Automorphic forms on $\mathrm {SO}(4)$

Abstract

We announce results of [F1] on automorphic forms on $\mathrm{SO}(4)$. An initial result is the proof by means of the trace formula that the functorial product of two automorphic representations $\pi_1$ and $\pi_2$ of the adèle group $\mathrm{GL}(2, \mathbf{A}_F)$ whose central characters $\omega_1$, $\omega_2$ satisfy $\omega_1 \omega_2 = 1$, exists as an automorphic representation $\pi_1 \boxtimes \pi_2$ of $\mathrm{PGL}(4, \mathbf{A}_F)$. The product is in the discrete spectrum if $\pi_1$ is inequivalent to a twist of the contragredient $\check{\pi}_2$ of $\pi_2$, and $\pi_1$, $\pi_2$ are not monomial from the same quadratic extension. If $\pi_2 = \check{\pi}_1$ then $\pi_1 \boxtimes \pi_2$ is the $\mathrm{PGL}(4, \mathbf{A}_F)$-module normalizedly parabolically induced from the $\mathrm{PGL}(3, \mathbf{A}_F)$-module $\operatorname{Sym}^2(\pi_1)$ on the Levi factor of the parabolic subgroup of type $(3,1)$. Finer results include the definition of a local product $\pi_{1v} \boxtimes \pi_{2v}$ by means of characters, injectivity of the global product, and a description of its image. Thus the product $(\pi_1, \pi_2) \mapsto \pi_1 \boxtimes \pi_2$ is injective in the following sense. If $\pi_1$, $\pi_2$, $\pi_1^0$, $\pi_2^0$ are discrete spectrum representations of $\mathrm{GL}(2, \mathbf{A})$ with central characters $\omega_1$, $\omega_2$, $\omega_1^0$, $\omega_2^0$ satisfying $\omega_1 \omega_2 = 1 = \omega_1^0 \omega_2^0$, and for each place $v$ outside a fixed finite set of places of the global field $F$ there is a character $\chi_v$ of $F_v^{\times}$ such that $\{ \pi_{1v}\chi_v, \pi_{2v}\chi_v^{-1} \} = \{ \pi_{1v}^0, \pi_{1v}^0 \}$, then there exists a character $\chi$ of $\mathbf{A}^{\times} / F^{\times}$ with $\{ \pi_1\chi, \pi_2\chi^{-1} \} = \{ \pi_1^0, \pi_2^0 \}$. In particular, starting with a pair $\pi_1$, $\pi_2$ of discrete spectrum representations of $\mathrm{GL}(2, \mathbf{A})$ with $\omega_1 \omega_2 = 1$, we cannot get another such pair by interchanging a set of their components $\pi_{1v}$, $\pi_{2v}$ and multiplying $\pi_{1v}$ by a local character and $\pi_{2v}$ by its inverse, unless we interchange $\pi_1$, $\pi_2$ and multiply $\pi_1$ by a global character and $\pi_2$ by its inverse. The injectivity of $(\pi_1, \pi_2) \mapsto \pi_1 \boxtimes \pi_2$ is a strong rigidity theorem for $\mathrm{SO}(4)$. The self contragredient discrete spectrum representations of $\mathrm{PGL}(4, \mathbf{A})$ of the form $\pi_1 \boxtimes \pi_2$ are those not obtained from the lifting from the symplectic group $\mathrm{PGSp}(2, \mathbf{A})$.