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

By Jacquet H., Langlands R.P.

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Extra resources for Automofphic forms on GL(2)

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If ω = µ1 µ2 then 1 a 0 0 a π = ω(a)I −1 −1 −1 so that π is equivalent to ω ⊗ π ˜ or to ω ⊗ ρ(µ−1 1 , µ2 ). It is easily seen that ω ⊗ ρ(µ1 , µ2 ) is equivalent −1 to ρ(ωµ−1 1 , ωµ1 ) = ρ(µ2 , µ1 ). −1 If µ1 µ2 = αF and π is the restriction of ρ to Bs (µ1 , µ2 ) then π ˜ is the representation on −1 −1 −1 −1 −1 B(µ−1 , µ )/B (µ , µ ) defined by ρ(µ , µ ) . Thus π is equivalent to the tensor product of f 1 2 1 2 1 2 ω = µ1 µ2 and this representation. The tensor product is of course equivalent to the representation on 1/2 −1/2 B(µ2 , µ1 )/Bf (µ2 , µ1 ).

4 Let λi , 1 ≤ i ≤ p, be p complex numbers. Let A be the space of all sequences {an }, n ∈ Z for which there exist two integers n1 and n2 such that an = λi an−i 1≤i≤p for n ≥ n1 and such that an = 0 for n ≤ n2 . Let A0 be the space of all sequences with only a finite number of nonzero terms. Then A/A0 is finite-dimensional. 2. 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.

To see this take π in the Kirillov form and observe Chapter 1 37 first of all that the map A : ϕ → ϕω is an isomorphism of V with another space V ′ on which GF acts by means of the representation π ′ = A(ω ⊗ π)A−1 . If b α x 0 1 belongs to BF and ϕ′ = ϕω then π ′ (b)ϕ′ (a) = ω(a){ω(α)ψ(ax)ϕ(αa)} = ψ(ax)ϕ′ (αa) so that π ′ (b)ϕ′ = ξψ (b)ϕ′ . By definition then π ′ is the Kirillov model of ω ⊗ π . Let ω1 be the restriction of ω to UF and let z1 = ω(̟). If ϕ′ = ϕω then ϕ′ (ν, t) = ϕ(νω1 , z1 t). Thus π ′ (w)ϕ′ (ν, t) = π(w)ϕ(νω1 , z1 t) = C(νω1 , z1 t)ϕ(v−1 ω1−1 ν0−1 , z0−1 z1−1 t−1 ).

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


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