wrong prediction.

As shown in Figure 4, by doubling Essential and reducing Negative knowledge,
ArT outperforms Self-talk in generating high-quality commonsense.

Figure 5 and 6 in Appendix show some examples of knowledge generated by ArT,
Self-talk and SEQA for COPA, SocialIQA and SCT. As can be seen, ArT generates
highly related knowledge on the basis of focusing on keyphrases of given
context. While Self-talk may generate meaningless or even noisy knowledge. It
is also noticed that SEQA could generate very reasonable pseudo-answers. And
to distinguish the relationship between these pseudo-answers and given
$options$ relies much on a sentence embedding extractor fine-tuned on labeled
NLI data.

###  5.4 Effect of Note Types

As we designed three types of notes: NP, VP and PNP, we only consider one type
at one time to show the effect of each in different benchmarks, as shown in
Table 6 (The PLM is GPT-2${}_{\textup{medium}}$).

Dataset | NP | VP | PNP | All / None
---|---|---|---|---
COPA | 63.6 | 63.2 | 62.4 | 64.8 / 62.4
SocialIQA | 45.0 | 44.8 | 45.3 | 45.6 / 44.3
SCT | 68.0 | 67.9 | 68.1 | 68.3 / 67.4
Table 6: Effect of each type of notes on different tasks.

It is noticed that each type of notes has a positive effect when applied
separately, and their combination works better. Note that COPA has no person
name phase (PNP), so PNP notes does not work on it. On SocialIQA, PNP works
best among three types of notes. This is reasonable since SocialIQA focus much
on human behavior in social interactions.

##  6 RT for Other Questions

Although RT is designed to enhance causal reasoning, we also explored if RT
has the potential to help other questions. To investigate this, we apply it on
other two datasets: SocialIQA and SCT. Note that in these tasks the opposite
question is hard to define, therefore we simply exchange the position of
$option$ and $context$, that is $ST^{i}_{O}=<C,Q,O^{i}>$ and
$ST^{i}_{R}=<O^{i},Q,C>$. We employ three scoring functions on basis of
GPT-2${}_{\textup{medium}}$ baseline, as shown in Table 7.

Functions | COPA | SocialIQA | SCT
---|---|---|---
$Score^{i}_{O}$ | 62.4 | 44.3 | 67.4
$Score^{i}_{R}$ | 63.2 | 42.5 | 62.8
$Score^{i}_{X}$ | 65.3 | 44.1 | 65.4
Table 7: Accuracy (%) of GPT-2${}_{\textup{medium}}$ baseline with different
scoring functions.

On all the tasks, $Score^{i}_{O}$ and $Score^{i}_{R}$ can obtain positive
results (much better than random selection). By simply averaging
($Score^{i}_{X}$), on COPA it can reach a higher score. On other tasks, it
falls into the middle of $Score^{i}_{O}$ and $Score^{i}_{R}$. Considering that
$Score^{i}_{R}$ shows a comparable performance with $Score^{i}_{O}$ by simply
exchanging the position of $option$ and $context$, developing a general method
for opposite question definition or designing a more exquisite method to
integrate $Score^{i}_{O}$ and $Score^{i}_{R}$ perhaps could make RT suitable
for questions beyond causal reasoning, which will be the key point of our
future work.

##  7 Conclusion

Commonsense QA has been a challenging task for it requires extra knowledge
beyond the given context. In consideration of the high resource consumption of
building knowledge bases (KBs) and the rarity of high-quality labeled data,
this work aims at addressing commonsense QA in a fully KBs-free and
unsupervised way. Inspired by the association process of human thinking, we
propose All-round Thinker (ArT), which first focuses on key parts in the given
context, and then generates highly related knowledge on such a basis in an
association way. Besides, a reverse thinking mechanism is introduced to
further enhance bidirectional inferring for causal reasoning as human will do.
We test ArT on three benchmarks: COPA, SocialIQA and SCT. ArT outperforms
previous advanced unsupervised models and shows stable performance on all
scales of PLM backbones.

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Figure 5: Top two most contributing generated knowledge for instances of COPA
and ScocialIQA. The correct options are underlined. Keyphrases extracted by
ArT are marked in red. Figure 6: Top two most contributing generated knowledge
for instances of SCT. The correct options are underlined. Keyphrases extracted
by ArT are marked in red.


# On the distinction of Iwahori-spherical discrete series representations

Paul Broussous

###### Abstract

Let $E/F$ be a quadratic unramified extension of non-archimedean local fields
and $\mathbb{H}$ a simply connected semisimple algebraic group defined and
split over $F$. We establish general results (multiplicities, test vectors) on
${\mathbb{H}}(F)$-distinguished Iwahori-spherical representations of
${\mathbb{H}}(E)$. For discrete series Iwahori-spherical representations of
${\mathbb{H}}(E)$, we prove a numerical criterion of
${\mathbb{H}}(F)$-distinction. As an application, we classify the
${\mathbb{H}}(F)$-distinguished discrete series representations of
${\mathbb{H}}(E)$ corresponding to degree $1$ characters of the Iwahori-Hecke
algebra.

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Iwahori-spherical representations: models and coefficients
  4. 4 Multiplicities
  5. 5 Test vectors
  6. 6 Generalized Poincaré series
  7. 7 A numerical criterion with application to degree $1$ characters

##  1 Introduction

This work is a contribution to the study of distinguished complex
representations for Galois symmetric spaces over a non-archimedean local
field. It may be viewed as a continuation of [6] where the authors prove a
particular case of Prasad’s conjecture on the distinction of the Steinberg
representation ([15], Conjecture 3, p. 77). However even though the Bruhat-
Tits building will be used here as a geometrical tool as in [6], our
fundamental techniques will be that of integration of matrix coefficients, in
the spirit of [5].

We restrict ourselves to the easy case where the symmetric space is of the
form ${\mathbb{H}}(E)/{\mathbb{H}}(F)$, where $E/F$ is a quadratic unramified
extension of non-archimedean local fields and $\mathbb{H}$ a connected
reductive group defined and split over $F$. For simplicity sake we shall
assume that $\mathbb{H}$ is semisimple and simply connected and that its root
system is irreducible. However the methods of this article work without those
restrictions.

Recall that a smooth complex representation $(\pi,\mathcal{V})$ of
${\mathbb{H}}(E)$ is said to be ${\mathbb{H}}(F)$-distinguished if the
intertwining space ${\rm Hom}_{{\mathbb{H}}(F)}\,(\mathcal{V},\mathbb{C})$ is
non-zero, where $\mathbb{C}$ is a complex line with the trivial action of
${\mathbb{H}}(F)$. In this artice we study the distinction of Iwahori-
spherical irreducible representations, i.e. representations
$(\pi,\mathcal{V})$ such that $\mathcal{V}^{I}\not=0$, where $I$ is some fixed
Iwahori subgroup of ${\mathbb{H}}(E)$. If $(\pi,\mathcal{V})$ is such a
representation, the Iwahori-Hecke algebra ${\mathcal{H}}_{\rm Iw}$ of
${\mathbb{H}}(E)$ relative to $I$ acts on $M=\mathcal{V}^{I}$ via an
irreducible representation $(r,M)$.

Our first result is the following.

###### Theorem 1.1

(Theorems 4.1 and 5.4) Let $(\pi,\mathcal{V})$ be an Iwahori-spherical
irreducible representation of ${\mathbb{H}}(E)$.

(i) We have ${\rm Dim}\,{\rm
Hom}_{{\mathbb{H}}(F)}\,(\mathcal{V},\mathbb{C})\leqslant{\rm
Dim}\,\mathcal{V}^{I}$

(ii) If $\Lambda\in{\rm Hom}_{{\mathbb{H}}(F)}(\mathcal{V},\mathbb{C})$ is
non-zero then there exists a ${\rm Galois}(E/F)$-stable Iwahori subgroup $J$
of $G$ and a vector $v\in\mathcal{V}^{J}$ such that $\Lambda(v)\not=0$.

The proofs of these results are based on a transitivity property of the action
of ${\mathbb{H}}(F)$ on the set of chambers of the Bruhat-Tits building of
${\mathbb{H}}(E)$, which was a crucial ingredient of [6].

We next assume moreover that $(\pi,\mathcal{V})$ is a discrete series
representation. Using the fact due to Zhang [19] that
${\mathbb{H}}(E)/{\mathbb{H}}(F)$ is a very strongly discrete symmetric space,
we establish a second result.

###### Theorem 1.2

(Theorem 5.3) Let $(\pi,\mathcal{V})$ be an irreducible Iwahori-spherical
discrete series representation of ${\mathbb{H}}(E)$. Let $J$ be some ${\rm
Gal}(E/F)$-stable Iwahori subgroup of ${\mathbb{H}}(E)$ and $\nu$ denote a
Haar measure on ${\mathbb{H}}(F)$.

Then $(\pi,\mathcal{V})$ is ${\mathbb{H}}(F)$-distinguished if, and only if,
there exist $v\in\mathcal{V}^{J}$, ${\tilde{v}}\in{\tilde{\mathcal{V}}}^{J}$,
such that

$\int_{{\mathbb{H}}(F)}c_{v,{\tilde{v}}}(h)\,d\nu(h)\not=0\ .$

where $\tilde{\mathcal{V}}$ is the contragredient representation and
$c_{v,{\tilde{v}}}$ is the coefficient of $\pi$ attached to $v$ and
$\tilde{v}$.

To turn the previous theorem into an effective criterion, we use the notion of
generalized Poincaré series attached to an Iwahori-Hecke algebra introduced by
Gyoja [11]. Let $(W,S)$ be a Coxeter system with associated Hecke algebra
${\mathcal{H}}$. Let $S_{i}$, $i=1,...,m$ be the intersections of the
conjugacy classes of $W$ with $S$. For each representation $(r,M)$ of
${\mathcal{H}}$, Gyoja attaches a power series $L(t_{1},...,t_{m},r)\in{\rm
End}(M)\otimes_{\mathbb{C}}\mathbb{C}[[t_{1},...,t_{m}]]$. Now assume that
${\mathcal{H}}={\mathcal{H}}_{\rm Iw}$ is the Iwahori-Hecke algebra of
${\mathbb{H}}(E)$ relative to $I$ and $(r,M)$ is the representation associated
to the ${\mathcal{H}}_{\rm Iw}$-module $\mathcal{V}^{I}$, for a representation
$(\pi,\mathcal{V})$ of ${\mathbb{H}}(E)$.

###### Theorem 1.3

(Theorem 7.1) Let $(\pi,\mathcal{V})$ be an irreducible Iwahori-spherical
discrete series representation of ${\mathbb{H}}(E)$. Then $\pi$ is
${\mathbb{H}}(F)$-distinguished if, and only if
$L(1/q_{o},...,1/q_{o},r)\in{\rm End}(M)$ is non-zero, where $q_{o}$ is the
size of the residue field of $F$.

Our last result is concerned with the case where the discrete series
$(\pi,\mathcal{V})$ corresponds to a degree $1$ character of
${\mathcal{H}}_{\rm Iw}$. Those representations where classified by A. Borel
in [2]. It is by using techniques borrowed from loc. cit that we prove the
following: the value of $L(t_{1},...,t_{m},r)$ occuring in Theorem 1.3 can be
calculated using a Macdonald’s Poincaré series (cf. [14]). Using Macdonald’s
formulas, we obtain:

###### Theorem 1.4

(Theorem 7.4) Let $(\pi,\mathcal{V})$ be an irreducible Iwahori-spherical
discrete series representation of ${\mathbb{H}}(E)$ such that
$\mathcal{V}^{I}$ is $1$-dimensional. If ${\mathbb{H}}$ is not of type
$G_{2}$, then $\pi$ is ${\mathbb{H}}(F)$-distinguished if, and only if, it is
the Steinberg representation. If ${\mathbb{H}}$ has type $G_{2}$ there are two
representations satisfying the assumptions of the theorem: the Steinberg
representation and another one; there are both
${\mathbb{H}}(F)$-distinguished.

In other words, the Steinberg representation of ${\mathbb{H}}(E)$ is the
unique representation satisfying the assumption of the theorem and which is
${\mathbb{H}}(F)$-distinguished, except in type $G_{2}$ where there is another
one.

The case of the Steinberg representation (originally D. Prasad’s conjecture)
was already established in [6] (where ${\mathbb{H}}$ is not necessarilly
assumed simply connected) and in characteristic $0$ Prasad’s conjecture was
entirely proved by Beuzart-Plessis [1].

Macdonald’s formulas [14] allow only to calculate Gyoja $L$-functions
$L(t_{1},...,t_{m},r)$ when $r$ is a degree $1$ character, this is why we are
not able to give a classification of distinguished discrete series
representations when ${\rm Dim}\,\mathcal{V}^{I}>1$.

In [16], Dipendra Prasad states a conjecture predicting the distinction of
irreducible smooth complex representations in the case of a general Galois
symmetric space ${\mathbb{H}}(E)/{\mathbb{H}}(F)$ (Conjecture 2 of loc. cit.).
An irreducible smooth representation $\pi$ of ${\mathbb{H}}(E)$ conjecturally
has a Langlands parameter $\phi_{\pi}$ : $W_{E}\times{\rm
SL}_{2}(\mathbb{C})\longrightarrow{}^{L}G$, where the Langlands dual ${}^{L}G$
is a semidirect product $W_{E}\ltimes\widehat{G}$, $\widehat{G}$ being the
complex dual of $G={\mathbb{H}}(E)$. Roughly speaking, the point of Prasad’s
conjecture is to relate the ${\mathbb{H}}(F)$-distinction of $\pi$ to
properties of the parameter $\varphi_{\pi}$. A precise formulation of this
conjecture in our case may be stated as follows (this formulation as well as
the two following remarks are due to Prasad himself).

###### Conjecture 1.5

(Prasad) Let $E/F$ be a quadratic unramified extension of nonarchimedean local
fields. Let ${\mathbb{H}}$ be a simply connected split simple group over $F$
if $-1$ belongs to the Weyl group of ${\mathbb{H}}$, and the unique quasi-
split (but not split) group over $F$ which splits over $E$ otherwise. (This
ensures that the opposition group $H^{\rm op}$ of $H={\mathbb{H}}(F)$, defined
in [16], is split over $F$.) A Iwahori-spherical discrete series
representation $\pi$ of $G={\mathbb{H}}(E)$ has, by the work of Kazhdan-
Lusztig, cf. [13], an $\rm L$-parameter $\phi:W_{E}\times{\rm
SL}_{2}(\mathbb{C})\rightarrow{}^{L}G$ which on $W_{E}$ factors through an
unramified extension of $E$. Assume that the group of connected components of
the centralizer of the corresponding unipotent element in $\widehat{G}$ is an
elementary abelian $2$-group (which is known to be the case for all groups of
type $B_{n},C_{n},D_{n}$). Then $\pi$ is distinguished by $H={\mathbb{H}}(F)$
if and only if

  1. 1.

The restriction of $\phi$ to $W_{E}$ is trivial,

  2. 2.

The representation $\pi$ is generic. (Under condition (1), genericity should
not depend on the choice of the Whittaker datum.)

Further, if the conditions (1) and (2) are satisfied, then ${\rm Dim}\,{\rm
Hom}_{H}\,(\pi,\mathbb{C})$ is the order of the centralizer of $\phi({\rm
SL}_{2}(\mathbb{C}))$ in $\hat{G}$, which by assumption is a finite elementary
abelian 2-group.

Remark 1. We relate the above Conjecture to that in [16]. In the present case,
$W_{F}\times{\rm SL}_{2}(\mathbb{C})$ and $W_{E}\times{\rm
SL}_{2}(\mathbb{C})$ can be replaced by $\langle{\rm
Fr}^{\mathbb{Z}}\rangle\times{\rm SL}_{2}(\mathbb{C})$ and $\langle{\rm
Fr}^{2\mathbb{Z}}\rangle\times{\rm SL}_{2}(\mathbb{C})$, where ${\rm Fr}$ is a
Frobenius element of $W_{F}$. Because of the hypothesis in the Conjecture,
$\pi_{0}(Z_{\widehat{G}}(\phi({\rm SL}_{2}(\mathbb{C}))))$ being an elementary
abelian $2$-group, $\langle{\rm Fr}^{2\mathbb{Z}}\rangle$ must land inside the
connected component of identity of $Z_{\widehat{G}}(\phi({\rm
SL}_{2}(\mathbb{C})))$, hence by the discreteness of the parameter $\phi$,
$Z_{\widehat{G}}(\phi({\rm SL}_{2}(\mathbb{C})))$ must be finite and therefore
an elementary abelian $2$-group, and the image of $\langle{\rm
Fr}^{2\mathbb{Z}}\rangle$ under $\phi$ must be trivial which is the conclusion
in part (1) of the Conjecture. Further, the natural map from the component
group of a lifted parameter $\tilde{\phi}$ for $H^{\rm op}$ to the component
group for $\phi$ is clearly an isomorphism, hence by the Conjecture of [16],
the only $\pi$ corresponding to the trivial character of the component group
can be distinguished which is the conclusion in part (2) of the Conjecture.

Remark 2. Here is the example of $G_{2}$ which appears in this work but for
which the hypothesis in the Conjecture above does not hold. There is a
discrete series representation $\pi$ of $G_{2}(E)$ which has a one dimensional
fixed vector for an Iwahori subgroup, which by Reeder [17], page 480,
corresponds to a subregular unipotent element with component group isomorphic
to $S_{3}$ with the representation of the component group, the $2$-dimensional
irreducible representation $\rho$ of $S_{3}$. It is thus a non-generic
representation of $G_{2}$ which is distinguished by $G_{2}(F)$ (by Theorem 7.4
of this paper). In this case, the representation $\rho$ of $S_{3}$ has a fixed
vector under any transposition in $S_{3}$, thus confirming the general
conjecture of [16] in an example with non-abelian component group.

It would be nice to check Prasad’s conjecture in our case. In this aim, one
would have to determine the Galois parameters of our representations, which
are not known in general.

The article is organized as follows. After introducing the general notation in
§1, in §2 we recall the models of Iwahori-spherical representations introduced
in [2] and explain how to deduce a very simple formula for the Iwahori-
spherical coefficients. The bound on the dimension of the space of
${\mathbb{H}}(F)$-invariant linear forms is given in §3 and the fact that
test-vectors can be chosen Iwahori-spherical is proved in §4. In §5 we recall
the definition of Gyoja’s generalized Poincaré series and we give Macdonald’s
formulas. Finally §6 is devoted to establishing the numerical criterion in
terms of Gyoja’s Poincaré series values and to applying it to the case of
degree $1$ characters of the Iwahori-Hecke algebra.

I warmly thank Nadir Matringe and Dipendra Prasad for their help in writing
this article.

##  2 Notation

Throughout the article we use the following notation.

If $X$ is a set, we denote by $|X|$ its cardinal. If a group $\Gamma$ acts on
$X$ and $\Gamma^{\prime}$ is a subset of $\Gamma$, $X^{\Gamma^{\prime}}$
denotes the subset of those $x\in X$ that are fixed by any element of
$\Gamma^{\prime}$.

We let $F$ denote a non-archimedean, non-discrete, locally compact field. We
denote by ${\mathfrak{o}}_{F}$ its ring of integers, by ${\mathfrak{p}}_{F}$
the maximal ideal of ${\mathfrak{o}}_{F}$, and we set
$k_{F}={\mathfrak{o}}_{F}/{\mathfrak{p}}_{F}$ (residue field),
$q_{0}=|k_{F}|=p^{r}$, $p$ being the characteristic of $k_{F}$ (residue
characteristic).

We fix an unramified quadratic extension $E/F$ and use the obvious pieces of
notation ${\mathfrak{o}}_{E}$, ${\mathfrak{p}}_{E}$, $k_{E}$. We denote by
$\theta$ the generator of the Galois group ${\rm Gal}(E/F)$. We set
$q=|k_{E}|$, so that $q=q_{0}^{2}$.

We fix a connected semisimple $F$-algebraic group $\mathbb{H}$ assumed to be
split over $F$ and simply connected and we write $H={\mathbb{H}}(F)$ for the
locally compact group of its $F$-rationnal points. We denote by
${\mathbb{G}}={\rm Res}_{E/F}{\mathbb{H}}$ the $F$-algebraic group obtained by
Weil’s restriction of scalars. It is semisimple and simply connected. We set
$G={\mathbb{G}}(F)={\mathbb{H}}(E)$.

The Galois group ${\rm Gal}(E/F)$ acts on $\mathbb{G}$ by $F$-rational
automorphisms of algebraic group. We still denote by $\theta$ the action of
$\theta$ on $\mathbb{G}$ and $G$ so that ${\mathbb{H}}={\mathbb{G}}^{\theta}$
and $H=G^{\theta}$. The pair $(G,H)$ (or the $G$-set $G/H$) is called a
(Galois) reductive symmetric space.

We let $X_{F}$ (resp. $X_{E}$) denote the Bruhat-Tits building of $\mathbb{H}$
over $F$ (resp. over $E$). Recall that $X_{E}$ identifies canonically with the
Bruhat-Tits building of ${\rm Res}_{E/F}({\mathbb{H}})$ over $F$. As
simplicial complexes $X_{F}$ and $X_{E}$ have dimension $d$, the $F$-rank of
$\mathbb{H}$. As a particular case of building functorialities, there is an
injection $j$ : $X_{F}\longrightarrow X_{E}$ satisfying the following
properties:

(a) $j$ is ${\rm Gal}(E/F)\ltimes H$-equivariant,

(b) the image of $j$ is $X_{E}^{{\rm Gal}(E/F)}$,

(c) $j$ is simplicial and maps chambers to chambers.

We identify $X_{F}$ as a subsimplicial complex of $X_{E}$ viewing $j$ as an
inclusion.

We fix a maximal $F$-split torus $\mathbb{T}$ of $\mathbb{H}$ and set
$T_{F}={\mathbb{T}}(F)$, $T_{E}={\mathbb{T}}(E)$. We assume that the relative
root system $\Phi=\Phi({\mathbb{T}},{\mathbb{H}})$ of $\mathbb{T}$ in
$\mathbb{H}$ is irreducible. Let $W^{\rm Aff}_{F}=N_{H}({T})/T_{F}^{o}$ be the
affine Weyl group of $H$ relative to $\mathbb{T}$, where $N_{H}(T_{F})$ is the
normalizer of $\mathbb{T}$ in $H$ and $T_{F}^{o}$ the maximal compact subgroup
of $T_{F}$. The group $W^{\rm Aff}_{E}$ is defined in the same way and the
inclusion $N_{H}(T_{F})\subset N_{G}(T_{E})$ induces an isomorphism of groups
$W_{F}^{\rm Aff}\simeq W_{E}^{\rm Aff}$. In the sequel we abbreviate $W^{\rm
Aff}=W_{F}^{\rm Aff}=W_{E}^{\rm Aff}$.

Let ${\mathcal{A}}_{F}$ (resp. ${\mathcal{A}}_{E}$ ) denote the appartment of
$X_{F}$ (resp. $X_{E}$) associated to $\mathbb{T}$. It follows from the
construction of $j$ that ${\mathcal{A}}_{F}={\mathcal{A}}_{E}$. Fix a chamber
$C_{o}$ of ${\mathcal{A}}_{F}$ and let $I^{o}_{F}$ (resp. $I_{E}^{o}$) denote
the Iwahori subgroup of $H$ (resp. the ${\rm Gal}(E/F)$-stable Iwahori
subgroup of $G$) fixing $C_{o}$. Since $\mathbb{H}$ is simply connected these
Iwahori subgroups are the (global) fixators of $C_{o}$. Let $S\subset
W_{F}^{\rm Aff}$ be the set of involutions corresponding to reflections in
${\mathcal{A}}_{F}$ relative to the walls of $C_{o}$. Then, since
${\mathbb{H}}$ is simply connected, $(I_{F}^{o},N_{H}(T_{F}))$ (resp.
$(I_{E}^{o},N_{G}(T_{E}))$) is a Tits system (or BN-pair) in $H$ (resp. in
$G$) with Coxeter system $(W^{\rm Aff},S)$. We denote by $l$ the length
function of $W$ relative to the generative subset $S$. Recall that

$|I_{E}^{o}wI_{E}^{o}/I_{E}^{o}|=q^{l(w)},\ w\in W^{\rm Aff}\ .$

Recall that the vertices of $X_{E}$ are naturally labelled by the elements of
$S$. If $C$ is a chamber of $X_{E}$, the wall of $C$ which does not contain
the unique vertex of type $s$ is said to be of type $s$. If $C$ and $D$ are
two chambers of $X_{E}$, we write $D\sim_{s}C$ if $D$ and $C$ are adjacent and
if the wall $D\cap C$ is of type $s$.

We let ${\mathcal{H}}_{\rm Iw}$ denote the Iwahori-Hecke algebra of $G$
relative to our choice $I_{E}^{o}$ of Iwahori subgroup. Recall that as a
complex vector space ${\mathcal{H}}_{\rm Iw}$ is the set of complex functions
of $G$ which are bi-invariant by $I_{E}^{o}$ and have compact support. It is
endowed with the convolution product defined by

$u\star v(g)=\int_{G}u(gx^{-1})v(x)\,d\mu(x)=\sum_{x\in I_{E}^{o}\backslash
G}u(gx^{-1})v(x),\ g\in G,\ u,v\in{\mathcal{H}}_{\rm Iw}$

where $\mu$ is the Haar measure on $G$ normalized by $\mu(I_{E}^{o})=1$.
Recall that $(e_{w})_{w\in W^{\rm Aff}}$, is a basis of ${\mathcal{H}}_{\rm
Iw}$, where for $w\in W^{\rm Aff}$, $e_{w}$ denote the caracteristic function
of $I_{E}^{o}wI_{E}^{o}$. Recall that we have the classical relations:

$\displaystyle(e_{s}+1)\star(e_{s}-q)=0,\ s\in S$ $\displaystyle e_{w}\star
e_{w^{\prime}}=e_{ww^{\prime}}\ {\rm if}\ l(ww^{\prime})=l(w)+l(w^{\prime}),\
w,\ w^{\prime}\in W^{\rm Aff}$

and that the $e_{s}$, $s\in S$, generate ${\mathcal{H}}_{\rm Iw}$ as an
algebra.

If $f\in{\mathcal{H}}_{\rm Iw}$, we define $\check{f}$ by
$\check{f}(x)=f(x^{-1})$, $x\in G$. The map $f\mapsto\check{f}$ is an
automorphism of the vector space ${\mathcal{H}}_{\rm Iw}$ satisfying
${\check{(}f\star g)}=\check{g}\star\check{f}$, $f,g\in{\mathcal{H}}_{\rm
Iw}$. If $(r,M)$ is a complex representation of ${\mathcal{H}}_{\rm Iw}$, we
define its contragredient $({\tilde{r}},{\tilde{M}})$ in the dual
${\tilde{M}}$ of $M$ by ${\tilde{r}}(f)={}^{t}r(\check{f})$,
$f\in{\mathcal{H}}_{\rm Iw}$, where t denotes the transposition.

In this article representations of $G$, $H$ or ${\mathcal{H}}_{\rm Iw}$ are
always assumed to be in complex vector spaces. If $(\pi,\mathcal{V})$ is a
smooth representation of $G$, $({\tilde{\pi}},{\tilde{\mathcal{V}}})$ denotes
its contragredient (i.e. its smooth dual). If $W$ is a complex vector space,
$W^{*}={\rm Hom}_{\mathbb{C}}(W,\mathbb{C})$ denotes its algebraic dual.

##  3 Iwahori-spherical representations: models and coefficients

In this section we abbreviate $I=I_{E}^{o}$.

Let $\mathcal{S}(G)$ denote the abelian category of smooth representations of
$G$ and $\mathcal{S}(G)_{\rm Iw}$ denote the full subcategory of those
representations $(\pi,\mathcal{V})$ that are generated by the fixed vector set
$\mathcal{V}^{I}$. Let ${\mathcal{H}}_{\rm Iw}{\rm-Mod}$ denote the category
of left ${\mathcal{H}}_{\rm Iw}$-modules. If $(\pi,\mathcal{V})$ is an object
of $\mathcal{S}(G)_{\rm Iw}$, then $\mathcal{V}^{I}$ is naturally a left
${\mathcal{H}}_{\rm Iw}$-module, so that we have a well defined functor

${\mathbf{M}}\ :\ \mathcal{S}(G)_{\rm Iw}\longrightarrow{\mathcal{H}}_{\rm
Iw}{\\!-}{\rm Mod},\ (\pi,\mathcal{V})\mapsto\mathcal{V}^{I}\ .$

It is now a well known fact ([2],[8]) that $\mathbf{M}$ is an equivalence of
categories; in other words the trivial character of $I$ is a type for $G$ in
the sense of [7]. Moreover if $(\pi,\mathcal{V})$ is admissible, we have
$\mathbf{M}({\tilde{\mathcal{V}}})=\mathbf{M}(\mathcal{V})\,\tilde{}$ : the
functor commutes with the operation of taking the contragredient.

A pseudo-inverse $\mathbf{V}$ for $\mathbf{M}$ may be constructed as follows
(cf. [2]§4). Let us fix a left ${\mathcal{H}}_{\rm Iw}$-module $M$ viewed as a
representation $(r,M)$ of ${\mathcal{H}}_{\rm Iw}$. Let $C_{c}(G/I)$ be the
space of functions on $G$ which are right $I$-invariant and have compact
support. This is naturally a right ${\mathcal{H}}_{\rm Iw}$-module and a
smooth representation of $G$ (by left translation) and both structures are
compatible. Hence we may form the tensor product
${\mathbf{V}}(M):=C_{c}(G/I)\otimes_{{\mathcal{H}}_{\rm Iw}}M$. Of course we
have an isomorphism of left ${\mathcal{H}}_{\rm Iw}$-modules
$M\simeq{\mathbf{V}}(M)^{I}$; it is explicitly given by $m\mapsto e_{1}\otimes
m$.

We shall need two other functorial constructions $\mathbf{P}$,
${\mathbf{P}}^{o}$, considered by Borel in loc. cit. §2. If $(r,M)$ is a
representation of ${\mathcal{H}}_{\rm Iw}$ we set

${\mathbf{P}}^{o}(M)=\\{f\in C(G/I,M)\ ;\ f\star r(u)=r(u).f,\
u\in{\mathcal{H}}_{\rm Iw}\\}\ ,$

where $C(G/I,M)$ denotes the space of $M$-valued functions on $G$ which are
right $I$-invariant. This space is acted upon by $G$ via left translation, but
the obtained representation of $G$ is not smooth in general; we denote by
${\mathbf{P}}(M)\subset{\mathbf{P}}^{o}(M)$ the subspace of smooth vectors. By
[2] Proposition 2.4, the map

$\nu_{o}\ :\ {\mathbf{P}}(M)^{I}={\mathbf{P}}^{o}(M)^{I}\longrightarrow M\ ,\
f\mapsto f(1)\ ,$

is an isomorphism of vector spaces. Moreover, for $m\in M$, the element
$f_{m}\in{\mathbf{P}}(M)^{I}$ mapped onto $m$ by $\nu_{o}$ is the function
$f_{e}$ on $G$ mapping any element of $IwI$, $w\in W^{\rm Aff}$, to
$q_{w}^{-1}\,r(\check{e_{w}}).m$, symbolically written by Borel as

$f_{m}=\sum_{w\in W^{\rm Aff}}e_{w}\ q_{w}^{-1}\,r(\check{e_{w}}).m\ .$ (1)

The tensor product $C_{c}(G/I)\otimes_{{\mathcal{H}}_{\rm Iw}}M$ may be viewed
as the quotient of $C_{c}(G/I)\otimes_{\mathbb{C}}M$ by the $G$-invariant
subspace generated by $(f\star u)\otimes m-f\otimes r(u).m$, $f\in
C_{c}(G/I)$, $u\in{\mathcal{H}}_{\rm Iw}$, $m\in M$. We may identify
$C_{c}(G/I)\otimes_{\mathbb{C}}M$ with the space $C_{c}(G/I,M)$ of $M$-valued
functions on $G$ which are right $I$-invariant. The bilinear $G$-invariant map

${\mathbf{P}}^{o}({\tilde{M}})\times C_{c}(G/I,M)\longrightarrow\mathbb{C},\
(f,g)\mapsto\sum_{x\in G/I}\langle f(x),g(x)\rangle_{M}$ (2)

induces an well-defined $G$-invariant bilinear map

${\mathbf{P}}^{o}({\tilde{M}})\times{\mathbf{V}}(M)\longrightarrow\mathbb{C}$

whence a $G$-intertwining operator

$\psi_{M}\ :\ {\mathbf{P}}^{o}({\tilde{M}})\longrightarrow{\mathbf{V}}(M)^{*}$

where $*$ denotes an algebraic dual. From [2] Proposition 2.6, we have the
following.

###### Proposition 3.1

Let $(r,M)$ be a representation of ${\mathcal{H}}_{\rm Iw}$. Then the
intertwining operator $\psi_{M}\ :\
{\mathbf{P}}^{o}({\tilde{M}})\longrightarrow{\mathbf{V}}(M)^{*}$ is an
isomorphism and restricts to an isomorphism
${\mathbf{P}}({\tilde{M}})\simeq{\mathbf{V}}(M)\tilde{}$. In particular if $M$
is finite dimensional we have an isomorphism of $G$-modules
${\mathbf{P}}(M)\simeq{\mathbf{V}}(M)$.

Let $(r,M)$ be a finite dimensional representation of ${\mathcal{H}}_{\rm Iw}$
and set $(\pi,\mathcal{V})={\mathbf{V}}(r,M)$. We make the canonical
identifications $\mathcal{V}^{I}=M$ and
${\tilde{\mathcal{V}}}^{I}={\tilde{M}}$. Recall that for $m\in M$ and
${\tilde{m}}\in{\tilde{M}}$, the corresponding coefficient $c_{m,{\tilde{m}}}$
of $\mathcal{V}$ is the complex function on $G$ defined by

$c_{m,{\tilde{m}}}(g)=\langle{\tilde{m}},\pi(g).m\rangle_{\mathcal{V}},\ g\in
G\ .$

Let us observe that by construction $c_{m,{\tilde{m}}}$ is $I$-bi-invariant.

###### Corollary 3.2

For $m\in M$, ${\tilde{m}}\in{\tilde{M}}$, $w\in W^{\rm Aff}$, the value of
$c_{m,{\tilde{m}}}$ on $IwI$ is
$q_{w}^{-1}\langle{\tilde{m}},r({e}_{w}).m\rangle_{M}$.

Proof. The vector $m\in\mathcal{V}$ corresponds to the function $v_{m}\in
C_{c}(G/I,M)$ which has support $I$ and constant value $m$ on $I$. The linear
form $\tilde{m}$ corresponds to the function $f_{\tilde{m}}\in
C(G/I,{\tilde{M}})$ whose value on $IwI$, $w\in W^{\rm Aff}$, is given by
$q_{w}^{-1}\,{\tilde{r}}(\check{e}_{w}).m$. Applying the pairing formula 2,
for $w\in W^{\rm Aff}$, we have $c_{m,{\tilde{m}}}(w)=\langle
q_{w}^{-1}\,{\tilde{r}}(\check{e}_{w}).{\tilde{m}},m\rangle_{M}=q_{w}^{-1}\,\langle{\tilde{m}},r(e_{w}).m\rangle$,
as required.

We now describe a geometric model for the space
${\mathbf{P}}^{o}({\tilde{M}})$ for a given representation $(r,M)$ of
${\mathcal{H}}_{\rm Iw}$. We let $\rm Ch(X_{E})$ denote the $G$-set of
chambers of $X_{E}$ and $\mathcal{F}(\rm Ch(X_{E}),{\tilde{M}})$ the space of
$\tilde{M}$-valued functions on $\rm Ch(X_{E})$. We let $\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})_{r}$ denote the subspace of $\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})$ formed of those functions $f$ satisfying for all
$C\in\rm Ch(X_{E})$ and for all $s\in S$:

${\tilde{r}}(e_{s}).f(C)=\sum_{D\sim_{s}C}f(D)\ .$ (3)

Since $G$ acts on $\rm Ch(X_{E})$ by preserving the labelling of vertices,
$\mathcal{F}(\rm Ch(X_{E}),{\tilde{M}})_{r}$ is a $G$-subspace of
$\mathcal{F}(\rm Ch(X_{E}),{\tilde{M}})_{r}$

###### Proposition 3.3

The spaces ${\mathbf{P}}^{o}({\tilde{M}})$ and $\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})_{r}$ are isomorphic as $G$-modules.

Proof. Since $G$ acts transitively on $\rm Ch(X_{E})$ and the global
stabilizer of $C_{o}$ is $I$, we may identify $C(G/I,{\tilde{M}})$ with
$\mathcal{F}(\rm Ch(X_{E}),{\tilde{M}})$. Since $\\{e_{s}\ ;\ s\in S\\}$
generates ${\mathcal{H}}_{\rm Iw}$ as an algebra, a function $f\in
C(G/I,{\tilde{M}})$ is in ${\mathbf{P}}^{o}({\tilde{M}})$ if and only if
$f\star e_{s}={\tilde{r}}(e_{s}).f$, for all $s\in S$. It follows that
${\mathbf{P}}^{o}({\tilde{M}})$ identifies with the space of functions $f$ :
$\rm Ch(X_{E})\longrightarrow{\tilde{M}}$ satisfying $f\star
e_{s}={\tilde{r}}(e_{s}).f$. Now our result follows from [2] §4 Equation (6) :

$f\star e_{s}(C)=\sum_{D\sim_{s}C}f(D),\ f\in\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}}),\ s\in S\ .$ (4)

##  4 Multiplicities

We fix an irreducible Iwahori-spherical representation $(\pi,\mathcal{V})$ of
$G$. It is of the form ${\mathbf{V}}(M)$, for some irreducible representation
$(r,M)$ of $G$. The aim of this section is to give a bound for the dimension
of the intertwining space ${\rm Hom}_{H}\,(\mathcal{V},\mathbb{C})$, where
$\mathbb{C}$ is acted upon via the trivial representation of $H$.

We shall prove the following.

###### Theorem 4.1

We have

${\rm Dim}_{\mathbb{C}}\,{\rm
Hom}_{H}\,({\mathbf{V}}(M),\mathbb{C})\leqslant{\rm Dim}_{\mathbb{C}}\,M\ .$

By Propositions 3.1 and 3.3, we have an isomorphism of vector spaces:

${\rm
Hom}_{H}\,({\mathbf{V}}(M),\mathbb{C})=({\mathbf{V}}(M)^{*})^{H}\simeq{\mathbf{P}}^{o}({\tilde{M}})^{H}\simeq\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})_{r}^{H}\ .$

Hence we are reduce to proving that ${\rm Dim}\,\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})_{r}^{H}\leqslant{\rm Dim}\,{\tilde{M}}$. In this aim we
prove:

###### Proposition 4.2

A map $f\in\mathcal{F}(\rm Ch(X_{E}),{\tilde{M}})_{r}^{H}$ is entirely
determined by the value $f(C_{o})$, i.e. the mapping $\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})_{r}^{H}\longrightarrow{\tilde{M}}$, $f\mapsto f(C_{o})$
is injective.

To prove the proposition we use the same tool that allowed to establish the
multiplicity $1$ property of the main result of [6] (Theorem 2):

###### Proposition 4.3

([6] Proposition A1) Let $d$ be a non-negative integer and $C$ be a chamber of
$X_{E}$ at combinatorial distance $d$ from $X_{F}$. Let $D_{o}$ be a chamber
of $X_{E}$ adjacent to $C$ and at combinatorial distance $d+1$ from $X_{F}$.
Set $A=D_{o}\cap C$. Let $\Delta$ be the set of chambers $D$ of $X_{E}$ at
combinatorial distance $d+1$ from $X_{F}$ and satisfying $D\cap C=A$. Then the
stabilizer $H_{C}$ of $C$ in $H$ acts transitively on $\Delta$.

We shall also need the following result due to Courtès (cf. the proof of
Theorem 6.1 of [9]).

###### Proposition 4.4

With the notation of the previous proposition, the cardinal of $\Delta$ is
either $q$ or $q-q_{o}$.

Proof of Proposition 4.2. Let $f\in\mathcal{F}(\rm
Ch(X_{E}),{\tilde{M}})_{r}^{H}$. For a non-negative integer $d$, we let $\rm
Ch(X_{E})_{d}$ denote the set of chambers in $X_{E}$ at combinatorial distance
from $X_{F}$ less than or equal to $d$. In particular $\rm Ch(X_{E})_{0}=\rm
Ch(X_{F})$, the set of chambers in $X_{F}$. We prove by induction on $d$, that
for all $d\geqslant 0$, the restriction of $f$ to $\rm Ch(X_{E})_{d}$ depends
only on $f(C_{o})$.

Our assertion holds for $d=0$ since $H$ acts transitively on $\rm Ch(X_{F})$.
Assume it holds for an integer $d$. Let $D$ be a chamber a combinatorial
distance $d+1$ from $X_{F}$. By definition there exists a chamber $C$ adjacent
to $D$ and at combinatorial distance $d$ from $X_{F}$. Let $A=D\cap C$ and
$\Delta$ as in Proposition 4.3. Let $s$ be the type of $A$. The set
$C_{s}=\\{E\in\rm Ch(X_{E})\ ;\ E\sim_{s}C\\}\cup\\{C\\}$ $=$ $\\{E\in\rm
Ch(X_{E})\ ;\ E\supset A\\}$ writes

$C_{s}=\Delta\sqcup\Delta^{\prime}$

where $\Delta^{\prime}$ is the subset of chambers at distance $d$ from
$X_{F}$. The value of $f$ on $\Delta^{\prime}$ only depends on $f(C_{o})$. By
Relation 3 we have

${\tilde{r}}(e_{s}).f(D)=\sum_{E\in\Delta\backslash\\{D\\}}f(E)+\sum_{E\in\Delta^{\prime}}f(E)=|\Delta\backslash\\{D\\}|\
f(D)+\sum_{E\in\Delta^{\prime}}f(E)$ (5)

since $H_{C}$ acts transitively on $\Delta$ and $f$ is $H$ invariant. Hence
using Proposition 4.4 we obtain:

$({\tilde{r}}(e_{s})-\lambda).f(D)=\sum_{E\in\Delta^{\prime}}f(E)$ (6)

where $\lambda=q-1$ or $q-q_{o}-1$. Because of the quadratic relation
$(e_{s}+1)(e_{s}-q)=0$, the spectrum of ${\tilde{r}}(e_{s})$ is contained in
$\\{-1,q\\}$. So the operator ${\tilde{r}}(e_{s})-\lambda$ is invertible and
$f(D)$ only depends on $f(C_{o})$ as required.

Theorem 4.1 has the following nice obvious consequence.

###### Theorem 4.5

Let $(\pi,\mathcal{V})$ be an irreducible Iwahori-spherical representation
satisfying ${\rm Dim}\,\mathcal{V}^{I}=1$. Then $\pi$ has the multiplicity $1$
property:

${\rm Dim}_{\mathbb{C}}\,{\rm Hom}_{H}\,(\mathcal{V},\mathbb{C})\leqslant 1\
.$

##  5 Test vectors

Let us fix an irreducible square integrable (or discrete series)
representation $(\pi,\mathcal{V})$ of $G$. By definition, for
$v\in\mathcal{V}$ and ${\tilde{v}}\in{\tilde{\mathcal{V}}}$, the coefficient
$c_{v,{\tilde{v}}}$ : $G\ni
g\mapsto\langle{\tilde{v}},\pi(g).v\rangle_{\mathcal{V}}$ lies in $L^{2}(G)$.
Fix a Haar measur $\nu$ on $H$.

###### Proposition 5.1

For all $v\in\mathcal{V}$, ${\tilde{v}}\in{\tilde{\mathcal{V}}}$, the integral

$\int_{H}c_{v,{\tilde{v}}}(h)\,d\nu(h)$

is absolutely convergent.

Proof. Indeed, by [10] Corollary 1.2, a Galois symmetric space as $G/H$ is
strongly discrete: the restriction to $H$ of a coefficient of an irreducible
discrete series representation of $G$ lies in $L^{1}(H)$.

Following Zhang [19], for $v\in\mathcal{V}$ and
${\tilde{v}}\in{\tilde{\mathcal{V}}}$, we set

${\mathcal{L}}(v,{\tilde{v}})=\int_{H}c_{v,{\tilde{v}}}(h)\,d\nu(h)\ .$

Then $\mathcal{L}$ :
$\mathcal{V}\times{\tilde{\mathcal{V}}}\longrightarrow\mathbb{C}$ is a
bi-$H$-invariant bilinear form. For ${\tilde{v}}\in{\tilde{\mathcal{V}}}$, we
set

${\mathcal{L}}_{\tilde{v}}\ :\ \mathcal{V}\longrightarrow\mathbb{C},\
v\mapsto{\mathcal{L}}(v,{\tilde{v}})\ .$

By construction ${\mathcal{L}}_{\tilde{v}}\in{\rm
Hom}_{H}\,(\mathcal{V},\mathbb{C})$, for all
${\tilde{v}}\in{\tilde{\mathcal{V}}}$ and we set
${\mathcal{H}}(\pi)=\\{{\mathcal{L}}_{\tilde{v}}\ ;\
{\tilde{v}}\in{\tilde{\mathcal{V}}}\\}\subset{\rm
Hom}_{H}\,(\mathcal{V},\mathbb{C})$.

By [19] Proposition 3.2, $G/H$ is very strongly discrete in the sense of loc.
cit. Definition 1.3; this means that the linear form

${\mathcal{C}}(G)\longrightarrow\mathbb{C},\ f\mapsto\int_{H}f(h)\,d\nu(h)$

is well-defined and continuous, where ${\mathcal{C}}(G)$ denotes the Schwarzt-
Harish-Chandra space of $G$ (cf. loc. cit. §2). As a consequence, by loc. cit.
Theorem 1.4, we have the following result.

###### Theorem 5.2

(Zhang) We have the equality ${\mathcal{H}}(\pi)={\rm
Hom}_{H}\,(\mathcal{V},\mathbb{C})$, i.e. any $H$-invariant linear form on
$\mathcal{V}$ is obtained by integration on $H$ against a smooth linear form
on $\mathcal{V}$.

The aim of this section is to prove the following.

###### Theorem 5.3

The discrete series representation $(\pi,\mathcal{V})$ is $H$-distinguished
if, and only if, there exist $v\in\mathcal{V}^{I}$ and
${\tilde{v}}\in{\tilde{\mathcal{V}}}^{I}$ such that
$\displaystyle\int_{H}c_{v,{\tilde{v}}}(h)\,d\nu(h)\not=0$.

This is in fact a consequence of the following result.

###### Theorem 5.4

Let $(\pi,\mathcal{V})$ be an admissible representation of $G$ generated by
$\mathcal{V}^{I}$ as a $G$-module (i. e. an admisible object of
${\mathcal{S}}(G)_{\rm Iw}$ that is not necessarily assumed to be a discrete
series representation). Let $\Lambda\in{\rm Hom}_{H}(\mathcal{V},\mathbb{C})$
be a non-zero $H$-invariant linear form on $\mathcal{V}$. Then there exists
$v\in\mathcal{V}^{I}$ such that $\Lambda(v)\not=0$.

Indeed, let us assume that this proposition holds. One implication in Theorem
5.3 being obvious, let us assume that $(\pi,\mathcal{V})$ is $H$-distinguished
and let us fix a non-zero $\Lambda\in{\rm Hom}_{H}(\mathcal{V},\mathbb{C})$.
By Theorem 5.2 there exists ${\tilde{v}}_{1}\in{\tilde{\mathcal{V}}}$ such
that $\Lambda={\mathcal{L}}_{{\tilde{v}}_{1}}$. Pick $v\in\mathcal{V}^{I}$
such that $\Lambda(v)={\mathcal{L}}(v,{\tilde{v}}_{1})\not=0$. The map
${\tilde{\Lambda}}$ : ${\tilde{v}}\mapsto{\mathcal{L}}(v,{\tilde{v}})$ is a
non-zero element of ${\rm Hom}_{H}\,({\tilde{\mathcal{V}}},\mathbb{C})$.
Applying Proposition 5.4 to $({\tilde{\pi}},{\tilde{\mathcal{V}}})$, there
exists ${\tilde{v}}\in{\tilde{\mathcal{V}}}^{I}$ such that
${\mathcal{L}}(v,{\tilde{v}})\not=0$ and we are done.

Proof of Theorem 5.4. Fix a basis $(v_{1},...,v_{r})$ of $\mathcal{V}^{I}$,
$r={\rm Dim}_{\mathbb{C}}\,\mathcal{V}^{I}$. We are going to define $r$
families $(v_{C}^{i})_{C\in\rm Ch(X_{E})})$ of vectors of $\mathcal{V}$
indexed by the chambers of $X_{E}$. Let $i=1,...,r$ and $C\in\rm Ch(X_{E})$.
There exists $g\in G$ such that $C=gC_{o}$. Set $v_{C}^{i}=gv_{i}$. Since
$v_{i}$ is fixed the stabilizer $I$ of $C_{o}$, the definition of $v_{C}^{i}$
does not depends on the choice of $g$.

###### Lemma 5.5

Le $C\in\rm Ch(X_{E})$ and $s\in S$. Write $C=g.C_{o}$, for some $g\in G$. For
$i=1,...,r$, we have the relation

$\pi(ge_{s}g^{-1}).v_{C}^{i}=\sum_{D\sim_{s}C}v_{D}^{i}$ (7)

Proof By equivariance it suffices to prove the relation when $C=C_{o}$ (and
$g=1$). Let $A$ be the wall of $C_{o}$ of type $s$. Recall that the stabilizer
of $A$ in $G$ is $K_{s}:=I\sqcup IsI$. By definition we have

$\displaystyle\pi(e_{s}).v_{i}$
$\displaystyle=\int_{G}e_{s}(g)\,\pi(g).v_{i}\,d\mu(g)$
$\displaystyle=\int_{IsI}\pi(g).v_{i}\,d\mu(g)$ $\displaystyle=\sum_{g\in
K_{s}/I}\pi(g).v_{i}-v_{i}$

Since the group $K_{s}$ acts transitively on the chamber $C$ containing $A$,
it follows that

$\pi(e_{s}).v_{i}=\sum_{C\supset A}v_{C}-v_{i}=\sum_{C\sim_{s}C_{o}}v_{C}$ (8)

as required.

Since $\mathcal{V}$ is generated by $\mathcal{V}^{I}$ as a $G$-module, it is
generated by the $v_{C}^{i}$, $C\in\rm Ch(X_{E})$, $i=1,...,r$ as a vector
space. So Theorem 5.4 follows from the following lemma whose proof is formally
the same as that of Proposition 4.2, the Relation 7 playing the role of
Relation 3.

###### Lemma 5.6

Let $\Lambda\in{\rm Hom}_{H}\,(\mathcal{V},\mathbb{C})$. Then if
$\Lambda(v_{C}^{i})=0$, for all $C\in\rm Ch(X_{F})$ and all $i=1,...,r$, we
have $\Lambda=0$.

N.B. This proof is inspired by the proof of Proposition 6.2 of [9].

##  6 Generalized Poincaré series

Recall that we abbreviate $W=W_{F}^{\rm Aff}=W_{E}^{\rm Aff}$.

The Poincaré series of the Coxeter system $(W,S)$ is the formal power series
$W(t)=\displaystyle\sum_{w\in W}t^{l(w)}\in\mathbb{Z}[[t]]$. This is the
generating function of the length of elements of $W$. It is known ([3],[18])
to be a rational function of $t$. We shall need a generalization of the
Poincaré series due to Macdonald and Gyoja.

We let $S_{1}$, …, $S_{m}$ be the non-empty intersections of $S$ with the
conjugacy classes of $W$. By [4] Proposition 5, page 16, for $w\in W$ the
number of elements of $S_{i}$ occuring in a reduced expression of $w$ depends
only $w$, we denote it by $l_{i}(w)$. We have $l(w)=l_{1}(w)+\cdots+l_{m}(w)$.
Let $t_{1}$, …, $t_{m}$ be indeterminates. For $w\in W$, we write
${\mathbf{t}}=(t_{1},...,t_{m})$ and
${\mathbf{t}}^{l(w)}:=t_{1}^{l_{1}(w)}\cdots t_{m}^{l_{m}(w)}$.

By [2] §3.3, the $S_{i}$, $i=1,...,m$ are the connected components of the
subgraph of the Coxeter graph ${\rm Cox}(W,S)$ obtained by erasing the
multiple edges. The number $m$ is so given as follows:

$m=1$ if $G$ is of type $A_{n}$ ($n\geqslant 2$), $D_{n}$ ($n\geqslant 3$) and
$E_{i}$ ($i=6,7,8$),

$m=2$ if $G$ is of type $A_{1}$, $B_{n}$ ($n\geqslant 2$), $G_{2}$ and
$F_{4}$,

$m=3$ if $G$ is of type $C_{n}$ ($n\geqslant 2$).

We choose the indexing in $(t_{i})_{i=1,..,m}$ as in [11], pages 173, 174.

Following Gyoja [11], for a representation $(r,M)$ of ${\mathcal{H}}_{\rm
Iw}$, define an element of ${\rm
End}_{\mathbb{C}}(M)\otimes_{\mathbb{C}}\mathbb{C}[[t_{1},...,t_{m}]]$ by

$L({\mathbf{t}},r)=\sum_{w\in W}r(e_{w})\,{\mathbf{t}}^{l(w)}\ .$ (9)

###### Theorem 6.1

([11] In any basis of $M$, the coefficients of $L({\mathbf{t}},r)$ are
rational functions of $t_{1}$, …, $t_{m}$.

When $(r,M)$ is the trivial representation of ${\mathcal{H}}_{\rm Iw}$, we set
$W({\mathbf{t}})=L({\mathbf{t}},r)$, that is
$W({\mathbf{t}})=\displaystyle\sum_{w\in W}{\mathbf{t}}^{l(w)}$. This rational
function has been computed by Macdonald ([14] Theorem 3.3). When $m=1$, then
$W({\mathbf{t}})=W(t)$ is the ordinary Poincaré series and Bott’s fomula gives

$W(t)=\prod_{i=1,..,l}\frac{1-t_{i}^{m_{i}+1}}{(1-t)(1-t_{i}^{m_{i}})}$

where $m_{1}$, …, $m_{l}$ are the exponents of the spherical Coxeter group
attached to $W$ (cf. [4] Chap. V, §6, Définition 2).

If $m=2$, the formulas are the following.

Type $A_{1}$,
$W(t_{1},t_{2})=\displaystyle\frac{(1+t_{1})(1+t_{2})}{1-t_{1}t_{2}}$.

Type $B_{n}$ ($n\geqslant 2$), $\displaystyle
W(t_{1},t_{2})=\frac{1-t_{1}^{n}}{(1-t_{1})^{n}}\,\prod_{i=1}^{n-1}(1-t_{1}^{2i})\,\prod_{i=0}^{n-1}\frac{1+t_{1}^{i}t_{2}}{1-t_{1}^{n-1+i}t_{2}}$.

Type $G_{2}$, $\displaystyle
W(t_{1},t_{2})=\frac{(1+t_{1})(1+t_{1}+t_{1}^{2})(1+t_{2})(1+t_{1}t_{2}+t_{1}^{2}t_{2}^{2})}{(1-t_{1}^{2}t_{2})(1-t_{1}^{3}t_{2}^{2})}$.

Type $F_{4}$, $W(t_{1},t_{2})=$

$\left(\prod_{i=1}^{3}\frac{(1-t_{1}^{i+1})(1+t_{1}^{i}t_{2})(1-t_{2}^{i})}{(1-t_{1})(1-t_{2})}\right)\,\frac{(1+t_{1}t_{2}^{2})(1+t_{1}^{2}t_{2}^{2})(1+t_{1}^{3}t_{2}^{3})}{(1-t_{1}^{3}t_{2}^{2})(1-t_{1}^{4}t_{2}^{3})(1-t_{1}^{5}t_{2}^{3})(1-t_{1}^{6}t_{2}^{5})}\
.$

Finally for $m=3$ we have the following formula.

Type $C_{n}$ ($n\geqslant 2$), $\displaystyle
W(t_{1},t_{2},t_{3})=\prod_{i=0}^{n-1}\frac{(1-t_{1}^{i+1})(1+t_{1}^{i}t_{2})(1+t_{1}^{i}t_{3})}{(1-t_{1})(1-t_{1}^{n-1+i}t_{2}t_{3})}$.

##  7 A numerical criterion with application to degree $1$ characters

Let $(\pi,\mathcal{V})$ be an irreducible discrete series representation of
$G$ and let us assume that it is Iwahori-spherical: $\mathcal{V}^{I}\not=0$.
Let $(r,M)$ denote the representation of ${\mathcal{H}}_{\rm Iw}$ in
$M=\mathcal{V}^{I}$. The following result is a numerical criterion to decide
whether or not $(\pi,\mathcal{V})$ is $H$-distinguished.

###### Theorem 7.1

(a) The representation $\pi$ is $H$-distinguished if, and only if, the
endomorphism $L({\mathbf{t}}_{o},r)\in{\rm End}_{\mathbb{C}}(M)$ is non-zero,
where ${\mathbf{t}}_{o}=\displaystyle(\frac{1}{q_{o}},...,\frac{1}{q_{o}})$.

(b) We have ${\rm Dim}\,{\rm Hom}_{H}\,(\pi,\mathbb{C})\geqslant{\rm
rank}\,L({\mathbf{t}}_{o},r)$.

Proof. (a) We abbreviate $J=I_{F}^{o}$. Recall that $JwJ\subset IwI$, for all
$w\in W$. Let $\nu$ the Haar measure on $H$ normalized by $\nu(J)=1$. Let
$m\in M$ and ${\tilde{m}}\in{\tilde{M}}$. We first calculate the integral
$\displaystyle\int_{H}c_{m,{\tilde{m}}}(h)\,d\nu(h)$. By the Bruhat-Tits
decomposition $\displaystyle H=\sqcup_{w\in W}JwJ$, we have:

$\displaystyle\int_{H}c_{m,{\tilde{m}}}(h)\,d\nu(h)$ $\displaystyle=\sum_{w\in
W}\int_{JwJ}c_{m,{\tilde{m}}}(h)\,d\nu(h)$ (10)
$\displaystyle=\frac{\nu(JwJ)}{q^{l(w)}}\sum_{w\in
W}\langle{\tilde{m}},r(e_{w}).m\rangle_{M}$ (11)
$\displaystyle=\frac{q_{o}^{l(w)}}{q^{l(w)}}\sum_{w\in
W}\langle{\tilde{m}},r(e_{w}).m\rangle_{M}$ (12)
$\displaystyle=\langle{\tilde{m}},\sum_{w\in
W}r(e_{w})\left(\frac{1}{q_{o}}\right)^{l(w)}.m\rangle_{M}$ (13)
$\displaystyle=\langle{\tilde{m}},L({\mathbf{t}}_{o},r).m\rangle_{M}$ (14)

where in (11) we used Corollary 3.2 and where we set
${\mathbf{t}}_{o}=\displaystyle(\frac{1}{q_{o}},...,\frac{1}{q_{o}})$.

By Proposition 5.3, $\pi$ is $H$-distinguished if and only if there exist
$m\in M$ and ${\tilde{m}}\in{\tilde{M}}$ such that
$\displaystyle\int_{H}c_{m,{\tilde{m}}}(h)\,d\nu(h)\not=0$. Thanks to the last
calculation this is indeed equivalent to $L({\mathbf{t}}_{o},r)\not=0$, as
required.

(b) It follows from (a), using the notation of §5, that for $m\in M$ and
${\tilde{m}}\in{\tilde{M}}$, we have
${\mathcal{L}}_{\tilde{m}}(m)=\langle{\tilde{m}},L({\mathbf{t}}_{o},r)\rangle_{M}$.
Let us put $s={\rm rank}\,L({\mathbf{t}}_{o},r)$. Then there exist
${\tilde{m}}_{1}$, …, ${\tilde{m}}_{s}\in{\tilde{M}}$ such that the
$({\mathcal{L}}_{{\tilde{m}}_{i}})_{\mid M}$, $i=1,...,s$, are linearly
independent. It follows that the ${\mathcal{M}}_{{\tilde{m}}_{i}}\in{\rm
Hom}_{H}\,(\pi,\mathbb{C})$, $i=1,...,s$, are linearly independent, and we are
done.

We now assume that the discrete series representation $(\pi,\mathcal{V})$
satisfies ${\rm Dim}\,M={\rm Dim}\,\mathcal{V}^{I}=1$, so that we may view $r$
as an algebra homomorphism ${\mathcal{H}}_{\rm Iw}\longrightarrow\mathbb{C}$.
Such discrete series where classified in [2] §5. We recall this
classification.

The character $r$ is constant on each $\\{e_{s}\ ;\ s\in S_{i}\\}$,
$i=1,...,m$ and because of the quadratic relations we have
$r(e_{s})\in\\{-1,q\\}$, $s\in S$. Following Borel loc. cit, for $i=1,...,m$
we set

$\epsilon_{i}=\left\\{\begin{array}[]{rl}1&{\rm if}\ r(e_{s})=q\\\ -1&{\rm
if}\ r(e_{s})=-1\end{array}\right.$

so that the representation $\pi$ is entirely characterized by the $m$-uple
$(\epsilon_{1},...,\epsilon_{m})$; we write
$\pi=\pi(\epsilon_{1},...,\epsilon_{m})$. Recall that
${\mathbf{S}}{\mathbf{t}}_{G}=\pi(-1,..,-1)$ is the Steinberg representation
of $G$.

###### Proposition 7.2

(Borel [2] §5.8) The discrete series representations of $G$ of the form
$\pi(\epsilon_{1},...,\epsilon_{m})$ are the Steinberg representation and

$\pi(-1,1)$ for types $B_{n}$ ($n\geqslant 3$), $F_{4}$, $G_{2}$,

$\pi(-1,-1,1)$, $\pi(-1,1,-1)$ for types $C_{2}$, $C_{3}$,

$\pi(-1,-1,1)$, $\pi(-1,1,-1)$, $\pi(-1,1,1)$ for type $C_{n}$ ($n\geqslant
4$).

###### Lemma 7.3

Assume that $\pi(\epsilon_{1},...,\epsilon_{m})$ is a discrete series
representations of $G$. We have
$L({\mathbf{t}}_{o},r)=W(\epsilon_{1}q_{o}^{\epsilon_{i}},...,\epsilon_{m}q_{o}^{\epsilon_{m}})$.
In particular $\pi(\epsilon_{1},...,\epsilon_{m})$ is $H$-distinguished if,
and only if,
$W(\epsilon_{1}q_{o}^{\epsilon_{i}},...,\epsilon_{m}q_{o}^{\epsilon_{m}})\not=0$.

Proof. Observe that for $i=1,...,m$ and $s\in S_{i}$, we have
$r(e_{s})=\epsilon_{i}q_{o}^{\epsilon_{i}+1}$, so that for $w\in W$ we have
$\displaystyle
r(e_{w})=\prod_{i=1}^{m}(\epsilon_{i}q_{o}^{\epsilon_{i}+1})^{l_{i}(w)}$.
Hence we have

$\displaystyle L({\mathbf{t}}_{o},r)$ $\displaystyle=\sum_{w\in
W}r(e_{w})(\frac{1}{q_{o}})^{l(w)}$ $\displaystyle=\sum_{w\in
W}\prod_{i=1}^{m}(\epsilon_{i}\frac{q_{o}^{\epsilon_{i}+1}}{q_{o}})^{l_{i}(w)}$
$\displaystyle=W(\epsilon_{i}q_{o}^{\epsilon_{i}},...,\epsilon_{i}q_{o}^{\epsilon_{i}})$

as required.

We may now state the main result of this article.

###### Theorem 7.4

The discrete series representation $\pi(\epsilon_{1},...,\epsilon_{m})$ is
$H$-distinguished if and only if it is a Steinberg representation, or if $H$
is of type $G_{2}$ and $(\epsilon_{1},\epsilon_{2})=(-1,1)$. When it is
distinguished it has the multiplicity $1$ property: $\displaystyle{\rm
Dim}\,{\rm Hom}_{H}\,(\pi(\epsilon_{1},...,\epsilon_{m}))=1$.

Proof. The multiplicity $1$ property follows from Theorem 4.1. The first
assertion is proved by a case-by-case calculation based on Gyoja’s formulas
given in §5. Note that in the case of the Steinberg representation, the
distinction property was originally D. Prasad’s conjecture ([15] Conjecture 3,
p. 77) and was proved in [6] (cf. [1] for a proof of Prasad’s conjecture in
the general case).

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Paul Broussous

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