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 offers upward thrust to a tree, and likelihood measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains specific 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 supplies upward push to a distinct foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For designated (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a distinct foundation, that inside of sure limits yield the genuine and the p-adic theories. this concept should be generalized variously. In illustration concept, it's the quantum normal linear crew 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 rework 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
2 γ-Measure Gives β-Measure Next we show that the γ-measure gives the β-measure. Let us denote by pr∗ the push forward from the set of measures on V ∗ (Qp ) onto the set of measures on P1 (Qp ). Then the measure pr∗ (τZαp ⊗ τZβp ) gives the β-measure. Actually it holds that (1:x) τZαp ⊗ τZβp = = Q∗ p d∗ a · φZp (a) |x|βp ζp (α)ζp (β) Q∗ p |a|α |ax|βp p φZp (ax) ζp (α) ζp (β) d∗ a · φZp (a · |1, x|p )|a|α+β p |x|βp |1, x|−α−β p ζp (α + β) ζp (α)ζp (β) 1 . 4 Remarks on the γ and β-Measure 31 Hence we have pr∗ (τZαp ⊗ τZβp ) = τpα,β .
6) x ∈Xn+1 Then we says that we have a Markov chain. If for any x ∈ Xn there exists a sequence x0 , x1 , . . , xn = x such that xj ∈ Xj and P (xj , xj+1 ) > 0, we say that x is reachable from x0 . We assume that every state x ∈ X is reachable from x0 . The function P can be extended as a function on X × X by giving 0 if two points x, x are not connected. Therefore we can regard P as a matrix over X × X. 6). We have the adjoint operator P ∗ , which acts on P ∗ µ(x ) := 1 (X), deﬁned by µ(x)P (x, x ).
Remember the projection from P1 (Qp ) onto P1 (Qp )/Z∗p . If we want to know the probability measure of an arrow in the tree of P1 (Qp ), we divide the probability of the projected arrow in P1 (Qp )/Z∗p by the number of the arrow of P1 (Qp ) corresponding to the given arrow in P1 (Qp )/Z∗p . For example if α = β = 1, it it easy to see that the probability of each arrow is given as in Fig. 2 (for the case p = 3). Note that if α = β = 1, the β-measure τp1,1 is the unique P GL2 (Zp )-invariant measure.