By H. Jacquet, R. P. Langlands
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Additional info for Automorphic Forms on GL(2): Part 1
It is one-dimensional and contains the function χ(detg). Moreover the GF -modules Bs (µ1 , µ2 ) and B(µ2 , µ1 )/Bf (µ2 , µ1 ) are equivalent as are the modules B(µ1 , µ2 )/Bs (µ1 , µ2 ) and Bf (µ2 , µ1 ). We start with a simple lemma. 1 Suppose there is a non-zero function f in B(µ1 , µ2 ) invariant under right translations −1/2 by elements of NF . Then there is a quasi-character χ such that µ1 = χαF f is a multiple of χ. 0 −1 1 0 1/2 and µ2 = χαF and Since NF AF 01 −1 0 NF is an open subset of GF the function f is determined by its value at .
The proposition follows. 23. Let π be an absolutely cuspidal representation and assume the largest ideal on which ψ is trivial is OF . Then, for all characters ν, Cn (ν) = 0 if n ≥ −1. Take a character ν and choose n1 such that Cn1 (ν) = 0. Then Cn (ν) = 0 for n = n1 . If ν = ν −1 ν0−1 then, as we have seen, C(ν, t)C(ν, t−1 z0−1 ) = ν0 (−1) so that Cn (ν) = 0 for n = n1 and Cn1 (ν)Cn1 (ν) = ν0 (−1)z0n1 . 11 take n = p = n1 + 1 to obtain η(σ −1 ν, n1 +1 )η(σ −1 ν, n1 +1 )C2n1 +2 (σ) = z0n1 +1 ν0 (−1) + (| | − 1)−1 z0 Cn1 (ν)Cn1 (ν).
Chapter 1 52 −1 If µ1 µ−1 2 is not αF or αF so that ρ(µ1 , µ2 ) is irreducible we let π(µ1 , µ2 ) be any representation in the class of ρ(µ1 , µ2 ). If ρ(µ1 , µ2 ) is reducible it has two constituents one finite dimensional and one infinite dimensional. A representation in the class of the first will be called π(µ1 , µ2 ). A representation in the class of the second will be called σ(µ1 , µ2 ). Any irreducible representation which is not absolutely cuspidal is either a π(µ1 , µ2 ) or a σ(µ1 , µ2 ).