By Serre J-P
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Additional info for Arbres, amalgames, SL2
Let (τ, Vτ ) be an irreducible, admissible, generic representation of GL2 (F ). Assume that L/F is either an unramiﬁed ﬁeld extension or L = F ⊕ F . Assume also that the conductor pn of τ satisﬁes n ≥ 1. Let the character χ0 of L× be such that χ0 F × = ωτ and χ0 ((1 + Pn ) ∩ o× L ) = 1. Let Λ be an unramiﬁed character of L× such that Λ F × = 1. Let the character χ of L× be deﬁned by ( 44). 4, normalized such that W # (η0 , s) = 1. Then the function K(s) deﬁned by ( 41) is given by K(s) = χL/F ( )n ωτ (c2 /d) ε(3s + 1, τ˜, ψ −c )2 ε(6s, ωτ−1 , ψ −c ) L(6s, χ F× L(1 − 6s, χ−1 )L(3s, τ × AI(Λ) × χ F× ) )L(3s + 1, τ × AI(Λ) × χ F× F× ) .
Proof. i) is obvious, since Φ# lies in Wm,l,l . 2 ii) Assume ﬁrst we are in Case A. By our hypotheses, 0 < l < l1 . Assume that G2 Φ ∈ IΦ (s, χ, χ0 , τ ) is K∞ -ﬁnite and satisﬁes (83) and (84). Then, evidently, the G2 Δ . 4, restriction of Φ to K∞ lies in Wm,l,l 2 CΦ# m,l,l2 ,j . 5, the weight j occurs in τ . Since τ has minimal weight l1 , this implies j ≤ −l1 or j ≥ l1 . The ﬁrst inequality leads to a contradiction, and the second inequality implies j = l1 . This proves the uniqueness in Case A.
5. The function Φ# m,l,l2 ,l1 belongs to IΦ (s, χ, χ0 , τ ) if and only if the weight l1 occurs in τ . Proof. As a subspace of IΦ (s, χ, χ0 , β1 × β2 ), the representation IΦ (s, χ, χ0 , τ ) consists of all functions Φ : G2 → C of the form G2 mi ∈ Mi (R), n ∈ N (R), k ∈ K∞ , Φ(m1 m2 nk) = δP (m1 m2 )s+1/2 χ(m1 )ϕ(m2 )J(k), G2 . It follows where ϕ lies in χ0 × τ , and where J is an appropriate function on K∞ that Φ ∈ IΦ (s, χ, χ0 , β1 × β2 ) lies in IΦ (s, χ, χ0 , τ ) if and only if the function m2 −→ Φ(m2 )δP (m2 )−s−1/2 M2 (R) # belongs to χ0 × τ .
Arbres, amalgames, SL2 by Serre J-P