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 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

Example text

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.

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