By H. Jacquet, R. P. Langlands

ISBN-10: 3540059318

ISBN-13: 9783540059318

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**Sample text**

Let V be the space on which the Kirillov model of π acts. For every W in W (π, ψ) there is a unique ϕ in V such that W a 0 0 1 = ϕ(a). Chapter 1 39 If π is itself the canonical model π(w)ϕ(a) = W where a 0 0 1 0 1 −1 0 w= w . If χ is any quasi-character of F × we set ϕ(a)χ(a) d× a ϕ(χ) = F× if the integral converges. If χ0 is the restriction of χ to UF then ϕ(χ) = ϕ χ0 , χ( ) . Thus if αF is the quasi-character αF (x) = |x| and the appropriate integrals converge s−1/2 ) = ϕ(1, q 1/2−s ) s−1/2 ω −1 ) = ϕ(ν0−1 , z0−1 q 1/2−s ) Ψ(e, s, W ) = ϕ(αF Ψ(e, s, W ) = ϕ(αF if ν0 is the restriction of ω to UF and z0 = ω(ϕ).

Let F be the space of continuous functions f on GF which satisfy a1 x 0 a2 f g = a1 f (g) a2 1/2 −1/2 for all g in GF , a1 , a2 in F × , and x in F . We observe that F contains B(αF , αF The Haar measure on GF if suitably normalized satisfies f (g) dg = GF NF AF GL(2,OF ) if a1 0 0 a2 a= a1 a2 ). GF acts on F. −1 f (nak) dn da dk . It follows easily from this that f (k) dk GL(2,OF ) is a GF -invariant linear form on F. There is also a positive constant c such that f (g) dg = c GF NF AF NF a1 a2 −1 f na 0 1 −1 0 n1 dn da dn1 .

10 we see that all of the sequences satisfying the recursion relation p λa ϕn−i (µ) ϕn (µ) = i=1 for n ≥ n1 . The integer n1 depends on ϕ. 1 is therefore a consequence of the following elementary lemma whose proof we postpone to Paragraph 8. 4 Let λi , 1 ≤ i ≤ p, be p complex numbers. Let A be the space of all sequences {an }, n ∈ Z for which there exist two integers n1 and n2 such that an = λi an−i 1≤i≤p for n ≥ n1 and such that an = 0 for n ≤ n2 . Let A0 be the space of all sequences with only a finite number of nonzero terms.

### Automorphic forms on GL(2) by H. Jacquet, R. P. Langlands

by Steven

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