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By Jacquet H., Langlands R.P.

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Example text

11 η(σ −1 ν, ̟ n )η(σ −1 ν, ̟ p )Cp+n (σ) σ is equal to −∞ z0p ν0 (−1)δn,p + (|̟| − 1)−1 z0ℓ+1 Cn−1−ℓ (ν)Cp−1−ℓ (ν) − z0−r Cn+r (ν)Cp+r (ν). −2−ℓ Remember that p−ℓ is the largest ideal on which ψ is trivial. Suppose first that ν = ν . Chapter 1 35 Take p = −ℓ and n > −ℓ. Then δ(n − p) = 0 and η(σ −1 ν, ̟ n )η(σ −1 ν, ̟ p ) = 0 unless σ = ν . Hence −∞ z0−r Cn+r (ν)C−ℓ+r (ν) Cn−ℓ (ν) = (|̟| − 1)−1 z0ℓ+1 Cn−1−ℓ (ν)C−2ℓ−1 (ν) − −2−ℓ which, since almost all of the coefficients C−ℓ+r (ν) in the sum are zero, is the relation required.

1 , ϕ2 ) = ZF \GF This is a positive invariant hermitian form on V . We have incidentally shown that π is square-integrable. Observe that even if the absolutely cuspidal representation π is not unitary one can choose a quasi-character χ such that χ ⊗ π is unitary. If π is unitary there is a conjugate linear map A : V → V defined by (ϕ1 , ϕ2 ) = ϕ1 , Aϕ2 . Clearly Aξψ (b) = ξψ (b)A for all b in BF . The map A0 defined by A0 ϕ(a) = ϕ(−a) has the same properties. We claim that A = λA0 with λ in C .

The rest of the theorem will now be proved for absolutely cuspidal representations. The remaining representations will be considered in the next chapter. We observe that since W (π, ψ) is invariant under right translations the assertions need only be established when g is the identity matrix e. If π is absolutely cuspidal then V = V0 = S(F × ) and W a0 01 = ϕ(a) is locally constant with compact support. Therefore the integrals defining Ψ(e, s, W ) and Ψ(e, s, W ) are absolutely convergent for all values of s and the two functions are entire.

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Automofphic forms on GL(2) by Jacquet H., Langlands R.P.

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