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

By Shai M. J. Haran

In this quantity the writer extra develops his philosophy of quantum interpolation among the genuine 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 likelihood measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains designated foundation for the Hilbert area L2(Zp,w). the genuine analogue of the p-adic integers is the period [-1,1], and a chance degree w on it provides upward push to a different 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 might be generalized variously. In illustration idea, it's the quantum normal linear staff GLn(q)that interpolates among the p-adic team GLn(Zp), and among its actual (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a comparable 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 specific sums.

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

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We will denote in future HN by Hp(N ) . ) The boundary space H is (α)β (α)β also written as Hp . Further we denote the basis ϕN,m of HN by ϕp(N ),m and (α)β (α)β the basis ϕm of H by ϕp,m . We call ϕp(N ),m the p-Hahn basis (an analogue of (α)β the Hahn polynomial) and ϕp,m the p-Jacobi basis (an analogue of the Jacobi polynomial). 5 p-Adic γ-Chain Let us consider the γ-measure. Take α → ∞ in either the symmetric β-chain or non-symmetry β-chain. We get the following tree in Fig. 4, called the p-adic γ-chain.

In this case we call this the “random walk”. Random means that the probability of each arrow is alway the same at any stage. But this is only α = β = 1. 4 Non-Symmetric p-Adic β-Chain The symmetric β-chain on P1 (Qp )/Z∗p is still too complicated for us. We next consider the chain on the tree P1 (Qp )/Z∗p Zp . Since this is not symmetric, we call this non-symmetric β-chain. Note that the tree of P1 (Qp )/Z∗p Zp is obtained by collapsing all of the paths corresponding to (pn : 1)Z∗p for n ≥ 0 of P1 (Qp )/Z∗p together.

Since we have a probability measure τ on ∂X, we have another Hilbert space H := 2 (∂X, τ ) = f : ∂X → C ||f ||H < ∞ , 1/2 where ||f ||H := (f, f )H and (·, ·)H is the inner product of H defined by (f, g)H := f (˜ x)g(˜ x)τ (d˜ x). ∂X There is also an unitary embedding map Hn → H for all n ≥ 0 defined by ϕ −→ ϕ˜ ∈ H; Hn ϕ(˜ ˜ x) := ϕ(xn ) with x ˜ = {xn } and this is an unitary embedding. The orthogonal projection from H onto Hn is given as follows; ϕ˜ −→ ϕ ∈ Hn ; H H0 z z P G H1 z z 1 τn (xn ) ϕ(xn ) := P G ··· { {  P x G Hn   _ ’’ H x P ϕ(˜ ˜ x)τ (d˜ x).

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