By Jacquet H., Langlands R.P.
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Then |ϕ(a)|2 da = 1. F× ′ ℓ If ϕ = π(w)ϕ and C(ν, t) = Cℓ (ν)t then ϕ′ (ǫ̟ m ) = δℓ−n,m Cℓ (ν)z0−n ν −1 (ǫ). Since |z0 | = 1 |ϕ′ (a)|2 da = |Cℓ (ν)|2 . F× Applying the first part of the lemma we see that, if |z| = 1, both |Cℓ (ν)|2 and |C(ν, z)|2 = |Cℓ (ν)|2 |z|2ℓ are 1. 22. Let π be an irreducible representation of GF . It is absolutely cuspidal if and only if for every vector v there is an ideal a in F such that π a 1 0 x 1 v dx = 0. It is clear that the condition cannot be satisfied by a finite dimensional representation.
If ω is an unramified quasi-character of F × the associated L-function is L(s, ω) = 1 . 1 − ω(̟) |̟|s It is independent of the choice of the generator ̟ of p. If ω is ramified L(s, ω) = 1. If ϕ belongs to 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 .
Although W (µ1 , µ2 ψ) is not irreducible we may still, for all W in this space, define the integrals Ψ(g, s, W ) = W a 0 0 1 g |a|s−1/2 d× a Ψ(g, s, W ) = W a 0 0 1 g |a|s−1/2 ω −1 (a) d× a. 5. In particular they converge to the right of some vertical line and if W = WΦ Ψ(e, s, W ) = Z(µ1 αsF , µ2 αsF , Φ) −1 s s Ψ(e, s, W ) = Z(µ−1 2 αF , µ1 αF , Φ). Moreover Ψ(g, s, W ) L(s, π) is a holomorphic function of s and Ψ(g, 1 − s, W ) Ψ(g, s, W ) = ε(s, π, ψ) . L(1 − s, π ˜) L(s, π) Therefore Φ(g, s, W ) = Ψ(g, s, W ) L(s, σ) Φ(g, s, W ) = Ψ(g, s, W ) L(s, σ) and are meromorphic functions of s and satisfy the local functional equation Φ(wg, 1 − s, W ) = ε(s, σ, ψ) Φ(g, s, W ).