Automorphic forms on SO(4)

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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, AF) whose central characters ω1, ω2 satisfy ω1ω 2 = 1, exists as an automorphic representation π1 Squared times π2 of PGL(4, AF). 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, AF)-module normalizedly parabolically induced from the PGL(3, AF)-module Sym21) on the Levi factor of the parabolic subgroup of type (3, 1). Finer results include the definition of a local product π Squared times π 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 χυ of Fυx such that {πχυ, πχ υ-1} = {π0, π0}, then there exists a character χof Ax/Fx with {π, π -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, π and multiplying π1υ by a local character and π 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).

Original languageEnglish
Pages (from-to)100-104
Number of pages5
JournalProceedings of the Japan Academy Series A: Mathematical Sciences
Issue number6
StatePublished - Jun 2004
Externally publishedYes


  • Automorphic representations
  • Liftings
  • Orthogonal group
  • Rigidity


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