Download Arithmetical Investigations: Representation Theory, by Shai M. J. Haran PDF

By Shai M. J. Haran

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 comprise the p-adic integers Zpwhich are the inverse restrict of the finite jewelry Z/pn. this provides upward thrust to a tree, and likelihood measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains detailed foundation for the Hilbert area L2(Zp,w). the genuine analogue of the p-adic integers is the period [-1,1], and a likelihood degree w on it provides upward thrust to a distinct foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For detailed (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a distinct foundation, that inside yes limits yield the true and the p-adic theories. this concept will be generalized variously. In illustration concept, it's the quantum basic linear workforce GLn(q)that interpolates among the p-adic team GLn(Zp), and among its actual (and advanced) analogue -the orthogonal On (and unitary Un )groups. there's a related quantum interpolation among the genuine and p-adic Fourier rework and among the true and p-adic (local unramified a part of) Tate thesis, and Weil specific sums.

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

Example text

Let us start from 48 3 Real Beta Chain and q-Interpolation α + 2i α + β + 2(N + 1) (i, j + 1) (i + 1, j + 1) β + 2j α + β + 2N β + 2j α + β + 2(N + 1) (i, j) α + 2i α + β + 2N (i + 1, j) Fig. 1. The real β-chain the origin (0, 0). We can go to (0, 1) and (1, 0) from the origin and give the probability on each walk by α and β respectably. To obtain the transition probability, we normalize by dividing by α + β. If we reach the point (i, j), we replace α by α + 2i and β by β + 2j, respectively (see Fig.

For p = η, we have ρη (v1 , v2 ) = | sin θ|, where θ is the angle between the lines v1 and v2 . Let us denote ρ∞ (x) (resp. ρ0 (x)) the distance with respect to ρp between (1 : x) ∈ P1 (Qp ) and ∞ = (0 : 1) (resp. 0 = (1 : 0)). Namely, ρ∞ (x) : = ρp (0 : 1), (1 : x) = |1, x|−1 p , ρ0 (x) : = ρp (1 : 0), (1 : x) = |1, x−1 |−1 p . 4) and for p = η ρ∞ (x) =(1 + x2 )− 2 , ρ0 (x) = (1 + x−2 )− 2 . 5) Finally we have max{ρ∞ (x), ρ0 (x)} = 1 2 2 ρ∞ (x) + ρ0 (x) = 1 and for all p ≥ η |x|p = if p = η, if p = η ρ0 (x) .

Then we can obtain the probability measure τn on Xn by τn (x) = τ (I(x)) := P (x0 → x1 ) · · · P (xn−1 → x) (x ∈ Xn ), where x0 → x1 → · · · → xn = x is the unique path from x0 to x. This can be also written as τn (x) = (P ∗ )n δx0 (x) where P ∗ is the adjoint of P and δx0 is the delta function at x0 (see the next section). Hence, for all n ≥ 0, we obtain the Hilbert space Hn := 2 (Xn , τn ) = f : Xn → C ||f ||Hn < ∞ , 1/2 where ||f ||Hn := (f, f )Hn and (·, ·)Hn is the inner product of Hn defined by (f, g)Hn := f (x)g(x)τn (x).

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