By H. Jacquet, R. P. Langlands

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Structure and Representation of Jordan Algebras

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Extra info for Automorphic forms on GL(2)

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If ϕ in V takes value in X1 and g belongs to PF then π(g)ϕ also takes values in X1 . Therefore all we need to do is show that if ϕ is in V1 then π(w)ϕ takes values in X1 . 10. To prove (iii) we show that the operators Cn (ν) are all scalar multiples of the identity. Because of (i) we need only show that every linear transformation of X which commutes with all the operators Cn (ν) is a scalar. Suppose T is such an operator. If ϕ belongs to V let Tϕ be the function from F × to X defined by T ϕ(a) = T ϕ(a) .

Suppose V is another such space of functions and π a representation of GF on V which is equivalent to π . We suppose of course that π (b)ϕ = ξψ (b)ϕ if b is in BF and ϕ is in V . Let A be an isomorphism of V with V such that Aπ(g) = π (g)A for all g . Let L be the linear functional L(ϕ) = Aϕ(1) on V . Then a 0 0 1 L π ϕ = Aϕ(a) so that A is determined by L. If we could prove the existence of a scalar λ such that L(ϕ) = λϕ(1) it would follow that Aϕ(a) = λϕ(a) for all a such that Aϕ = λϕ. This equality of course implies the theorem.

Let H0 be the subalgebra of the Hecke algebra formed by the functions which are invariant under left and right translations by elements of GL(2, OF ). Suppose the irreducible representation π acts on the space X and there is a non-zero vector x in X invariant under GL(2, OF ). If f is in H0 the vector π(f )x has the same property and is therefore a multiple λ(f )x of x. The map f → λ(f ) is a non-trivial homomorphism of H0 into the complex numbers. 10 Suppose π = π(µ1 , µ2 ) where µ1 and µ2 are unramified and λ is the associated homomorphism of H0 into C.