Exceptional theta functions and arithmeticity of modular forms on $G_{2}$

Let $G_{2}^{a}$ denote the the algebraic ${\mathbf{Q}}$ group of type $G_{2}$ that is split at every finite place and compact at the archimedean place. Let $H_{J}^{1}$ denote the simply connected group of type $E_{7}$ that is split at every finite place and the group $E_{7,3}$ at the archimedean place. There is a dual pair $\operatorname{Sp}_{6}\times G_{2}^{a}\subseteq H_{J}^{1}$, and a corresponding theta lift from $G_{2}^{a}$ to $\operatorname{Sp}_{6}$ using the automorphic minimal representation on $H_{J}^{1}$ [Kim93] that was studied by Gross-Savin [GS98]. This lift produces (in general) vector-valued holomorphic Siegel modular forms on $\operatorname{Sp}_{6}$. It makes sense to ask if, given an algebraic modular form on $G_{2}^{a}$, one can give an explicit Fourier expansion of its theta lift to $\operatorname{Sp}_{6}$. This is easy if the weight of the algebraic modular form is trivial, but is not immediate (at least to us) if the weight is nontrivial. Theorem 1.1.1 computes this Fourier expansion in the level one case.

We now setup this theorem. Let $\Theta$ denote the octonions over ${\mathbf{Q}}$ with positive definite norm. Denote by $J=H_{3}(\Theta)$ the $27$-dimensional exceptional cubic norm structure consisting of the $3\times 3$ Hermitian matrices with coefficients in $\Theta$. Let $V_{7}$ denote the $7$-dimensional space of trace $0$ octonions, let $W_{3}$ denote the standard representation of $\operatorname{GL}_{3}$ and set $V_{3}=\wedge^{2}W_{3}$. There is a projection $\mathcal{P}:J^{\vee}\simeq J\rightarrow V_{3}\otimes V_{7}$ given by taking the trace $0$ projections of the off-diagonal entries of an element of $J$. Consider the natural map

and denote by $\mathcal{P}_{k_{1},k_{2}}(T)$ the image of $(T)^{\otimes(k_{1}+2k_{2})}$ under this map.

Let $\omega_{1}$ denote the highest weight of the representation $V_{7}$ of $G_{2}$, and let $\omega_{2}$ denote the highest weight of the $14$-dimensional adjoint representation ${\mathfrak{g}}_{2}\subseteq\wedge^{2}V_{7}$. For non-negative integers $k_{1},k_{2}$, let $W(k_{1},k_{2})$ denote the representation of $G_{2}({\mathbf{C}})$ with highest weight $k_{1}\omega_{1}+k_{2}\omega_{2}$, embedded in $V_{7}^{\otimes k_{1}}\otimes(\wedge^{2}V_{7})^{\otimes k_{2}}\otimes{\mathbf{C}}$. This representation is the one generated by $v_{\omega_{1}}^{\otimes k_{1}}\otimes v_{\omega_{2}}^{\otimes k_{2}}$ where $v_{\omega_{1}}$ and $v_{\omega_{2}}$ are highest weight vectors for $V_{7}$ and ${\mathfrak{g}}_{2}$ for the same Borel subgroup of $G_{2}({\mathbf{C}})$. Given $\beta\in W(k_{1},k_{2})$, and $T\in J$, we can form the pairing

where we have shifted the $k_{2}\mapsto k_{2}+4$ in the exponent of $\det(W_{3})$.

Denote by $R_{\Theta}\subseteq\Theta$ Coxeter’s order of integral octonions, and set $J_{R}\subseteq J$ the elements whose diagonal entries are in ${\mathbf{Z}}$ and off-diagonal entries are in $R_{\Theta}$. Recall that Kim’s modular form on $H_{J}^{1}$ has Fourier expansion

where $a(0)=1$ and if $T$ is rank one then $a(T)=240\sigma_{3}(d_{T})$ where $d_{T}$ is the largest integer with $d_{T}^{-1}T\in J_{R}$.

Finally, set $\Gamma_{G_{2}}=G_{2}^{a}({\mathbf{Z}})$.

When $k_{2}>0$ [GS98] proves that $\Theta(\alpha)$ is a cusp form.

A simple restatement of Theorem 1.1.1 is as follows. Consider the projection map $J\rightarrow H_{3}({\mathbf{Q}})$ given by sending

Then if $T_{0}\in H_{3}({\mathbf{Q}})$ (Hermitian $3\times 3$ matrices with ${\mathbf{Q}}$ coefficients) is half-integral, the $T_{0}$ Fourier coefficient of $\Theta(\alpha)(Z)$ is

(The sum is finite and explicitly determinable.) We verify that the $a_{\Theta(\alpha)}(T_{0})$ live in the highest weight submodule

so that $\Theta(\alpha)$ really is a vector-valued Siegel modular form for the representation $V_{[k_{1},k_{2}]}$.

An important computational aspect of Theorem 1.1.1 is that one can use $\beta$ in the pairing $\{P_{k_{1},k_{2}}(T),\beta\}$ instead of $\alpha(1)=\frac{1}{|\Gamma_{G_{2}}|}\sum_{\gamma\in\Gamma_{G_{2}}}{\gamma\cdot\beta}$. This enables one to compute theta lifts much more quickly than if one had to use the algebraic modular form $\alpha(1)\in W(k_{1},k_{2})^{\Gamma_{G_{2}}}$. Moreover, even if one does not know a priori that $W(k_{1},k_{2})^{\Gamma_{G_{2}}}\neq 0$, the theorem still holds as stated. In particular, verifying that the right hand side of equation (1) is nonzero for a single $T_{0}$ shows that $\frac{1}{|\Gamma_{G_{2}}|}\sum_{\gamma\in\Gamma_{G_{2}}}{\gamma\cdot\beta}\neq 0$. The reader can, of course, easily check this claim directly.

The Fourier expansion in Theorem 1.1.1 can be seen as completely analogous to the Fourier expansion of classical pluriharmonic theta functions. Indeed, the $\beta\in W(k_{1},k_{2})$ becomes the pluriharmonic polynomial, and the rank one $T$’s become the lattice vectors over which one sums.

When the Siegel modular form $\Theta(\alpha)$ is a Hecke eigenform, it has Satake parameters $c_{p}$, one for each prime number $p$, which are semisimple elements in $\operatorname{Spin}_{7}({\mathbf{C}})$. (Because we work with level one forms, we blur the distinction between $\operatorname{Sp}_{6}$ and $\operatorname{PGSp}_{6}$.) It is proved by Gross-Savin [GS98], Maagard-Savin [MS97], and Gan-Savin [GS21] that the theta lift is functorial for spherical representations; in fact, it is functorial for all representations, see Gan-Savin [GS22]. In particular, these conjugacy classes $c_{p}$ are in $G_{2}({\mathbf{C}})\subseteq\operatorname{Spin}_{7}({\mathbf{C}})$. Thus Theorem 1.1.1 can produce numerous explicit examples of level one vector-valued Siegel modular forms all of whose Satake parameters are in $G_{2}({\mathbf{C}})$. We have taken the liberty of providing some small illustration of this, as follows. Let $\lambda_{1}=(12,8,8)$ and $\lambda_{2}=(14,10,8)$. Then it is known from Chenevier-Taibi [CT20] that the space of vector-valued, level one cuspidal Siegel modular forms of these weights are one dimensional.

Indeed, the proof of Corollary 1.1.2 is to produce a single $\beta_{1}\in W(0,4)$ and $\beta_{2}\in W(2,4)$, and a single $T_{0}$ so that the right hand side of (1) is nonzero for these $\beta_{i}$. It then follows that the $\Theta(\alpha)$ are nonzero level one Siegel modular forms, and thus by the dimension computation of [CT20] are the unique cuspidal level one eigenforms of weights $\lambda_{1}$ and $\lambda_{2}$. Producing many more such explicit examples would be possible. We remark that one can find conjectures here [BCFvdG17] of which small weight level one Siegel modular forms have their Satake parameters in $G_{2}({\mathbf{C}})$.

More than just a couple of examples, however, Theorem 1.1.1 produces an algorithm to determine if any fixed level one Siegel modular cusp form of most weights is a lift from $G_{2}^{a}$. To setup the result, note that for a weight $\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})$, it is known [Ibu02] that there exists explicitly determinable finite sets $\mathcal{C}_{\lambda}$ of half-integral symmetric matrices $T$ so that if $F$ is a cuspidal level one Siegel modular form of weight $\lambda$, and if $a_{F}(T)=0$ for all $T\in\mathcal{C}_{\lambda}$, then $F=0$.

Let us indicate now some of the ingredients that go into the proof of Corollary 1.1.3. First, let us clarify that the theta lifts $\Theta(\alpha)$ of level one algebraic modular forms $\alpha(1)\in W(k_{1},k_{2})^{\Gamma_{G_{2}}}$ that appear in Theorem 1.1.1 are defined as integrals

for a certain specific vector-valued element of $\Theta_{k_{1},k_{2}}$ in the minimal representation $\Pi_{min}$ on $H_{J}^{1}$. Here, in the integral, $\{\,,\,\}$ is a $G_{2}^{a}({\mathbf{R}})$-equivariant pairing valued in $S^{k_{1}}(V_{3})\otimes S^{k_{2}}(\wedge^{2}V_{3})$. Let $\{\beta_{1},\ldots,\beta_{N}\}$ be a spanning set of $W(k_{1},k_{2})$, and $\alpha_{j}=\frac{1}{|\Gamma_{G_{2}}|}\sum_{\gamma\in\Gamma_{G_{2}}}{\gamma\beta_{j}}$. One can use Theorem 1.1.1 to explicitly compute the Fourier coefficients of $\Theta(\alpha_{j})$ associated to the various $T^{\prime}s$ in $\mathcal{C}_{\lambda}$, where $\lambda=(k_{1}+2k_{2}+4,k_{1}+k_{2}+4,k_{2}+4)$. Using linear algebra, one can check algorithmically if $F$ is some linear combination of the $\Theta(\alpha)$’s. Thus Theorem 1.1.1 gives an algorithm to determine if the Siegel modular form $F$ is a $\Theta(\alpha)$ for some algebraic modular form $\alpha$ on $G_{2}^{a}$. Thus Corollary 1.1.3 will be proved if one knew the following claim:

We prove this claim. The proof uses some powerful ingredients: The Howe Duality theorem of Gan-Savin [GS21]; an analysis of the minimal representation, as provided by Gross-Savin [GS98], Magaard-Savin [MS97] and Gan-Savin [GS21]; the existence of enough nonvanishing Fourier coefficients of $F$ of a certain form, as proved by Böcherer-Das [BD21]; and another argument from Gross-Savin [GS98].

Our proof of Theorem 1.1.1 is very simple. Let ${\mathfrak{p}}_{J}$ be the complexification of the $-1$ eigenspace for the Cartan involution on the Lie algebra of $E_{7,3}=H_{J}^{1}({\mathbf{R}})$. Write ${\mathfrak{p}}_{J}={\mathfrak{p}}_{J}^{+}\oplus{\mathfrak{p}}_{J}^{-}$, the natural decomposition, so that ${\mathfrak{p}}_{J}^{-}$ annihilates the automorphic form on $E_{7,3}$ associated to any holomorphic modular form for this group. Let $\{X_{\alpha}\}_{\alpha}$ be a basis of ${\mathfrak{p}}_{J}^{+}$ and $\{X_{\alpha}^{\vee}\}_{\alpha}$ be the dual basis of ${\mathfrak{p}}_{J}^{+,\vee}\simeq{\mathfrak{p}}_{J}^{-}$. For an automorphic form $\varphi$ on $H_{J}^{1}$, set $D\varphi(g)=\sum_{\alpha}{X_{\alpha}\varphi\otimes X_{\alpha}^{\vee}}.$ For an integer $m\geq 0$, let $D^{m}\varphi=D\circ D\circ\cdots\circ D\varphi$, so that

If (abusing notation) $\Theta_{Kim}$ denotes the automorphic form on $H_{J}^{1}$ associated to Kim’s holomorphic modular form, we set $\Theta_{k_{1},k_{2}}(g)=D^{k_{1}+2k_{2}}\Theta_{Kim}(g)$.

With this definition, using Kim’s expansion of $\Theta_{Kim}$, we compute the Fourier expansion of $\Theta_{k_{1},k_{2}}(g)$. This is the main step in the proof of Theorem 1.1.1. We then use this to compute the Fourier expansion of $\Theta(\alpha)$. One obtains (the automorphic form associated to) a holomorphic function with the Fourier expansion as given in Theorem 1.1.1. We remark that the use of differential operators as we do has some overlap with the works [Ibu99, Cle22] of Ibukiyama and Clerc.

Denote by $G_{J}$ the group of type $E_{8}$ which is split at every finite place and $E_{8,4}$ at the archimedean place and by $G_{2}$ the split exceptional group of this Dynkin type. Analogous to the dual pair $\operatorname{Sp}_{6}\times G_{2}^{a}\subseteq H_{J}^{1}$ is the dual pair $G_{2}\times F_{4}^{I}\subseteq G_{J}$, where $F_{4}^{I}$ is a specific form of $F_{4}$ that is split at all finite places and compact at the archimedean place, defined as the stabilizer of the identity matrix $I\in J$. We compute the theta lifts of certain algebraic modular forms on $F_{4}^{I}$ to $G_{2}$, and obtain cuspidal quaternionic modular forms on $G_{2}$ together with their exact Fourier expansions.

More precisely, let $J^{0}$ denote the trace $0$ elements of $J$. In other words, $J^{0}$ consists of the $X\in J$ with $(I,X)_{I}=0$, where $(\,,\,)_{I}$ is the symmetric non-degenerate pairing on $J$ determined by $I$. There is a surjective $F_{4}^{I}$-equivariant map $\wedge^{2}J^{0}\rightarrow\mathfrak{f}_{4}$ from $\wedge^{2}J^{0}$ to the Lie algebra $\mathfrak{k}_{4}$ of $F_{4}^{I}$. Denote by $V_{\lambda_{3}}$ the kernel of this map. It is an irreducible representation of $F_{4}$ of dimension $273$. For an integer $m>0$, let $V_{m\lambda_{3}}\subseteq(\wedge^{2}J^{0})^{\otimes m}$ denote the irreducible representation of $F_{4}$ with highest weight $m\lambda_{3}$, generated by the tensor product of a highest weight vector of $V_{\lambda_{3}}$. It follows from the archimedean theta correspondence calculated in [HPS96] that algebraic modular forms on $F_{4}^{I}$ for the representation $V_{m\lambda_{3}}$ should lift to quaternionic modular forms on $G_{2}$ of integer weight $4+m$. We explicitly compute the Fourier expansion of this lift, and as a result, obtain exceptional “pluriharmonic” cuspidal quaternionic theta functions on $G_{2}$.

We setup the statement of the result. To do so, recall that the group $F_{4}^{I}$ has not one but two integral structures [Gro96], [EG96]. More precisely, if $F_{4}^{I}(\widehat{{\mathbf{Z}}})$ denotes one of these integral structures, then the double coset space $F_{4}^{I}({\mathbf{Q}})\backslash F_{4}^{I}({\mathbf{A}}_{f})/F_{4}^{I}(\widehat{{\mathbf{Z}}})$ has size two. Because of this, algebraic modular forms for $F_{4}^{I}$ can be described as follows. Denote by $M_{J}^{1}$ the subgroup of $\operatorname{GL}(J)$ fixing the cubic norm. Set $\Gamma_{I}$ to be the subgroup of $M_{J}^{1}$ preserving the lattice $J_{R}$ and fixing the element $I$. Recall the element $E\in J_{R}$ of norm one from [EG96] or [Gro96]. Let $\Gamma_{E}$ denote the subgroup of $M_{J}^{1}$ preserving the lattice $J_{R}$ and fixing the element $E$. Fix an element $\delta_{E}^{{\mathbf{Q}}}\in M_{J}^{1}({\mathbf{Q}})$ with $\delta_{E}^{\mathbf{Q}}E=I$. If $V$ is a representation of $F_{4}^{I}({\mathbf{R}})$, let $\Gamma_{E}$ act on $V$ via $\gamma\cdot_{E}v=(\delta_{E}^{\mathbf{Q}}v\delta_{E}^{{\mathbf{Q}},-1})v$, where the conjugation takes place in $M_{J}^{1}$. Then a level one algebraic modular form for $F_{4}^{I}$ can be considered as a pair

In the case of $V=V_{m\lambda_{3}}$, we rephrase this as follows. Let $(\,,\,)_{E}$ be the symmetric non-degenerate pairing on $J$ determined by $E$; one has

where $(\cdot,\cdot,\cdot)$ is the symmetric trilinear form on $J$ satisfying $(x,x,x)=6n(x)$, $6$ times the cubic norm on $J$. Set $J_{E}^{0}$ to be the perpendicular space to $E$ under the pairing $(\,,\,)_{E}$. One easily verifies that $\delta_{E}^{{\mathbf{Q}},-1}J^{0}=J_{E}^{0}$. Thus

and is $\Gamma_{E}$ invariant for the natural action of $\Gamma_{E}$. It will be convenient for us to consider the algebraic modular form to be the pair $(\alpha_{I},\alpha_{E})$.

Now, for $x,y,b\in J$ and $c\in J^{\vee}$, write

We use the same notation for the pairing $(\wedge^{2}J)^{\otimes m}\otimes(\wedge^{2}J)^{\otimes m}\rightarrow{\mathbf{C}}$ that extends this one via $\langle z_{1}\otimes\cdots\otimes z_{m},z_{1}^{\prime}\otimes\cdots\otimes z_{m}^{\prime}\rangle_{I}=\prod_{j=1}^{m}\langle z_{j},z_{j}^{\prime}\rangle_{I}$. Thus if $\beta\in V_{m\lambda_{3}}$, $b\in J$ and $c\in J^{\vee}$, we can compute the quantity $\langle\beta,(b\wedge c)^{\otimes m}\rangle\in{\mathbf{C}}$. We similarly define $\langle\,,\,\rangle_{E}$, by replacing the pairing $(\,,\,)_{I}$ with $(\,,\,)_{E}$.

Set $W_{J_{R}}={\mathbf{Z}}\oplus J_{R}\oplus J_{R}^{\vee}\oplus{\mathbf{Z}}$. For $w\in W_{J_{R}}$ of rank one, set $a(w)=\sigma_{4}(d_{w})$ where $d_{w}$ is the largest integer with $d_{w}^{-1}w\in W_{J_{R}}$. In [Pol20b], we proved that the minimal modular form $\Theta_{Gan}$ on $G_{J}$ has Fourier expansion

Here $N\supseteq Z\supseteq 1$ is the unipotent radical of the Heisenberg parabolic of $E_{8,4}$, $\Theta_{Gan,Z}$, $\Theta_{Gan,N}$ denote constant terms, and $W_{2\pi w}$ is the generalized Whittaker function of [Pol20a].

Now, if $w\in W_{J}$ and $m>0$, set $P_{m}(w)=(b\wedge c)^{\otimes m}\in(\wedge^{2}J)^{\otimes m}$ if $w=(a,b,c,d)$. Moreover, with this notation, let $p_{I}(w)$ and $p_{E}(w)$ be the binary cubic forms given as

We prove the following.

A simple restatement of Theorem 1.2.1 is as follows. If $w_{0}$ is an integral binary cubic form, then the $w_{0}$ Fourier coefficient of $\Theta(\alpha)$ is

These sums are finite.

The reason we do not state Theorem 1.2.1 in the case $m=0$ is because then the theta lifts will be non-cuspidal. In fact, the theta lifts obtained for $m=0$ are exactly the automorphic forms obtained in [GGS02, Section 10]. Thus Theorem 1.2.1 may be considered a generalization of [GGS02, Section 10].

Again, just like Theorem 1.1.1, the Fourier expansion given by Theorem 1.2.1 is completely parallel to the classical pluriharmonic theta functions: The $\beta^{\prime}s$ in $V_{m\lambda_{3}}$ are the pluriharmonic polynomial, and the sum over $w\in W_{J_{R}}$ of rank one is the sum over lattice vectors.

We now state a few corollaries of Theorem 1.2.1. For the first corollary, we can partially refine Theorem 1.0.1 in the case of level one.

For the second corollary, recall that it was proved in [cDD⁺22] that if $\pi$ is a cuspidal automorphic representation on $G_{2}({\mathbf{A}})$ that corresponds to a level one quaternionic modular form $\varphi_{\pi}$ of even weight $\ell$, then the completed standard $L$-function $\Lambda(\pi,Std,s)$ satisfies the exact functional equation $\Lambda(\pi,Std,s)=\Lambda(\pi,Std,1-s)$, so long as the $w_{0}=u^{2}v-uv^{2}$ Fourier coefficient of $\varphi_{\pi}$ is nonzero. At the time of the writing of [cDD⁺22], it was not known whether such $\pi$ exist. Using Theorem 1.2.1, one easily obtains the following.

This corollary is, in fact, an ingredient in the proof of Theorem 1.0.1.

The third corollary we state has to do with the Fourier expansion of a particular cuspidal quaternionic modular form. To setup this corollary, recall that Dalal [Dal21] has recently given an explicit formula for the dimension of the level one quaternionic cuspidal modular forms of weights at least $3$. From his dimension formula, one has that the first such nonzero cusp form appears in weight $6$, and the space of weight $6$ cuspidal quaternionic modular forms is one-dimensional, spanned by a cusp form $\Delta_{G_{2}}$. Combining the (proof of) Corollary 1.2.3 with Corollary 1.2.2, one obtains that $\Delta_{G_{2}}$ can be normalized to have integer Fourier coefficients, and that the Fourier coefficient associated to the cubic ring ${\mathbf{Z}}\times{\mathbf{Z}}\times{\mathbf{Z}}$ is nonzero.

Now, Benedict Gross has suggested that the Fourier coefficients of certain non-tempered cuspidal quaternionic modular forms on $G_{2}$ of weight $\ell$ should be related to square-roots of twists of $L$-values of classical modular forms of weight $2\ell$ by Artin motives associated to totally real cubic fields. As pointed out to the author by Mundy, the quaternionic modular form $\Delta_{G_{2}}$ should be one of these non-tempered lifts, to which Gross’s conjecture applies. Now, for $D$ congruent to $0$ or $1$ modulo $4$, denote by ${\mathbf{Z}}_{D}$ the quadratic ring of discriminant $D$. Then on the one hand, in the case of $\Delta_{G_{2}}$, Gross’s conjecture implies that the square Fourier coefficients $a_{\Delta_{G_{2}}}({\mathbf{Z}}\times{\mathbf{Z}}_{D})^{2}$ associated to the cubic ring ${\mathbf{Z}}\times{\mathbf{Z}}_{D}$ should be related to the central³³3We here use the classical normalization of $L$-functions, instead of the automorphic normalization. $L$-value $L(\Delta,D,6)$ of the twist of Ramanujan’s $\Delta$ function by the quadratic character associated to $D$. Denote by $\delta(z)\in S_{13/2}(\Gamma_{0}(4))^{+}$, $\delta(z)=\sum_{D\equiv 0,1\pmod{4}}{\alpha(D)q^{D}}$ the Shimura lift of $\Delta(z)$. On the other hand, following Waldspurger [Wal81], Kohnen-Zagier [KZ81] relate the squares of Fourier coefficients $\alpha(D)$ to the same $L$-value, $L(\Delta,D,6)$. It thus makes sense to ask, in light of Gross’s conjecture, if there is some relationship between $a_{\Delta_{G_{2}}}({\mathbf{Z}}\times{\mathbf{Z}}_{D})$ and $\alpha(D)$. It turns out, the numbers are equal:

It is pleasure to thank Wee Teck Gan, Nadya Gurevich, and Gordan Savin for engaging with the author in a “Research in Teams” project in Spring 2022 at the Erwin Schrodinger Institute, which helped to stimulate thinking about exceptional theta correspondences. We thank them for fruitful discussions. We thank Tomoyoshi Ibukiyama for sending us his note [Ibu02], and we also thank Gaetan Chenevier, Chao Li, Finley McGlade, Sam Mundy, Cris Poor and David Yuen for helpful discussions.

We begin by defining the group $H_{J}^{1}$. Thus let $J$ be our exceptional cubic norm structure, $W_{J}={\mathbf{Q}}\oplus J\oplus J^{\vee}\oplus{\mathbf{Q}}$. A typical element of $W_{J}$ we write as an ordered four-tuple $(a,b,c,d)$, so that $a,d\in{\mathbf{Q}}$, $b\in J$ and $c\in J^{\vee}$. We put on $W_{J}$ Freudenthal’s symplectic form, and quartic form. The group $H_{J}^{1}$ is defined as the algebraic ${\mathbf{Q}}$-group preserving these two forms. We let $H_{J}$ be the group that preserves the forms on $W_{J}$ up to similitude, and let $\nu:H_{J}\rightarrow\operatorname{GL}_{1}$ be this similitude.

The Siegel parabolic $P_{J}$ of $H_{J}^{1}$ is defined as the stabilizer of the line ${\mathbf{Q}}(0,0,0,1)$. Write $M_{J}$ for the group of linear automorphisms of $J$ that preserve the norm on $J$, up to scaling. Let $\lambda:M_{J}\rightarrow\operatorname{GL}_{1}$ be this scaling factor, and $M_{J}^{1}$ the kernel of $\lambda$. A Levi subgroup of $P_{J}$ can be defined as the subgroup that also stabilizes the line ${\mathbf{Q}}(1,0,0,0)$. This is isomorphic to the group of pairs $(\delta,m)\in\operatorname{GL}_{1}\times M_{J}$ with $\delta^{2}=\lambda(m)$. Such a pair acts on $W_{J}$ as $(a,b,c,d)\mapsto(\delta^{-1}a,\delta^{-1}m(b),\delta\widetilde{m}(c),\delta d)$. We write $M(\delta,m)$ for this group element of $H_{J}^{1}$.

We now explain how $\operatorname{GSp}_{6}\times G_{2}^{a}$ embeds in $H_{J}$. To accomplish this, we describe a linear isomorphism between $\wedge^{3}_{0}W_{6}\otimes\nu^{-1}\oplus W_{6}\otimes\Theta^{0}$ and $W_{J}$. Here $\Theta^{0}=V_{7}$ is the trace $0$ octonions, $W_{6}$ is the $6$-dimensional defining representation of $\operatorname{GSp}_{6}$, and $\wedge^{3}_{0}W_{6}$ is the kernel of the contraction map $\wedge^{3}W_{6}\rightarrow W_{6}\otimes\nu$. We let $e_{1},e_{2},e_{3},f_{1},f_{2},f_{3}$ be the standard symplectic basis of $W_{6}$.

• •

  If $v_{1},v_{2},v_{3},v_{1}^{\prime},v_{2}^{\prime},v_{3}^{\prime}\in\Theta^{0}$, we map $v_{1}^{\prime}e_{1}+v_{2}^{\prime}e_{2}+v_{3}^{\prime}e_{3}+v_{1}f_{1}+v_{2}f_{2}+v_{3}f_{3}$ to $(0,Y,Y^{\prime},0)$ where $Y=\left(\begin{array}[]{ccc}0&v_{3}&-v_{2}\\ -v_{3}&0&v_{1}\\ v_{2}&-v_{1}&0\end{array}\right)\in J$ and $Y^{\prime}=-\left(\begin{array}[]{ccc}0&v_{3}^{\prime}&-v_{2}^{\prime}\\ -v_{3}^{\prime}&0&v_{1}^{\prime}\\ v_{2}^{\prime}&-v_{1}^{\prime}&0\end{array}\right)$.

• •

  We map $f_{1}\wedge f_{2}\wedge f_{3}$ to $(1,0,0,0)$ and $e_{1}\wedge e_{2}\wedge e_{3}$ to $(0,0,0,1)$.

• •

  Set $f_{i}^{*}=f_{i+1}\wedge f_{i-1}$ and $e_{i}^{*}=e_{i+1}\wedge e_{i-1}$ (with indices taken modulo $3$). We map $\sum_{i,j}{b_{ij}f_{i}^{*}\wedge e_{j}}$ to $(0,(b_{ij}),0,0)$ and $\sum_{i,j}{c_{ij}e_{i}^{*}\wedge f_{j}}$ to $(0,0,(c_{ij}),0)$.

Via the natural action of $\operatorname{GSp}_{6}\times G_{2}^{a}$ on $\wedge^{3}_{0}W_{6}\otimes\nu^{-1}\oplus W_{6}\otimes\Theta^{0}$, we obtain an action of this group on $W_{J}$. It is clear that the action is faithful. To obtain the embedding into $H_{J}$, we must check that $\operatorname{GSp}_{6}\times G_{2}^{a}$ preserves the symplectic and quartic form on $W_{J}$, up to scaling: