Download e-book for iPad: Arithmetical Investigations: Representation Theory, by Shai M. J. Haran

By Shai M. J. Haran

ISBN-10: 0131402064

ISBN-13: 9780131402065

In this quantity the writer extra develops his philosophy of quantum interpolation among the true 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 distinctive foundation for the Hilbert house 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 push to a different foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For certain (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a unique foundation, that inside of convinced limits yield the true and the p-adic theories. this concept may be generalized variously. In illustration concept, it's the quantum basic linear workforce GLn(q)that interpolates among the p-adic crew GLn(Zp), and among its actual (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a comparable quantum interpolation among the true and p-adic Fourier rework and among the genuine and p-adic (local unramified a part of) Tate thesis, and Weil specific sums.

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

Example text

3 Boundary Now let us obtain the boundary of X. We first decide the Cauchy sequence. Let {xn } = {(in , jn )} be a Cauchy sequence of X. By the definition it means that, for any (i, j), {K((i, j), (in , jn ))} is a Cauchy sequence of R. In particular, let us pick (i, j) = (1, 0) and (0, 1). 3) of Martin kernel, the sequences K((1, 0), (in , jn )) = = (in + jn − 1)! jn ! jn ! (in + jn )! ζη (α + 2, β) in ζη (α, β) in + jn ζη (α + 2, β) and K((0, 1), (in , jn )) = jn ζη (α, β) . in + jn ζη (α, β + 2) should be Cauchy sequences.

The Martin kernel K(x, y), which is defined on X × X, is extended to X × ∂X as follows; For x ∈ X and {xn }/∼ ∈ ∂X, we define K(x, {xn }/∼) := lim K(x, xn ). ) This is welldefined. Fix a point y = {yn }/∼ ∈ ∂X. Let us write Kδy (x) = K(x, y). Then this is always Harmonic: P (x, x )K(x , y) = K(x, y) x→x If we take y1 = y2 , then we have Kδy1 = Kδy2 . More generally, for any probability measure µ on the boundary ∂X, the function Kµ (x) := K(x, y)µ(dy) ∂X is always a harmonic function. 1. For every harmonic function f ∈ Harm(X), there exists a probability measure µ ∈ M1 (∂X) such that f = Kµ .

K(x, y) = G(x0 , y) (x ∈ Xn ), (x ∈ Xn , y ∈ Xm ), The Martin kernel gives a metric. The sequence {yn } is a Cauchy sequence if {K(x, yn )} is a Cauchy sequence of R for all x and {yn } ∼ {yn } if {K(x, yn )} ∼ {K(x, yn )}. Then we obtain the compactification X = {Cauchy sequence of X}/∼ = X ∂X. Recall the theorem that every super-harmonic function f is equal to Kµ for some µ which is a probability measure on X ∂X. Here a function f is called superharmonic if P f ≥ f . If P f = f , we call f a harmonic function and µ is a measure supported only on the boundary ∂X.

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Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations by Shai M. J. Haran

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