Get Arithmetical Investigations: Representation Theory, PDF

By Shai M. J. Haran

ISBN-10: 0131402064

ISBN-13: 9780131402065

In this quantity the writer extra develops his philosophy of quantum interpolation among the true numbers and the p-adic numbers. The p-adic numbers include the p-adic integers Zpwhich are the inverse restrict of the finite jewelry Z/pn. this offers upward push to a tree, and chance measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains particular foundation for the Hilbert house L2(Zp,w). the true analogue of the p-adic integers is the period [-1,1], and a chance degree w on it provides upward thrust to a unique foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For particular (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a different foundation, that inside convinced limits yield the true and the p-adic theories. this concept could be generalized variously. In illustration concept, it's the quantum common linear staff GLn(q)that interpolates among the p-adic crew GLn(Zp), and among its genuine (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a related quantum interpolation among the true and p-adic Fourier remodel and among the genuine and p-adic (local unramified a part of) Tate thesis, and Weil particular sums.

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Extra resources for Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations

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For n ≥ 0, we denote by Xn the set Xn := {x ∈ X | d(x0 , x) = n}. Then X decomposes as the disjoint union of Xn ; X = n≥0 Xn . Note that X0 = {x0 } and Xn is a finite set. The boundary ∂X of X is defined by the inverse limit of sets Xn or as the collection of all paths starting from the root x0 , ˜ = {xn } xn ∈ Xn , d(xn , xn+1 ) = 1 . ∂X := lim Xn = x ←− For x ∈ Xn , we denote I(x) ⊂ ∂X, which is called the “interval” of x, by I(x) := x ˜ = {xn } ∈ ∂X | xn = x and give a topology in ∂X by regarding the family {I(x) | x ∈ X} as open base of ∂X.

X∈X This operator satisfies (i) µ ≥ 0 =⇒ P ∗ µ ≥ 0, P ∗µ = (ii) X µ= X µ(x). x∈X The Laplacian ∆ is given by the operator ∆ := 1 − P. The function f : X → [0, ∞) is called harmonic if ∆f ≡ 0, f (x0 ) = 1. ) Note that the constant function 1 is clearly harmonic. Up to a constant multiplication, this is equivalent to the equation f (x) = P (x, x )f (x ) x We denote by Harm(X) the collection of all harmonic functions. Notice that Harm(X) is convex. Namely, f0 , f1 ∈ Harm(X) =⇒ λ0 f0 + λ1 f1 ∈ Harm(X).

If we arrive at the point (i, j), we replace α α+β by α + 2i and β by β + 2j, respectively. Then the boundary is given by P1 (R)/{±1}. The harmonic measure is the measure τ (α)β = pr∗ τZαη (x) ⊗ τZβη (x) , the projection of the product of two real γ-measures down to a measure on the projective line. We call this the real β-chain. In Sect. 2 we introduce the q-zeta function ζq (s) = 1 = (q s ; q)∞ (1 − q s+n )−1 . n≥0 It interpolates the p-adic ζp (s) = (1 − p−s )−1 and the real ζη (s) = Γ 2s . Here there is a slight problem with the real limit: we have to introduce a factor (1 − q)s which destroy the periodicity of ζq (s).

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Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations by Shai M. J. Haran

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