# Counting Discrete, Level-$1$, Quaternionic Automorphic Representations on
$G_{2}$

Rahul Dalal

###### Abstract.

Quaternionic automorphic representations are one attempt to generalize to
other groups the special place holomorphic modular forms have among
automorphic representations of $\mathrm{GL}_{2}$. Here, we study quaternionic
automorphic representations on the exceptional group $G_{2}$. Using
“hyperendoscopy” techniques from a previous work, we develop for quaternionic,
$G_{2}$-representations an analog of the Eichler-Selberg trace formula for
classical modular forms. We then use this together with some techniques of
Chenevier, Renard, and Taïbi to compute dimensions of spaces of level-$1$
quaternionic representations. On the way, we prove a Jacquet-Langlands-style
result describing them in terms of classical modular forms and automorphic
representations on the compact-at-infinity form $G_{2}^{c}$.

The main technical difficulty is that the quaternionic discrete series that
quaternionic autmorphic representations are defined in terms of do not satisfy
a condition of being “regular”. Using some computations of Mundy, we show that
this miraculously does not matter for specifically the case of quaternionic
discrete series on $G_{2}$.

We hope that this project serves as a guiding example for anyone interested in
computing exact information about discrete-at-infinity automorphic
representations on arbitrary reductive groups instead of just classical ones.

###### Contents

  1. 1 Introduction
    1. 1.1 Context
    2. 1.2 Summary
    3. 1.3 Acknowledgements
    4. 1.4 Notation
  2. 2 $G_{2}$ and Quaternionic Discrete Series
    1. 2.1 Root System of $G_{2}$
      1. 2.1.1 Roots
      2. 2.1.2 Coroots
    2. 2.2 Quaternionic Discrete Series
      1. 2.2.1 Description
      2. 2.2.2 Their pseudocoefficients
  3. 3 Geometric Side/Application of Endoscopy
    1. 3.1 Notation
    2. 3.2 The Hyperendoscopy Formula
      1. 3.2.1 Preliminaries
      2. 3.2.2 Telescoping
      3. 3.2.3 Final Geometric Formulas and Method of Computation
  4. 4 Groups Contributing and Related Constants
    1. 4.1 Elliptic Endoscopy of $G_{2}$
    2. 4.2 Endoscopic Constants and Normalizations
      1. 4.2.1 The $\iota$
      2. 4.2.2 The transfer factors
      3. 4.2.3 The stabilizations
  5. 5 Real Endoscopic Transfers
    1. 5.1 Root Combinatorics
    2. 5.2 Endoscopic Characters
    3. 5.3 Final Formulas for Transfers
  6. 6 The $H=\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\pm 1$ term
    1. 6.1 Cohomological Representations of $H(\mathbb{R})$
    2. 6.2 Reduction to Modular Form Counts
    3. 6.3 Final Formula for $S^{H}$
  7. 7 A Jacquet-Langlands-style result
    1. 7.1 First Form
    2. 7.2 In terms of Modular Forms
  8. 8 Counts of forms
    1. 8.1 Formula in terms of $I^{G_{2}^{c}}$
    2. 8.2 Computing $I^{G_{2}^{c}}$
    3. 8.3 Table of Counts

##  1\. Introduction

###  1.1. Context

This work describes level-$1$, discrete, quaternionic automorphic
representations on $G_{2}$. Let $\mathcal{Q}_{1}(k)$ be the set of such
representations of weight $k$. For each $k>2$, we give a formula, (10), for
$|\mathcal{Q}_{1}(k)|$ in terms of counts of automorphic representations on
the compact-at-infinity inner form $G_{2}^{c}$ that were calculated by
Chenevier and Renard in [CR15]. We also give a Jacquet-Langlands-style result
(corollary 7.2.1) describing all elements of $\mathcal{Q}_{1}(k)$ in terms of
certain automorphic representations on $G_{2}^{c}$ and certain pairs of
classical modular forms.

Quaternionic automorphic representations were developed as a way to generalize
to other groups the special place holomorphic modular forms have among
automorphic representations of $\mathrm{GL}_{2}$. Just like holomorphic
modular forms, they are characterized by their infinite component being in a
particular nice class of discrete series representations: the quaternionic
discrete series of [GW96]. Just like modular forms they also have many
unexpected applications and connections to other areas of math. For example,
they have a nice theory of Fourier expansions with interesting arithmetic
content—this was described for $G_{2}$ in [GGS02] and generalized to all
exceptional groups in [Pol20]. They also somehow appear in certain string
theory computations involving black holes—see the end of chapter 15 in
[FGKP18] for example. Quaternionic forms have been studied a lot by Pollack:
see [Pol21] for an introductory article on them and [Pol18] for good
exposition specifically on $G_{2}$-quaternionic forms.

We study discrete, quaternionic representations on $G_{2}$ using the trace
formula. Since quaternionic discrete series appear in $L$-packets with non-
quaternionic members, this provides a great test case of the efficacy of the
techniques in [Dal19] developed to split $L$-packets with the trace formula.
More precisely, stabilizaiton lets us develop an “Eicher-Selberg”-style trace
formula for quaternionic discrete series. The counts of level one forms and
Jacquet-Langlands-style results come from computations with this. The
level-$1$ computation in particular also relies heavily on powerful shortcuts
developed in [CR15] and [Taï17] to get exact counts of level-$1$ automorphic
representations on classical groups through the invariant/stable trace
formulas.

Finally, as one technical point of interest, there is a particular miracle
about quaternionic discrete series on $G_{2}$ that crucially underpins this
result. A priori, [Dal19, Prop. 6.3.3] cannot be applied: such discrete series
are not regular, implying that there may not be a test function at infinity
whose trace picks out exactly a quaternionic discrete series without also
picking up some unwanted contributions from non-tempered representations.
However, it turns out that specifically quaternionic discrete series on
$G_{2}$ don’t get entangled in this way, even though other members of their
$L$-packet do. The proof of this depends on results about Adams-Johnson
packets for $G_{2}$ that Mundy developed for studying Eisenstein cohomology in
[Mun20]. The existence of this miracle suggests similar methods could be used
to study other irregular discrete series at infinity through the simple trace
formula.

The methods here should be able to also compute averages of Hecke eigenvalues.
We also hope that the computation here is an illuminating basic, starting
example for people interested in doing explicit computations with discrete-at-
infinity automorphic representations on general reductive groups instead of
just classical ones.

###  1.2. Summary

We summarize the method of computation. Proposition 2.2.1 shows that traces
against a pseudocoefficient of a quaternionic discrete series with weight
$k>2$ are $0$ against all other unitary representations. This allows us to get
a formula (2.2.2) for traces of finite-place test functions against the space
of all quaternionic representations of weight $k$.

Next, section 3 develops a general stabilized formula (2) for
$I_{\mathrm{geom}}$ applied to test functions like ours. The combination of
the results in §3 and formula (2.2.2) should be thought of as an analogue for
quaternionic, $G_{2}$-representations of the Eichler-Selberg trace formula for
classical modular forms. We work out what this formula reduces to in the
unramified case in section 4 using a computation of the endoscopy of $G_{2}$
in section 4.1. Instead of using formula (2) directly, we compare it applied
to $G_{2}$ to it applied to the compact real form $G_{2}^{c}$ to construct a
formula for $I_{\mathrm{spec}}^{G_{2}}$ involving just
$I_{\mathrm{spec}}^{G_{2}^{c}}$\- and $I_{\mathrm{spec}}^{H}$-terms. Here, $H$
is the endoscopic group $\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\pm 1$ of
$G_{2}$.

Section 5 then tells us which exact $I_{\mathrm{spec}}^{G_{2}^{c}}$\- and
$I_{\mathrm{spec}}^{H}$-terms appear by computing endoscopic transfers at
infinity. The difficult part of this computation is pinning down various signs
coming from transfer factors. As a last piece of the puzzle, section 6 uses
results about level-$1$ forms from [CR15] to reduce counts of forms on $H$ to
counts of classical modular forms.

Section 7 uses all these formulas to characterize representations in
$\mathcal{Q}_{k}(1)$ with $k>2$ in terms of automorphic representations on
$G_{2}^{c}$ and certain pairs of classical modular forms. Finally, we
substitute in values for the $I_{\mathrm{spec}}^{G_{2}^{c}}$-terms from [CR15]
and present a final table of dimensions: table 1, in section 8.

###  1.3. Acknowledgements

This work was done under the support of NSF RTG grant DMS-1646385. I would
like to thank Kimball Martin and Aaron Pollack for discussions at conferences
that eventually led to the consideration of this problem. The key proposition
2.2.1 that made this problem solvable grew out of email conversations with Sam
Mundy. Olivier Taïbi provided a great deal of help in pointing out many, many
tricks and previous results I could use to keep computations reasonably simple
and actually feasible. In particular, I am grateful for the suggestion to use
“method 2” described in section 3.2.3 that he passed on from Gaëtan Chenevier.
Alexander Bertoloni-Meli again provided much help, this time teaching me
things about transfer factors. The root system picture for $G_{2}$ was heavily
modified from an answer by user Heiko Oberdiek on TeX Stack Exchange.

###  1.4. Notation

Here is a list of notation used throughout:

The group $G_{2}$

  * •

$G_{2}$ is the standard exceptional Chevalley group defined over $\mathbb{Z}$.

  * •

$G_{2}^{c}$ is the unique inner form of $G_{2}$ over $\mathbb{Q}$. Recall that
$(G_{2}^{c})_{\mathbb{R}}$ is the compact real form.

  * •

$\alpha_{i},\epsilon_{i},\lambda_{i}$ are particular roots and coroots of
$G_{2}$ defined in section 2.1.

  * •

$s_{\lambda}$ is the simple reflection associated to root or coroot $\lambda$.

  * •

$\rho$ is half the sum of the positive roots of $G_{2}$.

  * •

$V_{\lambda}$ is the finite dimensional representation of $G_{2}$ of highest
weight $\lambda$.

  * •

$K$ is a choice of maximal compact subgroup
$\mathrm{SU}(2)\times\mathrm{SU}(2)/\pm 1$ of $G_{2}(\mathbb{R})$.

  * •

$K^{\infty}$ is the product of maximal compact subgroups
$G_{2}(\mathbb{Z}_{p})$ over all $p$.

  * •

$\Omega=\Omega_{\mathbb{C}}$ is the Weyl group of $G_{2}$.

  * •

$\Omega_{\mathbb{R}}$ is the Weyl group of $K$ as a subset of $\Omega$.

  * •

$H$ will often refer to the specific endoscopic group
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\pm 1$ of $G_{2}$.

  * •

$\mathcal{Q}_{k}(1)$ is the set of discrete, quaternionic automorphic
representations of $G_{2}$ of weight $k$ and level $1$ (see section 2.2).

General groups

  * •

$G_{\infty}=\operatorname{Res}^{F}_{\mathbb{Q}}G(\mathbb{R})$ for $G$ a
reductive group over number field $F$. Since most groups here are over
$\mathbb{Q}$, $G_{\infty}$ is usually $G(\mathbb{R})$.

  * •

$G^{S},G_{S}$ are more generally the standard upper- and lower-index notation
for $G(\mathbb{A}^{S}),G(\mathbb{A}_{S})$—leaving out the places in $S$ or
only including the places in $S$ respectively.

  * •

$\Omega(G)$ is the absolute Weyl group of $G$.

  * •

$\Omega_{\mathbb{R}}(G)$ is the subset of the Weyl group of $G_{\infty}$ with
respect to an elliptic maximal torus (if one exists) generated by elements of
$G_{\infty}$.

  * •

$K_{G}$ is a maximal compact subgroup of $G_{\infty}$.

  * •

$K^{\infty}_{G}$ for unramified $G$ is the product of chosen hyperspecial
subgroups at all finite places.

  * •

$\rho_{G}$ is half the sum of the positive roots of $G$.

  * •

$[G(F)],[G(F)]_{\mathrm{ss}},[G(F)]_{\mathrm{st}},[G(F)]_{\mathrm{ell}}$ are
the (semisimple, stable, elliptic) conjugacy classes of $G(F)$.

Real test functions

  * •

$\varphi_{\pi}$ for $\pi$ a discrete series representation of $G_{\infty}$ is
the pseudocoefficient defined in [CD90].

  * •

$\Pi_{\mathrm{disc}}(\lambda)$ is the discrete series $L$-packet corresponding
to dominant weight $\lambda$.

  * •

$\eta_{\lambda}$ is the Euler-Poincaré function from [CD90] for
$\Pi_{\mathrm{disc}}(\lambda)$. We normalize it to be the average of the
pseudocoefficients for $\pi\in\Pi_{\mathrm{disc}}(\lambda)$ instead of their
sum.

Trace Formula

  * •

$\mathcal{AR}(G)$, $\mathcal{AR}_{\mathrm{disc}}(G)$ is the set of (discrete)
automorphic representations on $G$.

  * •

$\mathcal{AR}_{\mathrm{ur}}(G)$ for $G$ unramified is the space of unramified
automorphic representations of $G$.

  * •

$m_{\mathrm{disc}}(\pi),m_{\mathrm{cusp}}(\pi)$ are the multiplicities of
automorphic representation $\pi$ in the discrete (cuspidal) subspace.

  * •

$I^{G}_{\mathrm{spec}},I^{G}_{\mathrm{geom}},I^{G}_{\mathrm{disc}}$ are the
distributions from Arthur’s invariant trace formula on $G$.

  * •

$S^{G}=S^{G}_{\mathrm{geom}}$ is the stable distribution defined in theorem
3.2.1.

Miscellaneous

  * •

$\mathbf{1}_{S}$ is the indicator function of set $S$.

  * •

$\mathcal{S}_{k}(1)$ is the set of normalized, classical, cuspidal eigenforms
on $\mathrm{GL}_{2}$ of level $1$ and weight $k$.

##  2\. $G_{2}$ and Quaternionic Discrete Series

###  2.1. Root System of $G_{2}$

####  2.1.1. Roots

We use notation from [LS93] to specify the root system of $G_{2}$. Let $K$ be
the maximal compact $\mathrm{SU}(2)\times\mathrm{SU}(2)/\pm 1$ of
$G_{2}(\mathbb{R})$. Choose a dominant chamber for $K$ and the choice of
simple roots of $G_{2}$ consistent with this. Let $\beta$ be the highest root
with respect to this and note that it is long.

Figure 1. Character lattice, roots, and choices of dominant chamber for
$G_{2}$$2\epsilon_{1}$$2\epsilon_{2}$$\lambda_{2}=\beta$$\alpha_{1}$$\lambda_{1}=\rho_{K}$$\alpha_{2}$••••••••••••••••••••••••••••••••••••••$\rho_{G_{2}}$

We now give explicit coordinates. As a mnemonic convention, roots indexed $1$
will be short and roots indexed $2$ will be long. Figure 1 displays all the
roots and shades our choices of dominant Weyl chambers with respect to both
$G_{2}$ and $K$. Compact roots at infinity are in red and non-compact in blue.

If the roots of the short and long $\mathrm{SU}_{2}$ are $2\epsilon_{1}$ and
$2\epsilon_{2}$ respectively, then the simple roots of $G_{2}$ are:

$\text{(short) }\alpha_{1}=-\epsilon_{1}+\epsilon_{2},\qquad\text{(long)
}\alpha_{2}=3\epsilon_{1}-\epsilon_{2}.$

The other positive roots are:

$\displaystyle\text{(short)
}2\epsilon_{1}=\alpha_{1}+\alpha_{2},\qquad\epsilon_{1}+\epsilon_{2}=2\alpha_{1}+\alpha_{2},$
$\displaystyle\text{(long) }2\epsilon_{2}=3\alpha_{1}+\alpha_{2},\qquad
3\epsilon_{1}+\epsilon_{2}=3\alpha_{2}+2\alpha_{2}.$

The fundamental weights are:

$\lambda_{1}=2\alpha_{1}+\alpha_{2},\qquad\lambda_{2}=3\alpha_{2}+2\alpha_{2}.$

Of course $\beta=\lambda_{2}$.

The Weyl group is generated by simple reflections:

$s_{\alpha_{1}}\begin{pmatrix}2\epsilon_{1}\\\
2\epsilon_{2}\end{pmatrix}=\begin{pmatrix}\epsilon_{1}+\epsilon_{2}\\\
3\epsilon_{1}-\epsilon_{2}\end{pmatrix},\qquad
s_{\alpha_{2}}\begin{pmatrix}2\epsilon_{1}\\\
2\epsilon_{2}\end{pmatrix}=\begin{pmatrix}-\epsilon_{1}+\epsilon_{2}\\\
3\epsilon_{1}+\epsilon_{2}\end{pmatrix}.$

Finally:

$\displaystyle\rho_{K}=\epsilon_{1}+\epsilon_{2}=2\alpha_{1}+\alpha_{2},$
$\displaystyle\rho_{G}=4\epsilon_{1}+2\epsilon_{2}=5\alpha_{1}+3\alpha_{2}.$

####  2.1.2. Coroots

Coroots will follow the opposite mnemonic: coroots indexed $1$ are long and
coroots indexed $2$ are short.

Let $T$ be a split maximal torus. Since $G_{2}$ has trivial center, $X^{*}(T)$
is the root lattice:

$X^{*}(T)=\\{a\epsilon_{1}+b\epsilon_{2}:a,b\in\mathbb{Z},a+b\in
2\mathbb{Z}\\}.$

Let $(\delta_{1},\delta_{2})$ be the dual basis to
$(2\epsilon_{1},2\epsilon_{2})$: i.e.
$(\delta_{i},\epsilon_{j})=1/2\mathbf{1}_{i=j}$. Then:

$X_{*}(T)=\\{a\delta_{1}+b\delta_{2}:a,b\in\mathbb{Z},a+b\in 2\mathbb{Z}\\}.$

Since $\epsilon_{1}$ and $\epsilon_{2}$ are perpendicular:

$\displaystyle(2\epsilon_{1})^{\vee}=2\delta_{1},$
$\displaystyle(2\epsilon_{2})^{\vee}=2\delta_{2}.$

More generally, the Weyl action gives:

$\displaystyle(\alpha_{1}^{\vee},2\epsilon_{1})=-1,\qquad(\alpha_{1}^{\vee},2\epsilon_{2})=3,$
$\displaystyle(\alpha_{2}^{\vee},2\epsilon_{1})=1,\qquad(\alpha_{2}^{\vee},2\epsilon_{2})=-1,$

so we get simple coroots:

$\displaystyle\alpha_{1}^{\vee}=-\delta_{1}+3\delta_{2},$
$\displaystyle\alpha_{2}^{\vee}=\delta_{1}-\delta_{2}.$

This reproduces that the coroot lattice is $X_{*}(T)$, implying that $G_{2}$
is simply connected. For completeness:

$\displaystyle\lambda_{1}^{\vee}=\delta_{1}+3\delta_{2},$
$\displaystyle\lambda_{2}^{\vee}=\delta_{1}+\delta_{2}.$

###  2.2. Quaternionic Discrete Series

####  2.2.1. Description

Recall the notation from [Dal19, §2.2.1] to discuss discrete series. In
particular, recall the two parametrizations of discrete series on
$G_{2}(\mathbb{R})$:

$\pi^{G_{2}}_{\lambda,\omega}=\pi^{G_{2}}_{\omega(\lambda+\rho_{G_{2}})}$

for $\lambda$ a dominant (but possibly irregular) weight of $G_{2}$ and
$\omega$ a Weyl-element that takes the $\Omega_{G_{2}}$ dominant chamber to
something $\Omega_{K}$-dominant—in other words, $1,s_{\alpha_{1}}$, or
$s_{\alpha_{2}}$. Note that $\pi^{G_{2}}_{\lambda,\omega}$ has infinitesimal
character $\lambda+\rho_{G_{2}}$. Recall that $\omega(\lambda+\rho_{G_{2}})$
is called the Harish-Chandra parameter of this discrete series.

The quaternionic discrete series of weight $k$ for $k\geq 2$ lies in the
$L$-packet

$\Pi_{\mathrm{disc}}((k-2)\beta).$

The members of this $L$-packet have Harish-Chandra parameters:

$(k-2)\beta+\rho_{G},\qquad s_{\alpha_{1}}((k-2)\beta+\rho_{G}),\qquad
s_{\alpha_{2}}((k-2)\beta+\rho_{G}).$

As in [GGS02], the quaternionic member is the one with minimal $K$-type
$\lambda_{B}=2k\epsilon_{2}$. We know that the discrete series
$\pi(\omega,\lambda)$ has minimal $K$-type

$\lambda_{B}=\omega(\lambda+2\rho_{G})-2\rho_{K}$

by [Kna01, Thm. 9.20]. Therefore the weight-$k$ quaternionic discrete series
$\pi_{k}$ is specifically $\pi(s_{\alpha_{2}},(k-2)\beta)$—computing,
$s_{\alpha_{2}}$ fixes $\rho_{K}$ so

$s_{\alpha_{2}}(\lambda+2\rho_{G})-2\rho_{K}=s_{\alpha_{2}}(\lambda+2\rho_{G}-2\rho_{K})=s_{\alpha_{2}}(\lambda+2\beta)=s_{\alpha_{2}}(k\beta)=2k\epsilon_{2}.$

This is the discrete series with Harish-Chandra parameter

$\lambda_{k,H}:=s_{\alpha_{2}}((k-2)\beta+\rho_{G}).$

####  2.2.2. Their pseudocoefficients

Recall from [CD90] the notion of pseudocoefficients $\varphi_{\pi}$ for
discrete series $\pi$. They satisfy:

$\operatorname{tr}_{\sigma}(\varphi_{\pi})=\begin{cases}1&\sigma=\pi\\\
0&\sigma\text{ basic, }\sigma\neq\pi\\\ ?&\text{else}\end{cases}$

for all unitary $\sigma$. Here a basic representation is a parabolic induction
of a discrete series or limit of discrete series. Recall also the Euler-
Poincaré functions $\eta_{\lambda}$ that we normalize to be the average of
pseudocoefficients over an $L$-packet. For a quick summary of properties of
these functions in the notation used here, see [Dal19, §2.2.2].

Let $\varphi_{k}$ be a pseudocoefficient for $\pi_{k}$. A priori, $\pi_{k}$ is
not a regular discrete series, so the trace against $\varphi_{k}$ may be non-
zero for certain non-tempered representations in addition to just $\pi_{k}$.
This could make it not work as a test function to pick out just automorphic
representations $\pi$ with $\pi_{\infty}=\pi_{k}$. However, this is
miraculously not a problem for specifically quaternionic discrete series.

######  Proposition 2.2.1.

Let $k>2$. Then for any unitary representation $\rho$ of $G_{2}(\mathbb{R})$:

$\operatorname{tr}_{\rho}(\varphi_{k})=\begin{cases}1&\rho=\pi_{k}\\\
0&\text{else}\end{cases}.$

###### Proof.

By the same argument of Vogan described in [Dal19, lem. 6.3.1], the trace is
$0$ unless $\operatorname{tr}_{\rho}(\eta_{(k-2)\beta})\neq 0$ for
$\eta_{(k-2)\beta}$ the Euler-Poincaré function. This is only possible if
$\rho$ appears in an appropriate $(\mathfrak{g},K)$-cohomology. By the main
result of [VZ84], the only representations that do so are the packet
$\Pi_{\lambda}((k-2)\beta)$ and something denoted
$A_{\mathfrak{q}}((k-2)\beta)$ for $\mathfrak{q}$ corresponding to the Levi
subgroup with roots $\pm\alpha_{1}$ and Weyl group
$\Omega_{L}=\\{s_{\alpha_{1}},1\\}$. This is because $k>2$ implies that this
is the only Levi such that $(k-2)\beta$ is fixed by $\Omega_{L}$. See also
[LS93][lem. 2.2].

If $\rho\in\Pi_{\lambda}((k-2)\beta)$ and
$\operatorname{tr}_{\rho}(\varphi_{k})=0$, then $\rho=\pi_{k}$ by definition.
It therefore suffices to exclude the case of $A_{\mathfrak{q}}((k-2)\beta)$.
We use that $\operatorname{tr}_{A_{\mathfrak{q}}((k-2)\beta)}(\varphi_{k})$ is
the coefficient of $\pi_{k}$ in the sum expansion

$A_{\mathfrak{q}}((k-2)\beta)=\sum_{\rho^{\prime}\text{
basic}}m(\rho^{\prime})\rho^{\prime}$

in the Grothendieck group and will show that this coefficient is $0$.

By [Mun20][thm. 6.4.4], $A_{\mathfrak{q}}$ is a Langlands quotient of a
discrete series on a $\mathrm{GL}_{2}$ Levi. Let the corresponding parabolic
induction be $I$. Since $\mathrm{GL}_{2}$ is a maximal proper Levi and the
infinitesimal character is regular, the other terms in the expansion need to
be discrete series. There have to be two of these since
$\operatorname{tr}_{A_{\mathfrak{q}}((k-2)\beta)}(\eta_{(k-2)\beta})=-2$ (see
[LS93][lem. 2.2] again). Call these $\rho_{1}$ and $\rho_{2}$ to make the
expansion into basics

$A_{\mathfrak{q}}((k-2)\beta)=I-\rho_{1}-\rho_{2}.$

The computation of the A-packet in [Mun20][thm. 6.4.4] shows that the
character

$A_{\mathfrak{q}}((k-2)\beta)-\pi_{k}=I-\rho_{1}-\rho_{2}-\pi_{k}$

is stable. By [She79][lem. 5.2], $I$ is stable so $\rho_{1}+\rho_{2}+\pi_{k}$
also has to be. Then, [She79][lem. 5.1] further implies that
$\rho_{1}+\rho_{2}+\pi_{k}$ is fully Weyl-invariant on an elliptic maximal
torus. Examining Harish-Chandra’s character formula for discrete series, this
is only possible if the three discrete series are exactly the three members
$\Pi_{\mathrm{disc}}((k-2)\beta)$. In particular,
$\pi_{k}\neq\rho_{1},\rho_{2}$ so we are done.

Beware that [Mun20] chooses dominant chambers for $K$ and $G$ in slightly
different relative position, making the notation there a bit different. ∎

######  Corollary 2.2.2.

Let $f^{\infty}$ be a compactly supported locally constant function on
$G_{2}(\mathbb{A}^{\infty})$ and $k>2$. Then

$\displaystyle I_{\mathrm{spec}}(\varphi_{k}\otimes f^{\infty})$
$\displaystyle=\sum_{\pi\in\mathcal{AR}_{\mathrm{disc}}(G_{2})}m_{\mathrm{disc}}(\pi)\delta_{\pi_{\infty}=\pi_{k}}\operatorname{tr}_{\pi^{\infty}}(f^{\infty})$
$\displaystyle=\sum_{\pi\in\mathcal{AR}_{\mathrm{cusp}}(G_{2})}m_{\mathrm{cusp}}(\pi)\delta_{\pi_{\infty}=\pi_{k}}\operatorname{tr}_{\pi^{\infty}}(f^{\infty}).$

###### Proof.

The statement for discrete representations is the same argument as [Dal19,
prop. 6.3.3] after we know proposition 2.2.1. Since $\pi_{\infty}=\pi_{k}$ is
necessarily discrete series for the non-zero terms, the main result of [Wal84]
shows that $m_{\mathrm{cusp}}(\pi)=m_{\mathrm{disc}}(\pi)$. ∎

We therefore have

(1) $|\mathcal{Q}_{k}(1)|=I_{\mathrm{spec}}(\varphi_{k}\otimes\mathbf{1}_{K})$

if we choose Gross’ canonical measure from [Gro97] at finite places. Note
again that this heavily depends on the miracle of proposition 2.2.1 and a
similar result does _not_ hold either for the other members of
$\Pi_{\mathrm{disc}}((k-2)\beta)$ or for the Euler-Poincaré function.

##  3\. Geometric Side/Application of Endoscopy

###  3.1. Notation

We will need to recall some extra notation related to general reductive group
$H$ over $F$:

  * •

$\Omega^{c}_{H}$ is the Weyl group generated by compact roots at infinity.

  * •

$d(H_{\infty})$ is the size of the discrete series $L$-packets of
$H_{\infty}$. Alternatively, $d(H_{\infty})=|\Omega_{H}|/|\Omega^{c}_{H}|$.

  * •

$k(H_{\infty})$ is the size of the group
$\mathfrak{K}=\ker(H^{1}(\mathbb{R},T_{\mathrm{ell}})\to
H^{1}(\mathbb{R},G_{\infty}))$ that appears in the theory of endoscopy for
$G_{\infty}$.

  * •

$q(H_{\infty})=\dim(H_{\infty}/K_{\infty}Z_{H_{\infty}})$ where $K_{\infty}$
is a maximal compact subgroup of $H_{\infty}$.

  * •

$H_{\infty}^{*}$ is the quasisplit inner form of $H_{\infty}$.

  * •

$\bar{H}_{\infty}$ is the compact form. If $H_{\infty}$ has an elliptic
maximal torus, this is inner.

  * •

$e(H_{\infty})$ is the Kottwitz sign $(-1)^{q(H^{*}_{\infty})-q(H_{\infty})}$.

  * •

$[H:M]=[H:M]_{F}=\dim(A_{M}/A_{G})$, where $A_{\star}$ is the maximal
$F$-split torus in the center of $\star$. We call this the index of $M$ in
$H$.

  * •

$\tau(H)$ is the Tamagawa number of $H$.

  * •

$\mathrm{Mot}_{H}$ is the Gross motive for $H$.

  * •

$L(\mathrm{Mot}_{H})$ is the value of the corresponding $L$-function at $0$
(or residue of the pole).

  * •

$\iota^{H}(\gamma)=\iota_{F}^{H}(\gamma)$ for $\gamma\in H(F)$ is the number
of connected components of $H_{\gamma}$ that have an $F$-point.

###  3.2. The Hyperendoscopy Formula

####  3.2.1. Preliminaries

We will use the hyperendoscopy formula of [Fer07] to compute
$I_{\mathrm{geom}}(\varphi_{k}\otimes f^{\infty})$. A priori, we need to apply
the general case of [Dal19, Thm. 4.2.3] since $G_{2}$ has endoscopy without
simply connected derived subgroup.

Let $\eta_{k}$ be the Euler-Poincaré function for
$\Pi_{\mathrm{disc}}((k-2)\beta)$. Let $\mathcal{HE}_{\mathrm{ell}}(G_{2})$ be
the set of non-trivial hyperendoscopic paths for $G_{2}$. Then, in the
notation of [Dal19, §4],

$I^{G_{2}}_{\mathrm{geom}}(\varphi_{k}\otimes
f^{\infty})=I^{G_{2}}_{\mathrm{geom}}(\eta_{k}\otimes
f^{\infty})+\sum_{H\in\mathcal{HE}_{\mathrm{ell}}(G_{2})}\iota(G,\mathcal{H})I^{\tilde{\mathcal{H}}}_{\mathrm{geom}}(((\eta_{k}-\varphi_{k})\otimes
f^{\infty})^{\tilde{\mathcal{H}}}),$

where the $\tilde{H}$ are choices of $z$-pair paths when they are needed.

####  3.2.2. Telescoping

Next, an unpublished result of Kottwitz summarized in [Mor10, §5.4] and proved
by other methods in [Pen19] stabilizes $I_{\mathrm{geom}}(\varphi\otimes
f^{\infty})$ when $\varphi$ satisfies a technical property of being stable-
cuspidal (as all terms on the right side above can be taken to be).

######  Theorem 3.2.1.

Let $\varphi$ be stable cuspidal on $G_{2}(\mathbb{R})$ and $f^{\infty}$ a
test function on $G(\mathbb{A}^{\infty})$. Then

$I^{G_{2}}_{\mathrm{geom}}(\varphi\otimes
f^{\infty})=\sum_{H\in\mathcal{E}_{\mathrm{ell}}(G_{2})}\iota(G,H)S^{\tilde{H}}_{\mathrm{geom}}((\varphi\otimes
f^{\infty})^{\tilde{H}}),$

where $\mathcal{E}_{\mathrm{ell}}(G_{2})$ is the set of elliptic endoscopic
groups for $G_{2}$ and the $\tilde{H}$ are $z$-extensions if necessary. The
transfers $(\varphi\otimes f^{\infty})^{\tilde{H}}$ depend on choices of
measures for $G$ and $H$.

The $S_{\mathrm{geom}}$ terms are defined by their values on Euler-Poincaré
functions:

$S^{H}_{\mathrm{geom}}(\eta_{\lambda}\otimes
f^{\infty})=\sum_{M\in\mathcal{L}^{\mathrm{cusp}}(H)}(-1)^{[H:M]}\frac{|\Omega_{M,F}|}{|\Omega_{H,F}|}\tau(M)\\\
\times\sum_{\gamma\in[M(\mathbb{Q})]_{\operatorname{\mathrm{st}},\mathrm{ell}^{\infty}}}|\iota^{M}(\gamma)|^{-1}\frac{e(\bar{M}_{\gamma,\infty})}{\operatorname{vol}(\bar{M}_{\gamma,\infty}/A_{\bar{M}_{\gamma},\infty})}\frac{k(M_{\infty})}{k(H_{\infty})}\Phi^{H}_{M}(\gamma,\lambda)SO_{\gamma}^{\infty}((f^{\infty})_{M}),$

choosing Tamagawa globally measure on all centralizers. The volume on
$\bar{M}_{\gamma,\infty}$ is transferred from that on $M_{\gamma,\infty}$ in
the standard way for inner forms so that the entire term doesn’t depend on a
choice of measure at infinity.

There’s an alternating sign in the hyperendoscopy formula: if $\mathcal{H}$ is
a hyperendoscopic path, then
$-\iota(G,\mathcal{H})\iota(\mathcal{H},H)=\iota(G,(\mathcal{H},H))$ for $H$
any endoscopic group of $\mathcal{H}$. Here, $(\mathcal{H},H)$ represents the
concatenation and $\mathcal{H}$ is overloaded to also refer to the last group
in $\mathcal{H}$.

In particular, substituting in the stabilization telescopes the hyperendoscopy
formula.

####  3.2.3. Final Geometric Formulas and Method of Computation

The final telescoped formula is:

(2) $I^{G_{2}}_{\mathrm{geom}}(\varphi_{k}\otimes
f^{\infty})=S^{G_{2}}_{\mathrm{geom}}(\eta_{k}\otimes
f^{\infty})+\sum_{\begin{subarray}{c}H\in\mathcal{E}_{\mathrm{ell}}(G_{2})\\\
H\neq
G_{2}\end{subarray}}\iota(G,H)S^{\tilde{H}}_{\mathrm{geom}}((\varphi_{k}\otimes
f^{\infty})^{\tilde{H}}).$

We recall:

$\iota(G,H)=|\Lambda(H,\mathcal{H},s,\eta)|^{-1}\frac{\tau(G)}{\tau(H)},$

where $\Lambda(H,\mathcal{H},s,\eta)$ is the image in
$\operatorname{Out}(\widehat{H})$ of the automorphisms of the endoscopic
quadruple.

There are two possible methods to proceed here. We will be using method 2 and
mention method 1 in case it is useful for anyone attempting a similar
computation on another group.

Method 1:

We can try calculate the $S_{\mathrm{geom}}$ terms directly from their
formula. We will need to choose Euler-Poincaré measure at $\bar{M}_{\gamma}$
times canonical measure for the orbital integrals (canonical measure is the
same for all inner forms). This adds an extra factor of

$d(\bar{M}_{\gamma,\infty})\frac{L(\mathrm{Mot}_{M_{\gamma}})}{e(\bar{M}_{\gamma,\infty})2^{\operatorname{rank}(M_{\gamma,\infty})}}$

by [ST16, lem. 6.2]. Since $d(H_{\infty})=1$ and
$\operatorname{vol}_{EP}(H_{\infty}/A_{H_{\infty}})=1$ for $H$ compact, this
expands the terms in (2) as:

$S^{H}_{\mathrm{geom}}(\eta_{\lambda}\otimes
f^{\infty})=\sum_{M\in\mathcal{L}^{\mathrm{cusp}}(H)}\left((-1)^{[H:M]}\frac{|\Omega_{M,F}|}{|\Omega_{H,F}|}\right)\left(\tau(M)\frac{k(M_{\infty})}{k(H_{\infty})}\right)\\\
\times\sum_{\gamma\in[M(\mathbb{Q})]_{\operatorname{\mathrm{st}},\mathrm{ell}^{\infty}}}2^{-\operatorname{rank}(M_{\gamma,\infty})}\Phi^{H}_{M}(\gamma,\lambda)\left(L(\mathrm{Mot}_{M_{\gamma}})|\iota^{M}(\gamma)|^{-1}SO_{\gamma}^{\infty}((f^{\infty})_{M})\right),$

where the stable orbital integrals are now computed using canonical measure on
centralizers.

The hardest terms here are the stable orbital integrals, the $L$-values, and
the characters $\Phi$. Note that since we are using
$f^{\infty}=\mathbf{1}_{K^{\infty}}$, the constant terms $(f^{\infty})_{M}$
are also indicator functions of hyperspecials.

The $L$-values may be computed as products of values of Artin $L$-functions by
explicitly describing the motives from [Gro97]. The terms $\Phi$ can be
reduced to linear combinations traces of $\gamma$ against finite dimensional
representations of $G_{2}$ by the algorithm on [Art89, pg. 273]. These can of
be computed by the Weyl character formula and its extension to irregular
elements stated in, for example, [CR15, prop. 2.3].

The stable orbital integrals unfortunately cause far more difficulty. They are
computed and listed in a table on [GP05, pg. 159]. First, they are interpreted
as orbital integrals on compact-at-infinity $G_{2}^{c}$. The spectral side of
the trace formula on $G_{2}^{c}$ is then possible to compute, allowing the
orbital integrals to be solved for once the coefficients in terms of
$L$-values are known.

Even using the previous work of [GP05], this method is horrendously
complicated.

Method 2:

Fortunately, there is a much simpler way to compute our desired count.
Recalling that $I^{G_{2}^{c}}$ is known from [CR15], we can compare the
expansions (2) for $G_{2}$ and $G_{2}^{c}$. The term for $S^{G_{2}}$ a appears
in the expansion for $I^{G_{2}^{c}}$ and can therefore be solved for and
substituted in the expansion for $I^{G_{2}}$. In total we get a formula

$I^{G_{2}}=I^{G_{2}^{c}}+\text{corrections},$

where the corrections are in terms of $S^{H}$ for smaller endoscopic $H$.

In the next section we will see that there aren’t actually that many $H$
appearing. Finally, section 6 will show that the terms for these $H$ are
easily computed through another trick. Method 2 also gives in section 7 a
Jacquet-Langlands-style result comparing quaternionic representations on
$G_{2}$ to representations on $G_{2}^{c}$.

##  4\. Groups Contributing and Related Constants

###  4.1. Elliptic Endoscopy of $G_{2}$

The elliptic endoscopic groups of $G_{2}$ are $G_{2}$, $\mathrm{PGL}_{3}$,
$\mathrm{SO}_{4}$, and potentially some tori. This is stated in a thesis
[Alt13] but not fully explained, so we work out the computation here for
reader convenience. We use notational conventions for endoscopy as in [Dal19,
§3].

By inspecting the root data, the conjugacy classes of centralizers
$\widehat{H}$ of a semisimple element in $\widehat{G}_{2}\simeq
G_{2}(\mathbb{C})$ are:

  * •

$G_{2}$ itself,

  * •

$\mathrm{SL}_{3}$ from the long roots,

  * •

$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\\{\pm 1\\}$ from a short and long root
that are orthogonal,

  * •

$\mathrm{GL}_{2}$ from a short root,

  * •

$\mathrm{GL}_{2}$ from a long root,

  * •

$\mathbb{G}_{m}^{2}$.

We compute the possible endoscopic pairs $(s,\rho)$ for each possibility.
Recall that, since $G_{2}$ is split, $\rho$ is a map from a Galois group to
$\operatorname{Out}(\widehat{H})\cap\Omega_{G_{2}}$.

Since $G_{2}$ has trivial center, the cohomology condition on $s$ is always
satisfied so we don’t bother checking it. Trivial center further gives that
the isomorphism class of the pair cannot change with $s$. Therefore the only
thing that depends on $s$ is whether we can exhibit one that is Galois
invariant when $\rho$ is non-trivial.

For each pair we will also compute the automorphism group $\Lambda$ that comes
up in the formula for $\iota(G,H)$.

$G_{2}$:

Then $\rho$ is trivial. This gives the trivial endoscopic group $G_{2}$. Since
only the trivial element of $\operatorname{Out}(\widehat{H})$ can be realized
in $\Omega_{G_{2}}$, $\Lambda=1$.

$\mathrm{SL}_{3}$:

There are two possibilities for $\rho$: trivial or sending a quadratic Galois
element to the outer automorphism of $\mathrm{SL}_{3}$:
$(\delta_{1},\delta_{2})\mapsto(\delta_{1},-\delta_{2})$ (fixing a long root).
We are forced to choose $s$ so that without loss of generality
$(\delta_{1}+\delta_{2})(s)=(\delta_{1}-\delta_{2})(s)=\zeta_{3}$ (two short
roots of $\widehat{G}_{2}$ at $120^{\circ}$). This is preserved by only the
trivial $\rho$.

This gives endoscopic group $\mathrm{PGL}_{3}$. Here,
$\operatorname{Out}(\widehat{H})$ is realized in $\Omega_{G_{2}}$ and commutes
with $\rho$ so $|\Lambda|=2$.

$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\\{\pm 1\\}$:

No outer automorphisms can be realized through conjugation in
$\widehat{G}_{2}$ so $\rho$ is trivial. This gives endoscopic group
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\\{\pm 1\\}$. Since
$\operatorname{Out}(\widehat{H})\cap\Omega_{G_{2}}$ is trivial, $\Lambda=1$.

Short $\mathrm{GL}_{2}$:

Without loss of generality assume the short root is $2\delta_{2}$. To be
elliptic $\rho$ needs to send a quadratic Galois element to the outer
automorphism $(\delta_{1},\delta_{2})\mapsto(-\delta_{1},\delta_{2})$ of
$\mathrm{GL}_{2}$. To have the right centralizer, $2\delta_{1}(s)=\alpha$ for
some $\alpha\neq 1$ and $2\delta_{2}(s)=1$. However, then
$(\delta_{1}+\delta_{2})(s)=\pm\sqrt{\alpha}$ which can’t be equal to it’s
inverse. Therefore $s$ can’t invariant under $\rho$. In total, there are no
such elliptic endoscopic groups.

Long $\mathrm{GL}_{2}$:

This is the same as the previous case and gives no elliptic endoscopy.

$\mathbb{G}_{m}^{2}$:

Here $\rho$ send Galois elements to any element of $\Omega_{G_{2}}$. Elliptic
means the action can have no invariants except zero in $X^{*}(T)$. To find
which $\rho$ have an invariant, regular $s$, we look through the possible
images of Galois: conjugacy classes of subgroups of $D_{12}$ that don’t fix
any line.

  * •

$C_{2}$: generated by $\alpha\mapsto-\alpha$: Then an invariant $s$ needs to
evaluate on all roots to $-1$ which is impossible.

  * •

$C_{3}$: Then three short roots in an orbit need to evaluate to the same value
on $s$. The other three short roots evaluate to the square, so we need
$\alpha_{1}(s)^{-1}=\alpha_{1}(s)^{2}\implies\alpha_{1}(s)=\zeta_{3}$, which
means that long roots evaluate to $1$, which is impossible.

  * •

$D_{4}$: impossible since $C_{2}$ is.

  * •

$D_{6}$: impossible since $C_{3}$ is

  * •

$C_{6}$: impossible since $C_{2},C_{3}$ are.

  * •

$D_{12}$: impossible since $C_{2},C_{3}$ are.

Therefore none are elliptic endoscopic groups.

If a group contributes to the stabilization applied to our test function, then
by the fundamental lemma, it needs to be unramified away from infinity. By
formulas for transfers of pseudocoefficients (see [Dal19, lem. 5.6.1]), it
needs to have an elliptic maximal torus at infinity. The only groups
contributing are therefore the $G_{2}$ and the
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\\{\pm 1\\}$.

###  4.2. Endoscopic Constants and Normalizations

####  4.2.1. The $\iota$

Let $H=\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\pm 1$ and let $G_{2}^{c}$ be the
unique non-split inner form of $G_{2}$ over $\mathbb{Q}$ which is compact at
infinity. Then,

$\displaystyle\iota(G_{2}^{c},H)=\iota(G_{2},H)=|\Lambda(H,\mathcal{H},s,\eta)|^{-1}\frac{\tau(G)}{\tau(H)}=1\cdot\frac{1}{2},$
$\displaystyle\iota(G_{2}^{c},G_{2})=1,$

by Kottwitz’s formula for Tamagawa numbers (note that
$\ker^{1}(\mathbb{Q},Z_{H})=\ker^{1}(\mathbb{Q},\\{\pm 1\\})=1$).

####  4.2.2. The transfer factors

We also need to fix transfer factors at all places to compute transfers. The
computations in [Taï17] demonstrate how to do so explicitly. First, they can
be chosen consistently by fixing a global Whittaker datum. The corresponding
local Whittaker datum determine the local transfer factors as in [KS99]. Since
$G_{2}$ is defined over $\mathbb{Z}$, we can choose a global datum that is
unramified/admissible at all finite places with respect to the
$G_{2}(\mathbb{Z}_{p})$ as in [Hal93, §7], so we can use the fundamental lemma
at all finite places.

All we will need to know about the Archimedean Whittaker datum for $G_{2}$ is
which element of $\Pi_{\mathrm{disc}}((k-2)\beta)$ it makes Whittaker-generic.
This will have to be $\pi_{(k-2)\beta,1}$ since our choice of dominant Weyl
chamber has all simple roots non-compact and is the only possible such choice
up to $\Omega_{K}$ (see the discussion before lemma 4.2.1 in [Taï17]. In fact,
there is only one possible conjugacy class of Whittaker datum at infinity by
considerations explained there).

####  4.2.3. The stabilizations

We fix canonical measure at finite places so that the fundamental lemma
directly gives
$\mathbf{1}_{K^{\infty}_{G_{2}}}^{H}=\mathbf{1}_{K^{\infty}_{H}}$. Recall that
EP-functions and pseudocoefficients are defined depending on measure so we
don’t need to fix measure at infinity.

Then, (2) gives

(3)
$I^{G_{2}}(\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)}\otimes\mathbf{1}_{K^{\infty}_{G_{2}}})\\\
=S^{G_{2}}(\eta^{G_{2}}_{(k-2)\beta}\otimes\mathbf{1}_{K^{\infty}_{G_{2}}})+\frac{1}{2}S^{H}((\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)})^{H}\otimes\mathbf{1}_{K^{\infty}_{H}}).$

A simple case of the discrete transfer formula in [Lab11, §IV.3] computes that
$(\eta^{G_{2}^{c}}_{(k-2)\beta})^{G_{2}}=\eta^{G_{2}}_{(k-2)\beta}$ (note that
$\Omega_{\mathbb{R}}(G_{2}^{c})\backslash\Omega_{\mathbb{C}}(G_{2}^{c})$ is
trivial so $\kappa$ is too), so

$I^{G_{2}^{c}}(\eta^{G_{2}^{c}}_{(k-2)\beta}\otimes\mathbf{1}_{K^{\infty}_{G_{2}^{c}}})=S^{G_{2}}(\eta^{G_{2}}_{(k-2)\beta}\otimes\mathbf{1}_{K^{\infty}_{G_{2}}})+\frac{1}{2}S^{H}((\eta^{G^{c}_{2}}_{(k-2)\beta})^{H}\otimes\mathbf{1}_{K^{\infty}_{H}}).$

Since type $A_{1}\times A_{1}$ has no non-trivial centralizer of full
semisimple rank, all elliptic endoscopy of
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\pm 1$ is non-split. Therefore, it is
ramified at some prime, so the transfers of $\mathbf{1}_{K^{\infty}_{H}}$
vanish, implying that $S^{H}=I^{H}$ on our test functions. Substituting one
stabilization into another finally gives:

(4)
$I^{G_{2}}(\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)}\otimes\mathbf{1}_{K^{\infty}_{G_{2}}})=I^{G_{2}^{c}}(\eta^{G_{2}^{c}}_{(k-2)\beta}\otimes\mathbf{1}_{K^{\infty}_{G_{2}^{c}}})\\\
-\frac{1}{2}I^{H}((\eta^{G^{c}_{2}}_{(k-2)\beta})^{H}\otimes\mathbf{1}_{K^{\infty}_{H}})+\frac{1}{2}I^{H}((\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)})^{H}\otimes\mathbf{1}_{K^{\infty}_{H}})$

under canonical measure at finite places.

This is our realization of method 2. There are three steps remaining to get
counts:

  1. (1)

Compute the transfers of EP-functions to $H$.

  2. (2)

Write the resulting $I^{H}(\eta_{\lambda}\otimes\mathbf{1}_{K_{H}})$ terms in
terms of counts of level-$1$, classical modular forms.

  3. (3)

Look up values for the $G_{2}^{c}$-term from [CR15].

##  5\. Real Endoscopic Transfers

Let $H$ again be the one endoscopic group we care about:
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\\{\pm 1\\}$. We want to compute
$(\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)})^{H}$ and
$(\eta^{G_{2}^{c}}_{(k-2)\beta})^{H}$. By our choice of transfer factors, we
may do so by the formulas in [Lab11, §IV.3].

As a choice for computation that doesn’t affect the final result, we realize
the roots of $H$ as $2\epsilon_{1}$ and $2\epsilon_{2}$. Orient $X^{*}(T)$ by
setting the 1st quadrant in $\epsilon_{1}$ and $\epsilon_{2}$ to be dominant.
The Weyl elements $\Omega(G,H)$ that send the $G$-dominant chamber to an
$H$-dominant one are $\\{1,s_{\alpha_{1}},s_{\alpha_{2}}\\}$.

###  5.1. Root Combinatorics

Since $\rho_{G}-\rho_{H}\in X^{*}(T)$, [Lab11, §IV.3] gives the transfer of
the pseudocoefficient of the quaternionic discrete series to $H$:

(5) $(\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)})^{H}=\\\
\kappa^{H}(s_{\alpha_{2}}^{-1})\eta^{H}_{(k-2)\beta+\rho_{G}-\rho_{H}}-\kappa^{H}(s_{\alpha_{1}}s_{\alpha_{2}}^{-1})\eta^{H}_{s_{\alpha_{1}}((k-2)\beta+\rho_{G})-\rho_{H}}-\eta^{H}_{s_{\alpha_{2}}((k-2)\beta+\rho_{G})-\rho_{H}}$

for some signs $\kappa$.

We compute that $\rho_{H}=\epsilon_{1}+\epsilon_{2}$. Then

$(k-2)\beta+\rho_{G}-\rho_{H}=(k-2)(3\epsilon_{1}+\epsilon_{2})+(3\epsilon_{1}+\epsilon)=3(k-1)\epsilon_{1}+(k-1)\epsilon_{2}.$

In addition,

$\displaystyle s_{\alpha_{1}}\rho_{G}=5\epsilon_{1}+\epsilon_{2},$
$\displaystyle\qquad s_{\alpha_{1}}\beta=\beta,$ $\displaystyle
s_{\alpha_{2}}\rho_{G}=\epsilon_{1}+3\epsilon_{2},$ $\displaystyle\qquad
s_{\alpha_{2}}\beta=2\epsilon_{2},$

so

$s_{\alpha_{1}}((k-2)\beta+\rho_{G})-\rho_{H}=(k-2)(3\epsilon_{1}+\epsilon_{2})+(4\epsilon_{1})=(3k-2)\epsilon_{1}+(k-2)\epsilon_{2}$

and

$s_{\alpha_{2}}((k-2)\beta+\rho_{G})-\rho_{H}=(k-2)(2\epsilon_{2})+(2\epsilon_{2})=2(k-1)\epsilon_{2}.$

###  5.2. Endoscopic Characters

It remains to compute the $\kappa$ terms in 5. These signs depend in a very
complicated way on the realization of $H$ and the exact transfer factors
chosen. They can be pinned down most easily by looking at endoscopic character
identities.

Let $\psi_{H}$ be a (discrete in our case) $L$-parameter for $H(\mathbb{R})$
and $\psi_{G}$ the composition with
${}^{L}\\!{H}\hookrightarrow{}^{L}\\!{G}_{2}$. The we have an identity of
traces over $L$-packets:

$S\Theta_{\psi_{H}}(f^{H})=\sum_{\pi\in\Pi_{\psi_{G}}}\langle\varphi_{H},\pi\rangle\Theta_{\pi}(f),$

where $f^{H}$ is a transfer of $f$, $\Theta_{\pi}$ is the Harish-Chandra
character, $S\Theta_{\psi_{H}}$ is the stable character corresponding to the
$L$-packet, $\Pi_{\varphi_{G}}$ is the $L$-packet corresponding to the
$L$-parameter, and $\langle\varphi_{H},\pi\rangle$ is a particular pairing
depending on transfer factors. See [Kal16, §1] for an exposition of how this
works in general.

If $\pi$ on $G_{2}$ is discrete series, Labesse’s formula tells us:

$(\varphi^{G_{2}}_{\pi})^{H}=\sum_{\lambda}\epsilon(\lambda,\pi)\eta^{H}_{\lambda}$

for some signs $\epsilon$ and weights $\lambda$. Let $\psi_{\lambda}$ be the
$L$-parameter corresponding to weight-$\lambda$ discrete series on $H$.
Plugging this formula into the character identity for $\psi_{\lambda}$ gives
that $\psi_{\lambda}$ is required to push forward to the parameter for $\pi$
and that $\epsilon(\lambda,\pi)=\langle\psi_{\lambda},\pi\rangle$.

The only fact we need now is that
$\epsilon(\lambda,\pi)=\langle\psi_{\lambda},\pi\rangle=1$ whenever $\pi$ is
the Whittaker-generic member of its $L$-packet. Therefore, in Labesse’s
formula for the _generic_ member $\pi_{1,(k-2)\beta}$,

$(\varphi_{\pi_{G_{2}}(1,(k-2)\beta)})^{H}=\eta^{H}_{(k-2)\beta+\rho_{G}-\rho_{H}}+\kappa^{H}(s_{\alpha_{1}})\operatorname{sgn}(s_{\alpha_{1}})\eta^{H}_{s_{\alpha_{1}}((k-2)\beta+\rho_{G})-\rho_{H}}\\\
+\kappa^{H}(s_{\alpha_{2}})\operatorname{sgn}(s_{\alpha_{2}})\eta^{H}_{s_{\alpha_{2}}((k-2)\beta+\rho_{G})-\rho_{H}},$

all the coefficients need to be $1$. The allows to solve

$\kappa^{H}(s_{\alpha_{1}})=\kappa^{H}(s_{\alpha_{2}})=-1$

for our choice of transfer factors. Right-$\Omega_{\mathbb{R}}$-invariance of
$\kappa$ then also gives that

$\kappa^{H}(s_{\alpha_{1}}s_{\alpha_{2}})=-1.$

###  5.3. Final Formulas for Transfers

Therefore, our final transfer is

(6)
$(\varphi_{\pi_{G_{2}}(s_{\alpha_{2}},(k-2)\beta)})^{H}=-\eta^{H}_{3(k-1)\epsilon_{1}+(k-1)\epsilon_{2}}+\eta^{H}_{(3k-2)\epsilon_{1}+(k-2)\epsilon_{2}}-\eta^{H}_{2(k-1)\epsilon_{2}}.$

Transfers from $G_{2}^{c}$ are easier. Here,
$\Omega_{\mathbb{R}}(G_{2}^{c})\backslash\Omega_{\mathbb{C}}(G_{2}^{c})$ is
trivial so the average value of $\kappa$ is $1$. Averaging Labesse’s formula
as in [Dal19, cor. 5.1.5] therefore gives:

(7)
$(\eta^{G_{2}^{c}}_{(k-2)\beta})^{H}=\eta^{H}_{3(k-1)\epsilon_{1}+(k-1)\epsilon_{2}}-\eta^{H}_{(3k-2)\epsilon_{1}+(k-2)\epsilon_{2}}-\eta^{H}_{2(k-1)\epsilon_{2}}.$

##  6\. The $H=\mathrm{SL}_{2}\times\mathrm{SL}_{2}/\pm 1$ term

Here we compute the terms $I^{H}(\eta_{\lambda}\otimes\mathbf{1}_{K_{H}})$ for
Euler-Poincaré functions $\eta_{\lambda}$. Any
$\lambda=a\epsilon_{1}+b\epsilon_{2}$ is a weight of $H$ if $a+b$ is even.
Note first that

$I^{H}(\eta^{H}_{\lambda}\otimes\mathbf{1}_{K_{H}})=\sum_{\pi\in\mathcal{AR}_{\mathrm{disc}}(H)}\operatorname{tr}_{\pi_{\infty}}(\eta^{H}_{\lambda})\operatorname{tr}_{\pi^{\infty}}(\mathbf{1}_{K_{H}})=\sum_{\begin{subarray}{c}\pi\in\mathcal{AR}_{\mathrm{disc}}(H)\\\
\pi\text{
unram.}\end{subarray}}\operatorname{tr}_{\pi_{\infty}}(\eta^{H}_{\lambda}),$

by Arthur’s simple trace formula and using our choice of canonical measure at
finite places.

To move forward, we need to understand automorphic reps on $H$ by relating
them to other groups. Consider the sequence

$1\to\pm 1\to\mathrm{SL}_{2}\times\mathrm{SL}_{2}\to H\to 1.$

It induces on local or global $F$:

$1\to\pm 1\to\mathrm{SL}_{2}\times\mathrm{SL}_{2}(F)\to H(F)\to
F^{\times}/(F^{\times})^{2}\to 1,$

using that $H^{1}(F,\pm 1)=F^{\times}/(F^{\times})^{2}$ and
$H^{1}(F,\mathrm{SL}_{2})=1$ for the $F$ we care about (the $\mathbb{R}$ case
of the second equality comes from the determinant exact sequence on
$\mathrm{GL}_{2}$). Let $H^{\prime}_{F}$ be the image of
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}(F)$.

As noted in a similar analysis for $\mathrm{SL}_{2}$ in [LL79], unitary
irreducibles for $H^{\prime}_{F}$ induce to semisimple representations of
$H(F)$.

###  6.1. Cohomological Representations of $H(\mathbb{R})$

Next, we recall that the infinite trace measures an Euler characteristic
against $(\mathfrak{h},K_{H,\infty})$-cohomology:

$\operatorname{tr}_{\pi_{\infty}}(\eta^{H}_{\lambda})=\chi(H^{*}(\mathfrak{h},K_{H,\infty},\pi_{\infty}\otimes
V_{\lambda})),$

where $\mathfrak{h}$ is the Lie algebra of $H(\mathbb{R})$ and $V_{\lambda}$
is the finite dimensional representation of weight $\lambda$. Using the
definition from [BW00, §5.1],

$H^{*}(\mathfrak{h},K_{H,\infty},\pi_{\infty}\otimes
V_{\lambda})=H^{*}(\mathfrak{h},K^{0}_{H,\infty},\pi_{\infty}\otimes
V_{\lambda})^{K_{H,\infty}/K^{0}_{H,\infty}},$

it suffices to consider the $\pi_{\infty}$ whose restrictions to
$H^{\prime}_{\infty}$ contain a component that is cohomological when pulled
back to $[\mathrm{SL}_{2}\times\mathrm{SL}_{2}](\mathbb{R})$. By Frobenius
reciprocity and semisimplicity of inductions, these are exactly the
irreducible constituents of
$\operatorname{Ind}_{H^{\prime}_{\infty}}^{H_{\infty}}\pi^{\prime}$ for
$\pi^{\prime}$ cohomological of $H^{\prime}_{\infty}$.

Next, $H^{\prime}_{\infty}$ is index $2$. Pick $h\in
H_{\infty}-H^{\prime}_{\infty}$ and let $\pi^{\prime(h)}$ be the
representation $\gamma\mapsto\pi^{\prime}(h^{-1}\gamma h)$. Define character

$\chi:H_{\infty}\mapsto H_{\infty}/H^{\prime}_{\infty}=\pm 1.$

There are two cases for $H^{\prime}_{\infty}$-cohomological $\pi^{\prime}$:

  1. (1)

$\pi^{\prime}\neq\pi^{\prime(h)}$: then
$\operatorname{Ind}_{H^{\prime}_{\infty}}^{H_{\infty}}\pi^{\prime}$ is
irreducible, and
$\operatorname{Res}_{H^{\prime}_{\infty}}^{H_{\infty}}\operatorname{Ind}_{H^{\prime}_{\infty}}^{H_{\infty}}\pi^{\prime}=\pi^{\prime}\oplus\pi^{\prime(h)}$.

  2. (2)

$\pi^{\prime}=\pi^{\prime(h)}$: then
$\operatorname{Ind}_{H^{\prime}_{\infty}}^{H_{\infty}}\pi^{\prime}=V\oplus(V\otimes\chi)$
for some irreducible $V$. Only one of these factors will have a subspace fixed
by $K_{H,\infty}$ so only one of them will have fixed chains and therefore be
cohomological. Also,
$\operatorname{Res}_{H^{\prime}_{\infty}}^{H_{\infty}}\operatorname{Ind}_{H^{\prime}_{\infty}}^{H_{\infty}}\pi^{\prime}=\pi^{\prime}\oplus\pi^{\prime}$.

Recalling a standard result, the cohomological representations of
$\mathrm{SL}_{2}(\mathbb{R})$ with respect to $\lambda$ are:

  * •

A discrete series $L$-packet $\pi_{\lambda,1}$, $\pi_{\lambda,s}$ (where
$\Omega_{\mathrm{SL}_{2}}=\\{1,s\\}$),

  * •

The trivial representation $\mathbf{1}_{\mathrm{SL}_{2}}$ if $\lambda=0$.

By the Künneth rule, cohomological representations of
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}(\mathbb{R})$ are exactly products of
those on $\mathrm{SL}_{2}(\mathbb{R})$. Those of $H^{\prime}_{\infty}$ are
exactly the $\mathrm{SL}_{2}\times\mathrm{SL}_{2}(\mathbb{R})$ ones that are
trivial on $\pm 1$—in other words, with $\lambda=a\epsilon_{1}+b\epsilon_{2}$
and $a+b$ even.

Consider such $\lambda$. There are three cases of inductions to consider to
compute the cohomological representations of $H$. Note that conjugation by
$h\in H_{\infty}-H^{\prime}_{\infty}$ switches both factors to the other
member of their $\mathrm{SL}_{2}$-$L$-packet if they are discrete series and
otherwise fixes the trivial representation.

  * •

$a,b\neq 0$: We look at the inductions of products of discrete series. This is
case (1) so the $4$ products pair up in sums that are $2$ members of an
$L$-packet. These are of course $\pi^{H}_{\lambda,1}$ and
$\pi^{H}_{\lambda,s}$ where $s$ is a length-$1$ element of $\Omega_{H}$:

$\displaystyle\pi^{H}_{\lambda,1}|_{H^{\prime}_{\infty}}$
$\displaystyle=(\pi_{a\epsilon_{1},1}\boxtimes\pi_{b\epsilon_{2},1})\oplus(\pi_{a\epsilon_{1},s}\boxtimes\pi_{b\epsilon_{2},s}),$
$\displaystyle\pi^{H}_{\lambda,s}|_{H^{\prime}_{\infty}}$
$\displaystyle=(\pi_{a\epsilon_{1},1}\boxtimes\pi_{b\epsilon_{2},s})\oplus(\pi_{a\epsilon_{1},s}\boxtimes\pi_{b\epsilon_{2},1}).$

  * •

Without loss of generality, $a=0,b\neq 0$: We also need to consider inductions
of $\mathbf{1}\boxtimes\pi_{b\epsilon_{2},\star}$. This is case (1) and both
induce to a single irreducible $\sigma^{H}_{\lambda}$:

$\sigma^{H}_{\lambda}|_{H^{\prime}_{\infty}}=(\mathbf{1}\boxtimes\pi_{b\epsilon_{2},1})\oplus(\mathbf{1}\boxtimes\pi_{b\epsilon_{2},s}).$

  * •

$a=b=0$: In addition to both the above, we need to consider the induction of
$\mathbf{1}_{\mathrm{SL}_{2}}\boxtimes\mathbf{1}_{\mathrm{SL}_{2}}$. This is
case (2). This trivial representation induces to
$\mathbf{1}_{H_{\infty}}\oplus\chi$ on $H_{\infty}$. The cohomological piece
is $\mathbf{1}_{H_{\infty}}$.

Grothendieck group relations stay true restricted to $H^{\prime}_{\infty}$ so
we can compute traces against $\eta_{\lambda}$. Recall that in
$\mathrm{SL}_{2}(\mathbb{R})$:

$\mathbf{1}=I-\pi_{0,1}-\pi_{0,s},$

where $I$ is some non-cohomological parabolically induced representation.

First, by our normalization

$\operatorname{tr}_{\pi^{H}_{\lambda,1}}(\eta^{H}_{\lambda})=\operatorname{tr}_{\pi^{H}_{\lambda,s}}(\eta^{H}_{\lambda})=1/2.$

Next, working in $H^{\prime}_{\infty}$:

$\mathbf{1}\boxtimes\pi_{\lambda,\star}=(I-\pi_{0,1}-\pi_{0,s})\boxtimes\pi_{\lambda,\star}=I\boxtimes\pi_{\lambda,\star}-\pi_{0,1}\boxtimes\pi_{\lambda,\star}-\pi_{0,s}\boxtimes\pi_{\lambda,\star},$

so

$\sigma^{H}_{\lambda}=\mathbf{1}\boxtimes\pi_{\lambda,1}+\mathbf{1}\boxtimes\pi_{\lambda,s}=I\boxtimes(\pi_{\lambda,1}+\pi_{\lambda,s})-\pi^{H}_{0+\lambda,1}-\pi^{H}_{0+\lambda,s},$

implying

$\operatorname{tr}_{\sigma^{H}_{\lambda}}(\eta^{H}_{\lambda})=-1.$

Finally

$\mathbf{1}\boxtimes\mathbf{1}=(I-\pi_{0,1}-\pi_{0,s})\boxtimes(I-\pi_{0,1}-\pi_{0,s})\\\
=I\boxtimes I-I\boxtimes(\pi_{0,1}+\pi_{0,s})-(\pi_{0,1}+\pi_{0,s})\boxtimes
I+\pi^{H}_{0+0,1}+\pi^{H}_{0+0,s},$

so

$\operatorname{tr}_{\mathbf{1}}(\eta^{H}_{\lambda})=1.$

In total, our $H$-term becomes a count

$\sum_{\pi\in\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(H)}w(\pi_{\infty}),$

where $w$ is a weight

$w^{H}(\pi_{\infty})=\begin{cases}0&\pi_{\infty}\text{ not cohomological}\\\
1/2&\pi_{\infty}\text{ one of the }\pi^{H}_{\lambda,*}\\\
-1&\pi_{\infty}\text{ one of the }\sigma^{H}_{\lambda,*}\\\
1&\pi_{\infty}\text{ trivial}\end{cases}.$

Call the cohomological cases type I, II, and III in order.

###  6.2. Reduction to Modular Form Counts

We now recall a result from [CR15]. Consider central isogeny $G\to G^{\prime}$
of algebraic groups over $\mathbb{Z}$. If
$\pi^{\prime}=\pi^{\prime}_{\infty}\otimes\pi^{\prime\infty}$ is an
unramified, discrete automorphic representation of $G^{\prime}$, let
$R(\pi^{\prime})$ be the set of unitary, admissible representations
$\pi=\pi_{\infty}\otimes\pi^{\infty}$ of $G(\mathbb{A})$ that satisfy:

  * •

$\pi^{\infty}$ is unramified with Satake parameters induced from those of
$\pi^{\prime\infty}$ through $\widehat{G}^{\prime}\to\widehat{G}$.

  * •

$\pi_{\infty}$ is a constituent of the restriction of $\pi^{\prime}_{\infty}$
through $G(\mathbb{R})\to G^{\prime}(\mathbb{R})$.

Note that the size of $R(\pi^{\prime})$ is the number of constituents of the
restriction of $\pi^{\prime}_{\infty}$.

######  Theorem 6.2.1 ([CR15, cor. 4.10]).

Assume that all $\pi\in\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(G)$ have
multiplicity one. Then the same holds for $G^{\prime}$ and the
$R(\pi^{\prime})$ with
$\pi^{\prime}\in\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(G^{\prime})$
partition $\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(G)$.

Make similar definitions of type I, II, and III for representations of
$[\mathrm{SL}_{2}\times\mathrm{SL}_{2}](\mathbb{R})$ and
$[\mathrm{GL}_{2}\times\mathrm{GL}_{2}](\mathbb{R})$. Since type I and II on
$H$ decompose into two constituents in $\mathrm{SL}_{2}\times\mathrm{SL}_{2}$
and type III decomposes into $1$, our count becomes
$\pi\in\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(\mathrm{SL}_{2}\times\mathrm{SL}_{2})$
weighted by

$w^{\mathrm{SL}_{2}\times\mathrm{SL}_{2}}(\pi_{\infty})=\begin{cases}1/4&\pi_{\infty}\text{
type I}\\\ -1/2&\pi_{\infty}\text{ type II}\\\ 1&\pi_{\infty}\text{ type
III}\end{cases}.$

Each
$\pi\in\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(\mathrm{SL}_{2}\times\mathrm{SL}_{2})$
lifts to a rep of
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}\times\mathbb{G}_{m}^{2}$. This group is
further isogenous to $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ so we apply the
theorem again. Type I on $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ decomposes
into $4$ constituents on
$\mathrm{SL}_{2}\times\mathrm{SL}_{2}\times\mathbb{G}_{m}^{2}$, type II into
$2$, and type III into $1$. Therefore, we get a count of
$\pi\in\mathcal{AR}_{\mathrm{disc},\mathrm{ur}}(\mathrm{GL}_{2}\times\mathrm{GL}_{2})$
weighted by

$w^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\pi_{\infty})=\begin{cases}1&\pi_{\infty}\text{
type I}\\\ -1&\pi_{\infty}\text{ type II}\\\ 1&\pi_{\infty}\text{ type
III}\end{cases}.$

Let $\mathcal{S}_{k}(1)$ be the set of normalized, level-$1$, weight-$k$
cuspidal eigenforms. If $\lambda=a\epsilon_{1}+b\epsilon_{2}$, then type I
representations on $\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ correspond to pairs
in $S_{a+2}(1)\times S_{b+2}(1)$. Type II is a single form times the trivial
representation and Type I is only the trivial representation.

###  6.3. Final Formula for $S^{H}$

Therefore, if

$S_{k}=|\mathcal{S}_{k}(1)|,$

we get:

(8)
$I^{H}(\eta^{H}_{a\epsilon_{1}+b\epsilon_{2}}\otimes\mathbf{1}_{K_{H}})=(S_{a+2}-\mathbf{1}_{a=0})(S_{b+2}-\mathbf{1}_{b=0}),$

using canonical measure at finite places. By a classical formula ([DS05, Thm.
3.5.2] for example),

$S_{a+2}=\begin{cases}0&a+2=2\text{ or }a+2\text{ odd}\\\
\lfloor\frac{a+2}{12}\rfloor-1&a+2\equiv 2\pmod{12}\\\
\lfloor\frac{a+2}{12}\rfloor&\text{else}\\\ \end{cases}.$

##  7\. A Jacquet-Langlands-style result

###  7.1. First Form

Generalizing (4) slightly and substituting in (6) and (7) gives:

(9) $I^{G_{2}}(\varphi_{\pi_{k}}\otimes
f^{\infty})=I^{G_{2}^{c}}(\eta^{G_{2}^{c}}_{(k-2)\beta}\otimes
f^{\infty})-I^{H}(\eta^{H}_{(3k-3)\epsilon_{1}+(k-1)\epsilon_{2}}\otimes(f^{\infty})^{H})\\\
+I^{H}(\eta^{H}_{(3k-2)\epsilon_{1}+(k-2)\epsilon_{2}}\otimes(f^{\infty})^{H}).$

for any unramified function $f^{\infty}$ (we use here that
$(G_{2}^{c})^{\infty}=(G_{2})^{\infty}$). This will let us describe the set
$\mathcal{Q}_{k}(1)$ for $k>2$ in terms of certain representations of
$G_{2}^{c}$ and $H$.

Choose $\pi=\pi_{k}\otimes\pi^{\infty}\in\mathcal{Q}_{k}(1)$. Since
$\pi^{\infty}$ is unramified, it can be described by a sequence of Satake
parameters: for each prime $p$, a semisimple conjugacy class
$c_{p}(\pi^{\infty})\in[\widehat{G_{2}}]_{\mathrm{ss}}$ (note that $G_{2}$ is
split so we don’t need to worry about the full Langlands dual and see [ST16,
§3.2] for full background).

The endoscopic datum for $H$ also gives an embedding
$\widehat{H}\hookrightarrow\widehat{G_{2}}$ (noting again that everything is
split) whose image contains a chosen maximal torus and therefore induces a map

$T^{G_{2}}_{H}:[\widehat{H}]_{\mathrm{ss}}\twoheadrightarrow[\widehat{G_{2}}]_{\mathrm{ss}}.$

The fibers of this map are $\Omega_{G_{2}}$-orbits of conjugacy classes in $H$
and have size $3$ at $G_{2}$-regular elements.

######  Proposition 7.1.1.

Let $k>2$ and $\pi^{\infty}$ an unramified representation of
$(G_{2})^{\infty}$. Then

$m^{G_{2}}_{\mathrm{disc}}(\pi_{k}\otimes\pi^{\infty})=m^{G_{2}^{c}}_{\mathrm{disc}}(V_{(k-2)\beta}\otimes\pi^{\infty})-\frac{1}{2}|S^{H}(\pi^{\infty},(3k-3)\epsilon_{1}+(k-1)\epsilon_{2})|\\\
+\frac{1}{2}|S^{H}(\pi^{\infty},(3k-2)\epsilon_{1}+(k-2)\epsilon_{2})|.$

Recall here that $V_{\lambda}$ is the finite dimensional representation of
$G_{2}^{c}$ with highest weight $\lambda$. Also, $S^{H}(\pi^{\infty},\lambda)$
is the set of
$\pi_{\infty}\otimes\pi_{1}^{\infty}\in\mathcal{AR}_{\mathrm{disc}}(H)$ such
that:

  * •

$\pi_{\infty}\in\Pi^{H}_{\mathrm{disc}}(\lambda)$,

  * •

For all $p$,
$c_{p}(\pi_{1}^{\infty})\in(T^{G_{2}}_{H})^{-1}(c_{p}(\pi^{\infty}))$.

###### Proof.

This is a standard Jacquet-Langlands-style argument. Through the Satake
isomorphism, each $f_{p}$ can be thought of as a function
$[\widehat{G_{2}}]_{\mathrm{ss}}\to\mathbb{C}$ through
$f_{p}(c_{p}(\pi))=\operatorname{tr}_{\pi_{p}}(f_{p})$. It is in fact a Weyl-
invariant regular function on a maximal torus in $\widehat{G_{2}}$. The full
version of the fundamental lemma (see the introduction to [Hal95] for example)
shows that

$f_{p}^{H}(c_{p})=f_{p}(T^{G_{2}}_{H}(c_{p}))$

for all $c_{p}\in\widehat{H}$.

There are only finitely many sequences $c_{p}(\pi_{1}^{\infty})$ and
$T^{G_{2}}_{H}(c_{p}(\pi_{1}^{\infty}))$ for $\pi_{1}^{\infty}$ the unramified
finite component of an automorphic representation either:

  * •

of $G_{2}$ with infinite part $\pi_{k}$,

  * •

of $G_{2}^{c}$ with infinite part $V_{(k-2)\beta}$,

  * •

or of $H$ with infinite part in
$\Pi_{\mathrm{disc}}((3k-3)\epsilon_{1}+(k-1)\epsilon_{2})$ or
$\Pi_{\mathrm{disc}}((3k-2)\epsilon_{1}+(k-2)\epsilon_{2})$.

Therefore we can choose an $f^{\infty}$ that is $0$ on all of these sequences
$c_{p}(\pi_{1}^{\infty})$ except $1$ on exactly the sequence
$c_{p}(\pi^{\infty})$ (this reduces to finding Weyl-invariant polynomials on
$(\mathbb{C}^{\times})^{2}$ that take specified values on certain Weyl
orbits). The result follows from plugging this $f^{\infty}$ into (9) and
noting that $m^{H}_{\mathrm{disc}}(\pi)$ is always $0$ or $1$. ∎

###  7.2. In terms of Modular Forms

We can use the argument from section 6.2 to reduce the $H$-multiplicity terms
to $\mathrm{GL}_{2}$-multiplicty ones. Since we already got multiplicity $1$
from comparing $H$ to $\mathrm{SL}_{2}\times\mathrm{SL}_{2}$ there, it will
end up being more convenient here to compare $H$ to
$\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}$.

First, we have a map on conjugacy classes

$T^{H}_{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}:[\widehat{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}]_{\mathrm{ss}}\twoheadrightarrow[\widehat{H}]_{\mathrm{ss}}.$

Since the first group is $\mathrm{SL}_{2}\times\mathrm{SL}_{2}(\mathbb{C})$,
the fibers of this map are of the form $\\{c,-c\\}$ for some
$c\in[\mathrm{SL}_{2}\times\mathrm{SL}_{2}(\mathbb{C})]_{\mathrm{ss}}$.
Composing then gives map

$T^{G_{2}}_{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}:[\widehat{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}]_{\mathrm{ss}}\twoheadrightarrow[\widehat{G_{2}}]_{\mathrm{ss}}.$

This allows us to define
$S^{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}(\pi^{\infty},\lambda)$ analogous
to $S^{H}(\pi^{\infty},\lambda)$ for all $\lambda=a\epsilon_{1}+b\epsilon_{2}$
with both $a$ and $b$ even. For indexing purposes, set it to be empty when $a$
and $b$ aren’t even.

Formula (8) gives us that
$S^{H}(\pi^{\infty},a\epsilon_{1}+b\epsilon_{2})=\emptyset$ also when $a$ and
$b$ aren’t both even. In addition, the restriction of discrete series
$\pi^{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}_{\lambda}$ to $H(\mathbb{R})$
has as components the entire $L$-packet $\Pi_{\mathrm{disc}}^{H}(\lambda)$.
Therefore, theorem 6.2.1 shows that the
$R^{H}_{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}(\pi^{\prime})$ for
$\pi^{\prime}\in
S^{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}(\pi^{\infty},\lambda)$ partition
$S^{H}(\pi^{\infty},\lambda)$. Since
$R^{H}_{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}$ is two-to-one, this gives

$|S^{H}(\pi^{\infty},\lambda)|=2|S^{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}}(\pi^{\infty},\lambda)|.$

Finally, $\mathrm{PGL}_{2}$ is a quotient of $\mathrm{GL}_{2}$ by a central
torus with trivial Galois cohomology, so automorphic representations on
$\mathrm{PGL}_{2}$ are just those on $\mathrm{GL}_{2}$ with all components
having trivial central character. Recalling injection

$\iota:[\mathrm{SL}_{2}\times\mathrm{SL}_{2}(\mathbb{C})]_{\mathrm{ss}}\hookrightarrow[\mathrm{GL}_{2}\times\mathrm{GL}_{2}(\mathbb{C})]_{\mathrm{ss}},$

this gives:

######  Corollary 7.2.1.

Let $k>2$ and $\pi^{\infty}$ an unramified representation of
$(G_{2})^{\infty}$. Then

$m^{G_{2}}_{\mathrm{disc}}(\pi_{k}\otimes\pi^{\infty})=m^{G_{2}^{c}}_{\mathrm{disc}}(V_{(k-2)\beta}\otimes\pi^{\infty})-|S^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\pi^{\infty},(3k-3)\epsilon_{1}+(k-1)\epsilon_{2})|\\\
+|S^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\pi^{\infty},(3k-2)\epsilon_{1}+(k-2)\epsilon_{2})|.$

Recall here that $V_{\lambda}$ is the finite dimensional representation of
$G_{2}^{c}$ with highest weight $\lambda$. Also,
$S^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\pi^{\infty},\lambda)$ is the set of
$\pi_{\infty}\otimes\pi_{1}^{\infty}\in\mathcal{AR}_{\mathrm{disc}}(\mathrm{GL}_{2}\times\mathrm{GL}_{2})$
such that:

  * •

$\pi_{\infty}$ is the discrete series
$\pi^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}_{\lambda}$,

  * •

For all $p$, $c_{p}(\pi_{1}^{\infty})=\iota(c^{\prime}_{p})$ for some
$c^{\prime}_{p}\in(T^{G_{2}}_{\mathrm{PGL}_{2}\times\mathrm{PGL}_{2}})^{-1}(c_{p}(\pi^{\infty}))$.
Here $\iota$ is the map
$[\mathrm{SL}_{2}\times\mathrm{SL}_{2}(\mathbb{C})]_{\mathrm{ss}}\hookrightarrow[\mathrm{GL}_{2}\times\mathrm{GL}_{2}(\mathbb{C})]_{\mathrm{ss}}$.

Of course, since all infinite terms in sight are discrete series, we may again
replace the $m_{\mathrm{disc}}$ by $m_{\mathrm{cusp}}$ using [Wal84].

Note of course that
$S^{\mathrm{GL}_{2}\times\mathrm{GL}_{2}}(\pi^{\infty},a\epsilon_{1}+b\epsilon_{2})=\emptyset$
unless both $a$ and $b$ are even. Therefore, we can interpret this as, for
$k>2$:

  * •

If $k$ is even: $\mathcal{Q}_{k}(1)$ is the corresponding set of
representations transferred from $G_{2}^{c}$ _in addition to_ representations
transferred from pairs of cuspidal eigenforms in
$\mathcal{S}_{3k-2}(1)\times\mathcal{S}_{k-2}(1)$.

  * •

If $k$ is odd: $\mathcal{Q}_{k}(1)$ is the corresponding set of
representations transferred from $G_{2}^{c}$ _except for_ representations that
are also transferred from pairs of cuspidal eigenforms in
$\mathcal{S}_{3k-3}(1)\times\mathcal{S}_{k-1}(1)$.

Results for level $>1$ would be a lot more complicated since formula (4) would
have many further hyperendoscopic terms and the comparison to
$\mathrm{GL}_{2}\times\mathrm{GL}_{2}$ would not work as nicely.

##  8\. Counts of forms

###  8.1. Formula in terms of $I^{G_{2}^{c}}$

To get counts instead of a list, combining formulas (1),(9), and (8) gives
that

$|\mathcal{Q}_{k}(1)|=I^{G_{2}^{c}}(\eta^{G_{2}^{c}}_{\lambda}\otimes\mathbf{1}_{K^{\infty}_{G_{2}^{c}}})-(S_{3k-1}-\mathbf{1}_{3k-3=0})(S_{k+1}-\mathbf{1}_{k-1=0})\\\
+(S_{3k}-\mathbf{1}_{3k-2=0})(S_{k}-\mathbf{1}_{k-2=0}).$

This finally becomes, for $k>2$:

(10)
$|\mathcal{Q}_{k}(1)|=I^{G_{2}^{c}}(\eta_{\lambda}\otimes\mathbf{1}_{K^{\infty}_{G_{2}^{c}}})\\\
+\begin{cases}\lfloor\frac{k}{4}\rfloor\left(\lfloor\frac{k}{12}\rfloor-1\right)&k\equiv
2\pmod{12}\\\ \lfloor\frac{k}{4}\rfloor\lfloor\frac{k}{12}\rfloor&k\equiv
0,4,6,8,10\pmod{12}\\\
-\left(\lfloor\frac{3k-1}{12}\rfloor-1\right)\left(\lfloor\frac{k+1}{12}\rfloor-1\right)&k\equiv
1\pmod{12}\\\
-\left(\lfloor\frac{3k-1}{12}\rfloor-1\right)\lfloor\frac{k+1}{12}\rfloor&k\equiv
5,9\pmod{12}\\\
-\lfloor\frac{3k-1}{12}\rfloor\lfloor\frac{k+1}{12}\rfloor&k\equiv
3,7,11\pmod{12}\\\ \end{cases}.$

###  8.2. Computing $I^{G_{2}^{c}}$

The group $G_{2}^{c}(\mathbb{R})$ is compact so the $I^{G_{2}^{c}}$ term takes
a very simple form: $L^{2}(G_{2}^{c}(\mathbb{Q})\backslash
G_{2}^{c}(\mathbb{A}))$ decomposes as a direct sum of automorphic
representations and the EP-functions $\eta_{\lambda}$ are just scaled matrix
coefficients of the finite-dimensional representations $V_{\lambda}$ with
highest weight $\lambda$ on $G_{2}^{c}(\mathbb{R})$. Therefore

$I^{G_{2}^{c}}(\eta_{\lambda}\otimes\mathbf{1}_{K^{\infty}_{G_{2}^{c}}})=\sum_{\pi\in\mathcal{AR}(G_{2}^{c})}\mathbf{1}_{\pi_{\infty}=V_{\lambda}}\operatorname{tr}_{\pi^{\infty}}(\mathbf{1}_{K^{\infty}_{G_{2}^{c}}}),$

which is just counting the number of unramifed automorphic reps of $G_{2}^{c}$
that have infinite component $V_{\lambda}$.

By standard results on unramified representations, taking
$K^{\infty}_{G_{2}^{c}}$ invariants sends each such $\pi$ to a linearly
independent copy of $V_{\lambda}$ that together span the
$V_{\lambda}$-isotypic component of

$L^{2}(G_{2}^{c}(\mathbb{Q})\backslash
G_{2}^{c}(\mathbb{A})/K^{\infty}_{G_{2}^{c}})=L^{2}(G_{2}^{c}(\mathbb{Z})\backslash
G_{2}^{c}(\mathbb{R}))\subseteq L^{2}(G_{2}^{c}(\mathbb{R})).$

By Peter-Weyl, $L^{2}(G_{2}^{c}(\mathbb{R}))$ has $V_{\lambda}$-isotypic
component $V_{\lambda}^{\oplus\dim V_{\lambda}}$. In fact, this component for
both the left- and right-actions is the same subspace. Therefore the number of
copies of $V_{\lambda}\subseteq L^{2}(G_{2}^{c}(\mathbb{Z})\backslash
G_{2}^{c}(\mathbb{R}))$ is
$\dim\left(V_{\lambda}^{G_{2}^{c}(\mathbb{Z})}\right)$ by a dimension count.

Summarizing:

(11)
$I^{G_{2}^{c}}(\eta_{\lambda}\otimes\mathbf{1}_{K^{\infty}_{G_{2}^{c}}})=\dim\left(V_{\lambda}^{G_{2}^{c}(\mathbb{Z})}\right).$

A PARI/GP 2.5.0 program in the online appendix to [CR15] computes this for all
$\lambda$ by pairing the trace character of $V_{\lambda}|_{G_{2}(\mathbb{Z})}$
with the trivial character.

###  8.3. Table of Counts

Table 1 gives values of $|\mathcal{Q}_{k}(1)|$ for $k=3$ to $52$ produced by
formula (10) and [CR15]’s table for formula (11). The lowest-weight example is
bolded, although this work does not rule out the existence of an example with
weight $2$ or weight $1$ (as defined by [Pol20, §1.1]).

Table 1. Counts of discrete, quaternionic automorphic representations of level $1$ on $G_{2}$. $k$ | $|\mathcal{Q}_{k}(1)|$ | $k$ | $|\mathcal{Q}_{k}(1)|$ | $k$ | $|\mathcal{Q}_{k}(1)|$ | $k$ | $|\mathcal{Q}_{k}(1)|$ | $k$ | $|\mathcal{Q}_{k}(1)|$
---|---|---|---|---|---|---|---|---|---
3 | 0 | 13 | 5 | 23 | 76 | 33 | 478 | 43 | 1792
4 | 0 | 14 | 13 | 24 | 126 | 34 | 610 | 44 | 2112
5 | 0 | 15 | 8 | 25 | 121 | 35 | 637 | 45 | 2250
6 | 1 | 16 | 23 | 26 | 175 | 36 | 807 | 46 | 2619
7 | 0 | 17 | 17 | 27 | 173 | 37 | 849 | 47 | 2790
8 | 2 | 18 | 37 | 28 | 248 | 38 | 1037 | 48 | 3233
9 | 1 | 19 | 30 | 29 | 250 | 39 | 1097 | 49 | 3447
10 | 4 | 20 | 56 | 30 | 341 | 40 | 1332 | 50 | 3938
11 | 1 | 21 | 50 | 31 | 349 | 41 | 1412 | 51 | 4201
12 | 9 | 22 | 83 | 32 | 460 | 42 | 1686 | 52 | 4780

.

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