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 genuine numbers and the p-adic numbers. The p-adic numbers include the p-adic integers Zpwhich are the inverse restrict of the finite earrings Z/pn. this provides upward push to a tree, and chance measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains distinct 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 offers upward push to a unique foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For specific (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a different foundation, that inside of yes limits yield the genuine and the p-adic theories. this concept might be generalized variously. In illustration idea, it's the quantum common linear staff GLn(q)that interpolates among the p-adic staff GLn(Zp), and among its actual (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a comparable 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 particular sums.

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Additional info for Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations

Example text

Our measure τpα,β is not invariant 0∗ under Z∗p Zp , but we can project it down to P1 (Qp )/Z∗p Zp , we denote the image (α)β measure by τp . Note that it is not symmetric in the two parameters α and β. Then we have a Markov chain on the tree P1 (Z/pn )/(Z/pn )∗ (Z/pn ), it is called P GL2 (Zp ) whose element are of the form n≥0 the p-adic β-chain. 1 Zp /Z∗p Every p-adic integer can be written as a power series in p and such a representation is unique. Namely we have Zp = lim Z/(pn ) = a0 + a1 p + a2 p2 + · · · 0 ≤ aj < p (j ≥ 0) .

This shows that the boundary is not totally disconnected, hence this is a non-tree (for any tree, the boundary is always totally disconnected). In this section we study the Markov chain on non-trees, which can have continuous boundary. 2 Harmonic Functions Let X = n≥0 Xn , X0 = {x0 } and Xn be a finite set for all n ≥ 0. We call X the state space. Let P : n≥0 Xn × Xn+1 → [0, 1] be a transition probability, that is, P satisfies P (x, x ) = 1 (x ∈ Xn ). 6) x ∈Xn+1 Then we says that we have a Markov chain.

2 Markov Chain on Non-Trees 41 Similarly we obtain the orthogonal decomposition of HN and H; β HN : = Hp(N ) = Cϕβp(N ),m , 0≤m≤N H := HZβp CϕβZp ,m , = m≥0 where ϕβp(N ),m (resp. ϕβZp ,m ) is the basis of HN (resp. H) defined by ϕβp(N ),0 = 1, (1 − p−β )pβ if 0 < i ≤ N, −1 if i = 0, ⎧ −β βm ⎪ if m − 1 < i ≤ N , ⎨(1 − p )p β ϕp(N ),m (i, j) = −pβ(m−1) if i = m − 1, ⎪ ⎩ 0 if 0 ≤ i < m − 1, ϕβp(N ),1 (i, j) = (m ≥ 2). and ϕβZp ,0 = φZp , ϕβZp ,m = pβm φpm Zp − pβ(m−1) φpm−1 Zp (m ≥ 1). We call ϕβZp ,m the p-Laguerre basis, it is the analogue of the Laguerre polynomial.

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