New PDF release: Basic Elements of Real Analysis (Undergraduate Texts in

By Murray H. Protter

Designed in particular for a quick one-term path in actual research together with such issues because the genuine quantity process, the speculation on the foundation of simple calculus, the topology of metric areas & limitless sequence. There are proofs of the fundamental theorems on limits at a speed that's planned & exact. DLC: Mathematical research.

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Chapter 1 39 If π is itself the canonical model π(w)ϕ(a) = W a 0 0 1 w where 0 1 −1 0 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). Thus if the theorem is valid the series ϕ(1, t) and ϕ(ν0−1 , t) have positive radii of convergence and define functions which are meromorphic in the whole t-plane.

It is a finite linear combination of powers of t and if it is not of the form indicated it has a zero at some point different form 0. C(νν0−1 , z0−1 t−1 ) is also a linear combination of powers of t and so cannot have a pole except at zero. To show that C(ν, t) has the required form we have only to show that C(ν, t)C(ν −1ν0−1 , z0−1 t−1 ) = ν0 (−1). 3) Choose ϕ in V0 and set ϕ = π(w)ϕ. We may suppose that ϕ (ν, t) = 0. The identity is obtained by combining the two relations ϕ (ν, t) = C(ν, t)ϕ(ν −1 ν0−1 , z0−1 t−1 ) and ν0 (−1)ϕ(ν −1 ν0−1 , t) = C(ν −1 ν0−1 , t)ϕ (ν, z0−1 t−1 ).

All that we have left to do is to show that the restiction of an absolutely cuspidal representation to GL(2, OF ) does not contain the trivial representation. Suppose the infinite-dimensional irreducible representation π is given in the Kirillov form with respect to an additive character ψ such that OF is the largest ideal on which ψ is trivial. Suppose the non-zero vector ϕ is invariant under GL(2, OF ). It is clear that if π a 0 0 a = ω(a)I then ω is unramified, that ϕ(ν, t) = 0 unless ν = 1 is the trivial character, and that ϕ(ν, t) has no pole at t = 0.

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