2.1 Octonion algebra

Let $F$ be a field of characteristic 0, and $D$ be a quaternion algebra over $F$. It is a four-dimensional associative and non-commutative algebra over $F$ which comes equipped with a conjugation map $x\mapsto \overline{x}$ with associated norm $N(x)=x\overline{x}=\overline{x}x$ and trace $\text{tr}(x)=x+\overline{x}$. Moreover, $N:D\rightarrow F$ is a non-degenerate quadratic form.

An octonion algebra $\mathbb{O}$ over $F$ is obtained by doubling the quaternion algebra $D$. More precisely, fix a non-zero element $\unicode[STIX]{x1D706}$ in $F$. As a vector space over $F$, $\mathbb{O}$ is a set of pairs $(a,b)$ of elements in $D$. The multiplication is defined by the formula

$$\begin{eqnarray}(a,b)\cdot (c,d)=(ac+\unicode[STIX]{x1D706}d\bar{b},\bar{a}d+cb).\end{eqnarray}$$

If $x=(a,b)$, then the conjugation map is $\bar{x}=(\bar{a},-b)$, so that $N(x)=x\cdot \bar{x}=N(a)-\unicode[STIX]{x1D706}N(b)$ is the norm and $\text{tr}(x)=x+\bar{x}=\text{tr}(a)$ the trace on $\mathbb{O}$. In particular, $\mathbb{O}$ is split if $\unicode[STIX]{x1D706}$ is a norm of an element in $D$. Every element $x$ of $\mathbb{O}$ satisfies its characteristic polynomial $t^{2}-\text{tr}(x)t+N(x)$. The automorphism group $\text{Aut}(\mathbb{O})$ of the $F$-algebra $\mathbb{O}$ is an exceptional group of the Lie type $\mathbf{G}_{2}$. It is a simple linear algebraic group of rank 2 which is both simply connected and adjoint. The algebra $D$ is naturally a subalgebra of $\mathbb{O}$, consisting of all $x=(a,0)$. Let $D^{1}$ be the group of norm-$1$ elements in $D$. Then any $g\in D^{1}$ acts as an automorphism of $\mathbb{O}$ by $g\cdot (a,b)=(a,b\bar{g})$ for all $(a,b)\in \mathbb{O}$. The subgroup $D^{1}\subset \text{Aut}(\mathbb{O})$ is precisely the pointwise stabilizer of the subalgebra $D\subset \mathbb{O}$.

2.2 Albert algebra

An Albert algebra is an exceptional 27-dimensional Jordan algebra $J$ over $F$. It can be realized as the set of matrices

$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}\unicode[STIX]{x1D6FC} & x & \bar{z}\\ \bar{x} & \unicode[STIX]{x1D6FD} & y\\ z & \bar{y} & \unicode[STIX]{x1D6FE}\end{array}\right),\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FE}\in F$ and $x,y,z\in \mathbb{O}$. The determinant $A\mapsto \det A$ defines a natural cubic form on $J$. Let $M$ be the similitude group of this cubic form. It is a reductive group of semisimple type $E_{6}$. The $M$-orbits in $J$ are classified by the rank of the matrix $A$. Without going into a general definition of the rank, we say that $A\neq 0$ has rank $1$ if $A^{2}=\text{tr}(A)\cdot A$. Explicitly, this means that the entries of $A$ satisfy the equalities

$$\begin{eqnarray}N(x)=\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD},\quad N(y)=\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE},\quad N(z)=\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FC},\quad \unicode[STIX]{x1D6FE}\bar{x}=yz,\quad \unicode[STIX]{x1D6FC}\bar{y}=zx,\quad \unicode[STIX]{x1D6FD}\bar{z}=xy.\end{eqnarray}$$

2.3 Dual pairs

Assume that $G$ is a reductive group over $F$, adjoint and of absolute type $E_{7}$, arising from the Albert algebra $J$ via the Koecher–Tits construction. For our purposes it will be more convenient to realize $G$ as a quotient, modulo one-dimensional center $C\cong F^{\times }$, of a reductive group $\tilde{G}$ acting on the 56-dimensional representation $W=F+J+J+F$. In particular, $G$ acts on the projective space $\mathbb{P}(W)$. Let $P$ be a maximal parabolic and $\bar{P}$ its opposite, defined as fixing the points $(1,0,0,0)$ and $(0,0,0,1)$ in $\mathbb{P}(W)$. Then $P=MN$, where $N$ is the unipotent radical and $M=P\cap \bar{P}$ a Levi subgroup. Then $M$ is isomorphic to the similitude group of the cubic form $\det$ on $J$, and $N\cong J$, as $M$-modules.

Recall that we have constructed $\mathbb{O}$ by doubling a quaternion subalgebra $D$. Let $J_{F}$ and $J_{D}$ be the subalgebras consisting of all elements in $J$ with off-diagonal entries in $F$ and $D$, respectively. Let $J_{0}=F$ be the scalar subalgebra of $J$. Consider a sequence of simple, simply connected groups

$$\begin{eqnarray}D^{1}\subset \text{Aut}(\mathbb{O})\subset \text{Aut}(J),\end{eqnarray}$$

where an element in $\text{Aut}(\mathbb{O})$ acts on the off-diagonal entries of elements in $J$. The pointwise stabilizers in $J$ of these three groups are, respectively,

$$\begin{eqnarray}J_{D}\supset J_{F}\supset J_{0}=F.\end{eqnarray}$$

Observe that $\text{Aut}(J)$ naturally acts on $W$, giving an embedding $\text{Aut}(J)\subset \tilde{G}$. The centralizers in $\tilde{G}$ of the three groups in the sequence are, respectively,

$$\begin{eqnarray}\tilde{G}_{D}\supset \text{GSp}_{6}(F)\supset \text{GL}_{2}(F).\end{eqnarray}$$

These three groups act on the 32-, 14- and four dimensional subspaces of $W$ obtained by replacing $J$ by $J_{D}$, $J_{F}$ and $J_{0}$, respectively. It is worth mentioning that the four-dimensional representation of $\text{GL}_{2}(F)$ is the symmetric cube of the standard two-dimensional representation, twisted by $\det ^{-1}$. The group $\tilde{G}_{D}$ acts on

$$\begin{eqnarray}W_{D}=F+J_{D}+J_{D}+F.\end{eqnarray}$$

It is worth noting that the action of $\tilde{G}_{D}$ on $W_{D}$ is not faithful (it has $\unicode[STIX]{x1D707}_{2}\subset D^{1}$ as its kernel). A detailed description of $\tilde{G}_{D}/\unicode[STIX]{x1D707}_{2}$ and its action on $W_{D}$ can be found in Pollack’s paper [Reference PollackPol17]. Let $G_{D}$ be the quotient of $\tilde{G}_{D}$ by the center $C\cong F^{\times }$ of $\tilde{G}$. Then $D^{1}\times G_{D}$ is a dual pair in $G$, as mentioned in the introduction.

Let $P_{D}=M_{D}N_{D}=G_{D}\cap P$. With the identification $N\cong J$ fixed, we have $N_{D}\cong J_{D}$. The group $P_{D}$ is a maximal parabolic subgroup of type $\mathbf{A}_{5}$.

3.1 Unitary model

Fix $\unicode[STIX]{x1D713}:F\rightarrow \mathbb{C}^{\times }$, a non-trivial additive character, unitary if $F=\mathbb{R}$. After identifying $N\cong J$ and $\bar{N}\cong J$ (note that the resulting actions of $M$ on $J$ are dual to each other), any $A\in J\cong \bar{N}$ defines a character of $N$ given by

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{A}(B)=\unicode[STIX]{x1D713}(\text{tr}(A\circ B))=\unicode[STIX]{x1D713}_{B}(A)\end{eqnarray}$$

for $B\in J\cong N$, where $A\circ B$ denotes the Jordan multiplication. Every unitary character of $N$ is equal to $\unicode[STIX]{x1D713}_{A}$ for some $A$. Let $\unicode[STIX]{x1D6FA}\subseteq J\cong \bar{N}$ be the set of rank-$1$ elements in $J$. A unitary model of the minimal representation is ${\mathcal{H}}=L^{2}(\unicode[STIX]{x1D6FA})$ [Reference SahiSah92]. Here only the action of the maximal parabolic $P=MN$ is obvious: the group $M$ acts geometrically,

$$\begin{eqnarray}\unicode[STIX]{x1D70B}(m)(f)(A)=\unicode[STIX]{x1D712}(m)f(m^{-1}A),\end{eqnarray}$$

for $f\in \unicode[STIX]{x1D6F1}$ and for some character $\unicode[STIX]{x1D712}:M\rightarrow \mathbb{R}^{\times }$, while $B\in J\cong N$ acts on $f$ by multiplying it by $\unicode[STIX]{x1D713}_{B}$.

3.2 Smooth model

We have the following theorem [Reference Kobayashi and SavinKS15].

Theorem 3.1. Let $\unicode[STIX]{x1D6F1}$ be the subspace of $G$-smooth vectors in the unitary minimal representation ${\mathcal{H}}$. Then

$$\begin{eqnarray}C_{c}^{\infty }(\unicode[STIX]{x1D6FA})\subset \unicode[STIX]{x1D6F1}\subset C^{\infty }(\unicode[STIX]{x1D6FA}).\end{eqnarray}$$

If $F$ is $p$-adic, then

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{N}\cong \unicode[STIX]{x1D6F1}/C_{c}^{\infty }(\unicode[STIX]{x1D6FA})\quad \text{as }M\text{-modules}.\end{eqnarray}$$

If $A\in J$ is non-zero, then any continuous functional $\ell$ on $\unicode[STIX]{x1D6F1}$ such that $\ell (B\cdot f)=\unicode[STIX]{x1D713}_{A}(B)\cdot \ell (f)$ for all $B\in N$ and $f\in \unicode[STIX]{x1D6F1}$ is equal to a multiple of the evaluation map $\unicode[STIX]{x1D6FF}_{A}(f)=f(A)$. In particular, $\ell =0$ if $A$ is not of rank $1$.

3.3 Spherical vector

It is not so easy to characterize the subspace $\unicode[STIX]{x1D6F1}\subset C^{\infty }(\unicode[STIX]{x1D6FA})$. However, we can describe a spherical vector in $\unicode[STIX]{x1D6F1}$ in the split case. The algebra $\mathbb{O}$ is obtained by doubling the matrix algebra $D=M_{2}(F)$ with $\unicode[STIX]{x1D706}=1$. Assume firstly that $F$ is a $p$-adic field. Let $O$ be the ring of integers in $F$ and $\unicode[STIX]{x1D71B}$ a uniformizing element. We have an obvious integral structure on $D$ (the lattice of integral matrices), and hence on $\mathbb{O}$, the integral lattice being the set of pairs $(a,b)$ where $a,b\in M_{2}(O)$. This lattice is a maximal order in $\mathbb{O}$. Now we have an integral structure on $J$ so that $J(O)$ is the set of elements $A\in J$ such that the diagonal entries are integral, and the off-diagonal entries are contained in the maximal order in $\mathbb{O}$. The greatest common divisor (GCD) of entries of $A\in J(O)$ is simply the largest power $\unicode[STIX]{x1D71B}^{n}$ dividing $A$, that is, such that $A/\unicode[STIX]{x1D71B}^{n}$ is in $J(O)$. We have the following theorem [Reference Savin and WoodburySW07].

Theorem 3.2. Assume that $G$ is split and $F$ a $p$-adic field. Assume the conductor of $\unicode[STIX]{x1D713}$ is $O$. Then the spherical vector in $\unicode[STIX]{x1D6F1}$ is a function $f^{\circ }\in C^{\infty }(\unicode[STIX]{x1D6FA})$ supported in $J(O)$. Its value at $A\in \unicode[STIX]{x1D6FA}$ depends on the GCD of entries of $A$. More precisely, if the GCD of the entries of $A$ is $\unicode[STIX]{x1D71B}^{n}$, and $q$ is the order of the residual field, then

$$\begin{eqnarray}f^{\circ }(A)=1+q^{3}+\cdots +q^{3n}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D6F1}$ is generated by $f^{\circ }$ as a $P$-module, and the action of $P$ on $\unicode[STIX]{x1D6F1}$ is easy to describe, this theorem gives us a good handle on $\unicode[STIX]{x1D6F1}$.

Assume now that $F=\mathbb{R}$; in this case, one has a similar result due to Dvorsky and Sahi [Reference Dvorsky and SahiDS99]. For every $a\in M_{2}(\mathbb{R})$, let $\Vert a\Vert ^{2}$ be the sum of squares of its entries. For $x=(a,b)\in \mathbb{O}$, let $\Vert x\Vert ^{2}=\Vert a\Vert ^{2}+\Vert b\Vert ^{2}$. Extend this to $A\in J$ by

$$\begin{eqnarray}\Vert A\Vert ^{2}=\unicode[STIX]{x1D6FC}^{2}+\unicode[STIX]{x1D6FD}^{2}+\unicode[STIX]{x1D6FE}^{2}+\Vert x\Vert ^{2}+\Vert y\Vert ^{2}+\Vert z\Vert ^{2}.\end{eqnarray}$$

Let $K_{3/2}(u)$ denote the modified Bessel function of the second kind. Recall that $K_{3/2}(u)$ is greater than 0, for $u>0$, and rapidly decreasing as $u\rightarrow +\infty$. Then we have the following result [Reference Dvorsky and SahiDS99, Theorem 0.1].

Theorem 3.3. Assume that $G$ is split and $F=\mathbb{R}$. Then the spherical vector in $\unicode[STIX]{x1D6F1}$ is a function $f^{\circ }\in C^{\infty }(\unicode[STIX]{x1D6FA})$ given by

$$\begin{eqnarray}f^{\circ }(A)=\Vert A\Vert ^{-3/2}K_{3/2}(\Vert A\Vert ).\end{eqnarray}$$

4.1 $N_{D}$-spectrum

A crucial step is to understand the $N_{D}$-spectrum of the minimal representation $\unicode[STIX]{x1D6F1}$. In this case we have an exact sequence of $P$-modules

$$\begin{eqnarray}0\rightarrow C_{c}^{\infty }(\unicode[STIX]{x1D6FA})\rightarrow \unicode[STIX]{x1D6F1}\rightarrow \unicode[STIX]{x1D6F1}_{N}\rightarrow 0.\end{eqnarray}$$

The characters of $N_{D}\cong J_{D}$ are identified with the elements in $J_{D}$ using the trace paring, as we did for $J$. We shall only need three characters, denoted by $\unicode[STIX]{x1D713}_{1}$, $\unicode[STIX]{x1D713}_{2}$ and $\unicode[STIX]{x1D713}_{3}$, corresponding to the elements

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}\pm 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{array}\right),\quad \left(\begin{array}{@{}ccc@{}}\pm 1 & 0 & 0\\ 0 & \pm 1 & 0\\ 0 & 0 & 0\end{array}\right)\quad \text{and}\quad \left(\begin{array}{@{}ccc@{}}\pm 1 & 0 & 0\\ 0 & \pm 1 & 0\\ 0 & 0 & \pm 1\end{array}\right)\end{eqnarray}$$

of rank 1, 2 and 3, respectively. We need to allow signs to capture all possible orbits of rank 1, 2 and 3 in the real case. The following lemma is one of the key ingredients in this paper, and we emphasize that we do not assume that $D$ is split here.

Proof. Let $\unicode[STIX]{x1D714}_{i}\subseteq \unicode[STIX]{x1D6FA}$ be the set of all $A\in \unicode[STIX]{x1D6FA}$ such that the restriction of $\unicode[STIX]{x1D713}_{A}$ to $N_{D}$ is equal to $\unicode[STIX]{x1D713}_{i}$. Because $\unicode[STIX]{x1D713}_{i}$ is not the trivial character, the set $\unicode[STIX]{x1D714}_{i}$ is (Zariski) closed in $\unicode[STIX]{x1D6FA}$. Hence,

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{N_{D},\unicode[STIX]{x1D713}_{i}}\cong C_{c}^{\infty }(\unicode[STIX]{x1D714}_{i}).\end{eqnarray}$$

It remains to determine each $\unicode[STIX]{x1D714}_{i}$. Let us start with $i=3$. Then $\unicode[STIX]{x1D714}_{3}$ consists of all $A\in \unicode[STIX]{x1D6FA}$ such that

$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}\pm 1 & x & -z\\ -x & \pm 1 & y\\ z & -y & \pm 1\end{array}\right),\end{eqnarray}$$

where $x=(0,a),y=(0,b)$ and $z=(0,c)$ for some $a,b,c\in D$. Since $A\in \unicode[STIX]{x1D6FA}$, we further have $A^{2}=\text{tr}(A)A$. Looking at the off-diagonal terms, we get the equations

$$\begin{eqnarray}yx=\pm z,\quad zy=\pm x\quad \text{and}\quad xz=\pm y.\end{eqnarray}$$

But the products $yx$, $zy$ and $xz$ have the second coordinate equal to 0. Hence $z=x=y=0$. But then $A$ cannot be a rank-$1$ matrix. Hence $\unicode[STIX]{x1D714}_{3}$ is empty, and this proves (i).

For (ii) we see analogously that $y=z=0$. Now $A$ has rank 1 if and only if the first $2\times 2$ minor is 0. This gives $x^{2}=\pm 1$. Writing this out, with $x=(0,a)$ we see that $\unicode[STIX]{x1D706}a\bar{a}=\pm 1$. Hence $\unicode[STIX]{x1D714}_{2}$ is identified with the set of all elements in $D$ with a fixed non-zero norm. This is a principal homogeneous space for $D^{1}$. This establishes (ii). In the last case it is easy to see that $x=y=z=0$.◻

We now derive a consequence. Let $\unicode[STIX]{x1D6E9}(1)$ be the maximal quotient of $\unicode[STIX]{x1D6F1}$ on which $D^{1}$ acts trivially; it is naturally a $G_{D}$-module. Lemma 4.1 implies that

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(1)_{N_{D},\unicode[STIX]{x1D713}_{3}}=0\quad \text{and}\quad \unicode[STIX]{x1D6E9}(1)_{N_{D},\unicode[STIX]{x1D713}_{2}}=\mathbb{C}.\end{eqnarray}$$

Let $I_{D}(s)$ be the degenerate principal series representation of $G_{D}$ attached to $P_{D}$ normalized as in [Reference WeissmanWei03]. In particular, the trivial representation is a quotient for $s=5$ and a submodule for $s=-5$. The inclusion $\unicode[STIX]{x1D6F1}\rightarrow I(-5)$ composed with the restriction of functions from $G$ to $G_{D}$ gives a non-zero $D^{1}$-invariant map $\unicode[STIX]{x1D6F1}\rightarrow I_{D}(-1)$, which clearly factors through $\unicode[STIX]{x1D6E9}(1)$. By [Reference WeissmanWei03] and [Reference Hanzer and SavinHS20], $I_{D}(-1)$ has a composition series of length 2. The unique irreducible submodule $\unicode[STIX]{x1D6F4}$ has $N_{D}$-rank 2. We have the following corollary.

Corollary 4.2. The above construction gives a surjective $G_{D}$-equivariant map

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(1)\rightarrow \unicode[STIX]{x1D6F4}\subset I_{D}(-1)\end{eqnarray}$$

whose kernel has $N_{D}$-rank no greater than $1$. If $D$ is a division algebra, then $\unicode[STIX]{x1D6E9}(1)\cong \unicode[STIX]{x1D6F4}$.

4.2 Local lifts for split $D$

We shall strengthen here the result of Corollary 4.2 by showing that $\unicode[STIX]{x1D6E9}(1)\cong \unicode[STIX]{x1D6F4}$ even when $D$ is split, in which case $G$ is also split.

Let $T\subset G$ be a maximal split torus, so we have the associated root groups. Furthermore, $D^{1}\cong \text{SL}_{2}$ and it is conjugated to a root $\text{SL}_{2}$. Without loss of generality, we can assume that $\text{SL}_{2}$ corresponds to the highest root for some choice of positive roots. Let $T_{1}=\text{SL}_{2}\cap T$. Then the centralizer of $T_{1}$ in $G$ is a Levi subgroup $L$ of semisimple type $\mathbf{D}_{6}$. The Levi subgroup $L$ is contained in two maximal parabolic subgroups: $Q=LU$ and its opposite $\bar{Q}=L\bar{U}$. The unipotent radical $U$ is a two-step unipotent group with the center $U_{1}$ given by the root group corresponding to the highest root. Similarly, the center of $\bar{U}$ is the root subgroup $\bar{U}_{1}$ corresponding to the lowest root. These two root groups $U_{1}$ and $\bar{U}_{1}$ generate $\text{SL}_{2}$. We identify $T_{1}\cong \text{GL}_{1}$ so that $x\in \text{GL}_{1}$ acts on $U/U_{1}$ as multiplication by $x$.

The conjugation action of $L$ on $U_{1}$ and $\bar{U}_{1}$ is given by a character and its inverse; this character is given by $x\mapsto x^{2}$ when restricted to $T_{1}\subset L$. Hence $G_{D}$ is the kernel of this character, which is the derived group of $L$. Since $G$ is of adjoint type, $G_{D}$ acts faithfully on $U/U_{1}$ (a 32-dimensional spin representation). Note that the representation $U/U_{1}$ is not $W_{D}$, the 32-dimensional representation of $\tilde{G}_{D}$, from § 2.3.

More precisely, recall that the center of $\text{Spin}_{12}$ can be identified with $\unicode[STIX]{x1D707}_{2}\times \unicode[STIX]{x1D707}_{2}$ in such a way that the outer automorphism exchanges the two $\unicode[STIX]{x1D707}_{2}$, and fixes the diagonal $\unicode[STIX]{x1D707}_{2}^{\unicode[STIX]{x1D6E5}}$. The quotient of $\text{Spin}_{12}$ by $\unicode[STIX]{x1D707}_{2}^{\unicode[STIX]{x1D6E5}}$ is the special orthogonal group $\text{SO}_{12}$. On the other hand, the quotient of $\text{Spin}_{12}$ by $\unicode[STIX]{x1D707}_{2}=\unicode[STIX]{x1D707}_{2}\times \{1\}$ and that by $\unicode[STIX]{x1D707}_{2}^{\prime }=\{1\}\times \unicode[STIX]{x1D707}_{2}$ are isomorphic (being isomorphic via the outer automorphism). Then one has

$$\begin{eqnarray}G_{D}\cong \text{Spin}_{12}/\unicode[STIX]{x1D707}_{2}\quad \text{and}\quad L\cong T_{1}\times _{\unicode[STIX]{x1D707}_{2}}G_{D}\cong \text{GL}_{1}\times _{\unicode[STIX]{x1D707}_{2}^{\prime }}(\text{Spin}_{12}/\unicode[STIX]{x1D707}_{2}),\end{eqnarray}$$

so that $L$ has connected center. On the other hand, the group $\tilde{G}_{D}$ from § 2.3 is given by

$$\begin{eqnarray}\tilde{G}_{D}\cong \text{GL}_{1}\times _{\unicode[STIX]{x1D707}_{2}}\text{Spin}_{12}.\end{eqnarray}$$

As we mentioned in § 2.3, the action of $\tilde{G}_{D}$ on $W_{D}$ is not faithful.

We now need a result on the restriction of $\unicode[STIX]{x1D6F1}$ to the maximal parabolic subgroup $Q=LU$. By [Reference Magaard and SavinMS97, Theorem 6.1], the space of $U_{1}$-coinvariants of $\unicode[STIX]{x1D6F1}$, an $L$-module, sits in an exact sequence

$$\begin{eqnarray}0\rightarrow C_{c}^{\infty }(\unicode[STIX]{x1D714})\rightarrow \unicode[STIX]{x1D6F1}_{U_{1}}\rightarrow \unicode[STIX]{x1D6F1}_{U}\rightarrow 0,\end{eqnarray}$$

where $\unicode[STIX]{x1D714}$ is the $L$-orbit of highest weight vectors in $\bar{U}/\bar{U}_{1}$. The action of $L$ on $C_{c}^{\infty }(\unicode[STIX]{x1D714})$ arises from the natural action of $L$ on $\unicode[STIX]{x1D714}$ twisted by an unramified character.

Let $Q_{D}=L_{D}U_{D}$ be a maximal parabolic subgroup in $G_{D}$ stabilizing the line through a point $v\in \unicode[STIX]{x1D714}$. Note that the Levi factor $L_{D}$ of $Q_{D}$ is also of type $\mathbf{A}_{5}$ (like that of $P_{D}$). The action of $Q_{D}$ on the line gives a homomorphism $\unicode[STIX]{x1D712}:Q_{D}\rightarrow \text{GL}_{1}$. Thus the stabilizer in $G_{D}\times \text{GL}_{1}$ of $v$ consists of all pairs $(g,x)$ such that $g\in Q_{D}$ and $\unicode[STIX]{x1D712}(g)=x$. Since $G_{D}\times \text{GL}_{1}$ acts transitively on $\unicode[STIX]{x1D714}$, it is easy to see that the following theorem holds.

Now we can prove the following result which strengthens Corollary 4.2 and which is needed later.

Proof. Let $\unicode[STIX]{x1D70B}$ be an irreducible representation of $\text{SL}_{2}$ and $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B})$ the corresponding big theta lift. We first note that $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B})$ is always non-trivial, as a simple consequence of Lemma 4.1. Moreover, $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B})_{N_{D},\unicode[STIX]{x1D713}_{2}}$ is isomorphic to $\unicode[STIX]{x1D70B}^{\vee }$, so that it is infinite-dimensional if and only if $\unicode[STIX]{x1D70B}$ is.

Let $J(s)$ be the principal series for $\text{SL}_{2}$ normalized so that the trivial representation is a quotient for $s=1$ and a submodule for $s=-1$. Likewise, let $J_{D}(s)$ denote the degenerate principal series associated to $Q_{D}$, normalized so that the trivial representation occurs at $J_{D}(\pm 5)$.

If $-s\neq 2,3$, then Theorem 4.3 implies by way of the Frobenius reciprocity that

$$\begin{eqnarray}\text{Hom}(\unicode[STIX]{x1D6E9}(J(-s)),\mathbb{C})\cong \text{Hom}_{\text{SL}_{2}}(\unicode[STIX]{x1D6F1},J(-s))\cong \text{Hom}(J_{D}(s),\mathbb{C})\end{eqnarray}$$

as $G_{D}$-modules. For generic $s$, both $J(-s)$ and $J_{D}(s)$ are irreducible and the above identity implies that

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(J(-s))\cong J_{D}(s)\end{eqnarray}$$

for such $s$. It follows from Lemma 4.1 that $J_{D}(s)_{N_{D},\unicode[STIX]{x1D713}_{2}}$ is infinite-dimensional for such $s$. However, since the restriction of $J_{D}(s)$ to $N_{D}$ is independent of $s$, it follows that $J_{D}(s)_{N_{D},\unicode[STIX]{x1D713}_{2}}$ is in fact infinite-dimensional for all $s$.

Now if $\unicode[STIX]{x1D70B}$ is a submodule of $J(-s)$ with $-s\neq 2,3$, then it follows that

$$\begin{eqnarray}\text{Hom}(\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B}),\mathbb{C})\cong \text{Hom}_{\text{SL}_{2}}(\unicode[STIX]{x1D6F1},\unicode[STIX]{x1D70B})\subset \text{Hom}_{\text{SL}_{2}}(\unicode[STIX]{x1D6F1},J(-s))\cong \text{Hom}(J_{D}(s),\mathbb{C}),\end{eqnarray}$$

so that $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B})$ is a quotient of $J_{D}(s)$. In particular, for $\unicode[STIX]{x1D70B}=1$ the trivial representation, we may take $s=1$ to deduce that $\unicode[STIX]{x1D6E9}(1)$ is a quotient of $J_{D}(1)$. Since we know that $\unicode[STIX]{x1D6E9}(1)_{N_{D},\unicode[STIX]{x1D713}_{2}}$ is one-dimensional whereas $J_{D}(1)_{N_{D},\unicode[STIX]{x1D713}_{2}}$ is infinite-dimensional, we conclude that $\unicode[STIX]{x1D6E9}(1)$ is isomorphic to the unique irreducible quotient of $J_{D}(1)$ which has $N_{D}$-rank 2. In particular, $\unicode[STIX]{x1D6E9}(1)$ is irreducible and isomorphic to $\unicode[STIX]{x1D6F4}$, the unique quotient of $I_{D}(1)$.◻

As a side remark, the representations $J_{D}(s)$ have $U_{D}$-rank 3. However, since $\unicode[STIX]{x1D6F1}$ has $N_{D}$-rank 2, it follows that the two parabolic subgroups $P_{D}$ and $Q_{D}$ are not conjugate in $G_{D}$. But the two principal series $I_{D}(s)$ and $J_{D}(s)$ share all small-rank subquotients – the trivial representation, the minimal representation and the rank-2 representation $\unicode[STIX]{x1D6F4}$ – as the above argument shows.

5.1 Global theta lifting

Let $\unicode[STIX]{x1D6F1}=\otimes \unicode[STIX]{x1D6F1}_{v}$ be the restricted tensor product of minimal representations over all local places $v$ of $F$, where $\unicode[STIX]{x1D6F1}_{v}\subset C^{\infty }(\unicode[STIX]{x1D6FA}_{v})$, as in Theorem 3.1. Every element in $\unicode[STIX]{x1D6F1}$ is a finite linear combination of pure tensors $f=\otimes f_{v}$, where $f_{v}=f_{v}^{\circ }$ for almost all places $v$. There is a unique (up to a non-zero scalar) embedding $\unicode[STIX]{x1D703}:\unicode[STIX]{x1D6F1}\rightarrow {\mathcal{A}}(G(F)\backslash G(\mathbb{A}))$ of $\unicode[STIX]{x1D6F1}$ into the space of automorphic functions of uniform moderate growth.

We restrict $\unicode[STIX]{x1D703}(f)$ to the dual pair $D^{1}\times G_{D}$ and, for every $h\in {\mathcal{A}}(D^{1}(F)\backslash D^{1}(\mathbb{A}))$, consider the function $\unicode[STIX]{x1D6E9}(f,h)$ on $G_{D}$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(f,h)(g_{D})=\int _{D^{1}(F)\backslash D^{1}(\mathbb{A})}\unicode[STIX]{x1D703}(f)(g_{D}g)\cdot \bar{h}(g)\,dg.\end{eqnarray}$$

If this is to be of any use, we require the function $\unicode[STIX]{x1D703}(f)(g_{D}g)\cdot \bar{h}(g)$ to be of rapid decay on $D^{1}(F)\backslash D^{1}(\mathbb{A})$ and of moderate growth on $G_{D}(F)\backslash G_{D}(\mathbb{A})$. This condition is clearly satisfied if $D^{1}$ is anisotropic or if $h$ is a cusp form. It is also satisfied for a regularized theta lift, to be constructed in the next section. Namely, for any finite place $v$, we will construct an element $z$ in the Bernstein center of $\text{SL}_{2}(F_{v})$, such that for any $f\in \unicode[STIX]{x1D6F1}$, the function $\unicode[STIX]{x1D703}(z\cdot f)(g_{1}g)$ is of rapid decay on $D^{1}(F)\backslash D^{1}(\mathbb{A})$ and of moderate growth on $G_{D}(F)\backslash G_{D}(\mathbb{A})$. (See Proposition 6.1, and the discussion of this particular dual pair thereafter.) In particular, in all these cases, the following integral is convergent:

$$\begin{eqnarray}\int _{N_{D}(F)\backslash N_{D}(\mathbb{A})}\int _{D^{1}(F)\backslash D^{1}(\mathbb{A})}|\unicode[STIX]{x1D703}(z\cdot f)(ng)\cdot \bar{h}(g)|\,dg\,dn.\end{eqnarray}$$

5.2 Fourier expansion

Let $\unicode[STIX]{x1D713}:\mathbb{A}/F\rightarrow \mathbb{C}^{\times }$ be a non-trivial character. Then any $A\in J(F)$ defines a character $\unicode[STIX]{x1D713}_{A}$ of $N(F)\backslash N(\mathbb{A})$ by $\unicode[STIX]{x1D713}_{A}(B)=\unicode[STIX]{x1D713}(\text{tr}(A\circ B))$ for all $B\in N(\mathbb{A})\cong J(\mathbb{A})$. For every $\unicode[STIX]{x1D711}\in {\mathcal{A}}(G(F)\backslash G(\mathbb{A}))$, let

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{A}(g)=\int _{N(F)\backslash N(\mathbb{A})}\unicode[STIX]{x1D711}(ng)\cdot \overline{\unicode[STIX]{x1D713}_{A}(n)}\,dn\end{eqnarray}$$

be the Fourier coefficient corresponding to $A$. We have a Fourier expansion

$$\begin{eqnarray}\unicode[STIX]{x1D703}(f)(g)=\unicode[STIX]{x1D703}(f)_{0}(g)+\mathop{\sum }_{A\in \unicode[STIX]{x1D6FA}(F)}\unicode[STIX]{x1D703}(f)_{A}(g).\end{eqnarray}$$

By uniqueness of local functionals (Theorem 3.1), for every $A\in \unicode[STIX]{x1D6FA}(F)$ there exists a non-zero scalar $c_{A}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D703}(f)_{A}(g)=c_{A}\mathop{\prod }_{v}(g_{v}\cdot f_{v})(A).\end{eqnarray}$$

This formula is particularly useful if $g_{v}\in M(F_{v})$, for then $(g_{v}\cdot f_{v})(A)=\unicode[STIX]{x1D712}_{v}(g_{v})\cdot f_{v}(g_{v}^{-1}\cdot A)$ for the character $\unicode[STIX]{x1D712}_{v}$ of $M(F_{v})$.

Let $\unicode[STIX]{x1D713}_{2}$ and $\unicode[STIX]{x1D713}_{3}$ be the rank-2 and rank-3 characters of $N_{D}(\mathbb{A})$, as in the local case. Recall that $x\in \mathbb{O}$ is a pair $x=(y,z)$ of elements in $D$, and $N(x)=N(y)-\unicode[STIX]{x1D706}N(z)$ for some $\unicode[STIX]{x1D706}\in F^{\times }$. Let $\unicode[STIX]{x1D711}_{N_{D},\unicode[STIX]{x1D713}_{i}}$ denote the global Fourier coefficient with respect to these two characters. Let $\unicode[STIX]{x1D714}_{2}(F)$ be the set of all rank-$1$ matrices

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}\pm 1 & x & 0\\ -x & \pm 1 & 0\\ 0 & 0 & 0\end{array}\right)\in J(F)\end{eqnarray}$$

such that $x=(0,a)$ and $\unicode[STIX]{x1D706}N(a)=\pm 1$ (for only one choice of sign, depending on $\unicode[STIX]{x1D713}_{2}$), that is, the $2\times 2$ minor is 0. Then we have a global version of Lemma 4.1.

Lemma 5.1. For every $f\in \unicode[STIX]{x1D6F1}$, $\unicode[STIX]{x1D703}(f)_{N_{D},\unicode[STIX]{x1D713}_{3}}=0$ and

$$\begin{eqnarray}\unicode[STIX]{x1D703}(f)_{N_{D},\unicode[STIX]{x1D713}_{2}}(g)=\mathop{\sum }_{B\in \unicode[STIX]{x1D714}_{2}(F)}\unicode[STIX]{x1D703}(f)_{B}(g).\end{eqnarray}$$

5.3 Non-vanishing of the theta lift

We shall prove non-vanishing of the (regularized) theta lift by computing the Fourier coefficient

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(f,h)_{N_{D},\unicode[STIX]{x1D713}_{2}}(1)=\int _{N_{D}(F)\backslash N_{D}(\mathbb{A})}\int _{D^{1}(F)\backslash D^{1}(\mathbb{A})}\unicode[STIX]{x1D703}(f)(ng)\cdot \bar{h}(g)\cdot \bar{\unicode[STIX]{x1D713}}_{2}(n)\,dg\,dn.\end{eqnarray}$$

Since this integral is absolutely convergent, we can reverse the order of integration. Then, using Lemma 5.1, we obtain

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(f,h)_{N_{D},\unicode[STIX]{x1D713}_{2}}(1)=\int _{D^{1}(F)\backslash D^{1}(\mathbb{A})}\mathop{\sum }_{B\in \unicode[STIX]{x1D714}_{2}(F)}\unicode[STIX]{x1D703}(f)_{B}(g)\cdot \bar{h}(g)\,dg.\end{eqnarray}$$

Lemma 5.2. Fix

$$\begin{eqnarray}A=\left(\begin{array}{@{}ccc@{}}\pm 1 & x & 0\\ -x & \pm 1 & 0\\ 0 & 0 & 0\end{array}\right)\in \unicode[STIX]{x1D714}_{2}(F),\end{eqnarray}$$

where $x=(0,a)$, $a\in D$ satisfies $\unicode[STIX]{x1D706}N(a)=\pm 1$.

For every automorphic form $h$ and every $f\in \unicode[STIX]{x1D6F1}$ we have

$$\begin{eqnarray}\int _{D^{1}(F)\backslash D^{1}(\mathbb{A})}\mathop{\sum }_{B\in \unicode[STIX]{x1D714}_{2}(F)}\unicode[STIX]{x1D703}(f)_{B}(g)\bar{h}(g)\,dg=c_{A}\int _{D^{1}(\mathbb{A})}f(g^{-1}A)\bar{h}(g)\,dg,\end{eqnarray}$$

where the second integral is absolutely convergent.

Proof. Since $\unicode[STIX]{x1D714}_{2}(F)$ is a principal homogeneous $D^{1}(F)$-space, the identity formally follows by unfolding the left-hand side and using the formula for $\unicode[STIX]{x1D703}(f)_{A}(g)$ as a product of local functionals given above. Hence it remains to discuss the issue of absolute convergence.

We may assume that $f=\bigotimes _{v}f_{v}$ is a pure tensor. For each place $v$, observe that if $g\in \text{SL}_{2}(F_{v})$, then $g^{-1}A$ is obtained from $A$ by replacing $x$ by $xg$. Hence, $g\mapsto g^{-1}A$ gives a closed embedding of $\text{SL}_{2}(F_{v})$ into $J(F_{v})$, with image contained in $\unicode[STIX]{x1D6FA}_{v}$. In particular, this image is bounded away from the vertex 0 of the cone $\unicode[STIX]{x1D6FA}_{v}$. Since $h$ is of moderate growth on $\text{SL}_{2}(F_{v})$, to show that the integral in question is absolutely convergent, we need to show that $g\mapsto f_{v}(g^{-1}A)$ is a Schwartz function on $\text{SL}_{2}(F_{v})$. For this, it suffices to show that as a function on the cone $\unicode[STIX]{x1D6FA}_{v}$, $f_{v}$ is rapidly decreasing toward infinity, as we shall explain below.

In greater detail, assume first that $v$ is a finite place. Due to $N_{v}$-smoothness, $f_{v}\in \unicode[STIX]{x1D6F1}_{v}$ is supported on a lattice in $J_{v}$ (and thus vanishes toward infinity). It follows that $g\mapsto f_{v}(g^{-1}A)$ is a compactly supported function on $\text{SL}_{2}(F_{v})$. Moreover, let $S$ be a finite set of places containing all archimedean places such that for $v\notin S$, all data is unramified: $D(F_{v})$ is split, $\unicode[STIX]{x1D706}\in O_{v}^{\times }$, $a\in \text{GL}_{2}(O_{v})$, $\unicode[STIX]{x1D713}_{v}$ has the conductor $O_{v}$, $f_{v}=f_{v}^{\circ }$, and $h$ is right $\text{SL}_{2}(O_{v})$-invariant. Here $O_{v}$ is the maximal order in $F_{v}$. It follows from Theorem 3.2 that $g\mapsto f_{v}^{\circ }(g^{-1}A)$ is the characteristic function of $\text{SL}_{2}(O_{v})$ for all $v\notin S$. Thus if we normalize the local measures so that $\text{vol}(\text{SL}_{2}(O_{v}))=1$ for all $v\notin S$, then

$$\begin{eqnarray}\int _{D^{1}(\mathbb{A})}|f(g^{-1}A)\bar{h}(g)|\,dg=\int _{D^{1}(\mathbb{A}_{S})}|f_{S}(g^{-1}A)\bar{h}(g)|\,dg,\end{eqnarray}$$

where the subscript $S$ denotes the product of the local data over all places $v\in S$.

Consider now the case where $v$ is a real place. We need to show that $C\mapsto f_{v}(C)$ is of rapid decay in $\Vert C\Vert$, where $C\in \unicode[STIX]{x1D6FA}(\mathbb{R})$. To that end, let $m_{v}\in M(\mathbb{R})$ such that $C=m_{v}^{-1}\cdot A$. Then, up to a non-zero constant $c$, independent of $C$,

$$\begin{eqnarray}f_{v}(C)=c\cdot \unicode[STIX]{x1D712}_{v}(m_{v})^{-1}\cdot \unicode[STIX]{x1D703}(f)_{A}(m_{v})\end{eqnarray}$$

for the character $\unicode[STIX]{x1D712}_{v}$ of $M(\mathbb{R})$. Now observe that $m_{v}$ can be taken to be a product of an element $k_{v}$ in a maximal compact subgroup of $M(\mathbb{R})$ and an element $z_{v}$ in $Z_{v}$, the identity component of the center of $M(\mathbb{R})$. We fix an isomorphism $\unicode[STIX]{x1D708}:Z_{v}\rightarrow \mathbb{R}^{+}$ such that the conjugation action of $z_{v}\in Z_{v}$ on $N(\mathbb{R})$ is given by multiplication by $\unicode[STIX]{x1D708}(z_{v})$. Now, in order to prove that $f_{v}$ is rapidly decreasing toward infinity, we shall give a global argument exploiting the automorphic form $\unicode[STIX]{x1D703}(f)$ (though a local proof is also possible). Namely, it suffices to show that $z_{v}\mapsto \unicode[STIX]{x1D703}(f)_{A}(z_{v}k_{v})$ is rapidly decreasing as $\unicode[STIX]{x1D708}(z_{v})\rightarrow \infty$, with bounds independent of $k_{v}$. This can be proved using the usual method of integration by parts, as in [Reference Moeglin and WaldspurgerMW95, p. 30, Lemma].

More precisely, if $X\in J\cong \mathfrak{n}$, then the $X$-derivative of the character $\unicode[STIX]{x1D713}_{A}$ is a multiple of $\unicode[STIX]{x1D713}_{A}$. Using the definition of the Fourier coefficient and integration by parts, one obtains that

$$\begin{eqnarray}\unicode[STIX]{x1D703}(f)_{A}(z_{v}k_{v})\text{ is a multiple of }(R_{Y}\cdot \unicode[STIX]{x1D703}(f))_{A}(z_{v}k_{v})\cdot \unicode[STIX]{x1D708}(z_{v})^{-1},\end{eqnarray}$$

where $Y=k_{v}^{-1}Xk_{v}$ and $R_{Y}$ denotes the right $Y$-derivative of the automorphic form $\unicode[STIX]{x1D703}(f)$. We can repeat this procedure to get any negative power of $\unicode[STIX]{x1D708}(z_{v})$. The rapid decay follows from the fact that $\unicode[STIX]{x1D703}(f)$ is of uniform moderate growth, and the fact that $Y=k_{v}^{-1}Xk_{v}$ is a linear combination of vectors in any fixed basis of $\mathfrak{n}$, with bounded coefficients, as $k_{v}$ runs over the maximal compact subgroup in $M(\mathbb{R})$.

Finally, suppose that $g\in \text{SL}_{2}(\mathbb{R})$ belongs to the double coset of the diagonal matrix $(\!\begin{smallmatrix}t & 0\\ 0 & 1/t\end{smallmatrix}\!)$, $t>0$, in the Cartan decomposition of $\text{SL}_{2}(\mathbb{R})$. If we assume for simplicity that $\unicode[STIX]{x1D706}=1$, so that $a$ in $x=(0,a)$ can be taken to be the identity matrix, then $\Vert xg\Vert ^{2}=t^{2}+1/t^{2}$ (on the nose) and $\Vert g^{-1}A\Vert =t+1/t$. In particular, $t<\Vert g^{-1}A\Vert <t+1$ for $t>1$. Hence, the rapid decay toward infinity of $f_{v}$ (as a function on $\unicode[STIX]{x1D6FA}_{v}$) implies that $g\mapsto f_{v}(g^{-1}A)$ has rapid decay on $\text{SL}_{2}(\mathbb{R})$, as desired.◻

We are now ready to prove the non-vanishing of the global theta lift. Assume firstly that $h$ is a cusp form. Then we have shown that

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(f,h)_{N_{D},\unicode[STIX]{x1D713}_{2}}(1)=\int _{D^{1}(\mathbb{A}_{S})}f_{S}(g^{-1}A)\bar{h}(g)\,dg\end{eqnarray}$$

for some large finite set of places. Since for every $v\in S$ the local $f_{v}$ can be an arbitrary compactly supported smooth function on $\unicode[STIX]{x1D6FA}_{v}$, the integral will not vanish for some choice of data. Now consider the regularized theta integral $\unicode[STIX]{x1D6E9}(z\cdot f,h)$, where $h$ is in an automorphic form, not necessarily cuspidal, and $z$ is an element of the Bernstein center of $\text{SL}_{2}(F_{v})$ for a particular fixed finite place $v$ (see the next section for the construction of $z$). The corresponding Fourier coefficient is

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(z\cdot f,h)_{N_{D},\unicode[STIX]{x1D713}_{2}}(1)=\int _{D^{1}(\mathbb{A})}(z\cdot f)(g^{-1}A)\bar{h}(g)\,dg.\end{eqnarray}$$

Let $K_{v}$ be a sufficiently small open compact subgroup of $\text{SL}_{2}(F_{v})$ such that $f_{v}$ is $K_{v}$-invariant. Then $z\cdot f_{v}=\unicode[STIX]{x1D6FC}\cdot f_{v}$, where $\unicode[STIX]{x1D6FC}$ is a $K_{v}$ bi-invariant, compactly supported function on $\text{SL}_{2}(F_{v})$. Let $\unicode[STIX]{x1D6FC}^{\vee }(g)=\bar{\unicode[STIX]{x1D6FC}}(g^{-1})$ and define $z^{\vee }\cdot h=\unicode[STIX]{x1D6FC}^{\vee }\cdot h$. Using the convergence guaranteed by Lemma 5.2,

$$\begin{eqnarray}\int _{D^{1}(\mathbb{A})}(z\cdot f)(g^{-1}A)\bar{h}(g)\,dg=\int _{D^{1}(\mathbb{A})}f(g^{-1}A)(\overline{z^{\vee }\cdot h})(g)\,dg,\end{eqnarray}$$

and this can again be arranged to be non-zero, provided $z^{\vee }\cdot h\neq 0$. Hence we have proved the following theorem.

Remark.

The main reason for introduction of the regularized theta lift is to be able to handle the lift of $h=1$ in the case where $D$ is split. In this case we can take all data to be the simplest possible: $\unicode[STIX]{x1D706}=1$, the matrix $A$ with $a=(0,x)$ with $x$ the identity matrix, etc. Then the non-vanishing of the theta lift is achieved with the spherical vector $f_{\infty }^{\circ }$ at any real place. Indeed, if $g\in \text{SL}_{2}(\mathbb{R})$ belongs to the double coset of the diagonal matrix $(\!\begin{smallmatrix}t & 0\\ 0 & 1/t\end{smallmatrix}\!)$, with $t>0$, in the Cartan decomposition of $\text{SL}_{2}(\mathbb{R})$, then $\Vert xg\Vert ^{2}=t^{2}+1/t^{2}$ and $\Vert g^{-1}A\Vert =t+1/t$. Write $u=t+1/t$ so that

$$\begin{eqnarray}du=\biggl(t-\frac{1}{t}\biggr)\frac{dt}{t}.\end{eqnarray}$$

Using the formula for the spherical vector given by Theorem 3.3 and the formula for the Haar measure on $\text{SL}_{2}(\mathbb{R})$ with respect to the Cartan decomposition, we have

$$\begin{eqnarray}\int _{\text{SL}_{2}(\mathbb{R})}f_{\infty }^{\circ }(g^{-1}A)\,dg=\int _{2}^{\infty }\frac{1}{2}\cdot u^{-1/2}K_{3/2}(u)\,du>0.\end{eqnarray}$$

It will be interesting to compute the value of this integral.

6.1 Moderate growth and rapid decay

Let $k$ be a number field and let $\mathbb{A}$ denote the corresponding ring of adèles. Let $G$ be a reductive group over $k$. In order to keep the notation simple, we shall assume that $G$ is split with a finite center. Fix a maximal split torus $T$ and a minimal parabolic subgroup $P$ containing $T$. Let $N$ be the unipotent radical of $P$. We have a root system $\unicode[STIX]{x1D6F7}$, obtained by $T$ acting on the Lie algebra $\mathfrak{g}$ of $G$ and a set of simple roots in $\unicode[STIX]{x1D6F7}$ corresponding to the choice of $P$.

If we fix a place $v$ of $k$, then $G_{v}$ will denote the group of $k_{v}$-points of $G$. Similarly, we shall use the subscript $v$ to denote various other subgroups of $G_{v}$. A smooth function $f$ on $G(\mathbb{A})$ is of uniform moderate growth if there exists an integer $m$ such that for every $X$ in the enveloping algebra of $\mathfrak{g}$ there exists a constant $c_{X}$ such that

$$\begin{eqnarray}|R_{X}f(g)|\leqslant c_{X}\Vert g\Vert ^{m},\end{eqnarray}$$

where $R_{X}$ denotes the action of the enveloping algebra on smooth functions obtained by the differentiation from the right and $\Vert g\Vert$ is a height function on $G$ defined in [Reference Moeglin and WaldspurgerMW95, p. 20]. Since there exists a constant $c$ such that $\Vert gh\Vert \leqslant c\Vert g\Vert \cdot \Vert h\Vert$ for all $g,h\in G(\mathbb{A})$, it is easy to see that the constants $c_{X}$ for the right translates $R_{h}f$ of $f$ are of moderate growth in $h$, or more precisely of growth $\Vert h\Vert ^{m+d}$ where $d$ is the degree of $X$.

Now assume that $v$ is a real or complex place of $k$. Let $P_{v}=M_{v}A_{v}N_{v}$ be the Langlands decomposition of $P_{v}$. For $\unicode[STIX]{x1D716}>0$, let $A_{v,\unicode[STIX]{x1D716}}$ be a cone in $A_{v}$ consisting of $a\in A_{v}$ such that $\unicode[STIX]{x1D6FC}(a)>\unicode[STIX]{x1D716}$ for all simple roots $\unicode[STIX]{x1D6FC}$. Let $A$ be the product of the $A_{v}$ and let $A_{\unicode[STIX]{x1D716}}$ be the product of the $A_{v,\unicode[STIX]{x1D716}}$ over all real and complex places $v$. Let $\unicode[STIX]{x1D714}_{N}$ be a compact set in $N(\mathbb{A})$ containing the identity element. Let $K$ be a product of maximal compact subgroups $K_{v}$ of $G_{v}$, where we have taken $K_{v}$ to be hyperspecial for all $p$-adic places. Then

$$\begin{eqnarray}S=\unicode[STIX]{x1D714}_{N}A_{\unicode[STIX]{x1D716}}K\end{eqnarray}$$

is a Siegel domain in $G(\mathbb{A})$. If $\unicode[STIX]{x1D714}_{N}$ is sufficiently large and $\unicode[STIX]{x1D716}$ is sufficiently small, then $G(\mathbb{A})=G(k)S$.

Let $\unicode[STIX]{x1D6F1}$ be an automorphic representation of $G$. Then any smooth $f\in \unicode[STIX]{x1D6F1}$ is of uniform moderate growth. In terms of the Siegel domain $S$, this means the following. Let $\unicode[STIX]{x1D70C}_{P}:A\rightarrow \mathbb{R}^{+}$ be the modular character. There exists an integer $m$ such that for every $X$ in the enveloping algebra of $\mathfrak{g}$, there exists a constant $c_{X}$ such that

$$\begin{eqnarray}|R_{X}f(nak)|\leqslant c_{X}\cdot \unicode[STIX]{x1D70C}_{P}(a)^{m}\end{eqnarray}$$

on $S$, where the constants $m$ and $c_{X}$ are not necessarily the same, but related to those above.

Now let $Q\supseteq P$ be a maximal parabolic with a unipotent radical $U\subseteq N$, corresponding to a simple root $\unicode[STIX]{x1D6FC}$. We have a standard Levi factor $L$ of $Q$ defined as the centralizer of a fundamental cocharacter $\unicode[STIX]{x1D712}:\mathbb{G}_{m}\rightarrow T$ (or a power of it). In any case, any element in $A_{v}$ is uniquely written as a product $\prod _{\unicode[STIX]{x1D712}}\unicode[STIX]{x1D712}(t_{\unicode[STIX]{x1D712}})$, over all fundamental cocharacters $\unicode[STIX]{x1D712}$, where $t_{\unicode[STIX]{x1D712}}\in \mathbb{R}^{+}$. The element $\prod _{\unicode[STIX]{x1D712}}\unicode[STIX]{x1D712}(t_{\unicode[STIX]{x1D712}})$ is contained in the cone $A_{v,\unicode[STIX]{x1D716}}$ if $t_{\unicode[STIX]{x1D712}}>\unicode[STIX]{x1D716}$ for all $\unicode[STIX]{x1D712}$. Let $f_{U}$ be the constant term of $f$ along $U$. Then, if $f$ has uniform moderate growth, by [Reference Moeglin and WaldspurgerMW95, p. 30, Lemma] for every positive integer $i$, there is a constant $c_{i}$ such that

$$\begin{eqnarray}|(f-f_{U})(nak)|\leqslant c_{i}\cdot \unicode[STIX]{x1D70C}_{P}(a)^{m}\cdot \unicode[STIX]{x1D6FC}^{-i}(a)\end{eqnarray}$$

on $S$. In particular, if $f_{U}=0$, then $f$ is rapidly decreasing in the variable $t_{\unicode[STIX]{x1D712}}$. If $f_{U}=0$ for all maximal parabolic subgroups, then $f$ is rapidly decreasing on $S$, and that is how the rapid decrease of cusp forms is established. The proof of [Reference Moeglin and WaldspurgerMW95, p. 30, Lemma] involves integration by parts, so it is easy to see that the constants $c_{i}$ for the right translates $R_{h}f$ of $f$ are of moderate growth in $h$, or more precisely of the growth $\Vert h\Vert ^{m^{\prime }}$ where $m^{\prime }$ depends on $i$: a larger $i$ will demand a larger $m^{\prime }$.

We highlight another important issue here. Assume that $f$ belongs to an automorphic representation $\unicode[STIX]{x1D70B}$. Then a Frechét space topology on $\unicode[STIX]{x1D70B}$ is given by the family of seminorms

$$\begin{eqnarray}\Vert f\Vert _{X}=\sup _{nak\in S}|R_{X}f(nak)|\cdot \unicode[STIX]{x1D70C}_{P}(a)^{-m},\end{eqnarray}$$

where $m$ depends on $\unicode[STIX]{x1D70B}$ and works for all $X$ in the enveloping algebra. Then [Reference Moeglin and WaldspurgerMW95, p. 30, Lemma] says that convergence in these seminorms implies convergence in the seminorm

$$\begin{eqnarray}\sup _{nak\in S}|(f-f_{U})(nak)|\cdot \unicode[STIX]{x1D70C}_{P}(a)^{-m}\cdot \unicode[STIX]{x1D6FC}^{i}(a).\end{eqnarray}$$

This observation will later imply that the regularized theta integral gives a continuous pairing.

6.2 Restricting to a subgroup

Let $(G_{1},G_{2})$ be a dual pair in $G$. Let $T_{1}$ be a maximal split torus in $G_{1}$ and fix a minimal parabolic subgroup $P_{1}$ containing $T_{1}$. Without loss of generality, we can assume that $T_{1}\subseteq T$ and $P_{1}\subseteq P$. Let $Q_{1}\supseteq P_{1}$ be a maximal parabolic subgroup of $G_{1}$. Let $\unicode[STIX]{x1D712}_{1}:G_{m}\rightarrow T_{1}$ be the corresponding fundamental cocharacter (or a multiple thereof) so that the centralizer of $\unicode[STIX]{x1D712}_{1}$ in $G_{1}$ is a Levi factor $L_{1}$ of $Q_{1}$. We make the following assumption.

This hypothesis holds in the following examples:

• ∙ the dual pair $G_{1}\times G_{2}=D^{1}\times G_{D}=\text{SL}_{2}\times G_{D}$ studied in this paper. Here $G_{1}=\text{SL}_{2}$ corresponds to the highest root and the highest weight is also a fundamental weight for $\mathbf{E}_{7}$ (the ambient group $G$).

• ∙ the split exceptional dual pairs in $G$ of type $\mathbf{E}_{n}$ where one member of the dual pair is the type $\mathbf{G}_{2}$; see [Reference Loke and SavinLS19]. In particular, this includes the case $\text{PGL}_{3}\times G_{2}$ treated in [Reference GanGan11].

The hypothesis has the following consequences.

• ∙ It implies that the cone $A_{1,\unicode[STIX]{x1D716}}$ sits as a subcone of $A_{\unicode[STIX]{x1D716}}$; in fact, it is a direct factor in the above cases. In particular, we have an inclusion of Siegel domains $S_{1}\subset S$.

• ∙ Given a fundamental cocharacter $\unicode[STIX]{x1D712}_{1}$ of $G_{1}$, the associated fundamental cocharacter $\unicode[STIX]{x1D712}$ of $G$ given by the hypothesis corresponds to a simple root and so determines a maximal parabolic subgroup $Q_{\unicode[STIX]{x1D712}_{1}}=L_{\unicode[STIX]{x1D712}_{1}}U_{\unicode[STIX]{x1D712}_{1}}$ of $G$. In the following, we will sometimes write $U=U_{\unicode[STIX]{x1D712}_{1}}$ to simplify notation.

Now let $v$ be a $p$-adic place and $z$ an element of the Bernstein center of $G_{1}(k_{v})$. Then $z\cdot \unicode[STIX]{x1D6F1}$ is naturally a $G_{1}(\mathbb{A})\times G_{2}(\mathbb{A})$-submodule of $\unicode[STIX]{x1D6F1}$. For a fixed cocharacter $\unicode[STIX]{x1D712}_{1}$ of $G_{1}$, with associated maximal parabolic $Q=LU$ of $G$, assume that

$$\begin{eqnarray}z\cdot \unicode[STIX]{x1D6F1}_{v}\subset \text{Ker}(\unicode[STIX]{x1D6F1}_{v}\longrightarrow (\unicode[STIX]{x1D6F1}_{v})_{U(k_{v})}).\end{eqnarray}$$

We claim that this implies that $(z\cdot f)_{U}=0$ on $G_{1}(\mathbb{A})\times G_{2}(\mathbb{A})$. Indeed, if $g\in G_{1}(\mathbb{A})\times G_{2}(\mathbb{A})$, then

$$\begin{eqnarray}(z\cdot f)_{U}(g)=(R_{g}(z\cdot f))_{U}(1)=(z\cdot R_{g}(f))_{U}(1)=0,\end{eqnarray}$$

where $R_{g}$ denotes the right translation by $g$. Here, the second equality holds since $z$ and $R_{g}$ commute, and the third equality holds since the projection of $z\cdot \unicode[STIX]{x1D6F1}$ on $\unicode[STIX]{x1D6F1}_{U}$ vanishes. Write $g=g_{1}\times g_{2}\in G_{1}(\mathbb{A})\times G_{2}(\mathbb{A})$ and assume that $g_{1}\in S_{1}$. Using the hypothesis that $S_{1}\subseteq S$ and the estimates for $|R_{g_{2}}(z\cdot f)-(R_{g_{2}}(z\cdot f))_{U}|$ on $S$ from the last subsection, it follows that

$$\begin{eqnarray}(z\cdot f)(g_{1}\times g_{2})=R_{g_{2}}(z\cdot f)(g_{1})\end{eqnarray}$$

is of moderate growth in both variables, and in the variable $g_{1}\in S_{1}$ it is rapidly decreasing in the direction of the fundamental cocharacter $\unicode[STIX]{x1D712}_{1}$. More precisely, we summarize the discussion in this subsection in the following proposition.

Proposition 6.1. Assume that:

1. (i) for every fundamental cocharacter $\unicode[STIX]{x1D712}_{1}$ of $G_{1}$, there is a fundamental cocharacter $\unicode[STIX]{x1D712}$ of $G$ such that $\unicode[STIX]{x1D712}_{1}$ is a multiple of $\unicode[STIX]{x1D712}$, which in turn determines a maximal parabolic subgroup $Q_{\unicode[STIX]{x1D712}_{1}}=L_{\unicode[STIX]{x1D712}_{1}}U_{\unicode[STIX]{x1D712}_{1}}$;

2. (ii) one can find an element $z$ in the Bernstein center of $G_{1}(k_{v})$ such that for every fundamental cocharacter $\unicode[STIX]{x1D712}_{1}$ of $G_{1}$, the natural projection of  $\unicode[STIX]{x1D6F1}_{v}$ to $(\unicode[STIX]{x1D6F1}_{v})_{U_{\unicode[STIX]{x1D712}_{1}}(k_{v})}$ vanishes on $z\cdot \unicode[STIX]{x1D6F1}_{v}$.

Then for every integer $n$, there exist an integer $m$ and a constant $c$ such that

$$\begin{eqnarray}|(z\cdot f)(g_{1}\times g_{2})|\leqslant c\Vert g_{1}\Vert ^{-n}\Vert g_{2}\Vert ^{m}\end{eqnarray}$$

for all $g_{1}\in S_{1}$ and $g_{2}\in G_{2}(\mathbb{A})$.

In the context of the above proposition, a small tradeoff here is that increasing $n$ can be obtained only by increasing $m$ at the same time. But this is still good enough to define a regularized theta lift which produces functions of moderate growth as output. To exploit the proposition, it remains then to construct an appropriate $z$. We also need to ensure that $z\cdot \unicode[STIX]{x1D6F1}_{v}\neq 0$, and this may not be always possible, as will be discussed in the next subsection.

6.3 Bernstein’s center

We work here locally over a $p$-adic field. Thus all our groups are local and we drop the subscript $v$. For simplicity, we shall discuss only the Bernstein center for the Bernstein component containing the trivial representation of $G_{1}$.

To that end, let $\hat{T}_{1}$ be the complex torus dual to $T_{1}$, and let $W(G_{1})$ be the Weyl group of $G_{1}$. The Bernstein center $Z(G_{1})$ of the said component is isomorphic to the algebra of $W(G_{1})$-invariant regular functions on $\hat{T}_{1}$. Similarly, the Bernstein center $Z(L_{1})$ of the Levi factor $L_{1}$ is isomorphic to the algebra of $W(L_{1})$-invariant regular functions on $\hat{T}_{1}$. In particular, we have a natural map $j:Z(G_{1})\rightarrow Z(L_{1})$. Let $\unicode[STIX]{x1D70B}$ be a smooth representation of $G_{1}$, and let $p:\unicode[STIX]{x1D70B}\rightarrow \unicode[STIX]{x1D70B}_{U_{1}}$ be the natural projection onto the normalized Jacquet module $\unicode[STIX]{x1D70B}_{U_{1}}$. Then, for every $z\in Z(G_{1})$ and $v\in \unicode[STIX]{x1D70B}$, we have

$$\begin{eqnarray}p(z\cdot v)=j(z)\cdot (p(v)).\end{eqnarray}$$

Now let $\unicode[STIX]{x1D6F1}$ be a smooth representation of $G$. Recall that we want to find a non-zero $z\in Z(G_{1})$ such that $z\cdot \unicode[STIX]{x1D6F1}$ is in the kernel of the projection of $\unicode[STIX]{x1D6F1}$ onto $\unicode[STIX]{x1D6F1}_{U}$. Since $\unicode[STIX]{x1D6F1}_{U}$ is an $L_{1}$-equivariant quotient of $\unicode[STIX]{x1D6F1}_{U_{1}}$, $j(z)\in Z(L_{1})$ acts on $\unicode[STIX]{x1D6F1}_{U}$ and hence we need to find $z\in Z(G_{1})$ such that $j(z)=0$ on $\unicode[STIX]{x1D6F1}_{U}$. This is always possible if $\unicode[STIX]{x1D6F1}$ is a finite-length $G$-module, in which case $\unicode[STIX]{x1D6F1}_{U}$ is a finite-length $L$-module. In particular, the center of $L$ acts finitely on $\unicode[STIX]{x1D6F1}_{U}$. Hence, the center of $L_{1}$ ($=$ the center of $L$) acts finitely on $\unicode[STIX]{x1D6F1}_{U}$ and the $Z(L_{1})$-spectrum of $\unicode[STIX]{x1D6F1}_{U}$ is contained in a proper subvariety of $\hat{T}_{1}$. In particular, any non-zero $W(G_{1})$-invariant function $z$ vanishing on the subvariety will have the desired property that $j(z)$ vanishes on $\unicode[STIX]{x1D6F1}_{U}$. Hence, a non-trivial $z$ with the desired property always exists.

A potential problem is that such a $z$ may kill the whole $\unicode[STIX]{x1D6F1}$. However, if $G$ is split, $G_{1}$ is the smaller member of the dual pair (also split) and $\unicode[STIX]{x1D6F1}$ the minimal representation, then the spherical matrix coefficient $\unicode[STIX]{x1D6F7}$ of $\unicode[STIX]{x1D6F1}$, when restricted to $G_{1}$, is typically contained in $L^{2-\unicode[STIX]{x1D716}}(G_{1})$ for some $\unicode[STIX]{x1D716}>0$. (This is easy to check in any given situation; see [Reference Loke and SavinLS19]). Thus, in such situations, it makes sense to integrate $\unicode[STIX]{x1D6F7}$ against spherical tempered functions of $G_{1}$, that is, to consider the spherical transform of $\unicode[STIX]{x1D6F7}$ on $G_{1}(k_{v})$. This integral will be non-zero for almost all tempered spherical functions, hence almost all spherical tempered representations of $G_{1}(k_{v})$ will appear as a quotient of $\unicode[STIX]{x1D6F1}$. (This argument for the non-vanishing of the theta lift of almost all irreducible spherical tempered representations holds over archimedean fields as well, as we shall exploit in Lemma 7.6 below.) Hence, if $z$ kills $\unicode[STIX]{x1D6F1}$, then $z$ kills all spherical tempered representations of $G_{1}(k_{v})$ and hence $z$ must be equal to 0. Therefore the desired regularization can be carried out in this case.

Let us look at our dual pair $G_{1}\times G_{2}=\text{SL}_{2}\times G_{D}$ in $G$ with $\unicode[STIX]{x1D6F1}$ the minimal representation. The Bernstein center is

$$\begin{eqnarray}Z(G_{1})=\mathbb{C}[x^{\pm 1}]^{S_{2}},\end{eqnarray}$$

where $S_{2}$ acts by permuting $x$ and $x^{-1}$. Let

$$\begin{eqnarray}z=(x-q^{2})(x^{-1}-q^{2})(x-q^{3})(x^{-1}-q^{3}),\end{eqnarray}$$

where $q$ is the order of the residual field. This element satisfies our requirement, since $j(z)$ vanishes on $\unicode[STIX]{x1D6F1}_{U}$ by Theorem 4.3, and the spherical matrix coefficient of $\unicode[STIX]{x1D6F1}$ is integrable when restricted to $\text{SL}_{2}$.

6.4 Global $\unicode[STIX]{x1D6E9}(1)$

Let $z$ be the element in the Bernstein center of $G_{1}=\text{SL}_{2}$, as in the previous subsection. We define $\unicode[STIX]{x1D6E9}(1)$ as the space of automorphic functions

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(f)(g_{D})=\int _{D^{1}(F)\backslash D^{1}(\mathbb{A})}\unicode[STIX]{x1D703}(z\cdot f)(g_{D}g)\,dg\quad \text{with }g_{D}\in G_{D}(\mathbb{A}),\end{eqnarray}$$

where we assume that $f_{\infty }$ is $K_{\infty }$-finite. (We assume this finiteness since in the next section we will determine the local lift at real places in the language of $(\mathfrak{g},K)$-modules.) We want to show that $\unicode[STIX]{x1D6E9}(1)\neq 0$, using Theorem 5.3. The input in the theta kernel is $h=1$, so the first thing is to show that $z^{\vee }\cdot 1\neq 0$. In the case at hand, $z^{\vee }$ is obtained from $z$ by replacing $x$ by $x^{-1}$ in the above expression for $z$. In particular, $z=z^{\vee }$. Moreover, $z$ acts on the trivial representation by the scalar obtained by substituting $x=q$, and this is non-zero. It remains to argue that we can arrange $f_{\infty }$ to be $K_{\infty }$-finite. This follows by the continuity of the regularized theta integral, which ensures that the non-vanishing for smooth $f$ implies the non-vanishing for $K_{\infty }$-finite vectors. Alternatively, by the remark following Theorem 5.3, non-vanishing can be achieved with $f_{\infty }=f_{\infty }^{\circ }$ the spherical vector.

7.1 Non-split $\mathbb{O}$

Assume firstly that $\mathbb{O}$ and hence $G$ is not split. Then the minimal representation of the adjoint group $G$, when restricted to the simply connected $G^{\prime }$, breaks up as $\unicode[STIX]{x1D6F1}=\unicode[STIX]{x1D6F1}_{1,0}\oplus \unicode[STIX]{x1D6F1}_{0,1}$, a sum of a holomorphic and an anti-holomorphic irreducible representation. This sum is the socle of the degenerate principal series $I(-5)$ (pulled back to $G^{\prime }$). The maximal compact subgroup $K^{\prime }$ is of semisimple type $E_{6}$, and has one-dimensional center $\text{U}(1)$ that acts on the Lie algebra $\mathfrak{g}$ with weights $-2$, $0$ and $2$. The weight-$2$ space is a 27-dimensional representation of $K^{\prime }$. Let $\unicode[STIX]{x1D714}$ be its highest weight. Then

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{1,0}=\bigoplus _{n\geqslant 0}V_{n\unicode[STIX]{x1D714}}(12),\end{eqnarray}$$

where 12 denotes a twist of the irreducible $K^{\prime }$-module $V_{n\unicode[STIX]{x1D714}}$ such that $\text{U}(1)$ acts with the weight $2n+12$ on it.

In this case, $D$ is necessarily non-split. Recalling that $G_{D}^{\prime }$ is the simply connected cover of $G_{D}$, its maximal compact subgroup is $K_{D}^{\prime }\cong \text{U}_{6}$. The socle of $I_{D}(-1)$, considered a representation of $G_{D}^{\prime }$, is a direct sum of three representations $\unicode[STIX]{x1D6F4}_{2,0}\oplus \unicode[STIX]{x1D6F4}_{1,1}\oplus \unicode[STIX]{x1D6F4}_{0,2}$, a holomorphic, spherical and anti-holomorphic representation. respectively [Reference SahiSah93, Theorem C]. The lowest and the highest $K_{D}^{\prime }$-types of $\unicode[STIX]{x1D6F4}_{2,0}$ and $\unicode[STIX]{x1D6F4}_{0,2}$ are one-dimensional with $\text{U}(1)$-weights 12 and $-12$, respectively. We have the following theorem.

Theorem 7.1. If  $\mathbb{O}$ is non-split (so $G$ is not split), we have

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{1,0}^{D^{1}}\cong \unicode[STIX]{x1D6F4}_{2,0}\quad \text{and}\quad \unicode[STIX]{x1D6F1}_{0,1}^{D^{1}}\cong \unicode[STIX]{x1D6F4}_{0,2}\end{eqnarray}$$

as $(\mathfrak{g}_{D},K_{D}^{\prime })$-modules. In particular, $\unicode[STIX]{x1D6E9}(1)\cong \unicode[STIX]{x1D6F4}_{2,0}\oplus \unicode[STIX]{x1D6F4}_{0,2}$ as $(\mathfrak{g}_{D},K_{D})$-modules.

7.2 Split $\mathbb{O}$ but non-split $D$

We move on to the case where $\mathbb{O}$ and hence $G$ is split. Let $K^{\prime }$ be a maximal compact subgroup of $G^{\prime }$ (the simply connected form of $G$), and $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ the corresponding Cartan decomposition of the complexification of the Lie algebra of $G$. Then $\mathfrak{k}$ is isomorphic to $\mathfrak{sl}_{8}$. Fixing this isomorphism, we see that as a $K^{\prime }\cong \text{SU}_{8}/\unicode[STIX]{x1D707}_{2}$-module, $\mathfrak{p}$ is isomorphic to $V_{\unicode[STIX]{x1D714}_{4}}$, where $\unicode[STIX]{x1D714}_{4}$ is the fourth fundamental weight. The minimal representation $\unicode[STIX]{x1D6F1}$ remains irreducible when pulled back to $G^{\prime }$ and is a direct sum of $K^{\prime }$-types $V_{n\unicode[STIX]{x1D714}_{4}}$, where $n=0,1,2,\ldots$ .

We have two cases depending on $D$. Assume in this subsection that $D$ is a division algebra. In this case $D^{1}\cong \text{SU}_{2}$ is compact, and embeds into $\text{SU}_{8}$ as a $2\times 2$ block. The centralizer of $\text{SU}_{2}$ in $K^{\prime }=\text{SU}_{8}/\unicode[STIX]{x1D707}_{2}$ is $K_{D}^{\prime }\cong \text{U}_{6}$. The minimal representation $\unicode[STIX]{x1D6F1}$ decomposes discretely when restricted to this dual pair. A simple application of the Gelfand–Zetlin rule shows that the $K_{D}^{\prime }$-types of $\unicode[STIX]{x1D6E9}(1)$ are multiplicity-free and the highest weights of the $K_{D}^{\prime }$-types which occur are

$$\begin{eqnarray}(x,x,0,0,y,y),\end{eqnarray}$$

where $x\geqslant 0\geqslant y$ are any two integers. Here we are using the standard description of highest weights for $\text{U}_{6}$ by sextuples of non-increasing integers. But these are precisely the $K_{D}^{\prime }$-types of the spherical submodule of $I_{D}(-1)$, that is, the $G_{D}^{\prime }$-constituent $\unicode[STIX]{x1D6F4}_{1,1}$ in [Reference SahiSah93, Theorem C]. In view of the natural non-zero $G_{D}$-equivariant map $\unicode[STIX]{x1D6E9}(1)\rightarrow I_{D}(-1)$, this proves the following theorem.

Theorem 7.2. When $\mathbb{O}$ is split but $D$ is non-split, one has

$$\begin{eqnarray}\unicode[STIX]{x1D6E9}(1)=\unicode[STIX]{x1D6F1}^{D^{1}}\cong \unicode[STIX]{x1D6F4}_{1,1}\end{eqnarray}$$

as $(\mathfrak{g}_{D},K_{D})$-modules.

7.3 Split $\mathbb{O}$ and split $D$

This is the most involved case. Let $(e,h,f)$ be an $\mathfrak{sl}_{2}$-triple spanning the complexified Lie algebra of $D^{1}=\text{SL}_{2}$. After conjugating by $G^{\prime }$, if necessary, we can assume that the triple is stable under the Cartan involution. Then $e\in \mathfrak{p}$ is a highest weight vector for the action of $K^{\prime }$, and $h\in \mathfrak{k}$. Let $\unicode[STIX]{x1D6E9}(1)$ be the maximal quotient of the $(\mathfrak{g},K^{\prime })$-module of the minimal representation such that the $\mathfrak{sl}_{2}$ triple acts trivially.

The proof of this result will take up the rest of this section. After conjugating by $K^{\prime }$, if necessary, we can assume that

$$\begin{eqnarray}h=\frac{1}{2}\left(\begin{array}{@{}cccccccc@{}}1 & & & & & & & \\ & 1 & & & & & & \\ & & 1 & & & & & \\ & & & 1 & & & & \\ & & & & -1 & & & \\ & & & & & -1 & & \\ & & & & & & -1 & \\ & & & & & & & -1\\ \end{array}\right)\in \mathfrak{sl}_{8}.\end{eqnarray}$$

Then $G_{D}^{\prime }$ is the centralizer of the $\mathfrak{sl}_{2}$-triple in $G^{\prime }$ and is isomorphic to $\text{Spin}(6,6)$ as an algebraic group. Let $\mathfrak{g}_{D}=\mathfrak{k}_{D}\oplus \mathfrak{p}_{D}$ be the corresponding Cartan decomposition. Then $\mathfrak{k}_{D}\cong \mathfrak{sl}_{4}\oplus \mathfrak{sl}_{4}$ sitting block diagonally in $\mathfrak{sl}_{8}$. The centralizer of $h$ in $\text{SU}_{8}/\unicode[STIX]{x1D707}_{2}$ is

$$\begin{eqnarray}K_{D}^{\prime }=\text{SU}_{4}\times \text{SU}_{4}/\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D707}_{2}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D6F1}$ be the $(\mathfrak{g},K^{\prime })$-module corresponding to the minimal representation of $G$. Then, as a $K^{\prime }$-module,

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}=\bigoplus _{n\geqslant 0}V_{n\unicode[STIX]{x1D714}_{4}}.\end{eqnarray}$$

We shall also need the following facts about the action of $e$ on $\unicode[STIX]{x1D6F1}$. From the formula for the tensor product $V_{\unicode[STIX]{x1D714}_{4}}\otimes V_{\mathbf{n}\unicode[STIX]{x1D714}_{4}}$ it follows that

$$\begin{eqnarray}e\cdot V_{n\unicode[STIX]{x1D714}_{4}}\subseteq V_{(n-1)\unicode[STIX]{x1D714}_{4}}\oplus V_{(n+1)\unicode[STIX]{x1D714}_{4}}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D6F1}$ is not a highest weight module, by [Reference Vogan, Carmona and VergneVog81, Lemma 3.4], $e$ is injective on $\unicode[STIX]{x1D6F1}$. The same results hold for $f$.

Let $\unicode[STIX]{x1D70B}$ be an irreducible $\mathfrak{sl}_{2}$-module such that $h$ acts semisimply and integrally. Let $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B})$ be the big theta lift of $\unicode[STIX]{x1D70B}$; it is a $(\mathfrak{g}_{D},K_{D}^{\prime })$-module. We shall now partially determine the structure of $K_{D}^{\prime }$-types of $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D70B})$. In order to state the result, we need some additional notation. A highest weight $\unicode[STIX]{x1D707}$ for $\text{SU}_{4}$ is represented by a quadruple $(x,y,z,u)$ of integers, such that $x\geqslant y\geqslant z\geqslant u$, and it is determined by the triple