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 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 specified foundation for the Hilbert area 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 thrust 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 unique foundation, that inside of yes limits yield the true and the p-adic theories. this concept might be generalized variously. In illustration thought, it's the quantum basic linear team GLn(q)that interpolates among the p-adic staff GLn(Zp), and among its genuine (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a comparable quantum interpolation among the genuine and p-adic Fourier remodel and among the true 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

Sample text

6) It is easy to see that the p-adic substitution p and real substitution η yield the (non-symmetric) p-adic and real β-chains, respectively; p N q-β-chain −→ the p-adic β-chain (N → ∞), 56 3 Real Beta Chain and q-Interpolation η N q-β-chain −→ the real β-chain (N → ∞). In this sense, the q-β-chain interpolates between the p-adic and the real βchain. Notice that if we take the limit N → ∞ in the p-adic substitution p N , the probability on the walk from (i, j) to (i + 1, j) vanishes unless j = 0.

Measure on Zp /Z∗p and an absolute value on V (Qp ) by ⎧ ⎨max{|x|p , |y|p } |x, y|p = |(x, y)|p := ⎩ |x|2 + |y|2 η η if p = η, if p = η (x, y) ∈ V (Qp ). For p = η, put V (Zp ) : = Zp × Zp , V ∗ (Zp ) : = (x, y) ∈ Zp |x, y|p = 1 . Then the projective line P1 (Qp ) is expressed as P1 (Qp ) = V ∗ (Qp )/Q∗p = V ∗ (Zp )/Z∗p . For all p ≥ η, there is a canonical distance function ρp : P1 (Qp ) × P1 (Qp ) → [0, 1] defined by ρp (x1 : y1 ), (x2 : y2 ) := |x1 y2 − y1 x2 |p |x1 , y1 |p · |x2 , y2 |p The function ρp is well-defined.

Cd ((x : y), Then it is easy to see that the stabilizer group Stab(0) of 0 = (1 : 0) is given by Stab(0) = 10 cd ∈ P GL(Zp ) c ∈ Zp , d ∈ Z∗p . This shows that Stab(0) is isomorphic to the semi-direct product Z∗p the isomorphic map Stab(0) 10 cd −→ (d, c) ∈ Z∗p Zp by Zp and, hence, we obtain a smaller quotient P1 (Zp )/Z∗p Zp of P1 (Zp ) than P1 (Zp )/Z∗p . From the tree of P1 (Zp )/Z∗p , we obtain the following tree corresponding to P1 (Zp )/Z∗p Zp since two elements (pn : 1) and (1 : 1) are equivalent under the action of Z∗p Zp for all n ∈ N (Figs.

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