\SL_3 dual pairs

Introduction

Let $F$ be a $p$-adic field. A simple, split Lie algebra $\mathfrak{e}$ of type $E_n$, $n = 6,7,8$, has a $\mathbb{Z}/3\mathbb{Z}$-grading

where $V_3$ is the standard representation of $\mathrm{SL}_3$ and, if $G$ is the split simply connected group corresponding to $\mathfrak{g}$, then $J$ is a representation of $G$. The pairs $(G,J)$ are tabulated here:

Note that $G$ acts on $J$ with a small kernel $\mu_m$, $m = 3,2,1$ ($= 9-n$). The group $G$ preserves a symmetric trilinear form $(x,y,z)$ on $J$. In fact, $G/\mu_m$ is the connected component of the group of linear transformations of $J$ preserving the trilinear form. The Lie bracket on $\mathfrak{e}$ can be written down using the trilinear form [22]. If we pick an element $e \in J$ such that $(e,e,e) \ne0$ (a generic element) then $J$ can be turned into a Freudenthal-Jordan algebra with the identity $e$ and the (cubic) norm $N_J$ such that $6N_J(x) = (x,x,x)$ and $e$ is the identity of $J$. The stabilizer of $e$ in $G$ is a group isomorphic to $H$, listed in the table above. The stabilizers of two generic elements $e \in J$ and $f \in J$ are conjugate in $G$ if and only if $N_J(e)/N_J(f)$ is a cube; see Proposition 1.1.

The purpose of this paper is to study the theta lift from $\mathrm{SL}_3$ to $G$ arising from the minimal representation $\Pi$ of $E_n$, with an application to rank-two symmetric spaces $G/H$. More precisely, fix an embedding of $H$ into $G$. Since $H$ fixes a line in both $J$ and $J^*$, the centralizer of $H$ in $\mathfrak{e}$ is

which we recognize as the exceptional Lie algebra of type $G_2$. Thus we have a see-saw of dual pairs

2020 Mathematics Subject Classification. 11F27, 22E50. This see-saw plays a key role for both the theta correspondence and its relationship with the analysis of symmetric spaces $G/H$. More precisely, if $(U,\psi)$ is a Whittaker pair for $\mathrm{SL}_3$, then

We note that conjugacy classes of Whittaker pairs for $\mathrm{SL}_3$ are also parameterized by classes of cubes in $F^\times$. Thus as $\psi$ varies, so does the group $H$ in $G$. With that caveat in mind, the above identity, via the standard see-saw argument, implies that, for any finite length representation $\pi$ of $G$,

where $\Theta(\pi) = (\Pi\otimes\pi^\vee)_G$ is the so-called big theta lift of $\pi$. In other words, understanding whether $\pi$ has an $H$-invariant functional reduces to computing $\Theta(\pi)_{U,\psi}$.

We now highlight some of the results obtained in this paper. The first one is Theorem 9.4:

Recall that an irreducible representation $\pi$ of $G$ is hermitian if $\overline{\pi} \cong\pi^\vee$. Clearly all unitarizable (in particular tempered) representations are hermitian. Other hermitian representations are described in terms of the corresponding standard modules, [3]. The second result is Theorem 9.6, a multiplicity-one statement for $H$-invariant functionals:

We derive both of these multiplicity-free results using the method of period ping-pong, see §9.1. As far as making the correspondence explicit, if $\sigma$ is tempered, then we determine the parameter of $\Theta(\sigma)$. In particular, it satisfies the expectations of BZSV [4] for $H$-distinguished representations.. We also have a rather complete description of $\Theta(\sigma)$ for non-tempered $\sigma$. For example, if $\sigma$ is (full) induced from the Borel subgroup, then $\Theta(\sigma)$ is the full degenerate principal series induced from a co-rank 2 parabolic subgroup with Levi of type $A_1 \times A_1 \times A_1$ if $G = \mathrm{SL}_6$, and type $D_4$ if $G = E_6$. We note that the symplectic period in $G = \mathrm{SL}_6$ has been studied in [18] and [20]; the main novelty of this theorem is thus the $E_6/F_4$ case.

Overview. We begin in Section 1 by discussing the possible embeddings of $H$ in $G$ and $\mathrm{SL}_3$ in $G_2$, as seen in diagram (2). In Section 2 we compute the key periods, including the Whittaker coinvariants of the minimal representation, as well as $\Pi_H = \Theta(1_H)$, the lift of the trivial representation of $H$ to $G_2$ and its restriction to $\mathrm{SL}_3$. Section 3 is devoted to a computation of the Jacquet module of the minimal representation with respect to a maximal parabolic in $\mathrm{SL}_3$; this is then used in Section 4 to compute the lifts of non-cuspidal representations. Section 5 is a self-contained treatment of the correspondence in $E_6$; the rest of the paper focuses on the correspondences in $E_7$ and $E_8$.

First, in Section 6, we narrow down the possibilities for cuspidal support of lifts from $\mathrm{SL}_3$ to $G$. The relevant Jacquet modules are then computed in Sections 7 and 8. Having done the necessary preparations, we are ready to play period ping-pong: we use it to prove our main results in Section 9. To complete the picture, we compute the lifts of tempered representations in Section 10. Appendix A features a discussion of $\mathrm{SL}_3$ orbits on $G_2$ flag varieties; this is used in crucial period computations in §2. In Appendix B, we summarize the theta correspondence between $\mathrm{GL}_2$ and a similitude spin group, which features in Jacquet module computations of § 4 and thus plays a prominent role throughout the paper.

Notation and conventions. We fix a local non-archimedean field $F$ of characteristic 0 and let $\overline{F}$ denote its algebraic closure. We often use the same symbol to denote a reductive group as well as its $F$-points; this abuse of notation is harmless, as there is never any ambiguity.

Unless otherwise stated, we work in the category of smooth (complex) representations. If $\pi$ is a representation of a $p$-adic group $G$, and $\psi$ a character of a subgroup $H \subset G$, we write $\pi_{H,\psi}$ for the space of twisted coinvariants of $\pi$ with respect to $(H,\psi)$; if $\psi$ is trivial, we write simply $\pi_H$. We use $\operatorname{ind}$ to denote compact (and in particular unnormalized parabolic) induction, and $\operatorname{Ind}$ to denote normalized parabolic induction.

We often refer to the big/full theta lift, as defined above. As is customary, we refer to the maximal semisimple quotient of big theta as the small theta lift.

Finally, to avoid introducing excessive notation, we recycle certain symbols (like $\Theta$, $P = MN$, etc.) throughout the paper. This is always explicitly announced, so there is no fear of confusion.

Some embeddings

Let $x \in J$ be an element of non-zero norm. Recall that the stabilizer of $x$ in $G$ is a group isomorphic to $H$, shown in Table 1. These stabilizers are parameterized by classes of cubes:

Proof. In [24], $G$-orbits on $J$ are classified over $\mathbb{Z}$. Extrapolating to fields, it follows that $G$ acts transitively on elements of a fixed non-zero norm. Since the stabilizers of $x$ and $ax$ are the same for $a \in F^\times$, and since $N_J(ax) = a^3N_J(x)$, we have one direction of the statement. We use Galois cohomology to complete the proof. The normalizer of $H$ in $G$ is $H \times\mu_3$, hence the orbits correspond to points in the kernel of the map

Since $H$ and $G$ are simply connected, their Galois cohomology vanishes. This implies the kernel of the above map is $H^1(F,\mu_3) = F^\times/(F^\times)^3$.

Let $H_a \subseteq G$ be the subgroup fixing an element $x \in J$ of norm $a$. As explained in the introduction, we have a see-saw of dual pairs