## Abstract

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 π_{1} and π_{2} of the adèle group GL(2, A_{F}) whose central characters ω_{1}, ω_{2} satisfy ω_{1}ω _{2} = 1, exists as an automorphic representation π_{1} Squared times π_{2} of PGL(4, A_{F}). The product is in the discrete spectrum if π_{1} is inequivalent to a twist of the contragredient π_{2} of π_{2}, and π_{1}, π_{2} are not monomial from the same quadratic extension. If π_{2} = π_{1} then π_{1} Squared times π_{2} is the PGL(4, A_{F})-module normalizedly parabolically induced from the PGL(3, A_{F})-module Sym^{2}(π_{1}) 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 (π_{1}, π_{2}) → π_{1} Squared times π_{2} is injective in the following sense. If π_{1}, π_{2}, π_{1}^{0}, π_{2}^{0} are discrete spectrum representations of GL(2, A) with central characters ω_{1}, ω_{2}, ω_{1}^{0}, ω_{2} ^{0} satisfying ω_{1}ω_{2} = 1 = ω_{1}^{0}ω_{2}^{0}, 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_{1υ}χ_{υ}, π_{2υ}χ _{υ}^{-1}} = {π_{1υ}^{0}, π_{1υ}^{0}}, then there exists a character χof A^{x}/F^{x} with {π_{1χ}, π _{2χ}^{-1}} = {π_{1}^{0}, π_{2}^{0}}. In particular, starting with a pair π_{1}, π_{2} of discrete spectrum representations of GL(2, A) with ω_{1}ω_{2} = 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 π_{1}, π_{2} and multiply π_{1} by a global character and π_{2} by its inverse. The injectivity of (π_{1}, π_{2}) → π_{1} Squared times π_{2} is a strong rigidity theorem for SO(4). The self contragredient discrete spectrum representations of PGL(4, A) of the form π_{1} Squared times π_{2} are those not obtained from the lifting from the symplectic group PGSp(2, A).

Original language | English |
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Pages (from-to) | 100-104 |

Number of pages | 5 |

Journal | Proceedings of the Japan Academy Series A: Mathematical Sciences |

Volume | 80 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2004 |

Externally published | Yes |

## Keywords

- Automorphic representations
- Liftings
- Orthogonal group
- Rigidity