11institutetext: De La Salle University Manila, Taft Ave., Malate, 1004
Manila, Philippines
11email:
<EMAIL_ADDRESS>22institutetext: University of the Philippines Diliman, Quezon Hall, 1101
Quezon City, Philippines
22email<EMAIL_ADDRESS>
# Exploiting News Article Structure for Automatic Corpus Generation of
Entailment Datasets

Jan Christian Blaise Cruz 11 Jose Kristian Resabal 22 James Lin 11 Dan John
Velasco 11 Charibeth Cheng 11

###### Abstract

Transformers represent the state-of-the-art in Natural Language Processing
(NLP) in recent years, proving effective even in tasks done in low-resource
languages. While pretrained transformers for these languages can be made, it
is challenging to measure their true performance and capacity due to the lack
of hard benchmark datasets, as well as the difficulty and cost of producing
them. In this paper, we present three contributions: First, we propose a
methodology for automatically producing Natural Language Inference (NLI)
benchmark datasets for low-resource languages using published news articles.
Through this, we create and release NewsPH-NLI, the first sentence entailment
benchmark dataset in the low-resource Filipino language. Second, we produce
new pretrained transformers based on the ELECTRA technique to further
alleviate the resource scarcity in Filipino, benchmarking them on our dataset
against other commonly-used transfer learning techniques. Lastly, we perform
analyses on transfer learning techniques to shed light on their true
performance when operating in low-data domains through the use of degradation
tests.

###### Keywords:

Low-resource Languages Automatic Corpus Creation Transformer Neural Networs.

##  1 Introduction

In recent years, Transformers [25] have begun to represent the state-of-the-
art not only in common NLP tasks where they have cemented their reputation,
but also in the context of tasks within low-resource languages. Using
Transformers, advancements have been done in various low-resource tasks,
including low-resource translation [8, 20], classification [21, 7],
summarization [14], and many more.

Transformers and transfer learning techniques in general owe their wide
adaptation in low-resource language tasks to the existence of abundant
unlabeled corpora available. While labeled datasets may be scarce to perform
tasks in these languages, unlabeled text is usually freely available and can
be scraped from various sources such as Wikipedia, news sites, book
repositories, and many more. Pretraining allows transfer learning techniques
to leverage learned priors from this unlabeled text to robustly perform
downstream tasks even when there is little task-specific data to learn from
[13, 5].

However, while it is possible to produce large pretrained models for transfer
learning in low-resource languages, there is a challenge in properly gauging
their performance in low-resource tasks. Most, if not all Transformers that
are pretrained and released open-source are evaluated with large, commonly-
used datasets. In low-resource languages, these datasets may not exist. Due to
this, it is often hard to properly benchmark a model’s true performance when
operating in low-data domains.

While it is possible to remedy this by constructing hard datasets in these
languages, added concerns have to be addressed.

Dataset construction is slow and cost-prohibitive. For hard tasks such as
various natural language inference and understanding tasks, datasets are
usually sized around 500,000 samples and more [1, 4, 27]. This would entail a
large enough budget to hire annotators to write text samples, and a different
set of annotators to write labels. This process is also slow and may take
months to finish. In that span of time, stronger techniques may have been
created that require more difficult datasets to accurately assess them. In
addition, once the dataset has been solved, harder datasets are needed to
properly gauge further, succeeding methods.

This creates a need for a method to produce benchmark datasets for low-
resource languages that is quick and cost effective, while still capable of
generating tasks that are challenging for high-capacity models such as
Transformers.

In this paper, we present the following contributions:

  * •

We propose an automatic method to generate Natural Language Inference (NLI)
benchmark datasets from a corpus of news articles.

  * •

We release NewsPH-NLI, the first sentence entailment benchmark dataset in the
low-resource Filipino language, created using the method we propose.

  * •

We produce pretrained Transformers based on the ELECTRA pretraining scheme to
further alleviate resource scarcity in Filipino.

  * •

We perform benchmarks and analyses on commonly-used transfer learning
techniques to properly and accurately gauge their true performance in low-data
domains.

Our method has a number of advantages. First, since our method is automatic,
it evades the issue of time and cost. This also allows datasets created this
way to be updated regularly as news is released everyday. Second, given that
news is freely available and published online even in low-resource languages,
text data for producing benchmark datasets will be easy to source. Lastly,
given that we generate sentence entailment tasks within the domain of news,
our method will produce sufficiently challenging datasets to properly gauge
the performance of large Transformers.

##  2 Methodology

In this section, we outline our experimental setups and methodology. First, we
describe our proposed methodology for producing benchmark NLI datasets in any
language, using our NewsPH-NLI dataset as an example. Second, we outline the
creation of ELECTRA models in Filipino. Lastly, we outline our methodology for
analysis using degradation tests.

###  2.1 NLI Datasets from News Articles

The creation of large datasets for NLI is often difficult, time-consuming, and
cost-prohibitive. It may also be not be feasible in low-data and low-resource
domains due to the lack of pre-encoded text data that can immediately be
annotated.

We propose the use of news articles for automatically creating benchmark
datasets for NLI because of two reasons. First, news articles commonly use
single-sentence paragraphing, meaning every paragraph in a news article is
limited to a single sentence [11, 10]. Second, straight news articles follow
the “inverted pyramid” structure, where every succeeding paragraph builds upon
the premise of those that came before it, with the most important information
on top and the least important towards the end [16, 2, 23]. A figure
illustrating the inverted pyramid can be found in Figure 1.

Figure 1:  Inverted pyramid structure of straight news articles. The most
important facts are at the top with succeeding paragraphs containing facts of
less and less importance. The structure of news articles makes succeeding
paragraphs build up on the information of prior paragraphs. Figure taken from
[22]

Due to the inverted pyramid structure, we can assume that every succeeding
paragraph (our “hypothesis”) entails the paragraph preceeding it (our
“premise”). This can be exploited to produce multiple samples of entailments
from a single news article. This way, using a corpus of straight news
articles, we can produce a large number of samples of entailments to make up
an NLI corpus.

Contradictions, on the other hand, are more difficult to produce. To
automatically make contradiction samples, we first randomly sample two news
articles from the pool of collected news articles, then randomly sample one
paragraph from each article to serve as our premise-hypothesis pair. To ensure
that the produced pair is a contradiction, we must first make sure that the
two randomly-sampled articles have two different topics. To do this, we train
a Doc2Vec [17] model on all of the collected news articles. Afterwards, we
then cluster the most similar articles. When sampling articles for
contradictions, we sample from two different clusters to ensure that the
topics are different.

One limitation of our proposed methodology is that it can only generate
entailments and contradictions, as “neutral” premise-hypothesis pairs can only
be obtained through manual annotation by humans. This lack of a third label
makes the generated datasets easier as compared to standard NLI datasets with
three labels. While a 2-label classification task is easier than a 3-label
classification task, the generated dataset will still be harder than a
standard single-sentence classification problem (like sentiment
classification) as the model will have to be able to encode inter-dependent
information between the sentence pairs.

In addition, there is a chance that an auto-generated dataset will have errors
that can only be identified when checked and studied by human annotators. As
the goal of the research is to produce a dataset with little-to-no human
supervision nor annotation, this human-based check is not done. Correctness is
instead ensured by thorough testing of the topic clustering model.

###  2.2 NewsPH-NLI

Using our proposed methodology, we automatically generate an NLI benchmark
dataset in Filipino we call the NewsPH-NLI dataset.

To create the dataset, we scrape news articles from all major Philippine news
sites online. We collect a total of 229,571 straight news articles, which we
then lightly preprocess to remove extraneous unicode characters and correct
minimal misspellings. No further preprocessing is done to preserve information
in the data.

We then use our proposed methodology. First, we create a Doc2Vec model (via
the Gensim111https://radimrehurek.com/gensim/ package) on our collected news
corpus, using Annoy222https://github.com/spotify/annoy as an indexer. We
remove Tagalog stopwords and use TF-IDF to filter the functions words (e.g.
“ng“ and “nang“) as these create noise. In testing, without the use of
stopword removal and TF-IDF filtering, clustering was difficult as most
articles were embedded closely due to their common usage of stopwords and
function words. After producing the Doc2Vec embeddings, we then cluster,
comparing two articles via the cosine similarity of the mean of their
projected vectors. We consider two articles to be dissimilar if their cosine
similarity is less than 0.65.

After clustering, we then take entailments by running through each article
iteratively, and produce contradictions by sampling from two randomly chosen
clusters. We shuffle the final set and randomly sample 600,000 premise-
hypothesis pairs to be part of the final dataset. We set this size for our
dataset in order to follow the size of the widely-used SNLI dataset [1].

From the full generated dataset, 420,000 of which form the training set, while
the remaining 80,000 are split evenly to produce the validation and test sets.
To generate the splits, we first sample 300,000 of both entailments and
contradictions using our methodology, shuffle the set, then split them
accordingly into training, validation, and testing sets.

###  2.3 ELECTRA Pretraining

We alleviate the resource scarcity of the Filipino language by producing
pretrained transformers. We chose the ELECTRA [3] pretraining method because
of the data efficiency of its pretraining task. While a large corpus of
unlabeled text is available in Filipino, this consolidated corpus is still far
smaller than the ones commonly used to pretrain English models. ELECTRA poses
an advantage over the widely-used BERT [9] in its ability to use pretraining
data more efficiently, as BERT only uses 15% of the training data for masked
language modeling per epoch, leading to data inefficiency. We surmise that
this increased data efficiency will provide improvements for tasks in low-
resource languages.

We produce four ELECTRA models: a cased and uncased model in the base size (12
layers, 768 hidden units, 12 attention heads), and a cased and uncased model
in the small size (12 layers, 256 hidden units, 4 attention heads). All our
models accept a maximum sequence length of 512.

Our models are pretrained using the WikiText-TL-39 dataset [5], producing a
SentencePiece333https://github.com/google/sentencepiece vocabulary of 320,000
subwords. We train the small models with a learning rate of 5e-4, batch size
of 128, and a generator hidden size 25% of the discriminator hidden size. For
the base models, we train with a learning rate of 2e-4, batch size of 256, and
a generator hidden size 33% of the discriminator hidden size. Models are
pretrained using the Adam [15] optimizer. We pretrain for a total of 1 million
steps for the base models and 766,000 steps for the small models, using the
first 10% of the total steps for linear learning rate warmup.

Pretraining was done using Tensor Processing Unit (TPU) v3 machines on Google
Cloud Platform, with small models finishing in four days and base variants
finishing in nine days.

###  2.4 Benchmarking

We then finetune to set initial benchmarks on the NewsPH-NLI using our ELECTRA
models, comparing their performance against another Transformer-based
finetuning technique (BERT) and an RNN-based finetuning technique (ULMFiT).
For Filipino versions of the aforementioned benchmark models, we use Tagalog-
BERT [6, 7] and Tagalog-ULMFiT [26].

For finetuning, small variants of ELECTRA use a learning rate of 2e-4. Base
variants of both ELECTRA and BERT use a learning rate of 5e-5. All
transformers were finetuned on the dataset for a total of 3 epochs using the
Adam optimizer, using the first 10% of the total steps for linear learning
rate warmup. For transformers, the standard separator token [SEP] was used to
convert sentence pairs into one single sequence.

ULMFiT follows a different finetuning protocol compared to the transformer
models. We first preprocess the data using the FastAI [12] tokenization
scheme. Sentence-pairs are turned into one sequence by using a special xxsep
token introduced in finetuning.

Finetuning was done in two stages: language model finetuning, and classifier
finetuning. For language model finetuning, we first finetune the last layer
for 1 epoch, leaving all other layers frozen, before unfreezing all layers and
finetuning for two epochs. We use a learning rate of 5e-2. For classifier
finetuning, we perform 5 epochs of finetuning, performing gradual unfreezing
[13] while reducing the learning rate from 1e-2 per epoch by a factor of 2.
All experiments with ULMFiT also used discriminative learning rates [13] and
cyclic learning rate schedules [24].

Finetuning, testing, and all other experiments were done on machines with
NVIDIA Tesla P100 GPUs. For small ELECTRA models, finetuning on the full
dataset takes three hours to finish. For base ELECTRA and BERT variants, full
finetuning finishes in five hours. For ULMFiT, it takes two hours

###  2.5 Degradation Tests

To further investigate the capacity and performance of these models especially
when operating in low-data environments, we run a number of degradation tests
[6].

Simply put, we reduce the amount of training data to a certain data percentage
(p%) of the full dataset while keeping the validation and testing data sizes
constant, then proceed to finetune a model. For each model, we perform
degradation tests at four different data percentages: 50%, 30%, 10%, and 1%.

For each degradation test, we log the test loss and test accuracy. In
addition, we take the accuracy degradation, which is described as:

$AD_{p\%}=Acc_{100\%}-Acc_{p\%}$

where $Acc_{100\%}$ refers to the accuracy of the model when finetuned on the
full dataset, and $Acc_{p\%}$ refers to the accuracy of the model when
finetuned on $p\%$ of the dataset. We also take the degradation percentage,
which is described as:

$DP_{p\%}=AD_{p\%}/Acc_{100\%}\times 100$

where $AD_{p\%}$ is the accuracy degradation of the model when finetuned at
$p\%$ data percentage. The degradation percentage measures how much of the
full performance of a model is lost when trained with less data, at a certain
data percentage $p\%$.

To compare which models are more robust to performance degradation in low-data
domains, we also measure degradation speed, which we define as the standard
deviation between the degradation percentages of a model for the 50%, 30%,
10%, and 1% setup. When the degradation percentages are more spread out, this
indicates that the model degrades faster as the number of training data is
reduced.

We perform degradation tests as a form of “stress test” to gauge the
performance and effectiveness of models when forced to work in low-data
domains. Most models in published literature show results as tested in
environments with abundant data. While this is an effective way to compare
performance against other models tested in a similar manner, it is not
representative of a model’s actual performance when adapting to low-data
domains, especially with low-resource languages.

##  3 Results and Discussion

###  3.1 Finetuning Results

Finetuning results show that ELECTRA outperforms both the Transformer baseline
(BERT) and the RNN baseline (ULMFiT). The best ELECTRA model (Small Uncased)
outperforms the best BERT model (Base Cased) by +3.75% accuracy, and
outperforms the ULMFiT model by +3.63%.

The ELECTRA models outperformed the BERT models on average by 3.01% accuracy
(average ELECTRA performance being 92.17% while average BERT performance is
only 89.16%). We hypothesize that the ELECTRA models perform better than the
BERT, with the small variants performing better than their larger BERT
counterparts despite the size and capacity difference, due to the pretraining
scheme. ELECTRA leverages pretraining data in a more data efficient way, using
all of the training data per batch to train the model. This is opposed to
BERT’s (particularly masked language modeling’s) inefficient use of
pretraining data, using only 15% of each batch to train the model. Since our
pretraining dataset is considerably smaller than most common English
pretraining datasets (39 million words in WikiText-TL-39 vs 2,500 million
words in the Bookcorpus dataset), a pretraining scheme that uses data more
efficiently will be able to learn more effectively.

Difference in performance among the ELECTRA variants is marginal at best, with
the difference in accuracy between the best ELECTRA model (Small Uncased) and
the weakest one (Base Cased) being only 1.22%. An interesting observation is
that the small variants both outperform their base variants, albeit
marginally. The small uncased model outperforms the base uncased model by
1.34%, while the small cased model outperforms the base cased model by 0.51%.
We hypothesize that this is due to the small models being easier to train,
given that there are less parameters to consider.

While the small variants outperform their base variants on the full dataset,
we hypothesize that the base models have an advantage in settings where there
is less data to learn from, since they have more effective capacity. We verify
this through our use of degradation tests shown in the next subsection.

Model | Val. Loss | Val. Acc. | Test Loss | Test Acc. | Seed
---|---|---|---|---|---
ELECTRA Tagalog Base Cased | 0.2646 | 91.74% | 0.2619 | 91.76% | 4567
ELECTRA Tagalog Base Uncased | 0.2502 | 91.98% | 0.2581 | 91.66% | 4567
ELECTRA Tagalog Small Cased | 0.1931 | 92.58% | 0.1959 | 92.27% | 1439
ELECTRA Tagalog Small Uncased | 0.1859 | 92.96% | 0.1894 | 93.00% | 45
BERT Tagalog Base Cased | 0.3225 | 88.81% | 0.3088 | 89.25% | 1111
BERT Tagalog Base Uncased | 0.3236 | 89.04% | 0.3257 | 89.06% | 6235
ULMFiT Tagalog | 0.2685 | 89.11% | 0.2589 | 89.37% | 42
Table 1:  Final Finetuning Results. The best ELECTRA model (Small Uncased)
outperforms the best BERT model (Base Cased) by +3.75% and the ULMFiT model by
+3.63%. An interesting observation is that the small ELECTRA models perform
marginally better than their base counterparts. We also report the random seed
used in our experiments for reproducibility with our released code.

A table summarizing the finetuning results can be found in Table 1.

###  3.2 Degradation Tests

In total, we perform four degradation tests per model variant, for a total of
28 degradation tests. Each model is finetuned with a fraction of the entire
NewsPH-NLI dataset (50%, 30%, 10%, and 1%), with the resulting performance
compared against the performance of the same model when finetuned with the
full dataset. A summary of all degradation tests can be found in Table 2.

Model | Data % | Test Loss | Test Acc | Acc. Deg. | Deg. % | Degradation Speed
---|---|---|---|---|---|---
ELECTRA Tagalog | 100% | 0.2619 | 91.76% |  |  |
Base Cased | 50% | 0.3184 | 90.56% | -1.20 | 1.31% |
| 30% | 0.3769 | 88.85% | -2.91 | 3.17% | 4.47
| 10% | 0.4467 | 86.23% | -5.53 | 6.03% |
| 1% | 0.5046 | 79.78% | -11.98 | 13.06% |
ELECTRA Tagalog | 100% | 0.2581 | 91.66% |  |  |
Base Uncased | 50% | 0.2920 | 90.85% | -0.81 | 0.88% |
| 30% | 0.3333 | 89.21% | -2.45 | 2.67% | 4.77
| 10% | 0.4041 | 87.20% | -4.46 | 4.87% |
| 1% | 0.5300 | 79.43% | -12.23 | 13.34% |
ELECTRA Tagalog | 100% | 0.1959 | 92.27% |  |  |
Small Cased | 50% | 0.2260 | 91.56% | -0.71 | 0.77% |
| 30% | 0.2504 | 90.13% | -2.14 | 2.32% | 5.69
| 10% | 0.3075 | 87.66% | -4.61 | 5.00% |
| 1% | 0.4873 | 78.09% | -14.18 | 15.37% |
ELECTRA Tagalog | 100% | 0.1894 | 93.00% |  |  |
Small Uncased | 50% | 0.2154 | 91.97% | -1.03 | 1.11% |
| 30% | 0.2439 | 90.86% | -2.14 | 2.30% | 6.92
| 10% | 0.2963 | 88.77% | -4.23 | 4.55% |
| 1% | 0.5303 | 75.91% | -17.09 | 18.38% |
BERT Tagalog | 100% | 0.3088 | 89.25% |  |  |
Base Cased | 50% | 0.3800 | 87.09% | -2.16 | 2.42% |
| 30% | 0.4394 | 86.25% | -3.00 | 3.36% | 2.49
| 10% | 0.5046 | 84.15% | -5.10 | 5.71% |
| 1% | 0.5285 | 81.33% | -7.92 | 8.87% |
BERT Tagalog | 100% | 0.3257 | 89.06% |  |  |
Base Uncased | 50% | 0.4126 | 87.15% | -1.91 | 2.14% |
| 30% | 0.4434 | 86.04% | -3.02 | 3.39% | 2.57
| 10% | 0.5232 | 83.65% | -5.41 | 6.07% |
| 1% | 0.5672 | 81.21% | -7.85 | 8.81% |
ULMFiT Tagalog | 100% | 0.2589 | 89.37% |  |  |
| 50% | 0.3093 | 86.82% | -2.55 | 2.85% |
| 30% | 0.3699 | 84.39% | -4.98 | 5.57% | 8.33
| 10% | 0.4840 | 79.07% | -10.30 | 11.53% |
| 1% | 0.8140 | 67.50% | -21.87 | 24.47% |
Table 2:  Degradation Test Results. “Acc. Deg.” refers to Accuracy
Degradation, the difference between the performance of the model when trained
with the full dataset and when trained with a smaller Data %. “Deg. %” refers
to Degradation Percentage, the percentage of the performance of the model when
trained with the full dataset that is lost when finetuned with a smaller Data
%. Degradation speed is the standard deviation of a model’s Degradation
Percentages, lower is better.

As we start to reduce the training data to 50%, the ELECTRA models remain more
resilient to performance degradation compared to the BERT models and ULMFiT.
We hypothesize this to be due to the more effective means of imparting learned
priors to the Transformer by its data-efficient pretraining scheme. At the 50%
data percentage, ELECTRA has only degraded by 1.02% on average, while BERT and
ULMFiT has degraded by 2.28% and 2.85% on average, respectively. This trend is
still evident at the 30% data percentage mark, with ELECTRA degrading by 2.62%
on average, while BERT and ULMFiT degrade by 3.38% and 5.57% on average,
respectively.

The trend begins to shift as we approach settings with even less data. As the
training data is reduced to 10% of the original (42,000 examples), we see that
ELECTRA starts to begin degrading faster, while BERT degrades at about the
same rate. ELECTRA has degraded by 5.11% on average. Meanwhile, BERT degrades
by 5.89% on average, which is only minimally larger than ELECTRA’s
degradation. The same is true on the extremely-low data 1% data percentage
mark, where ELECTRA has degraded by 15.04% on average, which is 6.2% higher
than BERT’s average degradation of 8.84%.

In extremely low-data domains, we see that BERT is more resilient to
performance degradation than ELECTRA is. ELECTRA is shown to degrade
exponentially as the number of training examples is reduced. As shown in
Figure 2, while BERT’s degradation on average remains relatively linear,
ELECTRA starts degrading faster and faster as we approach the 1% (4,200
examples) data percentage mark. When looking at degradation speeds, it is also
evident that ELECTRA degrades more (average degradation speed of 5.46) while
BERT degrades less (average degradation speed of 2.53).

Figure 2:  Per-technique degradation curves, averaging the performance of all
models belonging to one technique. ULMFiT still remains the easiest model to
degrade as the number of training examples reduce. ELECTRA starts to degrade
strongly after the 10% mark, while BERT remains to degrade slowly.

We hypothesize that this is a direct effect of their pretraining schemes.
ELECTRA is trained without a specific downstream task in mind, while BERT is
trained considering sentence-pair classification tasks, leading to its use of
next sentence prediction as a secondary pretraining task. Since BERT’s biases
are more adjusted to sentence-pair classification, we can hypothesize that it
should perform reliably well even when finetuned with little data, as it
already has an “idea” of how to perform the task.

In terms of per-model degradation, while the small ELECTRA models outperformed
their larger base counterparts in the full dataset, we show that the base
models are more resilient to degradation. As shown in Figure 3, this is more
evident as we approach the 1% data percentage mark, with the small uncased
model degrading 5.04% more than the base uncased model, and the small cased
model degrading 2.31% more than its base cased counterpart. In terms of speed,
we also see that the small models degrade more (average degradation speed of
6.31) than the base models (average degradation speed of 4.62).

Figure 3:  Per-model degradation curves. ULMFiT degrades the fastest out of
the three transfer learning techniques. ELECTRA begins to degrade at the 10%
data percentage mark. Both ELECTRA and ULMFiT degrade exponentially, while
BERT degrades nearly linearly, which is likely due to it’s pretraining scheme
designed for sentence-pair classification tasks.

###  3.3 Heuristics for Choosing Techniques for Low-Data Domains

Overall, the finetuning results and the degradation tests give us good
heuristics for choosing which techniques are appropriate for different tasks
when dealing with low-data domains.

ELECTRA is most effective in the general use-case. When there is a lot of
task-specific data to finetune a model, we see that ELECTRA is more effective
than the baselines BERT and ULMFiT. ELECTRA is also best to use when there is
little pretraining data available, as is with the case of the low-resource
language Filipino. Since there is less pretraining data, ELECTRA’s more data-
efficient pretraining scheme will impart more learned priors than BERT’s
masked language modeling objective will. From our results, we hypothesize that
the same will be true when compared with other Transformer-based pretraining
techniques that use the masked language modeling objective, such as RoBERTa
[18].

However, while ELECTRA is effective in the general case, this does not mean
that BERT will be deprecated anytime soon. BERT is very effective in the low-
data case, especially in tasks that deal with sentence-pair classification
such as natural language inference and sentence entailment. Since BERT’s
pretraining scheme is designed with sentence-pair classification tasks in
mind, it will perform well for such tasks even with little finetuning data as
it already has an idea how to perform these tasks due to its pretraining. As
we show with empirical results, BERT also degrades slower than ELECTRA, and
should be more robust for various tasks in low-data domains and low-resource
languages.

While both Transformer-based finetuning techniques outperform ULMFiT in the
degradation tests, this does not mean that RNN-based methods do not have a use
in current research dealing with language inference tasks. On the full
dataset, we see that ULMFiT performed with accuracy comparable to BERT, albeit
degrading the fastest on average on the degradation tests. ULMFiT’s fast
degradation is likely due to it being RNN-based, which has significantly less
representational capacity than the larger Transformers that leverage attention
used in the study. While this is the case, in settings where there is enough
data to finetune, ULMFiT (and other RNN-based transfer learning techniques)
will perform comparably to the Transformer-based techniques when tuned
properly. ULMFiT’s AWD-LSTM [19] backbone also enjoys the benefit of being
cheaper and faster to train. In cases where there is a lack of resources to
use Transformers effectively, RNN-based models will still suffice, assuming
there is an abundance of data.

##  4 Conclusion

In this paper, we proposed an automatic method for creating sentence
entailment benchmark datasets using news articles. Through our method,
datasets can be generated quickly and cost-efficiently, while ensuring that
they are challenging enough to accurately benchmark performance of high
capacity models. In addition, our method leverages the abundance of news
articles online, which allows datasets even in low-resource languages to be
created.

Using our method, we produce the first sentence entailment benchmark dataset
in Filipino which we call NewsPH-NLI. We also produce pretrained Transformers
based on the ELECTRA pretraining scheme, which we benchmark on our dataset
against two widely-used techniques, BERT and ULMFiT.

We shed light on the true performance of transfer learning techniques when
operating in low-data domains to solve a hard task. We show the importance of
the choice of pretraining task to the effectiveness of a Transformer when
finetuned with little data. We also show that while newer techniques
outperform older established ones, they may still perform worse when dealing
with low-resource languages.

For future work, we recommend further studies on automatic corpus generation
be done; particularly on correctness checking. The biggest disadvantage that
our method has is that to fully ensure correctness, humans will still have to
evaluate the resulting dataset. Should an automatic technique to verify
correctness be made, our dataset generation method will be more robust, and
can then be adapted to generate other tasks that require more human
supervision in creating, such as summarization and translation.

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There is a spectral sequence from

${\mathsf{S}}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(c\otimes\mathfrak{b}_{\geq
0}^{*}[-2])\otimes{H}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}_{\mathfrak{m}}{\mathsf{S}}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(qc\otimes\mathfrak{b}_{\geq
0})\otimes{\mathsf{S}}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(c\otimes(T^{*}M)_{\geq}[-1])$
(154)

to the cohomology of
${\Omega}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}_{c}\mathscr{T}$.

Under our assumption, the absolute values of the $(a,c,q)$-weights in
$c\otimes\mathfrak{b}_{\geq}^{*}$ are strictly less than $1$, while the
absolute values of the weight in $qc\otimes\mathfrak{b}_{\geq}$ are strictly
bigger than $1$. This implies that the weights in the first two factors in
(154) are less than $1$ and the corresponding Euler characteristics converge
to a rational function. Since the third factor in (154) is finite-dimensional,
it affect only finitely many weights and does not affect the convergence.

Since the Euler characteristic of (154) converges, we can forget about the
differentials in the spectral sequence. This gives (153). ∎

#### 4.3.5

######  Proposition 4.7.

The conditions of Lemma 4.6 are satisfied if $|a^{\alpha}|>1$ for all roots in
$\mathfrak{b}$ or if $x$ lies over the point $e\in\mathfrak{n}$ and $a\in
q^{h/2}\mathsf{C}_{\phi,\textup{cmp}}$.

###### Proof.

If

$|a^{\alpha}|>1$

for all roots then $\mathfrak{b}_{<}=0$ and there is nothing to check.

Since the stabilizer of $e$ in $\mathfrak{g}$ is contained in
$\mathfrak{p}(h)$, we conclude that it intersects $\mathfrak{b}_{<0}$
trivially. Therefore, the stabilizer of $x$ intersects $\mathfrak{b}_{<0}$
trivially and the result follows. ∎

#### 4.3.6

######  Corollary 4.8.

The equality (150) holds for functions $\psi$ analytic in the annulus
$1/R_{\mathsf{G}}<|x|<R_{\mathsf{G}}$ and nonvanishing outside of the critical
strip $|q|^{-1}<x<1$, where $R_{\mathsf{G}}$ is a sufficiently large number
that depends on $\mathsf{G}$.

###### Proof.

We make the substitution

$\hat{\psi}(\mathscr{T})=\psi(\mathscr{T})\otimes\bigotimes{\Omega}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}_{c_{i}}\mathscr{T}$

where all $c_{i}$ lie in the critical strip. The new genus $\psi(x)$ has
zeroes $c_{i}$ in the critical strip.

By Proposition 4.7 the insertion of
${\Omega}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}_{c_{i}}\mathscr{T}$
does not affect cohomological properness in either Theorem 6 or in the
integration over strata in Theorem 4, whence the conclusion. ∎

###  4.4 Proof of Theorems 5 and 2

####  4.4.1 Proof of Theorem 5

We first consider the K-theory case. The difference between the statement of
Corollary 4.8 and (37) is the domain of the integration, namely
$\widetilde{\mathbf{S}}_{e}$ versus $\mathbf{S}_{e}$. We recall that
$\mathscr{P}_{+}f$ is the restriction to $\widetilde{\mathbf{S}}_{e}$ of a
conjugation-invariant function on $\mathsf{G}$. Similarly, as observed in
Section 3.4.3, the density $\Psi_{e}$ is a restriction of a conjugation-
invariant functions. Thus the integrand in (150) is pulled back from
$\mathbf{S}_{e}$, while the Haar measure on $\mathbf{S}_{e}$ is pushed forward
from $\widetilde{\mathbf{S}}_{e}$, the equality (37) in the K-theory case.

#### 4.4.2

The cohomology case follows from the K-theory by taking a limit as explained
in Section 2.7. While this concludes the proof, it may be instructive to see
explicit formulas below.

#### 4.4.3

Since, in this paper, we focus on the more difficult K-theoretic computations,
it may be useful to spell out the formulas one obtains in the specialization
to cohomology.

The combination of Theorems 6 and 5 may be phrased as the following equality
of integrals involving general functions $\psi(s)$ and $f(s)$, analytic in a
sufficiently wide neighborhood of the imaginary axis. We define

$\Psi(s)=\frac{\psi(s)}{s}\,,\quad Z_{\psi}(s)=\Psi(-s)\Psi(s-1)$

and denote

$\textup{\small\cursive z}_{\psi}=-\psi(0)\psi(-1)\,.$

We require that:

  1. (1)

$\psi(s)$ is nonvanishing outside the vertical strip $-1<\Re s<0$ ,

  2. (2)

$Z_{\psi}(s)$ is real and positive for $s>1$ ,

  3. (3)

The integrals on both sides of (155) converge. For example, this can be
guaranteed by $f$ being a Paley-Wiener function and $\psi(s\pm 1)/\psi(s)$
growing at most polynomially in the imaginary direction.

In this setting, the combination of Theorems 6 and 5 implies the following
equality:

$\frac{1}{\textup{\small\cursive z}_{\psi}^{r}}\int_{\Re
x=x_{0}>\rho^{\vee}}\,\sum_{w}f(x)\,\overline{f}(-w^{-1}x)\prod_{\begin{subarray}{c}\alpha>0\\\
w^{-1}\alpha<0\end{subarray}}\frac{Z_{\psi}(\alpha(x))}{Z_{\psi}(1+\alpha(x))}\,dx=\\\
\\\
=\sum_{\phi}\int_{\frac{1}{2}\phi(h)+\operatorname{Lie}\mathsf{C}_{\phi,\textup{cmp}}}\left|\mathscr{P}_{+}f\right|^{2}\,|\Psi_{e}|\,dx\,.$
(155)

Here the summation is over the homomorphisms

$\phi:\mathfrak{sl}_{2}\to\mathfrak{g}\,,$

considered up to conjugation, and uses the image of the standard basis

$h=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\,,\quad e=\begin{pmatrix}0&1\\\
0&0\end{pmatrix}\quad e_{-}=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}\,,$

in $\mathfrak{g}$. The the integration is over a shift by
$\tfrac{1}{2}\phi(h)$ of the Lie algebra of the maximal compact subgroup of
the centralizer $\mathsf{C}_{\phi}$ of $\phi$ in $\mathsf{G}$.

The spectral projectors and the spectral measure in (155) are defined by

$\mathscr{P}_{+}f=\left(\sum_{w\in
W}w\right)\left(\prod_{\alpha<0}\frac{\Psi(\alpha(x))}{\Psi(\alpha(x)-1)}\right)f$

and

$\Psi_{e}=\psi(-1)^{-2r}\frac{\prod_{s\in\mathfrak{g}^{e_{-}}}\Psi(s-1)}{\prod_{s\in\mathfrak{g}^{\phi}}\psi(s)\prod_{s\in\mathfrak{g}^{e}/\mathfrak{g}^{\phi}}\Psi(s)}\,,$

where the notation means that we take the product over the weights
$s\in\left(\operatorname{Lie}\mathsf{C}_{\phi}\right)^{*}$ of the
corresponding representations.

####  4.4.4 Proof of Theorem 2

Theorem 5 produces an injection from LHS of (14) to the RHS. To prove the
surjectivity, it is enough to show the map has a dense image and, since it is
a map of $\mathbf{sH}$-modules it is enough to show it is nonzero on each
piece of the spectral decomposition. The latter claim follows from the
geometric definition of $\mathscr{P}_{+}$ as integration over the Spinger
fiber and the classical fact that $\left(\mathscr{B}^{e,h}\right)^{g}$ is
nonempty for any $g\in\mathsf{C}_{\phi,\textup{cmp}}$.

Indeed, any such $g$ is semisimple, as hence the Lie algebra
$\mathfrak{g}^{g}$, which contains $h$ and $e$, is reductive. Borel
subalgebras of $\mathfrak{g}^{g}$ parametrize a component of
$\mathscr{B}^{g}$. Since $h$ and $e$ generate a solvable Lie algebra, they
belong to some Borel subalgebra of $\mathfrak{g}^{g}$, which is thus fixed by
all three elements.

##  5 Example: Subregular in $G_{2}$

###  5.1 Generalities on the dCLP stratification

#### 5.1.1

The goal of this section is to present a discussion of a particularly
illustrative example — that of a subregular nilpotent in type $G_{2}$. From
the contour deformation point of view, it was dealt with in fine detail in [L]
and thus played a very important role in the development of the subject.

The geometry of all subregular Springer fibers being well-understood, see for
instance [Slodowy], the only novelty in this section is in how this geometry
matches Langlands’ computations. We begin with a few general reminders from
[dCLP] before specializing the discussion to the specific example. For the
convenience of the reader, we remind the proofs.

#### 5.1.2

Let $e$ and $h$ denote a fixed nilpotent element of $\mathfrak{g}$ and its
characteristic. Consider the grading of $\mathfrak{g}$ by the eigenvalues of
$\operatorname{ad}(h)$

$\mathfrak{g}=\bigoplus\mathfrak{g}_{i}\supset\bigoplus_{i\geq
0}\mathfrak{g}_{i}=\mathfrak{p}\,,$

and the corresponding parabolic subalgebra $\mathfrak{p}$. We denote by
$\mathsf{P}\subset\mathsf{G}$ the parabolic subgroup with Lie algebra
$\mathfrak{p}$. For instance,

$e=0\Leftrightarrow\mathfrak{p}=\mathfrak{g}\,.$

One notes that $\mathfrak{p}$ and $\mathsf{P}$ are canonically associated to
$e$.

#### 5.1.3

We have $e\in\mathfrak{g}_{2}$ and
$\operatorname{ad}(\mathsf{P})\,e\subset\mathfrak{g}_{\geq 2}$. The following
proposition is classical.

######  Proposition 5.1.

We have the following diagram

$\raisebox{30.0pt}{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
33.58443pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}^{\circ}_{\geq 2}\overset{\textup{\tiny
def}}{=}\operatorname{ad}(\mathsf{P})\,e\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.45036pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
57.58443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
0.0pt\raise-28.89406pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
57.58443pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathfrak{g}_{\geq
2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
66.66222pt\raise-20.3359pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\sigma(0)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
66.66222pt\raise-33.36627pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}^{\circ}_{2}\overset{\textup{\tiny
def}}{=}\operatorname{ad}(\mathsf{G}^{h})\,e\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
33.58443pt\raise-40.67181pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
59.7622pt\raise-40.67181pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
59.7622pt\raise-40.67181pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathfrak{g}_{2}}$}}}}}}}\ignorespaces}}}}\ignorespaces}\,,$
(156)

in which the horizontal maps are open embeddings and

$\sigma(q)\cdot x=q^{-1}\operatorname{ad}(q^{h/2})\,x\,.$ (157)

The fibers of $\sigma(0)$ are orbits of the unipotent radical
$\mathsf{P}_{\textup{uni}}$ isomorphic to

$\mathfrak{g}_{\geq 1}\big{/}\mathfrak{c}_{e,>0}\cong\mathfrak{g}_{>2}\,.$
(158)

Note that by construction

$\mathfrak{g}^{\circ}_{2}\cong\mathsf{G}^{h}/\mathsf{C}_{\phi}\,.$ (159)

###### Proof.

Decomposing $\mathfrak{g}$ with respect to the adjoint action of
$\phi(\mathfrak{sl}_{2})$, we see that the maps

$\mathfrak{g}_{i}\xrightarrow{\quad\operatorname{ad}(e)\quad}\mathfrak{g}_{i+2}\,,\quad
i\geq 0\,,$

are surjective. This shows that the horizontal maps in (156) are open
embeddings and also proves the isomorphism (158). Since the action of
$\sigma(q)$ preserves the $\mathsf{P}_{\textup{uni}}$-orbits and contracts
$\mathfrak{g}_{\geq 2}$ to $\mathfrak{g}_{2}$ as $q\to 0$, the remaining
claims follow.

∎

#### 5.1.4

The orbits of $\mathsf{P}$ on $\mathscr{B}$ are given by the Bruhat
decomposition

$\mathscr{B}=\bigsqcup_{w\in W^{h}\backslash W}\mathsf{P}\,w\,\mathfrak{n}\,,$
(160)

where it is convenient to take representatives of minimal length as in
Appendix A.3.3, and where we assume that $h\in\operatorname{Lie}\mathsf{A}$ as
in Appendix A.3.1. These Bruhat cells are the attracting manifolds for the
$q^{h}$-action on $\mathscr{B}$. By definition, the dCLP strata

$\mathscr{B}^{e}_{w}=\mathscr{B}^{e}\cap\mathsf{P}\,w\,\mathfrak{n}$ (161)

are the intersections of the Spinger fibers with the $\mathsf{P}$-orbits
(160).

######  Proposition 5.2 ([dCLP]).

The strata (161) are smooth.

###### Proof.

The Bruhat cells in (160) have the form $\mathsf{P}/\mathsf{P}_{w}$, where
$\mathsf{P}_{w}=\mathsf{P}\cap w\mathsf{B}w^{-1}$. The stabilizer
$\mathsf{P}_{w}$ acts in

$\mathfrak{g}_{\geq 2}\big{/}\left(\mathfrak{g}_{\geq 2}\cap
w\mathfrak{n}\right)$

and this defines a homogeneous vector bundle over $\mathsf{P}/\mathsf{P}_{w}$
with a section

$\mathsf{P}/\mathsf{P}_{w}\owns
g\mapsto\operatorname{ad}(g^{-1})e\bmod\left(w\mathfrak{n}\right)_{\geq 2}\,.$

This section is transverse because $\mathsf{P}$ acts on $\mathfrak{g}_{\geq
2}$ with an open orbit and its zero locus is precisely the stratum (161). ∎

#### 5.1.5

Recall from Appendix A.3.3 that

$\mathscr{B}^{h}=\bigsqcup_{w\in W^{h}\backslash
W}\mathscr{B}^{h}_{w}\,,\quad\mathscr{B}^{h}_{w}=\mathsf{G}^{h}\,w\,\mathfrak{n}\cong\mathsf{G}^{h}/\mathsf{B}^{h}\,,$
(162)

and recall the subset $W(e)$ introduced in Section 3.4.2.

######  Corollary 5.3.

We have:

  1. (1)

the fixed loci $\mathscr{B}^{h,e}=\bigsqcup\mathscr{B}^{h,e}_{w}$ are smooth;

  2. (2)

$\mathscr{B}^{e}_{w}\neq\varnothing\Leftrightarrow\mathscr{B}^{h,e}_{w}\neq\varnothing\Leftrightarrow
w\in W(e)$;

  3. (3)

the manifold $\mathscr{B}^{h,e}_{w}$ is cut out inside
$\mathscr{B}^{h}_{w}\cong\mathsf{G}^{h}/\mathsf{B}^{h}$ by a transverse
section

$\mathsf{G}^{h}/\mathsf{B}^{h}\owns
g\mapsto\operatorname{ad}(g^{-1})e\bmod\left(w\mathfrak{n}\right)_{2}$

of a homogeneous vector bundle;

  4. (4)

if $\mathscr{B}^{h,e}_{w}\neq\varnothing$ then

$\dim\mathscr{B}^{h,e}_{w}=\dim\mathsf{G}^{h}/\mathsf{B}^{h}-\dim\mathfrak{g}_{2}\big{/}\left(w\mathfrak{n}\right)_{2}\,,$
(163)

and $\mathscr{B}^{h,e}_{w}=\varnothing$ when this number is negative.

#### 5.1.6

The dimension formula (163) also follows from the identification

$\mathscr{B}^{h,e}_{w}/\mathsf{C}_{\phi}=\left(w\mathfrak{n}\right)_{2}^{\circ}/\mathsf{B}^{h}\,,$
(164)

because

$\dim\mathsf{C}_{\phi}=\dim\mathsf{G}^{h}-\dim\mathfrak{g}_{2}\,.$

Similarly, we have

$\displaystyle\dim\mathscr{B}^{e}_{w}$
$\displaystyle=\dim\mathsf{P}/\mathsf{P}_{w}-\dim\mathfrak{g}_{\geq
2}\big{/}\left(w\mathfrak{n}\right)_{\geq 2}$
$\displaystyle=\dim\mathfrak{g}_{[0,1]}-\dim\left(w\mathfrak{b}\right)_{[0,1]}\,,$
(165)

where the subscripts refer to the ranges of the grading.

###  5.2 Nilpotent elements in type $G_{2}$

#### 5.2.1

In the rest of this section, $\mathfrak{g}$ denotes the simple complex Lie
algebra of type $G_{2}$ and $\mathsf{G}$ the corresponding simple complex Lie
group (which is both $1$-connected and adjoint). Figure 3 explains how we
label the roots of $\alpha_{i}$ of $\mathfrak{g}$ so that

$\displaystyle\\{\alpha_{1},\alpha_{6}\\}$ $\displaystyle=\textup{simple
positive roots}\,,$ $\displaystyle\\{\alpha_{1},\alpha_{3},\alpha_{5}\\}$
$\displaystyle=\textup{short positive roots}\,,$
$\displaystyle\\{\alpha_{2},\alpha_{4},\alpha_{6}\\}$
$\displaystyle=\textup{long positive roots}\,.$

We identify $\mathfrak{h}^{\vee}\owns\alpha_{i}$ with its dual so that
$\|\alpha_{i}\|^{2}\in\\{2,6\\}$. Then

$\\{\alpha_{3},\alpha_{4}^{*}\\}\,,\quad\alpha_{4}^{*}=\tfrac{1}{3}\alpha_{4}\,,$

are the fundamental coroots.

Figure 3: The positive roots $\alpha_{i}$ of $\mathfrak{g}$ are labelled by
$\\{1,\dots,6\\}$. We normalize them so that $\|\alpha_{i}\|^{2}\in\\{2,6\\}$.
With this normalization, $3$ and $4^{*}$ are the fundamental coroots. The
positive cone is shaded. The black lines is where the characteristics
$\\{2\alpha_{2}^{*},2\alpha_{4}^{*},2\alpha_{6}^{*}\\}$ take value 2.

#### 5.2.2

Recall that the nilpotent elements in $\mathfrak{g}$ are classified as
follows, see [Coll]. First, there are those which are not distinguished, that
is, belong to a proper Levi subalgebra. This means they are either 0 or
conjugate to a root vector. The corresponding characteristic $h$ is thus
either 0 or an element in the dual roots system.

In addition the the principal nilpotent, the Lie algebra $\mathfrak{g}$ has
another distinguished class with the characteristic conjugate to

$h=2\alpha_{4}^{*}=\tfrac{2}{3}\alpha_{4}\,.$ (166)

This element in plotted in red in Figure 5. For the rest of this section, we
fix this $h$. It gives

$\mathfrak{g}_{2}=\mathfrak{n}_{2}=\textup{span}(e_{2},e_{3},e_{5},e_{6})\,.$

These are the roots that line on one of the black lines in Figure 3.

#### 5.2.3

We have

$\mathsf{G}^{h}\cong GL(2)$

where the $SL(2)$ subgroup corresponds to the root $\alpha_{1}$ and the center
corresponds to the coroot $h/2$. As a $\mathsf{G}^{h}$-module,

$\mathfrak{g}_{2}\cong{\det}^{-1}\otimes S^{3}\mathbb{C}^{2}\,,$

where $\mathbb{C}^{2}$ is the defining representation of $GL(2)$. The weights
of this module can be seen in Figure 5.

Since the center of $GL(2)$ acts with weight $1$ in this representation, the
action of $\mathsf{G}^{h}$ on $\mathfrak{g}_{2}$ reduces to action of $PSL(2)$
on

$\mathbb{P}\left(S^{3}\mathbb{C}^{2}\right)=S^{3}\mathbb{P}^{1}\,,$

that is, the action on unordered triples of points on $\mathbb{P}^{1}$. This
action has an open orbit corresponding to triples of disjoint points, with
stabilizer isomorphic to $S(3)$.

The preimage of this open orbit in $\mathfrak{g}_{2}$ is
$\mathfrak{g}_{2}^{\circ}$. It corresponds to cubic polynomials in
$S^{3}\mathbb{C}^{2}$ with distinct roots. For $e\in\mathfrak{g}_{2}^{\circ}$,
the stabilizer is $S(3)$ and the inclusion of the stabilizer

$S(3)=\mathsf{C}_{\phi}\xrightarrow{\quad\iota\quad}\mathsf{G}^{h}=GL(2)$
(167)

is the irreducible 2-dimensional representation of $S(3)$.

#### 5.2.4

Since $\mathfrak{g}_{2}=\mathfrak{n}_{2}$, we have
$\mathfrak{g}_{2}/\mathfrak{n}_{2}=0$ and hence

$\mathscr{B}^{e,h}_{1}=\mathsf{G}^{h}/\mathsf{B}^{h}\cong\mathbb{P}^{1}\,.$

For other elements in $W$, we have

$\displaystyle\left(s_{6}\mathfrak{n}\right)_{2}$
$\displaystyle=\textup{span}(e_{2},e_{3},e_{5})\,,$
$\displaystyle\left(s_{2}\mathfrak{n}\right)_{2}$
$\displaystyle=\textup{span}(e_{5},e_{6})\,,$

et cetera. In general, the dimension of $(w\mathfrak{n})_{2}$ for various $w$
correspond to the number of positive roots $\alpha$ on lines
$\alpha(w^{-1}h)=2$. Three of these lines are plotted in Figure 3, they
contain 4, 3, and 2 roots, respectively. Thus counting dimensions as in
Corollary 5.3, we see that

$\mathscr{B}^{e,h}_{w}=\varnothing\,,\quad w\notin\\{1,s_{6}\\}\,,$

while

$\mathscr{B}^{e,h}_{s_{6}}=\textup{3 points}\,.$

Indeed,

$\displaystyle\mathbb{P}((s_{6}\mathfrak{n})_{2}^{\circ})$
$\displaystyle\cong\textup{distinct triples in $S^{3}\mathbb{P}^{1}$
containing $\infty\in\mathbb{P}^{1}$}$ $\displaystyle\cong
S^{2}\mathbb{A}^{1}\,,$

and thus

$\mathscr{B}^{e,h}_{s_{6}}/S(3)\cong(s_{6}\mathfrak{n})_{2}^{\circ}\big{/}\mathsf{B}^{h}\cong\operatorname{pt}/S(2)\,,$

where we used the identification (164).

#### 5.2.5

Computing the dimensions, we see that $\mathscr{B}^{e}_{1}=\mathbb{P}^{1}$
while

$\mathscr{B}^{e}_{s_{6}}=\textup{3 copies of $\mathbb{A}^{1}$}\,.$

Altogether, the Spinger fiber $\mathscr{B}^{e}$ is the configuration of four
$\mathbb{P}^{1}$’s, in agreement with the general theory [Slodowy]. See Figure
4, where the arrows indicate the directions of attraction of $q^{h}$ as $q\to
0$.

Figure 4: The subregular Springer fiber of type $G_{2}$ is the configuration
of $\mathbb{P}^{1}$’s whose dual graph is the Dynkin diagram of type $D_{4}$.
The $h$-fixed locus, plotted in thick blue, is the union of
$\mathscr{B}_{e}^{1}\cong\mathbb{P}^{1}$ and one point in each of the other
components. These 3 components are permuted by $\mathsf{C}_{\phi}\cong S(3)$.

###  5.3 Spectral projectors and spectral measure

#### 5.3.1

Our next goal is to make explicit the spectral projectors and spectral measure
for $e$. We begin with choosing the coordinates $(x_{1},x_{2})$ on
$\mathsf{A}$ so that

$\mathsf{B}^{h}=\left\\{\begin{pmatrix}x_{1}&*\\\
0&x_{2}\end{pmatrix}\right\\}\subset GL(2)=\mathsf{G}^{h}\,.$ (168)

The adjoint weights of $\mathsf{A}$ are displayed in Figure 5. The Weyl group
$W^{h}\cong S(2)$ acts permuting $x_{1}$ and $x_{2}$.

$x_{1}/x_{2}$
---
$x_{1}^{2}/x_{2}$
---
$x_{1}$
---
$x_{1}x_{2}$
---
$h$
---
$x_{2}$
---
$x_{2}^{2}/x_{1}$
---
$x_{2}/x_{1}$
---
Figure 5: The characteristic $h$, the adjoint weights of $\mathsf{A}$, and the
positive cone. The dotted lines are where $h$ takes values $\pm 2$.

#### 5.3.2

The map (167) satisfies

$\iota((12))\sim\begin{pmatrix}1\\\
&-1\end{pmatrix}\,,\quad\iota((123))\sim\begin{pmatrix}\nu\\\
&\nu^{2}\end{pmatrix}\,,\quad\nu=e^{2\pi i/3}\,.$ (169)

Comparing the characters we conclude that the decomposition (142) takes the
form

$\mathfrak{g}=V^{2}\boxtimes(1+\mathbb{C}^{2})+V^{4}\boxtimes\operatorname{sgn}$
(170)

where $1$ and $\operatorname{sgn}$ are the two characters of $S(3)$, while
$\mathbb{C}^{2}$ is the irreducible 2-dimensional representation. In
particular,

$\displaystyle\operatorname{Slice}_{e}q^{-1}\mathfrak{g}$
$\displaystyle=q^{-2}(1+\mathbb{C}^{2})+q^{-3}\operatorname{sgn}\,,$
$\displaystyle\mathfrak{c}_{e}$
$\displaystyle=q(1+\mathbb{C}^{2})+q^{2}\operatorname{sgn}\,,$ (171)

as $\widehat{\mathsf{C}}_{e}$-modules. We conclude

$\Psi_{e}\Big{|}_{q^{h/2}S(3)}=\Psi\Big{(}(q^{-2}-q)(\mathbb{C}^{2}+1)+(q^{-3}-q^{2})\operatorname{sgn}-4q^{-1}\Big{)}\,.$
(172)

#### 5.3.3

The spectral projectors $\mathscr{P}_{\pm}$ are given by integration along the
Spinger fibers as in (125). They take a function $f(x_{1},x_{2})$ on the torus
$\mathsf{A}$, viewed as K-theory class on $\mathscr{B}_{e}$ to a conjugation-
invariant function on $S(3)$. The difference between $\mathscr{P}_{\pm}$ is in
the identification of $K_{\mathsf{G}}(\mathscr{B})$ with functions on the
torus $\mathscr{A}$. Since opposite Borel subgroups are used for this
identification, we have

$\mathscr{P}_{+}=\mathscr{P}_{-}\circ w_{0}\,,\quad
w_{0}(x_{1},x_{2})=(x_{1}^{-1},x_{2}^{-1})\,.$

Here $w_{0}$ is the longest element of the Weyl group.

By stratification and equivariant localization, the integral over
$\mathscr{B}^{e}$ is reduced to integrals over $\mathscr{B}^{h,e}_{w}$ for
$w=1,s_{6}$, giving the decomposition

$\mathscr{P}_{+}=\mathscr{P}_{+,1}+\mathscr{P}_{+,s_{6}}\,.$

We deal with them separately, starting with $w=1$.

#### 5.3.4

Restricted to $\mathscr{B}^{h}_{1}$, the virtual tangent bundle to
$\mathscr{B}_{e}$ is induced from the $\mathsf{B}^{h}$-module

$\displaystyle(1-q^{-1})\mathfrak{n}^{\vee}$
$\displaystyle=(1-q^{-1})(\mathfrak{g}_{-4}+\mathfrak{g}_{-2})+(1-q^{-1})x_{2}/x_{1}$
(173)
$\displaystyle=\underbrace{(1-q^{-1})(\mathfrak{g}_{-4}+\mathfrak{g}_{-2})+q^{-1}(2-\mathfrak{g}_{0})}_{\textup{trivial
bundle
$V_{0}$}}+\underbrace{x_{2}/x_{1}+q^{-1}x_{1}/x_{2}}_{T^{*}\mathbb{P}^{1}}\,.$

We observe that last two terms give the cotangent bundle to
$\mathsf{B}^{h,e}_{1}\cong\mathbb{P}^{1}$, while the remaining terms, being
$\mathsf{G}^{h}$-modules, induce a trivial virtual bundle $V_{0}$ over
$\mathbb{P}^{1}$ with the corresponding action. Computing their
$S(3)$-characters, we conclude

$V_{0}=-q^{-3}\operatorname{sgn}-q^{-2}(1+\mathbb{C}^{2})+2q^{-1}\,.$

Therefore

$\mathscr{P}_{+,1}\,f\Big{|}_{q^{h/2}S(3)}=\Psi(V_{0})\,\operatorname{\mathbf{D}}f\Big{|}_{q^{h/2}S(3)}$

where

$\displaystyle\operatorname{\mathbf{D}}f(x_{1},x_{2})$
$\displaystyle=\chi(f\otimes\psi(T^{*}\mathbb{P}^{1}))$
$\displaystyle=(1+(12))Z(x_{1}/x_{2})f(x_{1},x_{2})\,.$ (174)

The result in (174) is interpreted geometrically as an element of
$K_{GL(2)}(\operatorname{pt})$, which we subsequently restrict to
$q^{h/2}S(3)$.

Collecting all terms above, we obtain

$\Psi_{e}\left(\mathscr{P}_{+,1}f_{1}\right)\left(\mathscr{P}_{-,1}f_{2}\right)\Big{|}_{q^{h/2}S(3)}=\left.\frac{\operatorname{\mathbf{D}}f_{1}(x_{1},x_{2})\,\operatorname{\mathbf{D}}f_{2}(x_{1}^{-1},x_{2}^{-1})}{Z(q^{2}(1+\mathbb{C}^{2})+q^{3}\operatorname{sgn})}\right|_{q^{h/2}S(3)}\,.$
(175)

#### 5.3.5

Note that because of the pole of $Z(x_{1}/x_{2})$ at $x_{1}=x_{2}$, the value
of (174) at $q^{h/2}$, that is, at $(x_{1},x_{2})=(q,q)$, involves the
derivatives of $f$ at this point, as well as the subleading term in the
expansion of $Z$ at the pole.

Concretely, if

$Z(x)=\frac{z_{-1}}{1-x^{-1}}+z_{0}+\dots$

then

$\operatorname{\mathbf{D}}f(x_{1},x_{2})\Big{|}_{x_{1}=x_{2}}=(z_{-1}+2z_{0})f(x_{1},x_{1})+z_{-1}\left.\left(x_{1}\frac{\partial}{\partial
x_{1}}-x_{2}\frac{\partial}{\partial x_{2}}\right)f\right|_{x_{1}=x_{2}}\,.$

#### 5.3.6

We now consider $\mathscr{B}^{h,e}_{s_{6}}$, which consists of $3$ points
permuted by $S(3)$. Since this is a permutation representation, the character
of $(123)$ vanishes, the character of $(12)$ equals the contribution of unique
point fixed by this transposition, while the character of identity is 3 times
the contribution of a single point.

As a concrete nilpotent element $e$ we may take

$e=e_{2}+e_{5}\,,$

where the labeling of the roots is an in Figure 3, the normalization of the
root vectors being immaterial. This belongs to $\mathfrak{g}_{2}^{\circ}$, is
fixed by

$\mathsf{A}^{e}=\left\\{\begin{pmatrix}\varepsilon&0\\\
0&1\end{pmatrix}\right\\}\,,\quad\varepsilon=\pm 1\,.$ (176)

and is contained in $s_{6}\mathfrak{n}$, see Figure 6.

$x_{1}/x_{2}$
---
$x_{1}^{2}/x_{2}$
---
$x_{1}$
---
$x_{1}x_{2}$
---
$h$
---
$x_{2}$
---
$x_{2}^{2}/x_{1}$
---
$x_{2}/x_{1}$
---
$x_{1}/x_{2}^{2}$
---
Figure 6: Same as in Figure 5 but now $s_{6}\mathfrak{n}\in\mathscr{B}^{e}$ is
shaded.

The reflection $s_{6}$ acts by

$s_{6}\cdot(x_{1},x_{2})=(x_{1},x_{1}/x_{2})$

and we have to apply $s_{6}$ to both the K-theory class $f$ and to the
character of the virtual normal bundle (173) to compute the contribution of
the point $s_{6}\mathfrak{n}\in\mathscr{B}^{e,h}_{s_{6}}$. We obtain

$\displaystyle(1-q^{-1})T\mathscr{B}\Big{|}_{s_{6}\mathfrak{n}}$
$\displaystyle=2(q^{-1}-q^{-2})+(q-q^{-3})\varepsilon\,.$
$\displaystyle=V_{0}\Big{|}_{S(2)}+\varepsilon(q+q^{-2})\,.$ (177)

Therefore

$\displaystyle\mathscr{P}_{+,s_{6}}$ $\displaystyle f\Big{|}_{q^{h/2}}$
$\displaystyle=3\Psi(V_{0})Z(q^{2})f(q,1)\,,$ (178)
$\displaystyle\mathscr{P}_{+,s_{6}}$ $\displaystyle f\Big{|}_{q^{h/2}(12)}$
$\displaystyle=\Psi(V_{0})Z(-q^{2})f(-q,-1)\,,$ (179)
$\displaystyle\mathscr{P}_{+,s_{6}}$ $\displaystyle f\Big{|}_{q^{h/2}(123)}$
$\displaystyle=0\,.$ (180)

#### 5.3.7

We now have all the data needed to express the contribution of the subregular
nilpotent $e$ to the spectrum. The contributions consist of 3 isolated points
corresponding to the conjugacy classes of $S(3)$.

The identity $1\in S(3)$ corresponds to the point

$Wq^{h/2}=W(q,q)\in\mathsf{A}/W=\operatorname{Spec}\mathbf{sH}$

and contributes

$\|\delta_{W(q,q)}f\|_{\Psi}^{2}=\frac{1}{6}\frac{1}{Z(q^{2})^{3}Z(q^{3})}\left|\operatorname{\mathbf{D}}f\Big{|}_{(q,q)}+3Z(q^{2})f(q,1)\right|^{2}\,,$
(181)

to the spectrum, where $\delta_{W(q,q)}$ is the spectral projector. The
prefactor $\frac{1}{6}$ comes from the Haar measure on $S(3)$ and the
difference $3Z(q^{2})f(q,1)$ between (181) and (175) is the contribution of
(178) .

The transposition $(12)\in S(3)$ corresponds to the point
$(-q,q)\in\operatorname{Spec}\mathbf{sH}$ and it contributes

$\|\delta_{W(-q,q)}f\|_{\Psi}^{2}=\frac{1}{2}\frac{\left|Z(-1)(f(-q,q)+f(q,-q))+Z(-q^{2})f(-q,-1)\right|^{2}}{Z(q^{2})^{2}Z(-q^{2})Z(-q^{3})}\,.$
(182)

Finally, the cycle $(123)$ corresponds to the point $(q\nu,q\nu^{2})$ where
$\nu=e^{2\pi i/3}$, and it contributes

$\|\delta_{W(q\nu,q\nu^{2})}f\|_{\Psi}^{2}=\frac{1}{3}\frac{\left|Z(\nu^{-1})f(q\nu,q\nu^{2})+Z(\nu)f(q\nu^{2},q\nu)\right|^{2}}{Z(q^{2})Z(q^{2}\nu)Z(q^{2}\nu^{2})Z(q^{3})}\,.$
(183)

#### 5.3.8

In the number field situation, that is, in cohomology, only the point (181)
survives and becomes the point

$\tfrac{1}{2}h=\alpha^{*}_{4}=(1,1)\in\operatorname{Lie}\mathsf{A}/W\,.$

It is instructive to compare the results with Landlands’ computation.

The additive analog of the operator (174) is the operator

$\operatorname{\mathbf{D}_{a}}f=(1+(12))\,\xi(x_{1}-x_{2})f(x_{1},x_{2})\,.$
(184)

where

$\xi(x)=-\frac{1}{x}+a+\dots\,,$

in the notations introduces at the bottom of page 186 in [L]. This means that

$\operatorname{\mathbf{D}_{a}}f\Big{|}_{x_{1}=x_{2}}=-(\partial_{1}-\partial_{2})f(x_{1},x_{1})+2af(x_{1},x_{1})\,.$

Therefore

$\|\delta_{W(1,1)}f\|_{\Psi}^{2}=\frac{1}{6}\frac{\left|(\partial_{1}-\partial_{2})f(1,1)-2af(1,1)-3\xi(2)f(1,0)\right|^{2}}{\xi(2)^{3}\xi(3)}\,,$
(185)

where we have changed the sign under the absolute value sign to ease the
comparison with the matrix in Figure 7.

Figure 7: The spectral projector for the discrete spectrum of $G_{2}$
reproduced from page 195 of [L]. The point $\rho$, which corresponds to
$\rho^{\vee}$ in our notations, is the contribution of the regular nilpotent
considered in Section 3.6.

To match the coordinates with Langland’s coordinates, we note that Langlands’
roots $\beta_{i}$ are our dual roots $\alpha_{i}^{*}$, see the illustration on
on page 181 of [L]. In particular,

$\beta_{4}=(1,1)\,,\quad\beta_{2}=s_{6}\beta_{4}=(1,0)\,,$

and

$\beta_{1}=2\beta_{2}-\beta_{4}=(1,-1)\,.$

Since the derivative $D_{1}$ used by Langlands is the derivative in the
direction of $\beta_{1}$, we conclude

$D_{1}=\partial_{1}-\partial_{2}\,,$

completing the match.

##  Appendix A Appendix

###  A.1 Group cohomology

#### A.1.1

The cohomology of a complex algebraic group $\mathsf{G}$ with coefficients in
a representation $V$ is defined by

$H^{i}(\mathsf{G},V)=\operatorname{Ext}^{i}_{\mathsf{G}}(\mathbb{C},V)\,,$

where $\mathbb{C}$ is trivial representation and Ext’s are computed in the
category of algebraic $\mathsf{G}$-modules. These are the derived functors of
$\mathsf{G}$-invariants

$H^{0}(\mathsf{G},V)=V^{\mathsf{G}}\,.$

They may be computed using the usual cochain complex, see e.g. [FF], in which
the terms are given by morphisms $\mathsf{G}^{\times n}\to V$ of algebraic
varieties.

#### A.1.2

We define

$\chi(\mathsf{G},V)=\sum_{i}(-1)^{i}H^{i}(\mathsf{G},V)\,.$ (186)

This is linear in the $K_{\mathsf{G}}(\operatorname{pt})$-class of $V$ by the
long exact sequence of cohomology.

#### A.1.3

For a group extension

$1\to\mathsf{G}_{1}\to\mathsf{G}\to\mathsf{G}_{2}\to 1\,,$

we have

$\chi(\mathsf{G},V)=\chi(\mathsf{G}_{2},\chi(\mathsf{G}_{1},V))$ (187)

by the spectral sequence for derived functors of

$V^{\mathsf{G}}=\left(V^{\mathsf{G}_{1}}\right)^{\mathsf{G}_{2}}\,.$

Formula (187) reduces the computation of $\chi(\mathsf{G},V)$ to that for the
unipotent radical and the reductive quotient in (48).

#### A.1.4

We have (essentially by definition)

$\textup{$\mathsf{G}$ is reductive}\,\Leftrightarrow
H^{>0}(\mathsf{G},\,\cdot\,)=0\,,$

and if $\mathsf{U}$ is a unipotent group then

${H}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(\mathsf{U},V)={H}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(\mathfrak{u},V)\,,\quad\mathfrak{u}=\operatorname{Lie}\mathsf{U}\,,$
(188)

as a consequence of the Hochschild-Mostow spectral sequence, see [FF], Section
1.§3.

#### A.1.5

We recall that Lie algebra cohomology in (188) is succinctly computed by the
Chevalley-Eilenberg complex

${H}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(\mathfrak{u},V)={H}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}\left(V\to
V\otimes\mathfrak{u}^{\vee}\to
V\otimes\Lambda^{2}\mathfrak{u}^{\vee}\to\dots\right)\,.$ (189)

###  A.2 Cohomologically proper actions

#### A.2.1

Consider an action-preserving map

$(\mathsf{G}_{1},\mathsf{X}_{1})\xrightarrow{\quad(f_{\mathsf{G}},f_{\mathsf{X}})\quad}(\mathsf{G}_{2},\mathsf{X}_{2})$
(190)

with $f_{\mathsf{G}}$ surjective. It is cohomologically proper if
${\mathsf{R}}^{\raisebox{1.42262pt}{$\scriptscriptstyle\bullet$}}(f_{\mathsf{G}},f_{\mathsf{X}})_{*}\mathscr{F}$
is a bounded complex of coherent sheaves for any
$\mathscr{F}\in\operatorname{Coh}_{\mathsf{G}_{1}}(\mathsf{X}_{1})$. Here

$R^{0}(f_{\mathsf{G}},f_{\mathsf{X}})_{*}\mathscr{F}=\left(f_{\mathsf{X},*}\mathscr{F}\right)^{\ker
f_{\mathsf{G}}}\,.$

Obviously, cohomological proper maps are closed under composition and a proper
map is cohomologically proper.

#### A.2.2

By definition, an action of an algebraic group $\mathsf{G}$ on a scheme
$\mathsf{X}$ is cohomologically proper if the canonical map

$(\mathsf{G},\mathsf{X})\to(\\{1\\},\operatorname{pt})$

is cohomologically proper. In this case, we also say that the quotient stack
$\mathsf{X}/\mathsf{G}$ is cohomologically proper.

#### A.2.3

A nontrivial example of a cohomologically proper map is given by the following

######  Proposition A.1.

If the map $f_{\mathsf{X}}$ in (190) is affine, $\ker f_{\mathsf{G}}$ is
reductive, and

$(f_{\mathsf{X},*}\mathscr{O}_{\mathsf{X}_{1}})^{\ker
f_{\mathsf{G}}}=\mathscr{O}_{\mathsf{X}_{2}}$

then (190) is cohomologically proper

###### Proof.

It suffices to prove that for a finitely generated algebra $\mathscr{A}$ with
an action of a reductive group $\mathsf{G}$

$\dim\mathscr{A}^{\mathsf{G}}<\infty\Leftrightarrow\forall
V\,,\dim(\mathscr{A}\otimes V)^{\mathsf{G}}<\infty$

where $V$ ranges over finite-dimensional $\mathsf{G}$-modules.

Highest weights vectors of irreducible $\mathsf{G}$-submodules in
$\mathscr{A}$ span the graded algebra

$\mathscr{A}^{\mathsf{U}}=\bigoplus_{\lambda\in\,\textup{dominant
weights}}\mathscr{A}^{\mathsf{U}}_{\lambda}\,,$ (191)

where $\mathsf{U}\subset\mathsf{G}$ is a maximal unipotent subgroup. This
algebra is finitely generated by Theorem 3.13 in [VinPop]. Therefore

$\dim\mathscr{A}^{\mathsf{U}}_{0}<\infty\Leftrightarrow\forall\lambda\,,\dim\mathscr{A}_{\lambda}^{\mathsf{U}}<\infty\,,$

as was to be shown. ∎

#### A.2.4

######  Proposition A.2.

The conclusion of Proposition A.1 also holds for $\ker
f_{\mathsf{G}}=\mathbb{G}_{\mathbf{a}}$.

###### Proof.

Suffices to prove that

$\dim H^{1}(\mathbb{G}_{\mathbf{a}},\mathscr{A})\leq\dim
H^{0}(\mathbb{G}_{\mathbf{a}},\mathscr{A})\,,$

for any algebraic module $\mathscr{A}$, that is, any module of the form

$\mathscr{A}=\bigcup\mathscr{A}_{n}\,,\quad\dim\mathscr{A}_{n}<\infty\,.$

Here $\mathscr{A}_{n}$ is a finite-dimensional vector space with a nilpotent
operator, for which $H^{0}$ and $H^{1}$ are the kernel and the cokernel,
respectively. Therefore,

$\dim H^{1}(\mathbb{G}_{\mathbf{a}},\mathscr{A})\leq\lim\dim
H^{1}(\mathbb{G}_{\mathbf{a}},\mathscr{A}_{n})=\lim\dim
H^{0}(\mathbb{G}_{\mathbf{a}},\mathscr{A}_{n})=\dim
H^{0}(\mathbb{G}_{\mathbf{a}},\mathscr{A})\,.$

∎

######  Remark A.1.

For a general unipotent group, the conclusion of Proposition A.1 fails, which
has to do with the fact that a quotient of an affine variety by a unipotent
group is, in general, only quasiaffine. As a simplest example, one can take
the maps

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
38.50824pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{U\backslash
SL(2,\mathbb{C})/U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
99.73047pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
62.50824pt\raise-35.3159pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
66.11935pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 99.73047pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{U\backslash(\mathbb{C}^{2}\setminus\\{0\\})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
75.73047pt\raise-34.91562pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
62.50824pt\raise-39.0411pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{C}}$}}}}}}}\ignorespaces}}}}\ignorespaces\,,$
(192)

where $U$ is a maximal unipotent subgroup.

###  A.3 Flag varieties and Springer fibers

#### A.3.1

Let $\mathsf{G}$ be a complex semisimple Lie group with
$\operatorname{Lie}(\mathsf{G})=\mathfrak{g}$. Choose a Borel subgroup
$\mathsf{B}\subset\mathsf{G}$. We have

$\mathsf{B}=\mathsf{A}\ltimes\mathsf{U}$

where $\mathsf{U}$ is the unipotent radical and $\mathsf{A}$ a maximal torus.
We denote

$\mathfrak{b}=\operatorname{Lie}(\mathsf{B})\,,\quad\mathfrak{n}=\operatorname{Lie}\mathsf{U}\,.$

We denote by $r$ the rank of $\mathfrak{g}$ .

#### A.3.2

The flag variety

$\mathscr{B}\overset{\textup{\tiny
def}}{=}\mathsf{G}/\mathsf{B}=\\{\mathfrak{b}^{\prime}\subset\mathfrak{g}\\}=\\{\mathfrak{n}^{\prime}\subset\mathfrak{g}\\}$

may be seen as parametrizing Borel subalgebras
$\mathfrak{b}^{\prime}\subset\mathfrak{g}$, or the corresponding nilpotent
subalgebras
$\mathfrak{n}^{\prime}=[\mathfrak{b}^{\prime},\mathfrak{b}^{\prime}]$, with
the adjoint action of $\mathsf{G}$. Further,

$T^{*}\mathscr{B}=\\{(e,\mathfrak{n}^{\prime}),e\in\mathfrak{n}^{\prime}\\}\,.$
(193)

Here $e\in\mathfrak{g}$ is a nilpotent element. The map

$\mu:T^{*}\mathscr{B}\owns(e,\mathfrak{n}^{\prime})\mapsto e\in\mathfrak{g}$
(194)

becomes the moment map upon the identification
$\mathfrak{g}\cong\mathfrak{g}^{*}$ given by an invariant bilinear form. We
will always treat this map $\mathbb{C}^{\times}_{q}$-equivariantly, where
$\mathbb{C}^{\times}_{q}$ acts by $e\mapsto q^{-1}e$.

#### A.3.3

For any semisimple $h\in\mathfrak{g}$ we have

$\mathscr{B}^{h}=\bigsqcup_{w\in W^{h}\backslash
W}\mathsf{G}^{h}\,w\,\mathfrak{n}\,,$ (195)

where $w$ ranges over coset representatives of minimal length, that is, such
that

$w^{-1}\mathsf{B}^{h}w\subset\mathsf{B}\,.$

Each component in (195) parametrizes $h$-stable maximal nilpotent subalgebras
$\mathfrak{n}^{\prime}$ with the same dimensions of the graded components as
$w\cdot\mathfrak{n}$.

#### A.3.4

If $e\in\mathfrak{g}$ is a nilpotent element then

$\mathscr{B}^{e}=\mu^{-1}(e)$ (196)

is the Springer fiber of $e$. Being the zero locus of a vector field, it has a
natural derived scheme structure with the virtual tangent bundle
$(1-q^{-1})T\mathscr{B}$.

###  A.4 Normalization of measures

#### A.4.1

While the spectral decomposition is obviously insensitive to the precise
normalization of the measures, it is worth mentioning briefly the different
choices that contribute to the normalization in Langlands’ formula (5).

#### A.4.2

The Haar measure on $\mathsf{G}(\mathbb{A})$ is the pushforward of the measure

$d_{\textup{Haar}}(k)\times|a|^{-2\rho}d_{\textup{Haar}}(a)\times
d_{\textup{Haar}}(u)$ (197)

under the Iwasawa factorization map

$\mathsf{K}\times\mathsf{A}\times\mathsf{U}\to\mathsf{G}\,.$

There is a canonical way to normalize the first and the last factor in (197)
by

$\operatorname{vol}(\mathsf{K})=\operatorname{vol}(\mathsf{U}(\mathbb{A})/\mathsf{U}(\mathbb{F}))=1\,.$
(198)

Instead of the Iwasawa factorization, one can fix a measure on the Borel
subgroup $\mathsf{B}=\mathsf{A}\mathsf{U}$ and normalize the volume of the
flag manifold
$\operatorname{vol}(\mathsf{G}(\mathbb{A})/\mathsf{B}(\mathbb{A}))=1$. This is
a more natural and a more uniform normalization when working with Eisenstein
series and constant terms.

#### A.4.3

Because the group $\mathsf{U}$ is unipotent, the measure
$d_{\textup{Haar}}(u)$ normalized as in (198) equals the the Tamagawa measure

$d_{\textup{Haar}}(u)=d_{\textup{\raisebox{-2.84526pt}{\includegraphics[height=10.0pt]{Tama.png}}}}(u)\,.$

Recall that e.g. for the additive group of a number field we have

$\mathbb{A}/\mathbb{F}=\prod_{\textup{finite
$p$}}\mathscr{O}_{p}\times\left(\left.\prod_{p|\infty}F_{p}\right/\mathscr{O}_{\mathbb{F}}\right)$

and the Tamagawa measure of $\mathbb{A}/\mathbb{F}$ equals the square root of
the ratio of the absolute value of discriminant (from the infinite places) and
norms of the differents (from the finite places), hence $=1$. It is through
the relation

$\prod_{\textup{finite
$p$}}\operatorname{vol}(\mathscr{O}_{p})=\left|\textup{discr}\mathscr{O}_{F}\right|^{-1/2}\,,$

that the discriminant factor in completed zeta functions appears as product of
local contribution in (5).

#### A.4.4

For $\mathsf{A}$ factor in (197), we take the Tamagawa measure. Since
$\mathsf{A}\cong GL(1)^{r}$, it suffices to discuss the multiplicative group
$GL(1)$. Since

$H^{0}(GL(1,\overline{F}),\mathscr{O}^{\times})/\overline{F}^{\times}=\mathbb{Z}\,,\quad
H^{1}(GL(1,\overline{F}),\mathscr{O}^{\times})=0\,,$

where $\mathbb{Z}$ has the trivial Galois group action, we have [Ono]

$d_{GL(1),\textup{\raisebox{-2.84526pt}{\includegraphics[height=10.0pt]{Tama.png}}}}(x)=\prod_{\textup{finite
$p$}}\frac{1}{1-p^{-1}}\frac{dx}{\,\,\,|x|_{p}}\times\prod_{p|\infty}\frac{dx}{\,\,\,|x|_{p}}\,.$

#### A.4.5

Since we consider only unramified characters of $\mathsf{A}$, we effectively
work with the pushforward under the norm map

$\displaystyle
d_{\mathscr{Q},\textup{\raisebox{-2.84526pt}{\includegraphics[height=10.0pt]{Tama.png}}}}$
$\displaystyle=\left(GL(1,\mathbb{A})\xrightarrow{\quad\|\,\cdot\,\|\quad}\mathscr{Q}\right)_{*}d_{GL(1),\textup{\raisebox{-2.84526pt}{\includegraphics[height=10.0pt]{Tama.png}}}}$
$\displaystyle=\big{(}\operatorname{Res}_{s=1}\xi(s)ds\big{)}\,\frac{dq}{q}\,.$
(199)

Here the second line follows the presentation of $\xi(s)$ as an integral with
respect to $GL(1,\mathbb{A})$ as in Tate’s thesis [Tate]. In the case when
$\mathscr{Q}$ is discrete, the measure $\frac{dq}{q}$ is the counting measure.

#### A.4.6

By the formula (23) we have

$\operatorname{Res}_{s=1}\xi(s)ds=\begin{cases}\frac{1}{\ln
q}\,\textup{\small\cursive z}\,,&\operatorname{char}\mathbb{F}>0\\\
\textup{\small\cursive z}\,,&\operatorname{char}\mathbb{F}=0\,.\end{cases}$
(200)

Thus the natural measure on the dual group

$\mathscr{Q}^{\vee}=\mathbb{C}\left/\frac{2\pi i}{\ln
q}\,\mathbb{Z}\right.\,,$

where $q=1$ in the number field case in (1), is

$d_{\mathscr{Q}^{\vee},\textup{\raisebox{-2.84526pt}{\includegraphics[height=10.0pt]{Tama.png}}}}(s)=\frac{1}{2\pi
i\textup{\small\cursive z}}ds\,,$ (201)

in all cases. Note, in particular, this is correctly normalized in the $q\to
1$ limit.

This is the how the Haar measure (121) should be normalized to reproduce
actual $L^{2}$-norms in (5).

#### A.4.7

In [Lvol], Langlands compares the above normalization of the Haar measure on
$\mathsf{G}(\mathbb{A})$ with the Tamagawa measure for
$\mathsf{G}(\mathbb{A})$. This amount to computing the Tamagawa volumes of the
flag manifolds.

In the number field case, for the nonarchimedian places, point counts in flag
manifolds give

$\prod_{p\,\nmid\,\infty}\operatorname{vol}_{\textup{\raisebox{-2.84526pt}{\includegraphics[height=10.0pt]{Tama.png}}}}(\mathsf{G}(\mathbb{F}_{p})/\mathsf{B}(\mathbb{F}_{p}))=\left|\textup{discr}\mathscr{O}_{F}\right|^{-\frac{1}{2}\dim\mathsf{G}/\mathsf{B}}\prod_{i=1}^{r}\zeta_{\mathbb{F}}(m_{i}+1)^{-1}\,,$

where $m_{i}$ are the exponents of $\mathsf{G}$, with a similar answer in the
function field case. One should bear in mind that
$H^{1}(\mathsf{G}/\mathsf{B},\mathscr{O}^{\times})=\mathbb{Z}^{r}$ with a
trivial Galois action, which multiplies the point counts by $(1-p^{-1})^{r}$.

The archimedian contribution, concretely the volume of the flag manifold
computed on the big Bruhat cell, Langlands interprets as a particular value of
the Gindikin-Karpelevich formula. Altogether, the proportionality constant is
the residue at the regular nilpotent discussed in Section 3.6. Since the
residue of the Eisenstein series at that point is a constant function on
$\mathsf{G}$, Langlands computes the Tamagawa volume of
$\mathsf{G}(\mathbb{F})\backslash\mathsf{G}(\mathbb{A})$.

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