By H. Jacquet, R. P. Langlands
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Additional info for Automorphic Forms on GL(2): Part 1
If µ1 = χαF and µ2 = χαF the representations on B(µ1 , µ2 )/Bs(µ1 , µ2 ) and Bf (µ2 , µ1 ) are both equivalent to the representations g → χ(detg). The space W (µ1 , µ2 ; ψ) has been defined for all pairs µ1 , µ2 . 4 (i) For all pairs µ1 , µ2 W (µ1 , µ2 ; ψ) = W (µ2 , µ1 ; ψ) (ii) In particular if µ1 µ−1 = α−1 2 F the representation of GF on W (µ1 , µ2 ; ψ) is equivalent to ρ(µ1 , µ2 ). If Φ is a function on F 2 define Φι by Φι (x, y) = Φ(y, x). To prove the proposition we show that, if Φ is in S(F 2 ), µ1 (detg) |detg|1/2 θ µ1 , µ2 ; r(g)Φι = µ2 (detg) |detg|1/2 θ µ2 , µ1 ; r(g)Φ .
Since A = λA0 the hermitian form (ϕ1 , ϕ2 ) is equal to λ F× ϕ1 (a)ϕ2 (a) d× a. 2. Let π be an absolutely cuspidal representation of G F for which the quasi-character ω deﬁned by π a 0 0 a = ω(a)I is a character. (i) If π is in the Kirillov form the hermitian form F× ϕ1 (a)ϕ2 (a) d× a is invariant. (ii) If |z| = 1 then |C(ν, z)| = 1 and if Res = 1/2 | (s, π, ψ)| = 1. Since |z0 | = 1 the second relation of part (ii) follows from the first and the relation (s, π, ψ) = C(ν0−1 , q s−1/2 z0−1 ). If n is in Z and ν is a character of UF let m ϕ( ) = δn,m ν( )ν0 ( ) for m in Z and in UF .
It reduces to µ1 (a) × Φι (at, t−1 )µ1 (t)µ−1 2 (t) d t = µ2 (a) × Φ(at, t−1 )µ2 (t)µ−1 2 (t) d t. The left side equals µ1 (a) × Φ(t−1 , at)µ1 (t)µ−1 2 (t) d t which, after changing the variable of integration, one sees is equal to the right side. Chapter 1 52 −1 If µ1 µ−1 2 is not αF or αF so that ρ(µ1 , µ2 ) is irreducible we let π(µ1 , µ2 ) be any representation in the class of ρ(µ1 , µ2 ). If ρ(µ1 , µ2 ) is reducible it has two constituents one finite dimensional and one infinite dimensional.