By H. Jacquet, R. P. Langlands

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**Additional info for Automorphic Forms on GL(2): Part 1**

**Sample text**

10 we see that all of the sequences satisfying the recursion relation p ϕn (µ) = λa ϕn−i (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 λ i , 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.

Take x in F such that ψ(x) = 1 and ϕ in V . Then L(ϕ) = L ϕ − π 1 x 0 1 ϕ +L π 1 x 0 1 ϕ . Chapter 1 29 Since 1 x 0 1 ϕ−π ϕ is in V0 the right side is equal to λϕ(1) − λψ(x)ϕ(1) + ψ(x)L(ϕ) so that 1 − ψ(x) L(ϕ) = λ 1 − ψ(x) ϕ(1) which implies that L(ϕ) = λϕ(1). To prove the second lemma we have only to show that ϕ(1) = 0 implies L(ϕ) = 0. If we set ϕ(0) = 0 then ϕ becomes a locally constant function with compact support in F . Let ϕ be its Fourier transform so that ϕ(a) = ψ(ba) ϕ (−b) db. F Let Ω be an open compact subset of F × containing 1 and the support of ϕ.

By the Plancherel theorem for UF ν(−1)ϕn (ν)ϕn (ν −1 ). ϕ, ϕ = n ν The sum is in reality finite. It is easy to se that if b belongs to B ξψ (b)ϕ, ξψ (b)ϕ ) = ϕ, ϕ . Suppose π is given in the Kirillov form and acts on V . Let π , the Kirillov model of ω−1 ⊗ π , act on V . To prove part (i) we have only to construct an invariant non-degenerate bilinear form β on V × V . If ϕ belongs to V0 and ϕ belongs to V or if ϕ belongs to V and ϕ belongs to V0 we set β(ϕ, ϕ ) = ϕ, ϕ . If ϕ and ϕ are arbitrary vectors in V and V we may write ϕ = ϕ1 + π(w)ϕ2 and ϕ = ϕ1 + π (w)ϕ2 with ϕ, ϕ2 in V0 and ϕ1 , ϕ2 in V0 .