Automorphic forms on SO(4)

We announce results of [F1] on automorphic forms on SO(4). An initial result is the proof by means of the trace formula that the functorial product of two automorphic representations π₁ and π₂ of the adèle group GL(2, A_F) whose central characters ω₁, ω₂ satisfy ω₁ω ₂ = 1, exists as an automorphic representation π₁ Squared times π₂ of PGL(4, A_F). The product is in the discrete spectrum if π₁ is inequivalent to a twist of the contragredient π₂ of π₂, and π₁, π₂ are not monomial from the same quadratic extension. If π₂ = π₁ then π₁ Squared times π₂ is the PGL(4, A_F)-module normalizedly parabolically induced from the PGL(3, A_F)-module Sym²(π₁) on the Levi factor of the parabolic subgroup of type (3, 1). Finer results include the definition of a local product π_1υ Squared times π_2υ by means of characters, injectivity of the global product, and a description of its image. Thus the product (π₁, π₂) → π₁ Squared times π₂ is injective in the following sense. If π₁, π₂, π₁⁰, π₂⁰ are discrete spectrum representations of GL(2, A) with central characters ω₁, ω₂, ω₁⁰, ω₂ ⁰ satisfying ω₁ω₂ = 1 = ω₁⁰ω₂⁰, and for each place υ outside a fixed finite set of places of the global field F there is a character χ_υ of F_υ^x such that {π_1υχ_υ, π_2υχ _υ^-1} = {π_1υ⁰, π_1υ⁰}, then there exists a character χof A^x/F^x with {π_1χ, π _2χ^-1} = {π₁⁰, π₂⁰}. In particular, starting with a pair π₁, π₂ of discrete spectrum representations of GL(2, A) with ω₁ω₂ = 1, we cannot get another such pair by interchanging a set of their components π_1υ, π_2υ and multiplying π1υ by a local character and π_2υ by its inverse, unless we interchange π₁, π₂ and multiply π₁ by a global character and π₂ by its inverse. The injectivity of (π₁, π₂) → π₁ Squared times π₂ is a strong rigidity theorem for SO(4). The self contragredient discrete spectrum representations of PGL(4, A) of the form π₁ Squared times π₂ are those not obtained from the lifting from the symplectic group PGSp(2, A).