Algebras contained within H by Anderson J. M. PDF

By Anderson J. M.

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1 − ω( ) | |s of p. If ω is ramified L(s, ω) = 1. If ϕ belongs to It is independent of the choice of the generator S(F ) the integral Z(ωαsF , ϕ) = ϕ(α)ω(α) |α|s d× α F× is absolutely convergent in some half-plane Re s > s0 and the quotient Z(ωαsF , ϕ) L(s, ω) can be analytically continued to a function holomorphic in the whole complex plane. Moreover for a suitable choice of ϕ the quotient is 1. If ω is unramified and d× α = 0 UF one could take the characteristic function of OF . There is a factor ε(s, ω, ψ) which, for a given ω and ψ , is of the form abs so that if ϕ is the Fourier transform of ϕ Z(ω −1 α1−s Z(ωαsF , ϕ) F , ϕ) = ε(s, ω, ψ) .

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 ). The right side is equal to C(νω1 , z1 t)ϕ (ν −1 ν0−1 ω1−1 , z0−1 z1−2 t−1 ) so that when we replace π by ω ⊗ π we replace C(ν, t) by C(νω1 , z1 t). Suppose ψ (x) = ψ(bx) with b in F × is another non-trivial additive character. Then W (π, ψ ) consists of the functions b 0 0 1 W (g) = W g with W in W (π, ψ). The last identity of the following theorem is referred to as the local functional equation. It is the starting point of our approach to the Hecke theory.

If ϕ(1) = 0 then λb ψ(b) = 0 b∈S so that ϕ= λb ξψ 1 b 0 1 ϕ0 − ψ(b)ϕ0 It is clear that L(ϕ) = 0. The representation of the theorem will be called the Kirillov model. There is another model which will be used extensively. It is called the Whittaker model. Its properties are described in the next theorem. 14 (i) For any ϕ in V set Wϕ (g) = π(g)ϕ (1) so that Wϕ is a function in GF . Let W (π, ψ) be the space of such functions. The map ϕ → Wϕ is an isomorphism of V with W (π, ψ). Moreover Wπ(g)ϕ = ρ(g)Wϕ (ii) Let W (ψ) be the space of all functions W on GF such that 1 0 W x 1 g = ψ(x)W (g) for all x in F and g in G.

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Algebras contained within H by Anderson J. M.


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