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

By Jacquet H., Langlands R.P.

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This booklet is designed to introduce the reader to the speculation of semisimple Lie algebras over an algebraically closed box of attribute zero, with emphasis on representations. an outstanding wisdom of linear algebra (including eigenvalues, bilinear kinds, Euclidean areas, and tensor items of vector areas) is presupposed, in addition to a few acquaintance with the tools of summary algebra.

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Hier wird guy nun _u_ als _Geschwindigkeitspotential_ deuten, so dass

[formula] [formula] die Componenten der Geschwindigkeit sind, mit der eine

Flüssigkeit parallel zur [formula]-Ebene strömt. Wir mögen uns diese

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[formula]-Ebene verlaufen, oder auch uns vorstellen, dass die Flüssigkeit

als unendlich dünn

A scientific survey of all of the easy effects at the conception of discrete subgroups of Lie teams, provided in a handy shape for clients. The booklet makes the speculation obtainable to a large viewers, and should be a typical reference for a few years to return.

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**Extra resources for Automofphic forms on GL(2)**

**Sample text**

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 ).

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

by Charles

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