##  1 Experiments

Since the goal in UDA involves adapting a source annotated domain to an
unlabeled target domain, we use the challenging synthetic-to-real UDA
benchmarks for semantic segmentation to prove our main hypotheses as well as
compare our approach with state-of-the-art methods. In this set-up, models are
trained jointly with fully-annotated synthetic data as well as unlabeled real-
world data, both considered source and target domain respectively. The models
are then validated on an unseen portion of the target domain, measuring their
mean intersection-over-union (mIoU) (Everingham2014miou).

Considering that our approach aims to improve on the discrepancy loss proposed
in (Zhang2018ICASSP) by training each individual classifier with a different
set of features and using a meta-learning layer is to ensemble the
classifiers’ outputs, the first experiment focuses on analyzing the difference
between Multi-task tri-training (MTri) (Zhang2018ICASSP) and our method (see
Fig. LABEL:fig:eyecatch) during the first training stage. This comparison is
complemented with a (vanilla) single encoder-decoder (SED) network
architecture, often used in the literature for UDA (Li2019BDLCVPR;
yang2020fda; yang2020labeldriven; Zou2018eccv). The goal of this experiment is
to study how the meta-learner can balance the outputs of the classifiers with
respect to the classes, and how this compares to the aforementioned existing
UDA architectures. Additionally, we also show the distribution of the weights
for the meta-learner over all the classes and analyze the influence of entropy
minimization.

The second experiment compares our proposed UDA approach with current state-
of-the-art methods, putting our model into context with approaches that rely
on a combination of image translation, feature matching, entropy minimization
and SSL.

Finally, although most UDA methods focus only on analyzing their approach on
the standard benchmarks of adapting synthetic-to-real domains, they tend to
neglect the generalization capacity of the resulting model. For this reason,
we designed a third experiment to study how well our model can generalize to a
completely unseen dataset that was not used during the UDA training process.
The protocol for this experiment consists of first adapting GTA V
(Richter2016ECCV) to Cityscapes (Cordts2016Cityscapes), and then evaluating
the resulting model on WildDash (Zendel_2018_ECCV). To compare with other
state-of-the-art methods, we select those whose code is publicly available and
provide an evaluation script, and proceed to 1) reproduce their result on the
proposed synthetic-to-real benchmark and 2) evaluate the model on WildDash.

###  1.1 Datasets

Target dataset. Cityscapes (Cordts2016Cityscapes) is a large-scale and real
urban scene semantic segmentation dataset that provides $5000$ finely
annotated images split into three sets: train ($2975$), validation ($500$) and
test ($1525$). These sets are pixel-wise labeled, with a resolution of
$1024\times 2048$ pixels. The number of classes is $34$ but only $19$ are
officially considered in the evaluation protocol.

Source datasets. GTA V (Richter2016ECCV) is a synthetic dataset that contains
$24966$ labeled frames taken from a realistic open-world computer game called
Grand Theft Auto V (GTA V). The resolution of the images is $1052\times 1914$
pixels and most of the frames are vehicle-egocentric. All the classes are
compatible with the $19$ official classes of Cityscapes. SYNTHIA (synthia) is
a synthetic dataset consisting of driving scenes rendered from a virtual city.
We use the SYNTHIA-RAND-CITYSCAPES subset as source set, which contains $9400$
$1280\times 760$ images for training and $16$ common classes with Cityscapes,
and we evaluate the resulting model on these 16 classes.

Unseen dataset. WildDash (Zendel_2018_ECCV) is a real-world dataset containing
$4256$ finely annotated images in a pixel-wise manner, created with the
purpose of testing the robustness of models under different driving scenarios
(e.g. rain, road coverage, darkness, overexposure). These images have a
resolution of $1920\times 1080$ pixels and the labels are fully compatible
with Cityscapes.

Figure 1: weights after optimizing Eq. (LABEL:eq:arbiter) for the adaption GTA
V $\xrightarrow{}$ Cityscapes. Before starting SSL, $C_{1}$ and $C_{3}$ are
the most dominant predictors on the output space of $C_{m}$.

###  1.2 Implementation details

Table 1: Adapting from GTA V to Cityscapes. S1 and S2 indicate the training
stage, while R1 and R2 denote the first and second round of SSL, respectively.

width=0.95  GTA V $\xrightarrow{}\textbf{Cityscapes}$ Method road side. buil.
wall fence pole light sign veget. terr. sky person rider car truck bus train
motor bike mIoU DCAN (Wu2018acm) 85.0 30.8 81.3 25.8 21.2 22.2 25.4 26.6 83.4
36.7 76.2 58.9 24.9 80.7 29.5 42.9 2.5 26.9 11.6 41.7 DLOW (Gong_2019) 87.1
33.5 80.5 24.5 13.2 29.8 29.5 26.6 82.6 26.7 81.8 55.9 25.3 78.0 33.5 38.7 0.0
22.9 34.5 42.3 CLAN (Luo2018cvpr) 87.0 27.1 79.6 27.3 23.3 28.3 35.5 24.2 83.6
27.4 74.2 58.6 28.0 76.2 33.1 36.7 6.7 31.9 31.4 43.2 ABStruct (Chang_2019)
91.5 47.5 82.5 31.3 25.6 33.0 33.7 25.8 82.7 28.8 82.7 62.4 30.8 85.2 27.7
34.5 6.4 25.2 24.4 45.4 ADVENT (TH2018ADVENTCVPR) 89.4 33.1 81.0 26.6 26.8
27.2 33.5 24.7 83.9 36.7 78.8 58.7 30.5 84.8 38.5 44.5 1.7 31.6 32.4 45.5 BDL
(Li2019BDLCVPR) 91.0 44.7 84.2 34.6 27.6 30.2 36.0 36.0 85.0 43.6 83.0 58.6
31.6 83.3 35.3 49.7 3.3 28.8 35.6 48.5 FDA-MBT (yang2020fda) 92.5 53.3 82.4
26.5 27.6 36.4 40.6 38.9 82.3 39.8 78.0 62.6 34.4 84.9 34.1 53.1 16.9 27.7
46.4 50.45 PCEDA (Yang_2020) 91.0 49.2 85.6 37.2 29.7 33.7 38.1 39.2 85.4 35.4
85.1 61.1 32.8 84.1 45.6 46.9 0.0 34.2 44.5 50.5 Ours S1 ($C_{3}$) 87.22 36.57
81.26 28.65 17.82 35.55 32.58 29.11 83.46 30.39 77.06 62.48 28.78 81.49 22.75
22.85 0.06 29.85 35.14 43.32 Ours S1 ($C_{m}$) 85.29 35.57 81.69 29.93 20.24
35.53 36.63 35.94 83.24 28.1 81.75 63.75 29.18 81.8 23.44 24.58 4.67 31.0
47.26 45.24 Ours S2-R1 ($C_{3}$) 90.6 46.94 84.06 31.9 23.88 37.53 34.81 34.37
85.69 36.02 84.32 66.53 29.41 85.46 27.77 32.48 7.15 36.05 54.96 48.94 Ours
S2-R1 ($C_{m}$) 90.81 47.85 85.01 32.08 24.55 37.73 38.15 42.13 85.37 34.32
84.97 66.51 28.0 84.51 27.0 25.14 15.23 35.03 56.02 49.50 Ours S2-R2 ($C_{2}$)
92.27 51.59 86.19 35.28 26.84 36.73 35.68 42.4 86.84 37.3 85.49 66.9 27.6
85.75 32.26 32.85 20.59 33.89 58.1 51.29 Ours S2-R2 ($C_{m}$) 92.59 53.05
86.31 34.2 27.17 39.13 41.0 44.8 86.1 34.32 84.69 67.23 29.77 85.78 32.73 29.9
20.12 35.55 57.05 51.66 width=0.95  SYNTHIA
$\xrightarrow{}\textbf{Cityscapes}$ Method road side. buil. wall* fence* pole*
light sign veget. sky person rider car bus motor bike mIoU AdaptPatch
(Tsai2019ICCV) 82.4 38.0 78.6 8.7 0.6 26.0 3.9 11.1 75.5 84.6 53.5 21.6 71.4
32.6 19.3 31.7 40.0 AdaptSegNet (Tsai2018CVPR) 79.2 37.2 78.8 - - - 9.9 10.5
78.2 80.5 53.5 19.6 67.0 29.5 21.6 31.3 45.9* BDL (Li2019BDLCVPR) 86.0 46.7
80.3 - - - 14.1 11.6 79.2 81.3 54.1 27.9 73.7 42.2 25.7 45.3 51.4* CLAN
(Luo2018cvpr) 81.3 37.0 80.1 - - - 16.1 13.7 78.2 81.5 53.4 21.2 73.0 32.9
22.6 30.7 47.8* FDA-MBT (yang2020fda) 79.3 35.0 73.2 - - - 19.9 24.0 61.7 82.6
61.4 31.1 83.9 40.8 38.4 51.1 52.5* PCEDA (Yang_2020) 85.9 44.6 80.8 9.0 0.8
32.1 24.8 23.1 79.5 83.1 57.2 29.3 73.5 34.8 32.4 48.2 46.2 Ours ($C_{m}$)
81.90 41.88 78.21 3.38 0.02 44.76 24.82 27.17 86.59 85.18 68.74 30.55 84.65
24.42 20.12 40.77 46.45

Table 2: Adapting from SYNTHIA to Cityscapes. Total mIoU values with * are
reported only on 13 subclasses (excluding wall, fence and pole). Our method
achieves the best performance over all the 16 classes.

Table 3: Number of classes where each classifier outperforms the others on GTA
V $\xrightarrow{}$ Cityscapes. Although there is a clear dominance of $C_{3}$
before starting SSL, this trend tends to wear off as the self supervision
process unfolds.

width=0.75  classifier stage 1 SSL: round 1 SSL: round 2 $C_{1}$ 5 6 4 $C_{2}$
4 7 7 $C_{3}$ 10 6 8

Table 4: Influence of $T$ on GTA V $\xrightarrow{}$ Cityscapes during first
round of SSL. Using $T$ during SSL by feeding a transformation to its
corresponding classifier provokes a slight performance drop in the mIoU.

width=0.75  classifier stage 1 SSL without $T$ SSL with $T$ $C_{1}$ 41.99
48.87 47.9 $C_{2}$ 43.10 48.91 47.8 $C_{3}$ 43.32 48.94 48.0

Image translation module and discriminator. We have chosen CycleGAN
(Zhu2017Unpaired) as our image translation model $T$, as it is a commonly used
approach that can be trained efficiently. Its architecture provides two image
translations: the translation from the source to the target domain
$X^{t}_{\mathcal{S}}$ and the reconstruction back to the source set
$X^{r}_{\mathcal{S}}$. These two translations are combined with the original
source images to obtain three different representation
$X^{T_{1}}_{\mathcal{S}}=X_{\mathcal{S}}$,
$X^{T_{2}}_{\mathcal{S}}=X^{r}_{\mathcal{S}}$ and
$X^{T_{3}}_{\mathcal{S}}=X^{t}_{\mathcal{S}}$. Regarding the discriminator
$D$, feature alignment between target-like and target images is frequently
done when using CycleGAN’s as image translation module (Li2019BDLCVPR;
Hoffman18ICML), and therefore $X^{T_{c}}_{\mathcal{S}}$ turns into
$X^{T_{3}}_{\mathcal{S}}$ in Eq. (LABEL:eq:advtraining). We note that
potentially better performing image translations approaches exist but in our
work we opt for the commonly used approach CycleGAN.

Training protocols for MTri and SED. We have respected the protocol for MTri
as reported in (Zhang2018ICASSP), i.e. minimizing a cross-entropy loss for
semantic segmentation combined with a cosine distance loss for the discrepancy
between $C_{1}$ and $C_{2}$. As for the single encoder-decoder (SED) approach,
all our losses were implemented using one encoder and one classifier, while
using all three available transformations. In essence, the SED approach is
similar to our approach but does not use the ensemble approach with the three
classifiers.

Hardware and network architecture. In our experiments, we have implemented our
method using Tensorflow (tensorflow) and trained our model using a single
NVIDIA TITAN RTX with 24 GB memory. Regarding the segmentation network, we
have chosen ResNet101 (Kaiming2015CVPR) pretrained on ImageNet
(imagenet2009cvpr) as feature extractor for $E$. When it comes to the decoders
$C_{k},k\in\\{1,2,3\\}$, DeepLab-v2 (Deeplabv2) framework was used. For the
network $D$ we adopt a similar structure than (Radford2015ungan), which
consists of $5$ convolution layers with kernel size of $4\times 4$, stride $2$
and channel numbers {4, 8, 16, 32, 1}. Each layer uses Leaky-ReLU
(Maas2013RectifierNI) as activation function with a slope of $0.2$, except the
last layer that has no activation. Throughout the training process, we use SGD
(SGD) as optimizer with momentum of $0.9$, encoder and decoders follow a poly
learning rate policy, where the initial learning rate is set to $2.5e^{-4}$.
The discriminator is also optimized with SGD but with a fixed learning rate of
$1e^{-5}$. As for the entropy loss, we chose $\lambda_{ent}=0.005$ and
$\eta=2.0$ for all experiments. During the first stage the network is trained
for $150k$ iterations. Then we perform SSL until convergence is reached on
each round. We use a crop size of $512\times 1024$ during training, and we
evaluate on full resolution $1024\times 2048$ images from Cityscapes
validation split.

###  1.3 Our approach vs MTri vs single encoder-decoder (SED)

We can infer from Tab. LABEL:tab:abstudy1 that our method outperforms both SED
and MTri, and the difference is even more noticeable when comparing against
the meta-learner. $C_{m}$ performs better than the individual classifiers over
$14$ classes if we make the comparison with entropy minimization and over $11$
classes without entropy minimization. It is also important to mention that
there is a considerable correlation between the weights of the meta-learner
that are depicted in Fig. 1 and the performance of the meta-learner. If we
analyze the classes in Tab. LABEL:tab:abstudy1 where $C_{m}$ outperforms the
three classifiers (for instance: fence, traffic light, rider, bus, bike), we
can see also a pattern in Fig. 1 where the meta-learner tends to amplify the
contribution of the two best performing classifiers whilst penalizing the one
with the lowest mIoU. For some other classes such as traffic sign and truck,
it prefers to perform a mixture of weak classifiers, resulting in a final
prediction that outperforms the strongest member of the ensemble.

Figure 2: Effect of SSL for $C_{m}$. Retraining the meta-learner after each
round of SSL makes sure that it keeps outperforming the other classifiers with
a decreasing margin.

width=0.75  Method # encoders # classifiers mIoU ADVENT (TH2018ADVENTCVPR) 1 1
25.9 BDL (Li2019BDLCVPR) 1 1 26.57 FDA-MBT (yang2020fda) 3 3 31.07 Ours 1 3
31.2

Table 5: Generalization test on WildDash after adapting GTA V $\xrightarrow{}$
Cityscapes.

###  1.4 Our approach vs state-of-the-art methods

l XXXXX Image & [0.5pt] GT  [0.5pt] $C_{m}$

Figure 3: Qualitative comparison from GTA V to Cityscapes. The meta-learner
rebalances the predictions from $C_{1}$, $C_{2}$ and $C_{3}$ to achieve a
smoother output over all the classes, where less predominant classes such as
t. sign and pole have more presence on the pixel space of $C_{m}$.

The quantitative results for the adaption GTA V $\xrightarrow{}$ Cityscapes
can be seen in Tab. 1.2. When comparing it with state-of-the-art methods, we
can see that our approach outperforms PCEDA (Yang_2020), FDA-MBT (yang2020fda)
and BDL (Li2019BDLCVPR), three recent methods that use a combination of
strategies. Qualitative results in Fig. 3 shows satisfactory results on the
output space, leading to consistently clean predictions.

As for SYNTHIA $\xrightarrow{}$ Cityscapes, the mIoU values are shown in Tab.
1.2. We achieved competitive results over all 16 classes with respect to
state-of-the-art methods such as PCEDA (Yang_2020) and AdaptPatch
(Tsai2019ICCV), dominating on some difficult classes such as pole, traffic
light and traffic sign.

If we analyze the improvements during SSL, we see that the meta-learner
consistently scores better than the individual three classifiers, although its
gain diminishes with each round of SSL (see Fig. 2). This can be attributed to
the fact that all three classifiers are being optimized with the same images,
and thus the model is losing the capability to keep diversity among the
predictors. The results from Tab. 1.2 and Tab. 3 also show that the dominance
of the members of the ensemble can alternate since $C_{3}$ is the best
predictor after the first round (R1) and $C_{2}$ takes over after the second
one (R2). This suggests that some predictors can learn more than others, even
if they share the same input images, showing that all of them are equally
important. This can be better appreciated in Tab. 3 where $C_{2}$ stands out
after R1 and remains close to $C_{3}$ after R2 when analyzing the mIoU per
class during SSL.

Using the image translation module $T$ during the second stage by transforming
the target set into the closest transformation possible to each classifier,
i.e., transforming the target images to the source domain for the first two
classifiers while keeping them unaltered for the third one, leads to slightly
worse performance (see Tab. 4). This can be attributed to the fact that, since
SSL aims to close the gap for the target distribution, it is needed to keep
the inputs as similar as possible to those that the algorithm would receive
during inference.

###  1.5 Generalization capacity of our approach

The results of the proposed generalization test in Tab. 5 shows different UDA
methods along with their arrangement for the semantic segmentation network and
the corresponding mIoU performance on WildDash, after adapting GTA V to
Cityscapes. ADVENT (TH2018ADVENTCVPR) is a UDA approach that does not leverage
any image translation strategy, while BDL (Li2019BDLCVPR) and FDA-MBT
(yang2020fda) make use of one and three image translations respectively. If we
consider the amount of encoders and classifiers, we can notice that using a
single encoder-decoder gives a quite limiting generalization performance,
although BDL outperforms ADVENT. This slightly better performance of BDL can
be attributed to the usage of one image translation (transforming the source
to the target with CycleGAN) to increase the robustness of the model.

FDA-MBT uses three image representations, mapping the source annotated images
to the target using three different parameters for the image translation
module, and performing UDA by training each encoder-decoder segmentation model
with a specific representation. The reported performance of $31.07$ is the
result of averaging the three trained models, and although it is close to
ours, our semantic segmentation network takes up significantly fewer
parameters to train (one encoder and three classifiers). This makes our
approach attractive as it is a good trade-off between its number of parameters
and performance. More importantly, this experiment shows the advantageous
effect of using multiple image translations, as done in FDA-MBT and our
approach, on the generalizability of the trained models.


# Uniform bounds on the Harish-Chandra characters

Anna Szumowicz  Caltech, The Division of Physics, Mathematics and Astronomy,
1200 E California Blvd, Pasadena CA 91125<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathbf{G}}$ be a connected reductive algebraic group over a $p$-adic
local field $F$. In this paper we study the asymptotic behaviour of the trace
characters $\theta_{\pi}$ evaluated at a regular element $\gamma$ of
${\mathbf{G}}(F)$ as $\pi$ varies among supercuspidal representations of
${\mathbf{G}}(F)$. Kim, Shin and Templier [KST2016] conjectured that
$\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}$ tends to $0$ when $\pi$ runs over
irreducible supercuspidal representations of ${\mathbf{G}}(F)$ with unitary
central character and the formal degree of $\pi$ tends to infinity. For
${\mathbf{G}}$ semisimple we prove that the trace character is uniformly
bounded on $\gamma$ under the assumption, which is expected to hold true for
every ${\mathbf{G}}(F)$, that all irreducible supercuspidal representations of
${\mathbf{G}}(F)$ are compactly induced from an open compact modulo center
subgroup. Moreover, we give an explicit upper bound in the case of $\gamma$
ellitpic.

††

##  1\. Introduction

Let ${\mathbf{G}}$ be a connected reductive algebraic group over a $p$-adic
local field $F$. Denote $G:={\mathbf{G}}(F)$. Let $\pi$ be an admissible
representation of $G$. Denote by $\mathcal{C}^{\infty}_{c}(G)$ the set of
smooth compactly supported functions on $G$. The trace character
$\theta_{\pi}$ is the conjugation invariant distribution on
$\mathcal{C}_{c}^{\infty}(G)$ defined as

$\langle\theta_{\pi},f\rangle:={\rm tr}\,\pi(f)$

for $f\in\mathcal{C}_{c}^{\infty}(G)$. Harish-Chandra showed that
$\theta_{\pi}$ is represented by a locally constant function on the set of
regular semisimple elements which is locally $L^{1}$ on $G$. We will still
denote this function by $\theta_{\pi}$.

If $\pi$ is a discrete series representation, for example is irreducible
unitary supercuspidal, one can define the formal degree of $\pi$ denoted by
$\deg(\pi)$. It plays a similar role as the dimension of representations of
compact groups.

Denote by ${\rm{Irr}^{sc}}({\mathbf{G}}(F))$ the set of equivalence classes of
irreducible supercuspidal representations of $G$. Kim, Shin and Templier
proposed the following conjecture.

######  Conjecture 1.1 ([KST2016, Conjecture 4.1.]).

Fix a regular semisimple element $\gamma\in G$. Then,

$\lim_{\begin{subarray}{c}\pi\in{\rm{Irr}^{sc}}(G)\\\
\deg(\pi)\to\infty\end{subarray}}\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}=0$

where $\pi\in{\rm{Irr}^{sc}}({\mathbf{G}}(F))$ varies over representations
whose central character is unitary.

Kim, Shin and Templier [KST2016] proved Conjecture 1.1 in case of $\gamma$
topologically unipotent, $\pi$ constructed by Yu [Yu2001] and under certain
assumptions on $G$. In [KST2] they proved Conjecture 1.1 for $\gamma$ elliptic
and under certain assumption on tori in ${\mathbf{G}}$ which is satisfied for
example when every $F$-torus in ${\mathbf{G}}$ is tame and the characteristic
of the residue field of $F$ does not divide the order of the Weyl group of
${\mathbf{G}}$. An analogue of this conjecture for compact semisimple Lie
groups can be easily verified using the Weyl character formula. See also
[Witte] for partial results in the case of the general linear group.

As described in [KST2], bounds on the trace characters are motivated by their
relation with limit multiplicity formulas, Sato-Tate equidistribution and
bounds towards Ramanujan.

###  1.1. Uniform bounds

In this paper we prove Conjecture 1.1 for arbitrary semisimple connected
$p$-adic groups for irreducible supercuspidal representations of the form
${\operatorname{c-Ind}}_{J}^{G}\Lambda$ where $J$ is an open compact modulo
center subgroup of $G$. The condition is expected to hold true for any
irreducible supercuspidal representation. It has been proven that every
irreducible supercuspidal representation is a compact induction from an open
compact modulo center subgroup in the following cases: tamely ramified
$p$-adic group ${\mathbf{G}}$ when $p$ does not divide the order of the Weyl
group of ${\mathbf{G}}$ (Yu’s construction [Yu2001] and the exhaustion results
by Kim [Kim2007] and Fintzen [Fintzen]), for all supercuspidal representations
of ${\rm GL}_{n}(F)$ (Bushnell-Kutzko construction [BK]) and for the classical
groups (Stevens [Stevens]).
Kim, Shin and Templier in their work [KST2016], [KST2] towards Conjecture 1.1
rely on Yu’s construction of supercuspidal representations. Our methods are
different and avoid Yu’s construction, instead using the Arthur’s formula and
bounds on irreducible characters of compact $p$-adic groups from [Fraczyk]. We
will say that a representation $\pi$ of ${\mathbf{G}}(F)$ is compactly induced
if it is of the form
$\pi\cong{\operatorname{c-Ind}}_{J}^{{\mathbf{G}}(F)}\Lambda$ for some compact
modulo $Z({\mathbf{G}}(F))$ subgroup $J$ and some irreducible representation
$\Lambda$ of $J$.

######  Theorem 1.2.

Let ${\mathbf{G}}$ be a connected semisimple group over a non-Archimedean
local field $F$ of characteristic zero. Then, for every regular semisimple
element $\gamma\in{\mathbf{G}}(F)$ there exists a constant $\kappa_{\gamma}>0$
depending only on $\gamma$ and $G$ such that for every irreducible compactly
induced supercuspidal representation $\pi$ we have

$|\theta_{\pi}(\gamma)|\leqslant\kappa_{\gamma}.$

In particular, it implies the conjecture.

######  Corollary 1.3.

Let ${\mathbf{G}}$ be a connected semisimple group over a local non-
Archimedean field $F$ of characteristic zero. Then for every regular
semisimple element $\gamma\in{\mathbf{G}}(F)$ we have

$\lim_{\begin{subarray}{c}\pi\in{\rm{Irr}^{sc}}({\mathbf{G}}(F))\\\
\deg(\pi)\to\infty\end{subarray}}\frac{\theta_{\pi}(\gamma)}{\deg(\pi)}=0$

where each $\pi$ in the limit is assumed to be compactly induced.

For $\gamma$ ellitpic we prove an explicit upper bound.

######  Theorem 1.4.

Let ${\mathbf{G}}$ be a connected semisimple group over a non-Archimedean
local field $F$ of characteristic zero. Let $W$ be the Weyl group of
${\mathbf{G}}$. Then, for every regular elliptic element
$\gamma\in\mathbf{G}(F)$and every irreducible compactly induced supercuspidal
representation $\pi$ we have

$|\theta_{\pi}(\gamma)|\leqslant|\Delta(\gamma)|_{F}^{-1}2^{{\rm
dim}\,{\mathbf{G}}-{\rm rk}\,\mathbf{G}}\,|W|.$

.

###  1.2. Notation

Let $F$ be a local non-Archimedean field. Denote by $\mathcal{O}_{F}$ its ring
of integers and by $k_{F}$ its residue field. Write $\mathfrak{p}_{F}$ for the
maximal ideal of ${\mathcal{O}_{F}}$. Denote by $q$ the cardinality of
$k_{F}$. Fix a uniformizer $\varpi_{F}$ in $F$. Denote by $v_{F}$ the
associated additive valuation normalized as $v_{F}(x)=n$ for
$x=u\varpi_{F}^{n}\in F$ with $u$ an inveritble element in
${\mathcal{O}_{F}}$. Denote by $|\cdot|_{F}$ the corresponding absolute value
$|x|_{F}=q^{-v_{F}(x)}$. For a connected reductive group ${\mathbf{G}}$ we
write ${\rm rk}\,{\mathbf{G}}$ for the absolute rank of ${\mathbf{G}}$ and
$\Delta(\gamma)$ denotes the Weyl discriminant of a semisimple element
$\gamma$.

Fix a connected semisimple group ${\mathbf{G}}$ over a non-Archimedean local
field $F$ of characteristic $0$. We denote $G:={\mathbf{G}}(F)$. Denote by $Z$
the center of $G$. We fix a Haar measure $\mu$ on $G$ such that
$\mu({\mathbf{G}}({\mathcal{O}_{F}}))=1$. Write $Z(H)$ for the center of a
group $H$. We denote by $|S|$ the cardinality of a set $S$. If $\gamma$ is a
regular semisimple element in ${\mathbf{G}}$, then we denote by
$A_{M}(\gamma)$ the maximal split component of center of the Levi subgroup $M$
of a rational parabolic subgroup in ${\mathbf{G}}$ containing $\gamma$ and
minimal with this property. Denote by $A_{{\mathbf{G}}}$ the maximal split
component of the center of ${\mathbf{G}}$.

###  1.3. Outline of the proof of Theorem 1.2

In section 2 we define

$\Omega(K):=\\{g\in A_{M}(\gamma)\setminus G|\ \ g^{-1}\gamma g\in K\\}.$

The goal of section 2 is to show that the set $\Omega(K)$ is compact
(Proposition 2.1). Since ${\mathbf{G}}$ is semisimple and $\pi$ is compactly
induced we can write

$\pi\cong{\operatorname{c-Ind}}_{J}^{G}\Lambda={\operatorname{c-Ind}}_{K}^{G}\rho.$

for some maximal compact open subgroup $K$ of $G$ and its irreducible
representation $\rho$. Write $\chi_{\rho}$ for the trace of $\rho$ extended to
${\mathbf{G}}$ by $0$ outside of $K$. Using Arthur’s formula [Arthur1987] we
show

$|\theta_{\pi}(\gamma)|=\left|\int_{\Omega(K)}\chi_{\rho}(g^{-1}\gamma
g)v_{M}(g)dg\right|.$ (1.1)

Once we know that $\Omega(K)$ is compact the existence of an upper bound is
deduced from (1.1), the fact that $v_{M}$ is continuous and from the work of
Frączyk [Fraczyk] on upper bounds of irreducible characters of compact
$p$-adic groups (Proposition 3.1).

###  1.4. Acknowledgement

The author is grateful to Anne-Marie Aubert, Mikołaj Frączyk, Sug Woo Shin and
Bertrand Rémy for helpful discussions.

##  2\. Compactness

For a maximal compact open subgroup $K$ of $G$ we define

$\displaystyle\Omega(K):=\\{g\in A_{M}(\gamma)\setminus G|\ \ g^{-1}\gamma
g\in\ K\\}.$

One of the main ingredients of the proof of Theorem 1.2 is the following.

######  Proposition 2.1.

For any maximal compact open subgroup $K$ of $G$ the set $\Omega(K)$ is
compact.

Before the proof we establish some auxiliary results. Recall that a regular
semisimple element $\gamma\in G$ is called elliptic if $\gamma$ does not
belong to any proper parabolic subgroup of $G$.

######  Lemma 2.2.

Let ${\mathbf{H}}$ be a reductive group over a non-Archimedean local field
$F$. Denote $H:={\mathbf{H}}(F)$. Let $\gamma$ be an elliptic element in $H$.
Let $K_{H}$ be a maximal compact subgroup of $H$. Then, the set

$\\{h\in Z(H)\setminus H|\ \ h^{-1}\gamma h\in K_{H}\\}$

is compact.

###### Proof.

For the sake of contradiction assume that there exists a sequence $h_{l}\in
H$, $l\in{\mathbb{N}}$ which escapes to infinity and such that
$h_{l}^{-1}\gamma h_{l}\in K_{H}$. Let $\mathcal{B}({\mathbf{H}},F)$ be the
Bruhat-Tits building of ${\mathbf{H}}$. Every maximal compact subgroup of $H$
stabilizes some vertex in $\mathcal{B}({\mathbf{H}},F)$. Let $x$ be a point in
$\mathcal{B}({\mathbf{H}},F)$ which is stabilized by $K_{H}$. Then,

$\gamma h_{l}x=h_{l}x.$

Since the action of ${\mathbf{H}}$ on its building is proper, the sequence
$(h_{l}x)_{l}$ escapes to infinity. Consider the Satake-Berkovich
compactification $\overline{\mathcal{B}({\mathbf{H}},F)}$ of
$\mathcal{B}({\mathbf{H}},F)$ (see [R1], [R2],[Berkovich]). Passing to a
subsequence we can assume that the sequence $(h_{l}x)_{l}$ tends to a point in
the boundary of $\overline{\mathcal{B}({\mathbf{H}},F)}$, say $y$. Since the
action of ${\mathbf{H}}$ on its building is continuous, the element $\gamma$
stabilizes the point $y$. The stabilizer of a point in the boundary of
$\overline{\mathcal{B}({\mathbf{H}},F)}$ is contained in a proper parabolic
subgroup of $H$ (see e.g. [R1, Theorem 14.11.]), so $\gamma$ is contained in a
proper parabolic subgroup of $H$. This leads to a contradiction with the
assumption that $\gamma$ is elliptic.

∎

Recall that for ${\mathbf{G}}$ connected and an $F$-torus
$T\subseteq{\mathbf{G}}$ we can define a root subgroup in the following way. A
subgroup $U\subseteq{\mathbf{G}}$ is called a root subgroup (with respect to
$T$) if there exists an isomorphism $u\colon{\mathbf{G}}_{a}\to U$ and a root
$\alpha$ of ${\mathbf{G}}$ with respect to $T$ such that

$tu(x)t^{-1}=u(\alpha(t)x),$

for every $x\in{\mathbf{G}}_{a}$, $t\in T$. If $T$ is split then the root
$\alpha$ determines the root subgroup uniquely. We denote by $U_{\alpha}$ the
root subgroup associated to $\alpha$.

######  Lemma 2.3.

Let ${\mathbf{G}}$ be a split connected reductive group over a non-Archimedean
local field $F$. Assume that the maximal torus $T$ containing $\gamma$ is
split over $F$. Let $P$ be a minimal parabolic subgroup of ${\mathbf{G}}(F)$
containing $\gamma$. Denote by $N$ its unipotent radical. Let $\Delta^{+}$ be
the set of positive roots of $T$ associated to $P$ with order $\preceq$. Write
$\Delta^{+}=\\{\alpha_{1},\ldots,\alpha_{r}\\}$ with
$\alpha_{i}\preceq\alpha_{j}$ for every $i<j$. Denote by $U_{\alpha}$ the root
subgroup associated to a positive root $\alpha$. Then,

  1. (1)

$u_{\alpha}(x)^{-1}\gamma
u_{\alpha}(x)=u_{\alpha}\left((\alpha(\gamma)-1)x\right)\gamma$ for any
$u_{\alpha}(x)\in U_{\alpha}$;

  2. (2)

Let $m=\dim N$ and $n=u_{\alpha_{1}}(x_{1})\ldots u_{\alpha_{m}}(x_{m})$.
Then,

$n^{-1}\gamma
n=\prod_{i=0}^{m-1}u_{\alpha_{m-i}}\left((\alpha_{m-i}(\gamma)-1)x_{m-i}+W^{m}_{m-i}(\alpha_{1}(\gamma),\ldots,\alpha_{m}(\gamma),x_{1},\ldots,x_{m})\right)\gamma$

where $W^{m}_{m-i}$ is a polynomial in $2m$ variables with coefficients in
$F$, depending only on $i$, $m-i$ and $\alpha_{j}(\gamma)$, $x_{j}$ such that
$\alpha_{j}\prec\alpha_{i}$.

###### Proof.

$\mathbf{(1)}$ Follows from the formula

$\gamma u_{\alpha}(x)\gamma^{-1}=u_{\alpha}(\alpha(\gamma)x).$ (2.1)

$\mathbf{(2)}$ We prove the following statement.
Let $1\leqslant r\leqslant m$ and $n_{1}=u_{\alpha_{1}}(x_{1})\ldots
u_{\alpha_{r}}(x_{r})$. Then,

$n_{1}^{-1}\gamma
n_{1}=\prod_{i=0}^{m-1}u_{\alpha_{m-i}}((\alpha_{m-i}(\gamma)-1)x_{m-i}+W_{m-i}^{r}(\alpha_{1}(\gamma),\ldots,\alpha_{m}(\gamma),x_{1},\ldots,x_{m}))\gamma$

where $W_{m-i}^{r}$ is a polynomial in $2m$ variables with coefficients in
$F$, depending only on $r,i,m$ and $\alpha_{j}(\gamma),x_{j}$ such that
$\alpha_{j}\prec\alpha_{i}$ and we put $x_{s}=0$ for $r+1\leqslant s\leqslant
m$.

Proof is by induction on $r$. For $r=1$ it follows from $(1)$. Assume now that
the formula is true for $r=l$ and we want to prove it for $r=l+1$. By [Adler,
Proposition 1.2.3.], for any $\alpha,\beta\in\Delta^{+}$ we have

$[u_{\alpha}(x),u_{\beta}(y)]=\prod_{i,j>0}u_{i\alpha+j\beta}(c_{i,j;\alpha,\beta}xy)$
(2.2)

where $[g,h]$ denotes the commutator of elements $g,h$ and
$c_{i,j;\alpha,\beta}$ are integers dependent only on $\alpha$, $\beta$ and
$i,j$ and $u_{i\alpha i+j\beta}$ denotes the trivial map if $i\alpha+j\beta$
is not a root. Denote $n_{1}:=u_{\alpha_{1}}(x_{1})\ldots
u_{\alpha_{l}}(x_{l})$ and $n:=n_{1}u_{\alpha_{l+1}}(x_{l+1})$. Write
$\alpha(\gamma):=(\alpha_{1}(\gamma),\ldots,\alpha_{m}(\gamma))$,
$y:=(x_{1},\ldots,x_{l},0,\ldots,0)=:(y_{1},\ldots,y_{m})$ and
$z=(x_{1},\ldots,x_{l+1},0\ldots,0)=:(z_{1},\ldots,z_{m})$. By (2.1), repeated
use of (2.2) and the inductive step,

$\displaystyle n^{-1}\gamma n=u_{\alpha_{l+1}}(z_{l+1})^{-1}n_{1}^{-1}\gamma
n_{1}u_{\alpha_{l+1}}(z_{l+1})=$ $\displaystyle
u_{\alpha_{l+1}}(-z_{l+1})\prod_{i=0}^{m-1}u_{\alpha_{m-i}}\left((\alpha_{m-i}(\gamma)-1)y_{m-i}+W_{m-i}^{l}(\alpha(\gamma),y)\right)\gamma
u_{\alpha_{l+1}}(z_{l+1})=$ $\displaystyle
u_{\alpha_{l+1}}(-z_{l+1})\prod_{i=0}^{m-1}u_{\alpha_{m-i}}\left((\alpha_{m-i}(\gamma)-1)y_{m-i}+W_{m-i}^{l}(\alpha(\gamma),y)\right)u_{\alpha_{l+1}}(\alpha_{l+1}(\gamma)z_{l+1})\gamma=$
$\displaystyle=\prod_{i=0}^{m-1}u_{\alpha_{m-i}}\left((\alpha_{m-i}(\gamma)-1)z_{m-i}+W_{m-i}^{l+1}(\alpha(\gamma),z)\right)\gamma.$

∎

######  Lemma 2.4.

Let $\gamma$ be a non-elliptic element. Let $Q$ be a minimal proper parabolic
containing $\gamma$. Denote by $N_{Q}$ the unipotent radical of $Q$ and by $L$
the Levi subgroup of $Q$ containing $\gamma$. Then, for any compact subset
$\Omega$ of $L$ there exists a compact subset $E_{\Omega}\subseteq N_{Q}$ such
that $x^{-1}l^{-1}\gamma lx\not\in K$ for every $x\in N_{Q}\setminus
E_{\Omega}$ and every $l\in\Omega$.

###### Proof.

First suppose that the maximal split torus $T$ containing $\gamma$ is split
over $F$. For the sake of contradiction assume that there exist a compact
subset $\Omega$ of $L$ and a sequence $(n_{i})_{i}\subseteq N_{Q}$ escaping to
infinity such that $n_{i}^{-1}l_{i}^{-1}\gamma l_{i}n_{i}\in K$ for some
$(l_{i})_{i}\subseteq\Omega$. Write
$n_{i}=u_{\alpha_{1}}\left(x_{1}^{(i)}\right)\ldots
u_{\alpha_{m}}\left(x_{m}^{(i)}\right)$ where $\alpha_{1},\ldots,\alpha_{m}$
are as in Lemma 2.3 with $P=Q$ and $m=\dim N_{Q}$. Since the map

$F^{m}\ni(x_{1},\ldots,x_{m})\mapsto u_{\alpha_{1}}(x_{1})\ldots
u_{\alpha_{m}}(x_{m})\in N_{Q}$

is a homeomorphism, there exists $1\leqslant j\leqslant m$ such that the
sequence $\left(x_{j}^{(i)}\right)$ is not bounded. Since $\gamma$ is regular,
$\alpha_{i}(\gamma)\neq 1$ for any $\alpha_{i}\in\Delta^{+}$. For any $l_{i}$
the element $l_{i}^{-1}\gamma l_{i}$ is non-elliptic. By Lemma 2.3, the
sequence $n_{i}^{-1}l_{i}^{-1}\gamma l_{i}n_{i}$ is not bounded which leads to
a contradiction.

If $T$ is not split over $F$, then choose a finite extension $E/F$ such that
$T$ is split over $E$ and repeat the above argument for $\gamma\in Q(E)$.

∎

###### Proof of Proposition 2.1.

We consider $\gamma$ up to $G$-conjugation so without loss of generality we
can assume that $\gamma\in K$. If $\gamma$ is elliptic then $A_{M}(\gamma)$ is
trivial and Proposition 2.1 follows from Lemma 2.2. Assume now that $\gamma$
is non-elliptic. Then, there exists a proper parabolic subgroup of $G$
containing $\gamma$. Take $P$ to be a parabolic minimal with this property.
Denote by $L$ its Levi subgroup and by $N$ its unipotent radical. Denote by
$p\colon P\to P/N$ the canonical projection. For any subgroup $S$ of $P$
denote by $\overline{S}$ its image in $P/N$. By Iwasawa decomposition,

$\displaystyle\Omega(K)$ $\displaystyle=\\{g\in A_{M}(\gamma)\setminus G|\ \
g^{-1}\gamma g\in K\\}$ (2.3) $\displaystyle=\\{lnk|\ \ l\in
A_{M}(\gamma)\setminus L,n\in N,k\in K\ \ \textrm{and}\ \
k^{-1}n^{-1}l^{-1}\gamma lnk\in K\\}$ (2.4) $\displaystyle=\\{lnk|\ \ l\in
A_{M}(\gamma)\setminus L,n\in N,k\in K\ \ \textrm{and}\ \ n^{-1}l^{-1}\gamma
ln\in K\\}.$ (2.5)

Note that if $n^{-1}l^{-1}\gamma ln\in K$ then $l^{-l}\gamma l$ belongs to a
compact subgroup $p^{-1}(\overline{K\cap P})$. By Lemma 2.2 applied to $L$,
the set

$\Omega(K,P):=\\{l\in L|\ \ l^{-1}\gamma l\in p^{-1}(\overline{K\cap P})\\}$

is compact. By (2.5),

$\displaystyle\Omega(K)$ $\displaystyle=\\{lnk|\ \ l\in A_{M}(\gamma)\setminus
L,n\in N,k\in K,\ \ n^{-1}l^{-1}\gamma ln\in K\ \ {\rm and}\ \ l^{-1}\gamma
l\in p^{-1}(\overline{K\cap P})\\}$ $\displaystyle=\\{lnk|\ l\in
A_{M}(\gamma)\setminus\Omega(K,P),n\in N,k\in K\ \ {\rm and}\ \
n^{-1}l^{-1}\gamma ln\in K\\}.$

By Lemma 2.4 applied to $\Omega=\Omega(K,P)$, the set $\Omega(K)$ is compact.

∎

##  3\. Proof of Theorem 1.2

The second main ingredient of the proof of Theorem 1.2 is an upper bound on
the irreducible characters of compact $p$-adic groups [Fraczyk, Theorem 5.8.].
If $U$ is a compact subgroup of ${\mathbf{G}}(F)$ and $\rho$ is an irreducible
representation of $U$ then we denote by $\chi_{\rho}$ the character of $\rho$
extended to ${\mathbf{G}}(F)$ by $0$.

######  Proposition 3.1.

[Fraczyk, Theorem 5.8.] Let ${\mathbf{H}}$ be a connected reductive group over
a non-Archimedean local field $F$. Denote by $W$ the Weyl group of
${\mathbf{H}}$. For every compact subgroup $U\subseteq{\mathbf{H}}(F)$, every
irreducible (complex) representation $\rho$ of $U$ we have

$|\chi_{\rho}(\gamma)|\leqslant|\Delta(\gamma)|_{F}^{-1}2^{\dim{\mathbf{H}}-{\rm
rk}({\mathbf{H}})}|W|$

for every regular semisimple element $\gamma\in{\mathbf{H}}(F)$.

For readers’ convenience we prove the following, rather standard, fact.

######  Lemma 3.2.

The $\chi_{\rho}$ is a finite sum of matrix coefficients of $\pi$.

###### Proof.

Denote by $W$ the vector space associated to $\rho$. The trace of $\rho$ is
equal to

$\sum_{i}\langle w_{i},\rho(g)w_{i}\rangle$

for any orthonormal basis $\\{w_{i}\\}$ of $W$. Define
$f_{w_{i}}\in{\operatorname{c-Ind}}_{K}^{G}\rho$ as follows

$f_{w_{i}}(g)=\begin{cases}\rho(g)w_{i}&{\rm if}\ \ g\in K\\\ 0&{\rm
otherwise}.\end{cases}$

Write $\langle\cdot,\cdot\rangle$ for the inner product on
${\operatorname{c-Ind}}_{K}^{G}\rho$ given by $\langle
f,g\rangle=\int_{G}\langle f(h),g(h)\rangle dh$ for the Haar measure $dh$
which is $1$ on $K$. We have

$\chi_{\rho}(g)=\sum_{i}\langle f_{w_{i}},\pi(g)f_{w_{i}}\rangle$

for any $g\in G$. ∎

###### Proof of Theorem 1.2.

Let $\pi$ be an irreducible supercuspidal representation of $G$ of the form
${\operatorname{c-Ind}}_{J}^{G}\Lambda$ with an open compact modulo center
$J$. Since ${\mathbf{G}}$ is semisimple we can assume
$\pi={\operatorname{c-Ind}}_{K}^{G}\rho$ for a maximal compact open subgroup
$K$ of $G$ and an irreducible representation $\rho$ of $K$.

To estimate $\theta_{\pi}(\gamma)$ we use Arthur’s formula [Arthur1987,
Theorem]. Let $P$ be a parabolic subgroup of ${\mathbf{G}}(F)$ which contains
$\gamma$ and minimal with this property. Let $M$ be the Levi component of $P$
containing $\gamma$. The element $\gamma$ is $M$-elliptic over $F$, i.e. the
centralizer of $\gamma$ in $M$ is compact modulo $A_{M}(\gamma)$.
Recall now the definition of the weight function $v_{M}$ appearing in the
Arthur’s formula. Choose a special maximal compact subgroup $K_{1}$ of
${\mathbf{G}}(F)$ which is in a good position relative to $M$ (i.e. the vertex
of $K_{1}$ in the building of ${\mathbf{G}}$ belongs to the apartment of a
maximal split torus of $M$). Denote by $\mathcal{P}(M)$ the set of all
parabolic subgroups of ${\mathbf{G}}$ defined over $F$ with Levi component
$M$. Let $P^{\prime}\in\mathcal{P}(M)$ be a parabolic subgroup with Levi
decomposition $P^{\prime}=MN_{P^{\prime}}$. Any point $x\in{\mathbf{G}}(F)$
can be written in the form

$x=n_{P^{\prime}}(x)m_{P^{\prime}}(x)k_{P^{\prime}}(x),$

where $n_{P^{\prime}}(x)\in N_{P^{\prime}}(F)$, $m_{P^{\prime}}(x)\in M(F)$
and $k_{P^{\prime}}(x)\in K_{1}$. Write $X(M)$ for the module of rational
characters of $M$. Consider a map

$H_{M}\colon M(F)\to\mathbf{a}_{M}:={\rm Hom}(X(M)_{F},{\mathbb{R}})$

given by $\exp{(\langle H_{M}(m),\chi\rangle)}=|\chi(m)|$ for $m\in M(F)$,
$\chi\in X(M)_{F}$. We put

$H_{P^{\prime}}(x):=H_{M}(m_{P^{\prime}}(x)).$

The function $v_{M}$ is defined as the volume of the convex hull of the
projection of

$\\{-H_{P^{\prime}}(x)|\ \ P^{\prime}\in\mathcal{P}(M)\\}$

onto $\mathbf{a}_{M}/\mathbf{a}_{G}$. Remark that by definition, the function
$v_{M}$ is constant on right $K_{1}$-cosets, so in particular it is
continuous.

By Lemma 3.2 and [Arthur1987],

$\int_{A_{M}(\gamma)\setminus G}f(x^{-1}\gamma
x)v_{M}(x)d\mu(x)=(-1)^{\dim(A_{M}(\gamma)/A_{{\mathbf{G}}})}\theta_{\pi}(f)\theta_{\pi}(\gamma)$
(3.1)

for $f=\chi_{\rho}$. By Proposition 2.1, $\Omega(K)$ has finitely many right
cosets modulo $K$. By Frobenius reciprocity,

$\theta_{\pi}(\chi_{\rho})={\rm tr}\,\pi(\chi_{\rho})=\dim{\rm
Hom}_{K}(\rho,\pi)=\dim{\rm
Hom}_{G}({\operatorname{c-Ind}}_{K}^{G}\rho,\pi)=\dim{\rm
Hom}_{G}(\pi,\pi)=1.$

By (3.1),

$\displaystyle|\theta_{\pi}(\gamma)|$
$\displaystyle=\left|\int_{A_{M}(\gamma)\setminus G}\chi_{\rho}(g^{-1}\gamma
g)v_{M}(g)dg\right|=\left|\int_{\Omega(K)}\chi_{\rho}(g^{-1}\gamma
g)v_{M}(g)dg\right|$
$\displaystyle\leqslant\sum_{gK\in\Omega(K)/K}\int_{K}\left|\chi_{\rho}(k^{-1}g^{-1}\gamma
gk)\right||v_{M}(gk)|dk.$

Since the function $v_{M}$ is continuous and $\Omega(K)$ is compact, there
exists a constant $C>0$ dependent on $\gamma$ such that $|v_{M}(gk)|<C$ for
every $k\in K$, $g\in\Omega(K)$. Together with Proposition 3.1 it implies

$\displaystyle|\theta_{\pi}(\gamma)|$
$\displaystyle\leqslant\left(\sum_{g\in\Omega(K)/K}|\Delta(g^{-1}\gamma
g)|_{F}^{-1}C\right)2^{\dim{\mathbf{G}}-{\rm rk}({\mathbf{G}})}|W|$
$\displaystyle=|\Delta(\gamma)|_{F}^{-1}|\Omega(K)/K|C2^{\dim{\mathbf{G}}-\textrm{rk}({\mathbf{G}})}|W|=:\kappa_{\gamma}$

for some $\kappa_{\gamma}$ dependent only on $G$ and $\gamma$. We can take
$\kappa_{\gamma}$ to be independent of $K$ because there is only finitely many
$G$-conjugacy classes of maximal compact subgroups of $G$. ∎

##  4\. Explicit bound for $\gamma$ elliptic

In this section we compute an explicit bound on $\theta_{\pi}(\gamma)$ for
$\gamma$ elliptic.

######  Theorem 4.1.

Let ${\mathbf{G}}$ be a connected semisimple group over a non-Archimedean
local field $F$ of characteristic zero. Let $W$ be the Weyl group of
${\mathbf{G}}$. Then, for every regular elliptic $\gamma\in{\mathbf{G}}(F)$
and every irreducible compactly induced supercuspidal representation $\pi$ we
have

$|\theta_{\pi}(\gamma)|\leqslant|\Delta(\gamma)|_{F}^{-1}2^{\dim{\mathbf{G}}-{\rm
rk}\,{\mathbf{G}}}|W|.$

######  Corollary 4.2.

Let ${\mathbf{G}}$ be a connected semisimple group over a non-Archimedean
local field $F$ of characteristic zero. Let $W$ be the Weyl group of
${\mathbf{G}}$. Then, for every regular elliptic element
$\gamma\in{\mathbf{G}}(F)$ and every irreducible compactly induced
supercuspidal representation $\pi$ with
$\deg(\pi)\geqslant|\Delta(\gamma)|_{F}^{-1}$ we have

$|\theta_{\pi}(\gamma)|\leqslant|\Delta(\gamma)|_{F}^{-\frac{1}{2}}\deg(\pi)^{\frac{1}{2}}2^{\dim{\mathbf{G}}-{\rm
rk}\,{\mathbf{G}}}|W|.$

This goes towards [KST2, Conjecture 3.15. (iii)].
The proof of Theorem 4.1 is inspired by the proof of [Fraczyk, Theorem 5.8.].
However, for ${\mathbf{G}}$ non-compact some significant changes are needed.
In order to estimate $\theta_{\pi}$, we rely on Kazhdan’s orthogonality
relations (Lemma 4.4) and the following lemma.

Fix a regular elliptic element $\gamma\in{\mathbf{G}}(F)$. Let
$\mathbf{T}_{0}$ be an elliptic torus such that $\gamma\in\mathbf{T}_{0}(F)$.
Denote by ${\mathbf{G}}^{rs}(F)$ the set of regular semisimple elements in
${\mathbf{G}}(F)$. For any irreducible representation $\tau$ of a compact
group we write $\chi_{\tau}$ for its trace.

######  Lemma 4.3.

Let $S$ be a closed subset of $\mathbf{T}_{0}(F)\cap{\mathbf{G}}^{rs}(F)$.
Then, $\theta_{\pi}|_{S}$ is the restriction of a finite integral combination
of unitary characters of $\mathbf{T}_{0}(F)$.

###### Proof.

For any regular elliptic $g\in\mathbf{G}(F)$ write

$\Omega_{g}(K):=\\{x\in{\mathbf{G}}(F):\ \ x^{-1}gx\in K\\}.$

By Proposition 2.1, Arthur’s formula and the local constancy of
$\theta_{\pi}$, for every regular elliptic $g$ there exists an open
neighborhood $U_{g}$ of $g$ such that

$\theta_{\pi}(u_{g})=\sum_{xK\in\Omega_{g}(K)/K}\chi_{\rho}(x^{-1}u_{g}x)$

for every $u_{g}\in U_{g}$. Choose a finite cover $\\{U_{g_{i}}\\}_{i=1}^{n}$
of $S$ with $g_{i}\in S$. Let
$\mathcal{F}:=\bigcup_{i=1}^{n}\Omega_{g_{i}}(K)$. This set is left
$\mathbf{T}_{0}(F)$-invariant. Consider the representation

$\sigma:=\bigoplus_{x\in\mathbf{T}_{0}(F)\setminus\mathcal{F}/K}{\rm
Ind}_{\mathbf{T}_{0}(F)\cap K^{x^{-1}}}^{\mathbf{T}_{0}(F)}\rho^{x^{-1}}$

where $K^{x^{-1}}:=xKx^{-1}$ and $\rho^{x^{-1}}(a):=\rho(x^{-1}ax)$ for any
$a\in K^{x^{-1}}$. For every $g_{i}$ we have

$\displaystyle\chi_{\sigma}(g_{i})$
$\displaystyle=\sum_{x\in\mathbf{T}_{0}(F)\setminus\mathcal{F}/K}\ \
\sum_{\begin{subarray}{c}r\in\mathbf{T}_{0}(F)/\mathbf{T}_{0}(F)\cap
K^{x^{-1}},\\\ g_{i}\in
K^{x^{-1}}\end{subarray}}\chi_{\rho^{x^{-1}}}(g_{i})=\sum_{xK\in\Omega_{g_{i}}(K)/K}\chi_{\rho}(x^{-1}g_{i}x).$

Therefore $\theta_{\pi}|_{S}=\chi_{\sigma}|_{S}$. ∎

######  Lemma 4.4.

Let ${\mathbf{G}}$ be a semisimple group over a non-Archimedean local field
$F$. For any torus $\mathbf{T}$ write
$W_{\mathbf{T}}:=N(\mathbf{T}(F))/\mathbf{T}(F)$ where $N(\mathbf{T}(F))$
denotes the normalizer of $\mathbf{T}(F)$ in ${\mathbf{G}}(F)$. Then, for
every supercuspidal representation $\pi$ of ${\mathbf{G}}(F)$ we have

$\sum_{[\mathbf{T}]\subseteq{\mathbf{G}}}\frac{1}{|W_{\mathbf{T}}|}\int_{\mathbf{T}(F)\cap{\mathbf{G}}^{rs}(F)}|\Delta(t)|_{F}|\theta_{\pi}(t)|^{2}dt=1$

where $[\mathbf{T}]$ runs over all ${\mathbf{G}}(F)$-conjugacy classes of
maximal elliptic tori and $dt$ denotes the probabilistic Haar measure on
$\mathbf{T}(F)$.

###### Proof.

Follows from the Kazhdan’s orthogonality relations [SchneiderStuhler, Theorem
III.4.21] and [SchneiderStuhler, Theorem III.4.6.]. We apply [Kazhdan, §3,
Lemma 1.] and Weyl integration formula to determine the measure $dc$ in
[SchneiderStuhler, Theorem III.4.21]. ∎

###### Proof of Theorem 4.1.

Denote by $\mathbf{T}_{0}$ maximal split torus such that
$\gamma\in\mathbf{T}_{0}(F)$. Let $m:=\frac{1}{2}(\dim{\mathbf{G}}-{\rm
rk}\,{\mathbf{G}})$ and choose a set of positive roots
$\lambda_{1},\ldots,\lambda_{m}$ of $\mathbf{T}_{0}$ defined over the
algebraic closure of $F$. We have

$\Delta(t)=\prod_{j=1}^{m}(1-\lambda_{j}(t))(1-\lambda_{j}(t)^{-1}).$

Let

$Z_{i}:=\\{t\in\mathbf{T}_{0}(F)|\ \
|1-\lambda_{i}(t)|_{F}<|1-\lambda_{i}(\gamma)|_{F}\\}.$

Let $E$ be a finite extension of $F$ such that every character of
$\mathbf{T}_{0}$ is defined over $E$. Denote by $\mathcal{O}_{E}^{\times}$ the
group of all invertible elements in the ring of integers of $E$. By the
compactness of $\mathbf{T}_{0}(F)$,
$\lambda_{j}(\mathbf{T}_{0}(F))\subseteq\mathcal{O}_{E}^{\times}$ for every
$j\in\\{1,\ldots,m\\}$. By Pontryagin duality, for every
$j\in\\{1,\ldots,m\\}$ we can find a unitary character
$\alpha_{j}\colon\mathcal{O}_{E}^{\times}\to\mathbb{C}^{\times}$ such that
$\alpha_{j}(\lambda_{j}(\gamma))\neq 1$ and $\alpha_{j}(\lambda_{j}(z))=1$ for
every $z\in Z_{j}$. Define

$\beta(t):=\prod_{j=1}^{m}(1-\alpha_{j}(\lambda_{j}(t)))$

for every $t\in\mathbf{T}_{0}(F)$. Since $\beta(t)$ is a non-zero algebraic
integer, there exists a Galois automorphism $\kappa$ such that

$\alpha(t):=\prod_{j=1}^{m}(1-\kappa(\alpha_{j}(\lambda_{j}(t))))$

satisfies $|\alpha(t)|\geqslant 1$. Since $\alpha(z)=0$ for every
$z\in\bigcup_{j=1}^{m}Z_{j}$, for every $t\in\mathbf{T}_{0}(F)$ we have

$\frac{|\alpha(t)|^{2}}{|\Delta(t)|_{F}}\leqslant\frac{|\alpha(t)|^{2}}{|\Delta(\gamma)|_{F}}.$
(4.1)

Let $dt$ be the probabilistic Haar measure on $\mathbf{T}_{0}(F)$. By Lemma
4.4,

$\int_{\mathbf{T}_{0}(F)\cap{\mathbf{G}}^{rs}(F)}|\Delta(t)|_{F}|\theta_{\pi}(t)|^{2}dt\leqslant|W|.$
(4.2)

Putting (4.1) and (4.2) we get

$\int_{\mathbf{T}_{0}(F)\cap{\mathbf{G}}^{rs}(F)}|\alpha(t)|^{2}|\theta_{\pi}(t)|^{2}dt\leqslant\sup_{t\in\mathbf{T}_{0}(F)\cap{\mathbf{G}}^{rs}(F)}|\alpha(t)|^{2}|\Delta(\gamma)|_{F}^{-1}|W|\leqslant
2^{2m}|\Delta(\gamma)|_{F}^{-1}|W|.$ (4.3)

Define $R:=\\{t\in\mathbf{T}_{0}(F)\cap{\mathbf{G}}^{rs}(F)|\ \ \alpha(t)\neq
0\\}$. By Lemma 4.3 applied to $R$, $\theta_{\pi}$ is a finite integral
combination of unitary characters so

$\alpha\theta_{\pi}|_{R}=\sum_{\varphi}a_{\varphi}\varphi,$

where $a_{\varphi}\in\mathbb{Z}$ and each $\varphi$ is a unitary character of
$\mathbf{T}_{0}(F)$. By (4.3),

$\sum_{\varphi}a_{\varphi}^{2}=\int_{\mathbf{T}_{0}(F)\cap\mathbf{G}^{rs}(F)}|\alpha(t)|^{2}|\theta_{\pi}(t)|^{2}dt\leqslant
2^{2m}|\Delta(\gamma)|_{F}^{-1}|W|.$

Since $a_{\varphi}\in\mathbb{Z}$,

$|\theta_{\pi}(\gamma)|\leqslant|\alpha(\gamma)\theta_{\pi}(\gamma)|=\left|\sum_{\varphi}a_{\varphi}\varphi(\gamma)\right|\leqslant\sum_{\varphi}|a_{\varphi}|\leqslant
2^{\dim{\mathbf{G}}-{\rm rk}\,{\mathbf{G}}}|\Delta(\gamma)|_{F}^{-1}|W|.$

∎

## References