Arithmetical Investigations: Representation Theory, by Shai M. J. Haran

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 Zpwhich are the inverse restrict of the finite jewelry Z/pn. this offers upward thrust 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 area L2(Zp,w). the true analogue of the p-adic integers is the period [-1,1], and a chance degree w on it provides upward thrust to a distinct foundation for L2([-1,1],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 GLn(q)that interpolates among the p-adic staff GLn(Zp), and among its genuine (and advanced) analogue -the orthogonal On (and unitary Un )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.

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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.

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