By Shai M. J. Haran

In this quantity the writer additional develops his philosophy of quantum interpolation among the genuine numbers and the p-adic numbers. The *p*-adic numbers include the *p*-adic integers *Z _{p}*which are the inverse restrict of the finite jewelry

*Z/p*. this offers upward thrust to a tree, and chance measures w on

^{n}*Z*correspond to Markov chains in this tree. From the tree constitution one obtains distinct foundation for the Hilbert area

_{p}*L*(

_{2}*Z*). the true analogue of the

_{p},w*p*-adic integers is the period [-1,1], and a chance degree w on it provides upward thrust to a distinct foundation for

*L*([-1,1],

_{2}*w*) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For targeted (gamma and beta) measures there's a "quantum" or "

*q*-analogue" Markov chain, and a distinct foundation, that inside of 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 crew

*GL*(

_{n}*q*)that interpolates among the p-adic staff

*GL*(

_{n}*Z*), and among its genuine (and advanced) analogue -the orthogonal

_{p}*O*(and unitary

_{n}*U*)groups. there's a related 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 specific sums.

_{n}

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

**Example text**

4. 1 Quotients Zp /Z∗p and P1 (Qp )/Z∗p Zp 23 The action of the unit group Z∗p on the projective line P1 (Zp ) is given by P1 (Zp ) × Z∗p ((x : y), a) −→ (ax : y) = (x : a−1 y) ∈ P1 (Zp ). Similarly, the tree corresponding to the quotient P1 (Zp )/Z∗p can be obtained as the case of Zp /Z∗p . Further the group P GL2 (Zp ) also acts on P1 (Zp ) as follows; P1 (Zp ) × P GL2 (Zp ) ab ) −→ (ax + cy : bx + dy) ∈ P1 (Zp ). 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 .

8) Then we have (X + Y )N = 0≤k≤N N k Y N −k X k . q One can prove this formula by a simple induction on N .

If P f = f , we call f a harmonic function and µ is a measure supported only on the boundary ∂X. The set Harm(X) of all harmonic functions on X is divided as Harm(X) = Harm(X)ext Harm(X)non-ext and the boundary ∂X also decomposes as ∂X = ∂Xext ∂Xnon-ext . 34 2 Markov Chains Here a point y ∈ ∂X is called extream if Kδy = K(x, y) is extream harmonic function. Then there is one-to-one correspondence between the probability measures on ∂Xext and the harmonic functions on X. 1 Probability Measures on ∂X Let X be a tree and x0 ∈ X the root.