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

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 comprise the p-adic integers Zpwhich are the inverse restrict of the finite earrings Z/pn. this provides upward push to a tree, and chance measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains designated foundation for the Hilbert area L2(Zp,w). the genuine analogue of the p-adic integers is the period [-1,1], and a likelihood degree w on it supplies upward thrust to a unique foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For distinct (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a unique foundation, that inside of sure limits yield the true and the p-adic theories. this concept should be generalized variously. In illustration concept, it's the quantum basic linear staff GLn(q)that interpolates among the p-adic workforce GLn(Zp), and among its genuine (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a related 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 particular sums.

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

Example text

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. This means that either 1. There exists some i∞ and j∞ such that (in , jn ) ≡ (i∞ , j∞ ) for all n 0, or 2. in + jn → ∞ (n → ∞) and jn /in converges (in wide sense) in [0, ∞]. Conversely, as we will check below soon, the Martin kernel converges for all (i, j) whenever jn /in converges in the wide sense.

The name of the γ- measure is given from the gamma function. If p = η, τZβη is the Gaussian probability measure, which is invariant under Z∗η = {±1}. If p = η, then τZβp is also invariant under the action Z∗p , whence gives a probability measure on Zp /Z∗p . Notice that for all n ≥ 0, we have τZβp (pn Z∗p ) = p−nβ (1 − p−β ) (see Fig. 7). 1 The Projective Space P1 (Qp ) We next define a measure on the projective line P1 (Qp ), which is called β-measure. Let V (Qp ) := Qp × Qp be the plane and V ∗ (Qp ) := {(x, y) ∈ V (Qp ) | (x, y) = (0, 0)}.

This is a simple calculation. 3) again, we have K((i, j), (in , jn )) = (in + jn − i − j)! jn ! ζη (α, β) (in + jn )! (jn − j)! ζη (α + 2i, β + 2j) = in (in − 1) · · · (in − i + 1) · jn (jn − 1) · · · (jn − j + 1) ζη (α, β) (in + jn )(in + jn − 1) · · · (in + jn − i − j − 1) ζη (α + 2i, β + 2j) 1 1+ = 1 jn in 1+ jn +1 in −1 1· 1− ··· 1 1+ 1 in + jn jn +i−1 in −i+1 ··· 1 − 1 · 1+ 1 in jn 1+ in +1 jn −1 ··· i + j − 1 ζη (α + 2i, β + 2j) in + jn ζη (α, β) 1 1+ in +j−1 jn −j+1 . Therefore we have lim in +jn →∞ jn /in →|x|2 η K((i, j), (in , jn )) = 1 1 + |x|2η i 1 1 + |x|−2 η j ζη (α, β) .

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