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Chapter 1 32 Let NF be the group of matrices of the form 1 x 0 1 with x in F and let B be the space of functions on GF invariant under left translations by elements of NF . B is invariant under right translations and the question of whether or not a given irreducible representation π is contained in B arises. The answer is obviously positive when π = χ is onedimensional for then the function g → χ(detg) is itself contained in B. Assume that the representation π which is given in the Kirillov form acts on B.
Suppose ψ (x) = ψ(bx) with b in F × is another non-trivial additive character. Then W (π, ψ ) consists of the functions b 0 0 1 W (g) = W g with W in W (π, ψ). The last identity of the following theorem is referred to as the local functional equation. It is the starting point of our approach to the Hecke theory. 18 Let π be an irreducible inﬁnite-dimensional admissible representation of G F . (i) If ω is the quasi-character of GF deﬁned by a 0 π 0 a = ω(a)I then the contragredient representation π is equivalent to ω −1 ⊗ π.
4) r These two expressions are equal for all choice of n, p, ρ, and ν . 4) reduces to η(ρν −1 , −m− Since r ) is zero unless r = −m− . )z0−p−m− Cn−m− (ν)Cp−m− (ρ−1 ν0−1 ). 3) is equal to ρ−1 ν(−1) η(σ −1 ν, n ) η(ρ−1 σ −1 ν0−1 p )z0−p Cp+n (σ). σ ρ−1 ν0−1 Replacing ρ by we obtain the first part of the proposition. If ρ = ν then δ(ρν −1 ) = 1. Moreover, as is well-known and easily verified, η(ρν−1 , r≥− , η(ρν −1 , and η(ρν −1 , r − −1 r ) = 1 if ) = | |(| | − 1)−1 ) = 0 if r ≤ − − 2. 4) is equal to ν0 (−1)δn,pI+(| |−1)−1 z0−p+ +1 Cn− −1 (ν)Cn− −1 −1 ν0 )− −1 (ν −∞ r=− −2 The second part of the proposition follows.